From 22e73f16aea7c6b6195aa5481f831ebb083a7c27 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Wed, 8 Nov 2023 23:01:50 +0000 Subject: [PATCH] build based on b3603ae --- previews/PR238/.documenter-siteinfo.json | 2 +- previews/PR238/export/export/index.html | 2 +- previews/PR238/identifiability/identifiability/index.html | 8 ++++---- previews/PR238/index.html | 4 ++-- previews/PR238/input/input/index.html | 2 +- previews/PR238/ioequations/ioequations/index.html | 2 +- previews/PR238/search_index.js | 2 +- previews/PR238/tutorials/discrete_time/index.html | 6 ++++-- previews/PR238/utils/elimination/index.html | 2 +- previews/PR238/utils/global_identifiability/index.html | 2 +- previews/PR238/utils/local_identifiability/index.html | 2 +- previews/PR238/utils/ode/index.html | 6 +++--- previews/PR238/utils/power_series_utils/index.html | 2 +- previews/PR238/utils/primality/index.html | 2 +- previews/PR238/utils/reparametrization/index.html | 2 +- previews/PR238/utils/util/index.html | 4 ++-- previews/PR238/utils/wronskian/index.html | 2 +- 17 files changed, 27 insertions(+), 25 deletions(-) diff --git a/previews/PR238/.documenter-siteinfo.json b/previews/PR238/.documenter-siteinfo.json index 3cb2df524..2596d292f 100644 --- a/previews/PR238/.documenter-siteinfo.json +++ b/previews/PR238/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-11-08T23:01:07","documenter_version":"1.1.2"}} \ No newline at end of file +{"documenter":{"julia_version":"1.9.3","generation_timestamp":"2023-11-08T23:01:35","documenter_version":"1.1.2"}} \ No newline at end of file diff --git a/previews/PR238/export/export/index.html b/previews/PR238/export/export/index.html index 061ca9505..45c6a0b76 100644 --- a/previews/PR238/export/export/index.html +++ b/previews/PR238/export/export/index.html @@ -3,4 +3,4 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub

Exporting to Other Systems

Here we put some helpful utilities to export your code to other identifiability software.

StructuralIdentifiability.print_for_maple — Function
print_for_maple(ode, package)

Prints the ODE in the format accepted by maple packages

  • SIAN (https://github.com/pogudingleb/SIAN) if package=:SIAN
  • DifferentialAlgebra if package=:DifferentialAlgebra
  • DifferentialThomas if package=:DifferentialThomas
source
+
GitHub

Exporting to Other Systems

Here we put some helpful utilities to export your code to other identifiability software.

StructuralIdentifiability.print_for_maple — Function
print_for_maple(ode, package)

Prints the ODE in the format accepted by maple packages

  • SIAN (https://github.com/pogudingleb/SIAN) if package=:SIAN
  • DifferentialAlgebra if package=:DifferentialAlgebra
  • DifferentialThomas if package=:DifferentialThomas
source
diff --git a/previews/PR238/identifiability/identifiability/index.html b/previews/PR238/identifiability/identifiability/index.html index 67d02fc02..27c59b609 100644 --- a/previews/PR238/identifiability/identifiability/index.html +++ b/previews/PR238/identifiability/identifiability/index.html @@ -3,12 +3,12 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub

Functions to Assess Identifiability

Assessing All Types of Identifiability

StructuralIdentifiability.assess_identifiability — Function
assess_identifiability(ode; funcs_to_check = [], p=0.99)

Input:

  • ode - the ODE model
  • funcs_to_check - list of functions to check identifiability for; if empty, all parameters and states are taken
  • p - probability of correctness.

Assesses identifiability of a given ODE model. The result is guaranteed to be correct with the probability at least p. The function returns a dictionary from the functions to check to their identifiability properties (one of :nonidentifiable, :locally, :globally).

source
assess_identifiability(ode::ModelingToolkit.ODESystem; measured_quantities=Array{ModelingToolkit.Equation}[], funcs_to_check=[], p = 0.99)

Input:

  • ode - the ModelingToolkit.ODESystem object that defines the model
  • measured_quantities - the output functions of the model
  • funcs_to_check - functions of parameters for which to check the identifiability
  • p - probability of correctness.

Assesses identifiability (both local and global) of a given ODE model (parameters detected automatically). The result is guaranteed to be correct with the probability at least p.

source

Assessing Local Identifiability

StructuralIdentifiability.assess_local_identifiability — Function
function assess_local_identifiability(ode::ModelingToolkit.ODESystem; measured_quantities=Array{ModelingToolkit.Equation}[], funcs_to_check=Array{}[], p::Float64=0.99, type=:SE)

Input:

  • ode - the ODESystem object from ModelingToolkit
  • measured_quantities - the measurable outputs of the model
  • funcs_to_check - functions of parameters for which to check identifiability
  • p - probability of correctness
  • type - identifiability type (:SE for single-experiment, :ME for multi-experiment)

Output:

  • for type=:SE, the result is a dictionary from each parameter to boolean;
  • for type=:ME, the result is a tuple with the dictionary as in :SE case and array of number of experiments.

The function determines local identifiability of parameters in funcs_to_check or all possible parameters if funcs_to_check is empty

The result is correct with probability at least p.

type can be either :SE (single-experiment identifiability) or :ME (multi-experiment identifiability). The return value is a tuple consisting of the array of bools and the number of experiments to be performed.

source
assess_local_identifiability(ode::ODE{P}; funcs_to_check::Array{<: Any, 1}, p::Float64=0.99, type=:SE) where P <: MPolyElem{Nemo.fmpq}

Checks the local identifiability/observability of the functions in funcs_to_check. The result is correct with probability at least p.

Call this function if you have a specific collection of parameters of which you would like to check local identifiability.

type can be either :SE (single-experiment identifiability) or :ME (multi-experiment identifiability). If the type is :ME, states are not allowed to appear in the funcs_to_check.

source
function assess_local_identifiability(
+
GitHub
Functions to Assess Identifiability
Assessing All Types of Identifiability
StructuralIdentifiability.assess_identifiability — Function
assess_identifiability(ode; funcs_to_check = [], p=0.99)
Input:
  • ode - the ODE model
  • funcs_to_check - list of functions to check identifiability for; if empty, all parameters  and states are taken
  • p - probability of correctness.
Assesses identifiability of a given ODE model. The result is guaranteed to be correct with the probability at least p. The function returns a dictionary from the functions to check to their identifiability properties  (one of :nonidentifiable, :locally, :globally).
source
assess_identifiability(ode::ModelingToolkit.ODESystem; measured_quantities=Array{ModelingToolkit.Equation}[], funcs_to_check=[], p = 0.99)
Input:
  • ode - the ModelingToolkit.ODESystem object that defines the model
  • measured_quantities - the output functions of the model
  • funcs_to_check - functions of parameters for which to check the identifiability
  • p - probability of correctness.
Assesses identifiability (both local and global) of a given ODE model (parameters detected automatically). The result is guaranteed to be correct with the probability at least p.
source
Assessing Local Identifiability
StructuralIdentifiability.assess_local_identifiability — Function
function assess_local_identifiability(ode::ModelingToolkit.ODESystem; measured_quantities=Array{ModelingToolkit.Equation}[], funcs_to_check=Array{}[], p::Float64=0.99, type=:SE)
Input:
  • ode - the ODESystem object from ModelingToolkit
  • measured_quantities - the measurable outputs of the model
  • funcs_to_check - functions of parameters for which to check identifiability
  • p - probability of correctness
  • type - identifiability type (:SE for single-experiment, :ME for multi-experiment)
Output:
  • for type=:SE, the result is a dictionary from each parameter to boolean;
  • for type=:ME, the result is a tuple with the dictionary as in :SE case and array of number of experiments.
The function determines local identifiability of parameters in funcs_to_check or all possible parameters if funcs_to_check is empty
The result is correct with probability at least p.
type can be either :SE (single-experiment identifiability) or :ME (multi-experiment identifiability). The return value is a tuple consisting of the array of bools and the number of experiments to be performed.
source
assess_local_identifiability(ode::ODE{P}; funcs_to_check::Array{<: Any, 1}, p::Float64=0.99, type=:SE) where P <: MPolyElem{Nemo.fmpq}
Checks the local identifiability/observability of the functions in funcs_to_check. The result is correct with probability at least p.
Call this function if you have a specific collection of parameters of which you would like to check local identifiability.
type can be either :SE (single-experiment identifiability) or :ME (multi-experiment identifiability). If the type is :ME, states are not allowed to appear in the funcs_to_check.
source
function assess_local_identifiability(
     dds::ModelingToolkit.DiscreteSystem; 
     measured_quantities=Array{ModelingToolkit.Equation}[], 
     funcs_to_check=Array{}[], 
     known_ic=Array{}[],
-    p::Float64=0.99)
Input:
  • dds - the DiscreteSystem object from ModelingToolkit
  • measured_quantities - the measurable outputs of the model
  • funcs_to_check - functions of parameters for which to check identifiability (all parameters and states if not specified)
  • known_ic - functions (of states and parameter) whose initial conditions are assumed to be known
  • p - probability of correctness
Output:
  • the result is a dictionary from each function to to boolean;
The result is correct with probability at least p.
source
Assessing Global Identifiability
StructuralIdentifiability.assess_global_identifiability — Function
assess_global_identifiability(ode::ODE{P}, p::Float64=0.99; var_change=:default) where P <: MPolyElem{fmpq}
Input:
  • ode - the ODE model
  • known - a list of functions in states which are assumed to be known and generic
  • p - probability of correctness
  • var_change - a policy for variable change (:default, :yes, :no), affects only the runtime
Output:
  • a dictionary mapping each parameter to a boolean.
Checks global identifiability for parameters of the model provided in ode. Call this function to check global identifiability of all parameters automatically.
source
assess_global_identifiability(ode, [funcs_to_check, p=0.99, var_change=:default])
Input:
  • ode - the ODE model
  • funcs_to_check - rational functions in parameters
  • known - function in parameters that are assumed to be known and generic
  • p - probability of correctness
  • var_change - a policy for variable change (:default, :yes, :no),               affects only the runtime
Output:
  • array of length length(funcs_to_check) with true/false values for global identifiability       or dictionary param => Bool if funcs_to_check are not given
Checks global identifiability of functions of parameters specified in funcs_to_check.
source
Finding Identifiable Functions
StructuralIdentifiability.find_identifiable_functions — Function
find_identifiable_functions(ode::ODE; options...)
Finds all functions of parameters/states that are identifiable in the given ODE system.
Options
This functions takes the following optional arguments:
  • with_states: When true, also reports the identifiabile functions in the   ODE states. Default is false.
  • simplify: The extent to which the output functions are simplified. Stronger simplification may require more time. Possible options are:
    • :standard: Default simplification.
    • :weak: Weak simplification. This option is the fastest, but the output functions can be quite complex.
    • :strong: Strong simplification. This option is the slowest, but the output
    functions are nice and simple.
    • :absent: No simplification.
  • p: A float in the range from 0 to 1, the probability of correctness. Default is 0.99.
  • seed: The rng seed. Default value is 42.
Example
using StructuralIdentifiability
+    p::Float64=0.99)
Input:
  • dds - the DiscreteSystem object from ModelingToolkit
  • measured_quantities - the measurable outputs of the model
  • funcs_to_check - functions of parameters for which to check identifiability (all parameters and states if not specified)
  • known_ic - functions (of states and parameter) whose initial conditions are assumed to be known
  • p - probability of correctness
Output:
  • the result is a dictionary from each function to to boolean;
The result is correct with probability at least p.
source
Assessing Global Identifiability
StructuralIdentifiability.assess_global_identifiability — Function
assess_global_identifiability(ode::ODE{P}, p::Float64=0.99; var_change=:default) where P <: MPolyElem{fmpq}
Input:
  • ode - the ODE model
  • known - a list of functions in states which are assumed to be known and generic
  • p - probability of correctness
  • var_change - a policy for variable change (:default, :yes, :no), affects only the runtime
Output:
  • a dictionary mapping each parameter to a boolean.
Checks global identifiability for parameters of the model provided in ode. Call this function to check global identifiability of all parameters automatically.
source
assess_global_identifiability(ode, [funcs_to_check, p=0.99, var_change=:default])
Input:
  • ode - the ODE model
  • funcs_to_check - rational functions in parameters
  • known - function in parameters that are assumed to be known and generic
  • p - probability of correctness
  • var_change - a policy for variable change (:default, :yes, :no),               affects only the runtime
Output:
  • array of length length(funcs_to_check) with true/false values for global identifiability       or dictionary param => Bool if funcs_to_check are not given
Checks global identifiability of functions of parameters specified in funcs_to_check.
source
Finding Identifiable Functions
StructuralIdentifiability.find_identifiable_functions — Function
find_identifiable_functions(ode::ODE; options...)
Finds all functions of parameters/states that are identifiable in the given ODE system.
Options
This functions takes the following optional arguments:
  • with_states: When true, also reports the identifiabile functions in the   ODE states. Default is false.
  • simplify: The extent to which the output functions are simplified. Stronger simplification may require more time. Possible options are:
    • :standard: Default simplification.
    • :weak: Weak simplification. This option is the fastest, but the output functions can be quite complex.
    • :strong: Strong simplification. This option is the slowest, but the output
    functions are nice and simple.
    • :absent: No simplification.
  • p: A float in the range from 0 to 1, the probability of correctness. Default is 0.99.
  • seed: The rng seed. Default value is 42.
Example
using StructuralIdentifiability
 
 ode = @ODEmodel(
     x0'(t) = -(a01 + a21) * x0(t) + a12 * x1(t),
@@ -21,7 +21,7 @@
 # prints
 3-element Vector{AbstractAlgebra.Generic.Frac{Nemo.fmpq_mpoly}}:
  a12 + a01 + a21
- a12*a01
source
find_identifiable_functions(ode::ModelingToolkit.ODESystem; measured_quantities=[], options...)
Finds all functions of parameters/states that are identifiable in the given ODE system.
Options
This functions takes the following optional arguments:
  • measured_quantities - the output functions of the model.
Example
using StructuralIdentifiability
+ a12*a01
source
find_identifiable_functions(ode::ModelingToolkit.ODESystem; measured_quantities=[], options...)
Finds all functions of parameters/states that are identifiable in the given ODE system.
Options
This functions takes the following optional arguments:
  • measured_quantities - the output functions of the model.
Example
using StructuralIdentifiability
 using ModelingToolkit
 
 @parameters a01 a21 a12
@@ -39,4 +39,4 @@
 # prints
 2-element Vector{Num}:
          a01*a12
- a01 + a12 + a21
source

+ a01 + a12 + a21
source
diff --git a/previews/PR238/index.html b/previews/PR238/index.html index 35d0cd916..a751cc8c9 100644 --- a/previews/PR238/index.html +++ b/previews/PR238/index.html @@ -23,10 +23,10 @@ Official https://julialang.org/ release Platform Info: OS: Linux (x86_64-linux-gnu) - CPU: 2 × Intel(R) Xeon(R) CPU E5-2673 v3 @ 2.40GHz + CPU: 2 × Intel(R) Xeon(R) Platinum 8171M CPU @ 2.60GHz WORD_SIZE: 64 LIBM: libopenlibm - LLVM: libLLVM-14.0.6 (ORCJIT, haswell) + LLVM: libLLVM-14.0.6 (ORCJIT, skylake-avx512) Threads: 1 on 2 virtual cores
A more complete overview of all dependencies and their versions is also provided.
Status `~/work/StructuralIdentifiability.jl/StructuralIdentifiability.jl/docs/Manifest.toml`
   [47edcb42] ADTypes v0.2.4
   [a4c015fc] ANSIColoredPrinters v0.0.1
diff --git a/previews/PR238/input/input/index.html b/previews/PR238/input/input/index.html
index 933a60b5c..fe746f0d9 100644
--- a/previews/PR238/input/input/index.html
+++ b/previews/PR238/input/input/index.html
@@ -9,4 +9,4 @@
     x1'(t) = a * x1(t) + u(t),
     x2'(t) = b * x2(t) + c*x1(t)*x2(t),
     y(t) = x1(t)
-)

Here,

source
StructuralIdentifiability.ODE — Type

The main structure that represents input ODE system.

Stores information about states (x_vars), outputs (y_vars), inputs (u_vars), parameters (parameters) and the equations.

This structure is constructed via @ODEmodel macro.

source
StructuralIdentifiability.set_parameter_values — Function
set_parameter_values(ode, param_values)

Input:

  • ode - an ODE as above
  • param_values - values for (possibly, some of) the parameters as dictionary parameter => value

Output:

  • new ode with the parameters in param_values plugged with the given numbers
source

Create Compartmental Model

StructuralIdentifiability.linear_compartment_model — Function
linear_compartment_model(graph, inputs, outputs, leaks)

Input: defines a linear compartment model with nodes numbered from 1 to n by

  • graph - and array of integer arrays representing the adjacency lists of the graph
  • inputs - array of input nodes
  • outputs - array of output nodes
  • leaks - array of sink nodes

Output:

  • the corresponding ODE system in the notation of https://doi.org/10.1007/s11538-015-0098-0
source
+)

Here,

source
StructuralIdentifiability.ODE — Type

The main structure that represents input ODE system.

Stores information about states (x_vars), outputs (y_vars), inputs (u_vars), parameters (parameters) and the equations.

This structure is constructed via @ODEmodel macro.

source
StructuralIdentifiability.set_parameter_values — Function
set_parameter_values(ode, param_values)

Input:

  • ode - an ODE as above
  • param_values - values for (possibly, some of) the parameters as dictionary parameter => value

Output:

  • new ode with the parameters in param_values plugged with the given numbers
source

Create Compartmental Model

StructuralIdentifiability.linear_compartment_model — Function
linear_compartment_model(graph, inputs, outputs, leaks)

Input: defines a linear compartment model with nodes numbered from 1 to n by

  • graph - and array of integer arrays representing the adjacency lists of the graph
  • inputs - array of input nodes
  • outputs - array of output nodes
  • leaks - array of sink nodes

Output:

  • the corresponding ODE system in the notation of https://doi.org/10.1007/s11538-015-0098-0
source
diff --git a/previews/PR238/ioequations/ioequations/index.html b/previews/PR238/ioequations/ioequations/index.html index b39e74071..8855eecc6 100644 --- a/previews/PR238/ioequations/ioequations/index.html +++ b/previews/PR238/ioequations/ioequations/index.html @@ -3,4 +3,4 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub

Finding Input-Output Equations

StructuralIdentifiability.find_ioequations — Function
find_ioequations(ode, [var_change_policy=:default])

Finds the input-output equations of an ODE system Input:

  • ode - the ODE system
  • var_change_policy - whether to perform automatic variable change, can be one of :default, :yes, :no

Output:

  • a dictionary from “leaders” to the corresponding input-output equations; if an extra projection is needed, it will be the value corresponding to rand_proj_var
source

Reducing with respect to Input-Output Equations

StructuralIdentifiability.PBRepresentation — Type

The structure for storing a projection-based representation of differential ideal (see Section 2.3 https://arxiv.org/abs/2111.00991). Contains the following fields:

  • y_names - the names of the variables with finite order in the profile (typically, outputs)
  • u_names - the names of the variables with infinite order in the profile (typically, inputs)
  • param_names - the names of the parameters
  • profile - the profile of the PB-representation (see Definition 2.13) as a dict from y_names with finite orders to the orders
  • projections - the corresponding projections (see Definition 2.15) as a dict from y_names to the projections
source
StructuralIdentifiability.pseudodivision — Function
pseudodivision(f, g, x)

Computes the result of pseudodivision of f by g as univariate polynomials in x Input:

  • f - the polynomial to be divided
  • g - the polynomial to divide by
  • x - the variable for the division

Output: the pseudoremainder of f divided by g w.r.t. x

source
StructuralIdentifiability.diffreduce — Function
diffreduce(diffpoly, pbr)

Computes the result of differential reduction of a differential polynomial diffpoly with respect to the charset defined by a PB-representation pbr Input:

  • diffpoly - a polynomial representing a differential polynomial to be reduced
  • pbr - a projection-based representation

Output: the result of differential reduction of diffpoly by pbr considered as a characteristic set (see Remark 2.20 in the paper)

source
+
GitHub

Finding Input-Output Equations

StructuralIdentifiability.find_ioequations — Function
find_ioequations(ode, [var_change_policy=:default])

Finds the input-output equations of an ODE system Input:

  • ode - the ODE system
  • var_change_policy - whether to perform automatic variable change, can be one of :default, :yes, :no

Output:

  • a dictionary from “leaders” to the corresponding input-output equations; if an extra projection is needed, it will be the value corresponding to rand_proj_var
source

Reducing with respect to Input-Output Equations

StructuralIdentifiability.PBRepresentation — Type

The structure for storing a projection-based representation of differential ideal (see Section 2.3 https://arxiv.org/abs/2111.00991). Contains the following fields:

  • y_names - the names of the variables with finite order in the profile (typically, outputs)
  • u_names - the names of the variables with infinite order in the profile (typically, inputs)
  • param_names - the names of the parameters
  • profile - the profile of the PB-representation (see Definition 2.13) as a dict from y_names with finite orders to the orders
  • projections - the corresponding projections (see Definition 2.15) as a dict from y_names to the projections
source
StructuralIdentifiability.pseudodivision — Function
pseudodivision(f, g, x)

Computes the result of pseudodivision of f by g as univariate polynomials in x Input:

  • f - the polynomial to be divided
  • g - the polynomial to divide by
  • x - the variable for the division

Output: the pseudoremainder of f divided by g w.r.t. x

source
StructuralIdentifiability.diffreduce — Function
diffreduce(diffpoly, pbr)

Computes the result of differential reduction of a differential polynomial diffpoly with respect to the charset defined by a PB-representation pbr Input:

  • diffpoly - a polynomial representing a differential polynomial to be reduced
  • pbr - a projection-based representation

Output: the result of differential reduction of diffpoly by pbr considered as a characteristic set (see Remark 2.20 in the paper)

source
diff --git a/previews/PR238/search_index.js b/previews/PR238/search_index.js index da229708d..c999a7a9f 100644 --- a/previews/PR238/search_index.js +++ b/previews/PR238/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"utils/ode/#Functions-to-work-with-the-ODE-structure","page":"ODE Tools","title":"Functions to work with the ODE structure","text":"","category":"section"},{"location":"utils/ode/","page":"ODE Tools","title":"ODE Tools","text":"Modules = [StructuralIdentifiability]\nPages = [\"ODE.jl\", \"submodels.jl\"]\nOrder = [:function]","category":"page"},{"location":"utils/ode/#StructuralIdentifiability.__preprocess_ode-Tuple{ModelingToolkit.AbstractTimeDependentSystem, Array{<:Tuple{String, var\"#s6\"} where var\"#s6\"<:SymbolicUtils.BasicSymbolic}}","page":"ODE Tools","title":"StructuralIdentifiability.__preprocess_ode","text":"function __preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{Tuple{String, SymbolicUtils.BasicSymbolic}})\n\nInput:\n\nde - ModelingToolkit.AbstractTimeDependentSystem, a system for identifiability query\nmeasured_quantities - array of input function in the form (name, expression)\n\nOutput:\n\nODE object containing required data for identifiability assessment\nconversion dictionary from the symbols in the input MTK model to the variable involved in the produced ODE object\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability._get_var-Tuple{Any}","page":"ODE Tools","title":"StructuralIdentifiability._get_var","text":"For an expression of the form f'(t) or f(t) returns (f, true) and (f, false), resp\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability._reduce_mod_p-Tuple{Nemo.QQMPolyRingElem, Int64}","page":"ODE Tools","title":"StructuralIdentifiability._reduce_mod_p","text":"_reduce_mod_p(f, p)\n\nReduces a polynomial/rational function over Q modulo p\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability.power_series_solution-Union{Tuple{P}, Tuple{T}, Tuple{ODE{P}, Dict{P, T}, Dict{P, T}, Dict{P, Vector{T}}, Int64}} where {T<:AbstractAlgebra.FieldElem, P<:AbstractAlgebra.MPolyRingElem{T}}","page":"ODE Tools","title":"StructuralIdentifiability.power_series_solution","text":"power_series_solution(ode, param_values, initial_conditions, input_values, prec)\n\nInput:\n\node - an ode to solve\nparam_values - parameter values, must be a dictionary mapping parameter to a value\ninitial_conditions - initial conditions of ode, must be a dictionary mapping state variable to a value\ninput_values - power series for the inputs presented as a dictionary variable => list of coefficients\nprec - the precision of solutions\n\nOutput:\n\ncomputes a power series solution with precision prec presented as a dictionary variable => corresponding coordinate of the solution\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability.preprocess_ode-Tuple{ModelingToolkit.AbstractTimeDependentSystem, Array{Symbolics.Equation}}","page":"ODE Tools","title":"StructuralIdentifiability.preprocess_ode","text":"function preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{ModelingToolkit.Equation})\nfunction preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{SymbolicUtils.BasicSymbolic})\n\nInput:\n\nde - ModelingToolkit.AbstractTimeDependentSystem, a system for identifiability query\nmeasured_quantities - array of output functions (as equations of just functions)\n\nOutput:\n\nODE object containing required data for identifiability assessment\nconversion dictionary from the symbols in the input MTK model to the variable involved in the produced ODE object\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability.reduce_ode_mod_p-Tuple{ODE{<:AbstractAlgebra.MPolyRingElem{Nemo.QQFieldElem}}, Int64}","page":"ODE Tools","title":"StructuralIdentifiability.reduce_ode_mod_p","text":"reduce_ode_mod_p(ode, p)\n\nInput: ode is an ODE over QQ, p is a prime number Output: the reduction mod p, throws an exception if p divides one of the denominators\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability.set_parameter_values-Union{Tuple{P}, Tuple{T}, Tuple{ODE{P}, Dict{P, T}}} where {T<:AbstractAlgebra.FieldElem, P<:AbstractAlgebra.MPolyRingElem{T}}","page":"ODE Tools","title":"StructuralIdentifiability.set_parameter_values","text":"set_parameter_values(ode, param_values)\n\nInput:\n\node - an ODE as above\nparam_values - values for (possibly, some of) the parameters as dictionary parameter => value\n\nOutput:\n\nnew ode with the parameters in param_values plugged with the given numbers\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability.find_submodels-Union{Tuple{ODE{P}}, Tuple{P}} where P<:AbstractAlgebra.MPolyRingElem","page":"ODE Tools","title":"StructuralIdentifiability.find_submodels","text":"find_submodels(ode)\n\nThe function calculates and returns all valid submodels given a system of ODEs.\n\nInput:\n\node - an ODEs system to be studied\n\nOutput: \n\nA list of submodels represented as ode objects\n\nExample:\n\n>ode = @ODEmodel(x1'(t) = x1(t)^2, \n x2'(t) = x1(t) * x2(t), \n y1(t) = x1(t), \n y2(t) = x2(t))\n>find_submodels(ode)\n ODE{fmpq_mpoly}[\n \n x1'(t) = a(t)*x2(t)^2 + x1(t)\n y1(t) = x1(t)\n ]\n\n\n\n\n\n","category":"method"},{"location":"utils/primality/#Primality-Checks","page":"Primality Checks","title":"Primality Checks","text":"","category":"section"},{"location":"utils/primality/","page":"Primality Checks","title":"Primality Checks","text":"Pages=[\"primality.md\"]","category":"page"},{"location":"utils/primality/","page":"Primality Checks","title":"Primality Checks","text":"StructuralIdentifiability.check_primality","category":"page"},{"location":"utils/primality/#StructuralIdentifiability.check_primality","page":"Primality Checks","title":"StructuralIdentifiability.check_primality","text":"check_primality(polys::Dict{fmpq_mpoly, fmpq_mpoly}, extra_relations::Array{fmpq_mpoly, 1})\n\nThe function checks if the ideal generated by the polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals.\n\nThe extra_relations allows adding more polynomials to the generators (not affecting the saturation).\n\n\n\n\n\ncheck_primality(polys::Dict{fmpq_mpoly, fmpq_mpoly})\n\nThe function checks if the ideal generated by the polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals.\n\n\n\n\n\n","category":"function"},{"location":"tutorials/identifiable_functions/#Globally-Identifiable-Functions","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"","category":"section"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"In addition to assessing identifiabuility of given functions of parameters or states, StructuralIdentifiability.jl provides the function find_identifiable_functions which finds a set of identifiable functions such that any other identifiable function can be expressed using them. This allows to find out what actually is identifiable and what does non-identifiability in the model at hand looks like.","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"For example, consider the following model[1].","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"using StructuralIdentifiability # hide\nLLW1987 = @ODEmodel(\n x1'(t) = -p1 * x1(t) + p2 * u(t),\n x2'(t) = -p3 * x2(t) + p4 * u(t),\n x3'(t) = -(p1 + p3) * x3(t) + (p4 * x1(t) + p2 * x2(t)) * u(t),\n y1(t) = x3(t)\n)","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"Several decades ago, this model was introduced to demonstrate the presence of nontrivial unidentifiability in nonlinear systems of ODEs. Nowadays, we can automatically find the identifiable combinations of parameters:","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"using StructuralIdentifiability # hide\nfind_identifiable_functions(LLW1987)","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"From these expressions, we see that the values of p1 and p3 are not identifiable but an unordered pair of numbers {p1, p3} is uniquely determined since p1 + p3 and p1 * p3 are known. Furthermore, we see that, for fixed input and output, p2 and p4 can take infinitely many values but knowing one of them, we would also be able to determine the other.","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"Moreover, we can find generators of all identifiable functions in parameters and states:","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"find_identifiable_functions(LLW1987, with_states = true)","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"By default, find_identifiable_functions tries to simplify the output functions as much as possible, and it has simplify keyword responsible for the degree of simplification. The default value is :standard but one could use :strong to try to simplify further (at the expense of heavier computation) or use :weak to simplify less (but compute faster).","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"[1]: Y. Lecourtier, F. Lamnabhi-Lagarrigue, and E. Walter, A method to prove that nonlinear models can be unidentifiable, Proceedings of the 26th Conference on Decision and Control, December 1987, 2144-2145;","category":"page"},{"location":"tutorials/identifiability/#Identifiability-of-Differential-Models-(Local-and-Global)","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"","category":"section"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Recall that we consider ODE models in the state-space form","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"begincases\nmathbfx(t) = mathbff(mathbfx(t) mathbfp mathbfu(t))\nmathbfy(t) = mathbfg(mathbfx(t) mathbfp mathbfu(t))\nendcases","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"where mathbfx(t) mathbfy(t), and mathbfu(t) are time-dependent states, outputs, and inputs, respectively, and mathbfp are scalar parameters. We will call that a parameter or a states (or a function of them) is identifiable if its value can be recovered from time series for inputs and outputs. Typically, two types of identifiability are distinguished","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"local identifiability: the value can be recovered up to finitely many options;\nglobal identifiability: the value can be recovered uniquely.","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Note that in the case of states, term observability it typically used. We will use identifiability for both states and parameters for brevity and uniformity. While the notion of global identifiability is much more precise, assessing local identifiability is typically much faster, and can be performed for the models whose global identifiability analysis is out of reach.","category":"page"},{"location":"tutorials/identifiability/#Local-identifiability","page":"Identifiability of Differential Models (Local and Global)","title":"Local identifiability","text":"","category":"section"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"We consider the Lotka-Volterra model:","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"begincases\nx_1(t) = a x_1(t) - b x_1(t) x_2(t) + u(t)\nx_2(t) = -c x_2(t) + d x_1(t) x_2(t)\ny(t) = x_1(t)\nendcases","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"The local identifiability of all parameters and states in this model can be assessed as follows","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"using StructuralIdentifiability\n\node = @ODEmodel(\n x1'(t) = a * x1(t) - b * x1(t) * x2(t) + u(t),\n x2'(t) = -c * x2(t) + d * x1(t) * x2(t),\n y(t) = x1(t)\n)\n\nassess_local_identifiability(ode)","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"We see that x_1(t) is locally identifiable (no surprises, this is an output), a c and d are identifiable as well. On the other hand, x_2(t) and b are nonidentifiable. This can be explained by the following scaling symmetry","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"x_2(t) to lambda x_2(t) quad b to fracblambda","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"which preserves input and output for every nonzero lambda. The algorithm behind this call is the one due to Sedoglavic[1].","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Function assess_local_identifiability has several optional parameters","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"funcs_to_check a list of specific functions of parameters and states to check identifiability for (see an example below). If not provided, the identifiability is assessed for all parameters and states.\np (default 099) is the probability of correctness. The algorithm can, in theory, produce wrong result, but the probability that it is correct is guaranteed to be at least p. However, the probability bounds we use are quite conservative, so the actual probability of correctness is likely to be much higher.\ntype (default :SE). By default, the algorithm checks the standard single-experiment identifiability. If one sets type = :ME, then the algorithm checks multi-experiment identifiability, that is, identifiability from several experiments with independent initial conditions (the algorithm from [2] is used).","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Note that the scaling symmetry given above suggests that b x_2(t) may in fact be identifiable. This can be checked using funcs_to_check parameter:","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"assess_local_identifiability(ode, funcs_to_check = [b * x2])","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Indeed!","category":"page"},{"location":"tutorials/identifiability/#Global-identifiability","page":"Identifiability of Differential Models (Local and Global)","title":"Global identifiability","text":"","category":"section"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"One can obtain more refined information about a model using assess_identifiability function. We will showcase it using the Goodwin oscillator model [3].","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"using StructuralIdentifiability\n\node = @ODEmodel(\n x1'(t) = -b * x1(t) + 1 / (c + x4(t)),\n x2'(t) = alpha * x1(t) - beta * x2(t),\n x3'(t) = gama * x2(t) - delta * x3(t),\n x4'(t) = sigma * x4(t) * (gama * x2(t) - delta * x3(t)) / x3(t),\n y(t) = x1(t)\n)\n\nassess_identifiability(ode)","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"As a result, each parameter/state is assigned one of the labels :globally (globally identifiable), :locally (locally but not globally identifiable), or :nonidentifiable (not identifiable, even locally). The algorithm behind this computation follows [4].","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Similarly to assess_local_identifiability, this function has optional parameters:","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"funcs_to_check a list of specific functions of parameters and states to check identifiability for (see an example below). If not provided, the identifiability is assessed for all parameters and states. Note that the computations for states may be more involved than for the parameters, so one may want to call the function with funcs_to_check = ode.parameters if the call assess_identifiability(ode) takes too long.\np (default 099) is the probability of correctness. Same story as above: the probability estimates are very conservative, so the actual error probability is much lower than 1%. Also, currently, the probability of correctness does not include the probability of correctness of the modular reconstruction for Groebner bases. This probability is ensured by an additional check modulo a large prime, and can be neglected for practical purposes.","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Using funcs_to_check parameter, one can further inverstigate the nature of the lack of identifiability in the model at hand. For example, for the Goodwin oscillator, we can check if beta + delta and beta * delta are identifiable:","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"assess_identifiability(ode, funcs_to_check = [beta + delta, beta * delta])","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"And we see that they indeed are. This means, in particular, that the reason why beta and delta are not identifiable is because their values can be exchanged. One may wonder how could we guess these functions beta + delta, beta * delta. In fact, they can be just computed using find_identifiable_functions function as we will explain in the next tutorial. Stay tuned!","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"[1]: A. Sedoglavic, A probabilistic algorithm to test local algebraic observability in polynomial time, Journal of Symbolic Computation, 2002.","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"[2]: A. Ovchinnikov, A. Pillay, G. Pogudin, T. Scanlon, Multi-experiment Parameter Identifiability of ODEs and Model Theory, SIAM Journal on Applied Algebra and Geometry, 2022.","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"[3]: D. Gonze, P. Ruoff, The Goodwin Oscillator and its Legacy, Acta Biotheoretica, 2020.","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"[4]: R. Dong, C. Goodbrake, H. Harrington, G. Pogudin, Differential elimination for dynamical models via projections with applications to structural identifiability, SIAM Journal on Applied Algebra and Geometry, 2023.","category":"page"},{"location":"export/export/#Exporting-to-Other-Systems","page":"Exporting to Other Systems","title":"Exporting to Other Systems","text":"","category":"section"},{"location":"export/export/","page":"Exporting to Other Systems","title":"Exporting to Other Systems","text":"Here we put some helpful utilities to export your code to other identifiability software.","category":"page"},{"location":"export/export/","page":"Exporting to Other Systems","title":"Exporting to Other Systems","text":"print_for_maple\nprint_for_DAISY\nprint_for_GenSSI\nprint_for_COMBOS","category":"page"},{"location":"export/export/#StructuralIdentifiability.print_for_maple","page":"Exporting to Other Systems","title":"StructuralIdentifiability.print_for_maple","text":"print_for_maple(ode, package)\n\nPrints the ODE in the format accepted by maple packages\n\nSIAN (https://github.com/pogudingleb/SIAN) if package=:SIAN\nDifferentialAlgebra if package=:DifferentialAlgebra\nDifferentialThomas if package=:DifferentialThomas\n\n\n\n\n\n","category":"function"},{"location":"export/export/#StructuralIdentifiability.print_for_DAISY","page":"Exporting to Other Systems","title":"StructuralIdentifiability.print_for_DAISY","text":"print_for_DAISY(ode)\n\nPrints the ODE in the format accepted by DAISY (https://daisy.dei.unipd.it/)\n\n\n\n\n\n","category":"function"},{"location":"export/export/#StructuralIdentifiability.print_for_GenSSI","page":"Exporting to Other Systems","title":"StructuralIdentifiability.print_for_GenSSI","text":"print_for_GenSSI(ode)\n\nPrints the ODE in the format accepted by GenSSI 2.0 (https://github.com/genssi-developer/GenSSI)\n\n\n\n\n\n","category":"function"},{"location":"export/export/#StructuralIdentifiability.print_for_COMBOS","page":"Exporting to Other Systems","title":"StructuralIdentifiability.print_for_COMBOS","text":"print_for_COMBOS(ode)\n\nPrints the ODE in the format accepted by COMBOS (http://biocyb1.cs.ucla.edu/combos/)\n\n\n\n\n\n","category":"function"},{"location":"utils/util/#Other-Helpful-Functions","page":"Other Utilities","title":"Other Helpful Functions","text":"","category":"section"},{"location":"utils/util/","page":"Other Utilities","title":"Other Utilities","text":"Pages=[\"util.md\"]","category":"page"},{"location":"utils/util/","page":"Other Utilities","title":"Other Utilities","text":"Modules = [StructuralIdentifiability]\nPages = [\"util.jl\"]","category":"page"},{"location":"utils/util/#StructuralIdentifiability.compare_rational_func_by-Union{Tuple{By}, Tuple{Any, Any, By}, Tuple{Any, Any, By, Any}} where By","page":"Other Utilities","title":"StructuralIdentifiability.compare_rational_func_by","text":"compare_rational_func_by(f, g, by)\n\nReturns \n\n-1 if f < g,\n0 if f = g, and \n1 if f > g.\n\nFunctions' numerators and denominators are compared using by.\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.decompose_derivative-Tuple{String, Array{String}}","page":"Other Utilities","title":"StructuralIdentifiability.decompose_derivative","text":"decompose_derivative(varname, prefixes)\n\nDetermines if it is possible to represent the varname as a_number where a is an element of prefixes If yes, returns a pair (a, number), otherwise nothing\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.dennums_to_fractions-Union{Tuple{Array{Vector{T}, 1}}, Tuple{T}} where T","page":"Other Utilities","title":"StructuralIdentifiability.dennums_to_fractions","text":"dennums_to_fractions(dennums)\n\nReturns the field generators represented by fractions.\n\nInput: an array of arrays of polynomials, as in [[f1, f2, f3, ...], [g1, g2, g3, ...], ...]\n\nOutput: an array of fractions [f2/f1, f3/f1, ..., g2/g1, g3/g1, ...]\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.eval_at_dict-Union{Tuple{P}, Tuple{P, Dict{P, <:AbstractAlgebra.RingElem}}} where P<:AbstractAlgebra.MPolyRingElem","page":"Other Utilities","title":"StructuralIdentifiability.eval_at_dict","text":"eval_at_dict(f, d)\n\nEvaluates a polynomial/rational function on a dictionary of type var => val and missing values are replaced with zeroes\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.extract_coefficients-Union{Tuple{P}, Tuple{P, Vector{P}}} where P<:AbstractAlgebra.MPolyRingElem","page":"Other Utilities","title":"StructuralIdentifiability.extract_coefficients","text":"extract_coefficients(poly, variables)\n\nInput:\n\npoly - multivariate polynomial\nvariables - a list of variables from the generators of the ring of p\n\nOutput:\n\ndictionary with keys being tuples of length lenght(variables) and values being polynomials in the variables other than those which are the coefficients at the corresponding monomials (in a smaller polynomial ring)\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.fractions_to_dennums-Tuple{Any}","page":"Other Utilities","title":"StructuralIdentifiability.fractions_to_dennums","text":"fractions_to_dennums(fractions)\n\nReturns the field generators represented by lists of denominators and numerators.\n\nInput: an array of fractions, as in [f2/f1, f3/f1, ..., g2/g1, g3/g1, ...]\n\nOutput: an array of arrays of polynomials, [[f1, f2, f3, ...], [g1, g2, g3, ...], ...]\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.gen_tag_name","page":"Other Utilities","title":"StructuralIdentifiability.gen_tag_name","text":"gen_tag_name(base; stop_words)\ngen_tag_names(n, base; stop_words)\n\nGenerates a string which will not collide with the words in stop_words.\n\nArguments\n\nn: Generates a sequence of unique strings of length n\nbase: A string or a vector of strings, the base for the generated sequence\nstop_words: A vector of strings, stop words\n\n\n\n\n\n","category":"function"},{"location":"utils/util/#StructuralIdentifiability.make_substitution-Union{Tuple{P}, NTuple{4, P}} where P<:AbstractAlgebra.MPolyRingElem","page":"Other Utilities","title":"StructuralIdentifiability.make_substitution","text":"make_substitution(f, var_sub, val_numer, val_denom)\n\nSubstitute a variable in a polynomial with an expression\n\nInput:\n\nf - the polynomial\nvar_sub - the variable to be substituted\nvar_numer - numerator of the substitution expression\nvar_denom - denominator of the substitution expression\n\nOutput:\n\npolynomial - result of substitution\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.parent_ring_change-Tuple{AbstractAlgebra.MPolyRingElem, AbstractAlgebra.MPolyRing}","page":"Other Utilities","title":"StructuralIdentifiability.parent_ring_change","text":"parent_ring_change(poly, new_ring)\n\nConverts a polynomial to a different polynomial ring Input\n\npoly - a polynomial to be converted\nnew_ring - a polynomial ring such that every variable name appearing in poly appears among the generators\n\nOutput:\n\na polynomial in new_ring “equal” to poly\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.switch_ring-Tuple{AbstractAlgebra.MPolyRingElem, AbstractAlgebra.MPolyRing}","page":"Other Utilities","title":"StructuralIdentifiability.switch_ring","text":"switch_ring(v, ring)\n\nFor a variable v, returns a variable in ring with the same name\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.uncertain_factorization-Tuple{AbstractAlgebra.MPolyRingElem{Nemo.QQFieldElem}}","page":"Other Utilities","title":"StructuralIdentifiability.uncertain_factorization","text":"uncertain_factorization(f)\n\nInput:\n\nf - polynomial with rational coefficients\n\nOutput:\n\nlist of pairs (div, certainty) where\ndiv's are divisors of f such that f is their product with certain powers\nif certainty is true, div is Q-irreducible\n\n\n\n\n\n","category":"method"},{"location":"utils/wronskian/#Wronskian-Tools","page":"Wronskian Tools","title":"Wronskian Tools","text":"","category":"section"},{"location":"utils/wronskian/","page":"Wronskian Tools","title":"Wronskian Tools","text":"Modules = [StructuralIdentifiability]\nPages = [\"wronskian.jl\"]","category":"page"},{"location":"utils/wronskian/#StructuralIdentifiability.get_max_below-Tuple{StructuralIdentifiability.ExpVectTrie, Vector{Int64}}","page":"Wronskian Tools","title":"StructuralIdentifiability.get_max_below","text":"get_max_below(t, vect)\n\nInput:\n\nt - a trie with exponent vectors\nvect - yet another exponent vector\n\nOutput:\n\na pair (d, v) where v is a vector in the trie which is componentwise ≤ vect and the difference d is as small as possible\n\n\n\n\n\n","category":"method"},{"location":"utils/wronskian/#StructuralIdentifiability.massive_eval-Tuple{Any, Any}","page":"Wronskian Tools","title":"StructuralIdentifiability.massive_eval","text":"massive_eval(polys, eval_dict)\n\nInput:\n\npolys - a list of polynomials\neval_dict - dictionary from variables to the values. Missing values are treated as zeroes\n\nOutput:\n\na list of values of the polynomials\n\nEvaluates a list of polynomials at a point. Assumes that multiplications are relatively expensive (like in truncated power series) so all the monomials are precomputed first and the values of monomials of lower degree are cached and used to compute the values of the monomials of higher degree\n\n\n\n\n\n","category":"method"},{"location":"utils/wronskian/#StructuralIdentifiability.monomial_compress-Tuple{Any, ODE}","page":"Wronskian Tools","title":"StructuralIdentifiability.monomial_compress","text":"monomial_compress(io_equation, ode)\n\nCompresses an input-output equation for the rank computation Input:\n\nio_equation - input-output equation\node - the corresponding ODE model\n\nOutput:\n\npair (coeffs, terms) such that:\nsum of coeffs[i] * terms[i] = io_equation\ncoeffs involve only parameters, terms involve only inputs and outputs\nlength of the representation is the smallest possible\n\n\n\n\n\n","category":"method"},{"location":"utils/wronskian/#StructuralIdentifiability.wronskian-Union{Tuple{P}, Tuple{Dict{P, P}, ODE{P}}} where P<:AbstractAlgebra.MPolyRingElem","page":"Wronskian Tools","title":"StructuralIdentifiability.wronskian","text":"wronskian(io_equations, ode)\n\nInput:\n\nio_equations - a set of io-equations in the form of the Dict as returned by find_ioequations\node - the ODE object\n\nOutput:\n\na list of Wronskians evaluated at a point modulo prime\n\nComputes the Wronskians of io_equations\n\n\n\n\n\n","category":"method"},{"location":"utils/global_identifiability/#Global-Identifiability-Tools","page":"Global Identifiability Tools","title":"Global Identifiability Tools","text":"","category":"section"},{"location":"utils/global_identifiability/","page":"Global Identifiability Tools","title":"Global Identifiability Tools","text":"Pages=[\"global_identifiability.md\"]","category":"page"},{"location":"utils/global_identifiability/","page":"Global Identifiability Tools","title":"Global Identifiability Tools","text":"CurrentModule=StructuralIdentifiability","category":"page"},{"location":"utils/global_identifiability/","page":"Global Identifiability Tools","title":"Global Identifiability Tools","text":"StructuralIdentifiability.RationalFunctionField\nStructuralIdentifiability.field_contains\nStructuralIdentifiability.get_degree_and_coeffsize","category":"page"},{"location":"utils/global_identifiability/#StructuralIdentifiability.RationalFunctionField","page":"Global Identifiability Tools","title":"StructuralIdentifiability.RationalFunctionField","text":"RationalFunctionField\n\nA subfield of the field of rational functions over the rationals.\n\nExample\n\nusing Nemo\nusing StructuralIdentifiability: RationalFunctionField\n\nR, (x, y, z) = QQ[\"x\", \"y\", \"z\"]\n\n# Constructs a subfield generated by x / y, y / z\nrff = RationalFunctionField([x // y, y // z])\n\n# Constructs a subfield generated by y / x, 1 / x, z / y\nrff = RationalFunctionField([[x, y, R(1)], [y, z]])\n\n\n\n\n\n","category":"type"},{"location":"utils/global_identifiability/#StructuralIdentifiability.field_contains","page":"Global Identifiability Tools","title":"StructuralIdentifiability.field_contains","text":"field_contains(field, ratfuncs, p)\n\nChecks whether given rational function field field contains given rational functions ratfuncs (represented as a list of lists). The result is correct with probability at least p\n\nInputs:\n\nfield - a rational function field\nratfuncs - a list of lists of polynomials. Each of the lists, say, [f1, ..., fn], defines generators f2/f1, ..., fn/f1.\np real number from (0, 1)\n\nOutput:\n\na list L[i] of bools of length length(rat_funcs) such that L[i] is true iff the i-th function belongs to field\n\n\n\n\n\n","category":"function"},{"location":"utils/global_identifiability/#StructuralIdentifiability.get_degree_and_coeffsize","page":"Global Identifiability Tools","title":"StructuralIdentifiability.get_degree_and_coeffsize","text":"get_degree_and_coeffsize(f)\n\nfor f being a polynomial/rational function over rationals (QQ) returns a tuple (degree, max_coef_size)\n\n\n\n\n\n","category":"function"},{"location":"utils/reparametrization/#Model-reparametrization","page":"Model reparametrization","title":"Model reparametrization","text":"","category":"section"},{"location":"utils/reparametrization/","page":"Model reparametrization","title":"Model reparametrization","text":"StructuralIdentifiability.reparametrize_global","category":"page"},{"location":"utils/reparametrization/#StructuralIdentifiability.reparametrize_global","page":"Model reparametrization","title":"StructuralIdentifiability.reparametrize_global","text":"reparametrize_global(ode, options...)\n\nFinds a reparametrization of ode in terms of globally identifiabile functions.\n\nReturns a tuple (new_ode, new_vars, implicit_relations), such that:\n\nnew_ode is the reparametrized ODE system\nnew_vars is a mapping from the new variables to the original ones\nrelations is the array of implicit algebraic relations among new_vars. Usually, relations is empty.\n\nOptions\n\nThe function accepts the following optional arguments.\n\nseed: A float in the range from 0 to 1, random seed (default is seed = 42). \np: The probability of correctness (default is p = 0.99).\n\nExample\n\nusing StructuralIdentifiability\n\node = @ODEmodel(\n x1'(t) = a * x1(t) - b*x1(t)*x2(t),\n x2'(t) = -c * x2(t) + d*x1(t)*x2(t),\n y(t) = x1(t)\n)\n\nnew_ode, new_vars, relations = reparametrize_global(ode)\n\nThen, we have the following:\n\n# new_ode\nX2'(t) = X1(t)*X2(t)*a2 - X2(t)*a1\nX1'(t) = -X1(t)*X2(t) + X1(t)*a3\ny1(t) = X1(t)\n\n# new_vars\nDict{Nemo.QQMPolyRingElem, AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}} with 6 entries:\n X2 => b*x2\n y1 => y\n X1 => x1\n a2 => d\n a3 => a\n a1 => c\n\nNotice that the new_ode is fully identifiabile, and has 1 less parameter compared to the original one.\n\n\n\n\n\n","category":"function"},{"location":"utils/power_series_utils/#Power-Series-Utilities","page":"Power Series Tools","title":"Power Series Utilities","text":"","category":"section"},{"location":"utils/power_series_utils/","page":"Power Series Tools","title":"Power Series Tools","text":"Pages =[\"power_series_utils.md\"]","category":"page"},{"location":"utils/power_series_utils/","page":"Power Series Tools","title":"Power Series Tools","text":"Modules = [StructuralIdentifiability]\nPages = [\"power_series_utils.jl\"]","category":"page"},{"location":"utils/power_series_utils/#StructuralIdentifiability._matrix_inv_newton_iteration-Union{Tuple{T}, Tuple{AbstractAlgebra.MatElem{T}, AbstractAlgebra.MatElem{T}}} where T<:(AbstractAlgebra.AbsPowerSeriesRingElem{<:AbstractAlgebra.FieldElem})","page":"Power Series Tools","title":"StructuralIdentifiability._matrix_inv_newton_iteration","text":"_matrix_inv_newton_iteration(M, Minv)\n\nPerforms a single step of Newton iteration for inverting M with Minv being a partial result\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_diff-Tuple{AbstractAlgebra.AbsPowerSeriesRingElem{<:AbstractAlgebra.RingElem}}","page":"Power Series Tools","title":"StructuralIdentifiability.ps_diff","text":"ps_diff(ps)\n\nInput:\n\nps - (absolute capped) univariate power series\n\nOutput:\n\nthe derivative of ps\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_integrate-Tuple{AbstractAlgebra.AbsPowerSeriesRingElem{<:AbstractAlgebra.FieldElem}}","page":"Power Series Tools","title":"StructuralIdentifiability.ps_integrate","text":"ps_integrate(ps)\n\nInput:\n\nps - (absolute capped) univariate power series\n\nOutput:\n\nthe integral of ps without constant term\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_matrix_homlinear_de-Union{Tuple{T}, Tuple{AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:T}}, Tuple{AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:T}, Int64}} where T<:AbstractAlgebra.FieldElem","page":"Power Series Tools","title":"StructuralIdentifiability.ps_matrix_homlinear_de","text":"ps_matrix_homlinear_de(A, Y0, prec)\n\nInput:\n\nA - a square matrix with entries in a univariate power series ring\nY0 - a square invertible matrix over the base field\n\nOutput:\n\nmatrix Y such that Y' = AY up to precision of A - 1 and Y(0) = Y0\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_matrix_inv","page":"Power Series Tools","title":"StructuralIdentifiability.ps_matrix_inv","text":"ps_matrix_inv(M, prec)\n\nInput:\n\nM - a square matrix with entries in a univariate power series ring it is assumed that M(0) is invertible and all entries having the same precision\nprec - an integer, precision, if -1 then defaults to precision of M\n\nOutput:\n\nthe inverse of M computed up to prec\n\n\n\n\n\n","category":"function"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_matrix_linear_de-Union{Tuple{T}, Tuple{AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:T}}, Tuple{AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:T}, Int64}} where T<:AbstractAlgebra.FieldElem","page":"Power Series Tools","title":"StructuralIdentifiability.ps_matrix_linear_de","text":"ps_matrix_linear_de(A, B, Y0, prec)\n\nInput:\n\nA, B - square matrices with entries in a univariate power series ring\nY0 - a matrix over the base field with the rows number the same as A\n\nOutput:\n\nmatrix Y such that Y' = AY + B up to precision of A - 1 and Y(0) = Y0\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_matrix_log-Tuple{AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{<:AbstractAlgebra.FieldElem}}}","page":"Power Series Tools","title":"StructuralIdentifiability.ps_matrix_log","text":"ps_matrix_log(M)\n\nInput:\n\nM - a square matrix with entries in a univariate power series ring it is assumed that M(0) is the identity\n\nOutput:\n\nthe natural log of M\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_ode_solution-Union{Tuple{P}, Tuple{T}, Tuple{Vector{P}, Dict{P, T}, Dict{P, Vector{T}}, Int64}} where {T<:AbstractAlgebra.FieldElem, P<:AbstractAlgebra.MPolyRingElem{T}}","page":"Power Series Tools","title":"StructuralIdentifiability.ps_ode_solution","text":"ps_ode_solution(equations, ic, inputs, prec)\n\nInput:\n\nequations - a system of the form A(x u mu)x - B(x u mu) = 0, where A is a generically nonsingular square matrix. Assumption: A is nonzero at zero\nic - initial conditions for x's (dictionary)\ninputs - power series for inputs represented as arrays (dictionary)\nprec - precision of the solution\n\nOutput:\n\npower series solution of the system\n\n\n\n\n\n","category":"method"},{"location":"identifiability/identifiability/#Functions-to-Assess-Identifiability","page":"Functions to Assess Identifiability","title":"Functions to Assess Identifiability","text":"","category":"section"},{"location":"identifiability/identifiability/#Assessing-All-Types-of-Identifiability","page":"Functions to Assess Identifiability","title":"Assessing All Types of Identifiability","text":"","category":"section"},{"location":"identifiability/identifiability/","page":"Functions to Assess Identifiability","title":"Functions to Assess Identifiability","text":"assess_identifiability","category":"page"},{"location":"identifiability/identifiability/#StructuralIdentifiability.assess_identifiability","page":"Functions to Assess Identifiability","title":"StructuralIdentifiability.assess_identifiability","text":"assess_identifiability(ode; funcs_to_check = [], p=0.99)\n\nInput:\n\node - the ODE model\nfuncs_to_check - list of functions to check identifiability for; if empty, all parameters and states are taken\np - probability of correctness.\n\nAssesses identifiability of a given ODE model. The result is guaranteed to be correct with the probability at least p. The function returns a dictionary from the functions to check to their identifiability properties (one of :nonidentifiable, :locally, :globally).\n\n\n\n\n\nassess_identifiability(ode::ModelingToolkit.ODESystem; measured_quantities=Array{ModelingToolkit.Equation}[], funcs_to_check=[], p = 0.99)\n\nInput:\n\node - the ModelingToolkit.ODESystem object that defines the model\nmeasured_quantities - the output functions of the model\nfuncs_to_check - functions of parameters for which to check the identifiability\np - probability of correctness.\n\nAssesses identifiability (both local and global) of a given ODE model (parameters detected automatically). The result is guaranteed to be correct with the probability at least p.\n\n\n\n\n\n","category":"function"},{"location":"identifiability/identifiability/#Assessing-Local-Identifiability","page":"Functions to Assess Identifiability","title":"Assessing Local Identifiability","text":"","category":"section"},{"location":"identifiability/identifiability/","page":"Functions to Assess Identifiability","title":"Functions to Assess Identifiability","text":"assess_local_identifiability","category":"page"},{"location":"identifiability/identifiability/#StructuralIdentifiability.assess_local_identifiability","page":"Functions to Assess Identifiability","title":"StructuralIdentifiability.assess_local_identifiability","text":"function assess_local_identifiability(ode::ModelingToolkit.ODESystem; measured_quantities=Array{ModelingToolkit.Equation}[], funcs_to_check=Array{}[], p::Float64=0.99, type=:SE)\n\nInput:\n\node - the ODESystem object from ModelingToolkit\nmeasured_quantities - the measurable outputs of the model\nfuncs_to_check - functions of parameters for which to check identifiability\np - probability of correctness\ntype - identifiability type (:SE for single-experiment, :ME for multi-experiment)\n\nOutput:\n\nfor type=:SE, the result is a dictionary from each parameter to boolean;\nfor type=:ME, the result is a tuple with the dictionary as in :SE case and array of number of experiments.\n\nThe function determines local identifiability of parameters in funcs_to_check or all possible parameters if funcs_to_check is empty\n\nThe result is correct with probability at least p.\n\ntype can be either :SE (single-experiment identifiability) or :ME (multi-experiment identifiability). The return value is a tuple consisting of the array of bools and the number of experiments to be performed.\n\n\n\n\n\nassess_local_identifiability(ode::ODE{P}; funcs_to_check::Array{<: Any, 1}, p::Float64=0.99, type=:SE) where P <: MPolyElem{Nemo.fmpq}\n\nChecks the local identifiability/observability of the functions in funcs_to_check. The result is correct with probability at least p.\n\nCall this function if you have a specific collection of parameters of which you would like to check local identifiability.\n\ntype can be either :SE (single-experiment identifiability) or :ME (multi-experiment identifiability). If the type is :ME, states are not allowed to appear in the funcs_to_check.\n\n\n\n\n\nfunction assess_local_identifiability(\n dds::ModelingToolkit.DiscreteSystem; \n measured_quantities=Array{ModelingToolkit.Equation}[], \n funcs_to_check=Array{}[], \n known_ic=Array{}[],\n p::Float64=0.99)\n\nInput:\n\ndds - the DiscreteSystem object from ModelingToolkit\nmeasured_quantities - the measurable outputs of the model\nfuncs_to_check - functions of parameters for which to check identifiability (all parameters and states if not specified)\nknown_ic - functions (of states and parameter) whose initial conditions are assumed to be known\np - probability of correctness\n\nOutput:\n\nthe result is a dictionary from each function to to boolean;\n\nThe result is correct with probability at least p.\n\n\n\n\n\n","category":"function"},{"location":"identifiability/identifiability/#Assessing-Global-Identifiability","page":"Functions to Assess Identifiability","title":"Assessing Global Identifiability","text":"","category":"section"},{"location":"identifiability/identifiability/","page":"Functions to Assess Identifiability","title":"Functions to Assess Identifiability","text":"assess_global_identifiability","category":"page"},{"location":"identifiability/identifiability/#StructuralIdentifiability.assess_global_identifiability","page":"Functions to Assess Identifiability","title":"StructuralIdentifiability.assess_global_identifiability","text":"assess_global_identifiability(ode::ODE{P}, p::Float64=0.99; var_change=:default) where P <: MPolyElem{fmpq}\n\nInput:\n\node - the ODE model\nknown - a list of functions in states which are assumed to be known and generic\np - probability of correctness\nvar_change - a policy for variable change (:default, :yes, :no), affects only the runtime\n\nOutput:\n\na dictionary mapping each parameter to a boolean.\n\nChecks global identifiability for parameters of the model provided in ode. Call this function to check global identifiability of all parameters automatically.\n\n\n\n\n\nassess_global_identifiability(ode, [funcs_to_check, p=0.99, var_change=:default])\n\nInput:\n\node - the ODE model\nfuncs_to_check - rational functions in parameters\nknown - function in parameters that are assumed to be known and generic\np - probability of correctness\nvar_change - a policy for variable change (:default, :yes, :no), affects only the runtime\n\nOutput:\n\narray of length length(funcs_to_check) with true/false values for global identifiability or dictionary param => Bool if funcs_to_check are not given\n\nChecks global identifiability of functions of parameters specified in funcs_to_check.\n\n\n\n\n\n","category":"function"},{"location":"identifiability/identifiability/#Finding-Identifiable-Functions","page":"Functions to Assess Identifiability","title":"Finding Identifiable Functions","text":"","category":"section"},{"location":"identifiability/identifiability/","page":"Functions to Assess Identifiability","title":"Functions to Assess Identifiability","text":"find_identifiable_functions","category":"page"},{"location":"identifiability/identifiability/#StructuralIdentifiability.find_identifiable_functions","page":"Functions to Assess Identifiability","title":"StructuralIdentifiability.find_identifiable_functions","text":"find_identifiable_functions(ode::ODE; options...)\n\nFinds all functions of parameters/states that are identifiable in the given ODE system.\n\nOptions\n\nThis functions takes the following optional arguments:\n\nwith_states: When true, also reports the identifiabile functions in the ODE states. Default is false.\nsimplify: The extent to which the output functions are simplified. Stronger simplification may require more time. Possible options are:\n:standard: Default simplification.\n:weak: Weak simplification. This option is the fastest, but the output functions can be quite complex.\n:strong: Strong simplification. This option is the slowest, but the output\nfunctions are nice and simple.\n:absent: No simplification.\np: A float in the range from 0 to 1, the probability of correctness. Default is 0.99.\nseed: The rng seed. Default value is 42.\n\nExample\n\nusing StructuralIdentifiability\n\node = @ODEmodel(\n x0'(t) = -(a01 + a21) * x0(t) + a12 * x1(t),\n x1'(t) = a21 * x0(t) - a12 * x1(t),\n y(t) = x0(t)\n)\n\nfind_identifiable_functions(ode)\n\n# prints\n3-element Vector{AbstractAlgebra.Generic.Frac{Nemo.fmpq_mpoly}}:\n a12 + a01 + a21\n a12*a01\n\n\n\n\n\nfind_identifiable_functions(ode::ModelingToolkit.ODESystem; measured_quantities=[], options...)\n\nFinds all functions of parameters/states that are identifiable in the given ODE system.\n\nOptions\n\nThis functions takes the following optional arguments:\n\nmeasured_quantities - the output functions of the model.\n\nExample\n\nusing StructuralIdentifiability\nusing ModelingToolkit\n\n@parameters a01 a21 a12\n@variables t x0(t) x1(t) y1(t) [output = true]\nD = Differential(t)\n\neqs = [\n D(x0) ~ -(a01 + a21) * x0 + a12 * x1, \n D(x1) ~ a21 * x0 - a12 * x1, y1 ~ x0\n]\nde = ODESystem(eqs, t, name = :Test)\n\nfind_identifiable_functions(de, measured_quantities = [y1 ~ x0])\n\n# prints\n2-element Vector{Num}:\n a01*a12\n a01 + a12 + a21\n\n\n\n\n\n","category":"function"},{"location":"utils/elimination/#Elimination","page":"Elimination","title":"Elimination","text":"","category":"section"},{"location":"utils/elimination/","page":"Elimination","title":"Elimination","text":"Pages=[\"elimination.md\"]","category":"page"},{"location":"utils/elimination/","page":"Elimination","title":"Elimination","text":"Modules = [StructuralIdentifiability]\nPages = [\"elimination.jl\"]","category":"page"},{"location":"utils/elimination/#StructuralIdentifiability.Bezout_matrix-Union{Tuple{P}, Tuple{P, P, P}} where P<:AbstractAlgebra.MPolyRingElem","page":"Elimination","title":"StructuralIdentifiability.Bezout_matrix","text":"Bezout_matrix(f, g, var_elim)\n\nCompute the Bezout matrix of two polynomials f, g with respect to var_elim\n\nInputs:\n\nf - first polynomial\ng - second polynomial\nvar_elim - variable, of which f and g are considered as polynomials\n\nOutput:\n\nM::MatrixElem - The Bezout matrix\n\n\n\n\n\n","category":"method"},{"location":"utils/elimination/#StructuralIdentifiability.Sylvester_matrix-Union{Tuple{P}, Tuple{P, P, P}} where P<:AbstractAlgebra.MPolyRingElem","page":"Elimination","title":"StructuralIdentifiability.Sylvester_matrix","text":"Sylvester_matrix(f, g, var_elim)\n\nCompute the Sylvester matrix of two polynomials f, g with respect to var_elim Inputs:\n\nf - first polynomial\ng - second polynomial\nvar_elim - variable, of which f and g are considered as polynomials\n\nOutput:\n\nM::MatrixElem - The Sylvester matrix\n\n\n\n\n\n","category":"method"},{"location":"utils/elimination/#StructuralIdentifiability.choose-Union{Tuple{P}, Tuple{Vector{P}, Any}} where P<:(AbstractAlgebra.MPolyRingElem{<:AbstractAlgebra.FieldElem})","page":"Elimination","title":"StructuralIdentifiability.choose","text":"choose(polys, generic_point_generator)\n\nInput:\n\npolys - an array of distinct irreducible polynomials in the same ring\ngeneric_point_generator - a generic point generator as described above for one of polys\n\nOutput:\n\nthe polynomial that vanishes at the generic_point_generator\n\n\n\n\n\n","category":"method"},{"location":"utils/elimination/#StructuralIdentifiability.eliminate_var-Union{Tuple{P}, Tuple{P, P, P, Any}} where P<:(AbstractAlgebra.MPolyRingElem{<:AbstractAlgebra.FieldElem})","page":"Elimination","title":"StructuralIdentifiability.eliminate_var","text":"eliminate_var(f, g, var_elim, generic_point_generator)\n\nEliminate a variable from a pair of polynomials\n\nInput:\n\nf and g - polynomials\nvar_elim - variable to be eliminated\ngeneric_point_generator - a generic point generator object for the factor of the resultant of f and g of interest\n\nOutput:\n\npolynomial - the desired factor of the resultant of f and g\n\n\n\n\n\n","category":"method"},{"location":"utils/elimination/#StructuralIdentifiability.simplify_matrix-Union{Tuple{AbstractAlgebra.MatElem{P}}, Tuple{P}} where P<:AbstractAlgebra.MPolyRingElem","page":"Elimination","title":"StructuralIdentifiability.simplify_matrix","text":"simplify_matrix(M)\n\nEliminate GCD of entries of every row and column\n\nInput:\n\nM::MatrixElem - matrix to be simplified\n\nOutput:\n\nM::MatrixElem - Simplified matrix\nextra_factors::Vector{AbstractAlgebra.MPolyElem} - array of GCDs eliminated from M.\n\n\n\n\n\n","category":"method"},{"location":"tutorials/discrete_time/#Identifiability-of-Discrete-Time-Models-(Local)","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"","category":"section"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"Now we consider a discrete-time model in the state-space form","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"begincases\nmathbfx(t + 1) = mathbff(mathbfx(t) mathbfp mathbfu(t))\nmathbfy(t) = mathbfg(mathbfx(t) mathbfp mathbfu(t))\nendcases","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"where mathbfx(t) mathbfy(t), and mathbfu(t) are time-dependent states, outputs, and inputs, respectively, and mathbfp are scalar parameters. As in the ODE case, we will call that a parameter or a states (or a function of them) is identifiable if its value can be recovered from time series for inputs and outputs (in the generic case, see Definition 3 in [1] for details). Again, we will distinguish two types of identifiability","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"local identifiability: the value can be recovered up to finitely many options;\nglobal identifiability: the value can be recovered uniquely.","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"Currently, StructuralIdentifiability.jl allows to assess only local identifiability for discrete-time models, and below we will describe how this can be done. As a running example, we will use the following discrete version of the SIR model: begincases S(t + 1) = S(t) - beta S(t) I(t)\nI(t + 1) = I(t) + beta S(t) I(t) - alpha I(t)\nR(t + 1) = R(t) + alpha I(t)\ny(t) = I(t) endcases","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"where the observable is I, the number of infected people. We start with creating a system as a DiscreteSystem from ModelingToolkit:","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"using ModelingToolkit\nusing StructuralIdentifiability\n\n@parameters α β\n@variables t S(t) I(t) R(t) y(t)\nD = Difference(t; dt = 1.0)\n\neqs = [D(S) ~ S - β * S * I, D(I) ~ I + β * S * I - α * I, D(R) ~ R + α * I]\n@named sir = DiscreteSystem(eqs)","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"Once the model is defined, we can assess identifiability by providing the formula for the observable:","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"assess_local_identifiability(sir; measured_quantities = [y ~ I])","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"For each parameter or state, the value in the returned dictionary is true (1) if the parameter is locally identifiable and false (0) otherwise. We see that R(t) is not identifiable, which makes sense: it does not affect the dynamics of the observable in any way.","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"In principle, it is not required to give a name to the observable, so one can write this shorter","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"assess_local_identifiability(sir; measured_quantities = [I])","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"The assess_local_identifiability function has two important keyword arguments:","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"funcs_to_check is a list of functions for which one want to assess identifiability, for example, the following code will check if β * S is locally identifiable.","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"assess_local_identifiability(sir; measured_quantities = [I], funcs_to_check = [β * S])","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"p is the probability of correctness (default value 0.99, i.e., 99%). The underlying algorithm is a Monte-Carlo algorithm, so in principle it may produce incorrect result but the probability of correctness of the returned result is guaranteed to be at least p (in fact, the employed bounds are quite conservative, so in practice incorrect result is almost never produced).","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"The implementation is based on a version of the observability rank criterion and will be described in a forthcoming paper.","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"[1]: S. Nõmm, C. Moog, Identifiability of discrete-time nonlinear systems, IFAC Proceedings Volumes, 2004.","category":"page"},{"location":"tutorials/creating_ode/#Creating-ODE-System","page":"Creating ODE System","title":"Creating ODE System","text":"","category":"section"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"Most of the algorithms in StructuralIdentifiability.jl take as input system of ordinary differential equations (ODEs) in the state space form, that is:","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"begincases\nmathbfx(t) = mathbff(mathbfx(t) mathbfp mathbfu(t))\nmathbfy(t) = mathbfg(mathbfx(t) mathbfp mathbfu(t))\nendcases","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"which involves","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"a vector mathbfx(t) of the state variables of the system,\na vector mathbfu(t) of extermal inputs,\na vector mathbfp of scalar parameters,\na vector mathbfy(t) of outputs (i.e., observations),\nand vectors of rational functions mathbff and mathbfg (for discussion of the non-rational case, see this issue).","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"In the standard setup, inputs and outputs are assumed to be known, and the goal is to assess identifiability of parameters and/or states from the input-output data. In the case of states, this property is also called observability.","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"There are two ways to define such a system to be processed using StructuralIdentifiability.jl. We will demonstrate them using the following example system (Wright's population model of two mutualist species with control[1]):","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"begincases\nx_1(t) = r_1 x_1(t)(1 - c_1 x_1(t)) + fracbeta_1 x_1(t)x_2(t)chi_1 + x_2(t) + u(t)\nx_2(t) = r_2 x_2(t)(1 - c_2 x_2(t)) + fracbeta_2 x_1(t)x_2(t)chi_2 + x_1(t)\ny(t) = x_1(t)\nendcases","category":"page"},{"location":"tutorials/creating_ode/#Defining-the-model-using-@ODEmodel-macro","page":"Creating ODE System","title":"Defining the model using @ODEmodel macro","text":"","category":"section"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"One way to define the model is to use the @ODEmodel macro provided by the StructuralIdentifiability.jl package.","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"using StructuralIdentifiability\n\node = @ODEmodel(\n x1'(t) =\n r1 * x1(t) * (1 - c1 * x1(t)) + beta1 * x1(t) * x2(t) / (chi1 + x2(t)) + u(t),\n x2'(t) = r2 * x2(t) * (1 - c2 * x2(t)) + beta2 * x1(t) * x2(t) / (chi2 + x1(t)),\n y(t) = x1(t)\n)","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"Then one can, for example, assess identifiability of the parameters and states by","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"assess_identifiability(ode)","category":"page"},{"location":"tutorials/creating_ode/#Defining-using-ModelingToolkit","page":"Creating ODE System","title":"Defining using ModelingToolkit","text":"","category":"section"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"Alternatively, one can use ModelingToolkit: encode the equations for the states as ODESystem and specify the outputs separately. In order to do this, we first introduce all functions and scalars:","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"using StructuralIdentifiability, ModelingToolkit\n\n@parameters r1, r2, c1, c2, beta1, beta2, chi1, chi2\n@variables t, x1(t), x2(t), y(t), u(t)\n\nD = Differential(t)","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"And then defined the system and separately the outputs:","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"eqs = [\n D(x1) ~ r1 * x1 * (1 - c1 * x1) + beta1 * x1 * x2 / (chi1 + x2) + u,\n D(x2) ~ r2 * x2 * (1 - c2 * x2) + beta2 * x1 * x2 / (chi2 + x1),\n]\n\nmeasured_quantities = [y ~ x1]\n\node_mtk = ODESystem(eqs, t, name = :mutualist)","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"Then, for example, the identifiability of parameters and states can be assessed as follows:","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"assess_identifiability(ode_mtk, measured_quantities = measured_quantities)","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"[1]: D. H. Wright, A Simple, Stable Model of Mutualism Incorporating Handling Time, The American Naturalist, 1989, 134(4).","category":"page"},{"location":"ioequations/ioequations/#Finding-Input-Output-Equations","page":"Input-Output Equation tools","title":"Finding Input-Output Equations","text":"","category":"section"},{"location":"ioequations/ioequations/","page":"Input-Output Equation tools","title":"Input-Output Equation tools","text":"find_ioequations","category":"page"},{"location":"ioequations/ioequations/#StructuralIdentifiability.find_ioequations","page":"Input-Output Equation tools","title":"StructuralIdentifiability.find_ioequations","text":"find_ioequations(ode, [var_change_policy=:default])\n\nFinds the input-output equations of an ODE system Input:\n\node - the ODE system\nvar_change_policy - whether to perform automatic variable change, can be one of :default, :yes, :no\n\nOutput:\n\na dictionary from “leaders” to the corresponding input-output equations; if an extra projection is needed, it will be the value corresponding to rand_proj_var\n\n\n\n\n\n","category":"function"},{"location":"ioequations/ioequations/#Reducing-with-respect-to-Input-Output-Equations","page":"Input-Output Equation tools","title":"Reducing with respect to Input-Output Equations","text":"","category":"section"},{"location":"ioequations/ioequations/","page":"Input-Output Equation tools","title":"Input-Output Equation tools","text":"PBRepresentation\npseudodivision\ndiffreduce\nio_switch!","category":"page"},{"location":"ioequations/ioequations/#StructuralIdentifiability.PBRepresentation","page":"Input-Output Equation tools","title":"StructuralIdentifiability.PBRepresentation","text":"The structure for storing a projection-based representation of differential ideal (see Section 2.3 https://arxiv.org/abs/2111.00991). Contains the following fields:\n\ny_names - the names of the variables with finite order in the profile (typically, outputs)\nu_names - the names of the variables with infinite order in the profile (typically, inputs)\nparam_names - the names of the parameters\nprofile - the profile of the PB-representation (see Definition 2.13) as a dict from y_names with finite orders to the orders\nprojections - the corresponding projections (see Definition 2.15) as a dict from y_names to the projections\n\n\n\n\n\n","category":"type"},{"location":"ioequations/ioequations/#StructuralIdentifiability.pseudodivision","page":"Input-Output Equation tools","title":"StructuralIdentifiability.pseudodivision","text":"pseudodivision(f, g, x)\n\nComputes the result of pseudodivision of f by g as univariate polynomials in x Input:\n\nf - the polynomial to be divided\ng - the polynomial to divide by\nx - the variable for the division\n\nOutput: the pseudoremainder of f divided by g w.r.t. x\n\n\n\n\n\n","category":"function"},{"location":"ioequations/ioequations/#StructuralIdentifiability.diffreduce","page":"Input-Output Equation tools","title":"StructuralIdentifiability.diffreduce","text":"diffreduce(diffpoly, pbr)\n\nComputes the result of differential reduction of a differential polynomial diffpoly with respect to the charset defined by a PB-representation pbr Input:\n\ndiffpoly - a polynomial representing a differential polynomial to be reduced\npbr - a projection-based representation\n\nOutput: the result of differential reduction of diffpoly by pbr considered as a characteristic set (see Remark 2.20 in the paper)\n\n\n\n\n\n","category":"function"},{"location":"ioequations/ioequations/#StructuralIdentifiability.io_switch!","page":"Input-Output Equation tools","title":"StructuralIdentifiability.io_switch!","text":"io_switch(pbr)\n\nIn a single-output pb-representation pbr makes the leading variable to be the first of the inputs\n\n\n\n\n\n","category":"function"},{"location":"utils/local_identifiability/#Local-Identifiability-Tools","page":"Local Identifiability Tools","title":"Local Identifiability Tools","text":"","category":"section"},{"location":"utils/local_identifiability/","page":"Local Identifiability Tools","title":"Local Identifiability Tools","text":"Pages=[\"local_identifiability.md\"]","category":"page"},{"location":"utils/local_identifiability/","page":"Local Identifiability Tools","title":"Local Identifiability Tools","text":"CurrentModule=StructuralIdentifiability","category":"page"},{"location":"utils/local_identifiability/","page":"Local Identifiability Tools","title":"Local Identifiability Tools","text":"StructuralIdentifiability.differentiate_solution\nStructuralIdentifiability.differentiate_output","category":"page"},{"location":"utils/local_identifiability/#StructuralIdentifiability.differentiate_solution","page":"Local Identifiability Tools","title":"StructuralIdentifiability.differentiate_solution","text":"differentiate_solution(ode, params, ic, inputs, prec)\n\nInput:\n\nthe same as for power_series_solutions\n\nOutput:\n\na tuple consisting of the power series solution and a dictionary of the form (u, v) => power series, where u is a state variable v is a state or parameter, and the power series is the partial derivative of the function u w.r.t. v evaluated at the solution\n\n\n\n\n\n","category":"function"},{"location":"utils/local_identifiability/#StructuralIdentifiability.differentiate_output","page":"Local Identifiability Tools","title":"StructuralIdentifiability.differentiate_output","text":"differentiate_output(ode, params, ic, inputs, prec)\n\nSimilar to differentiate_solution but computes partial derivatives of prescribed outputs returns a dictionary of the form y_function => Dict(var => dy/dvar) where dy/dvar is the derivative of y_function with respect to var.\n\n\n\n\n\n","category":"function"},{"location":"#StructuralIdentifiability.jl","page":"Home","title":"StructuralIdentifiability.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"StructuralIdentifiability.jl is a comprehensive toolbox for assessing identifiability parameters.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To install StructuralIdentifiability.jl, use the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(\"StructuralIdentifiability\")","category":"page"},{"location":"#Citation","page":"Home","title":"Citation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"@article{structidjl,\n author = {Dong, R. and Goodbrake, C. and Harrington, H. and Pogudin G.},\n title = {Differential Elimination for Dynamical Models via Projections with Applications to Structural Identifiability},\n journal = {SIAM Journal on Applied Algebra and Geometry},\n url = {https://doi.org/10.1137/22M1469067},\n year = {2023}\n volume = {7},\n number = {1},\n pages = {194-235}\n}","category":"page"},{"location":"#Feature-Summary","page":"Home","title":"Feature Summary","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"StructuralIdentifiability.jl can assess local and global identifiability of ODE models. In addition to these straightforward identifiability queries on individual parameters, the package can distinguish between single- and multi-experiment identifiability.","category":"page"},{"location":"#Feature-List","page":"Home","title":"Feature List","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Local identifiability checks\nGlobal identifiability checks\nAssessment of identifiable functions of parameters and states\nModel reparametrization (experimental)","category":"page"},{"location":"#Contributing","page":"Home","title":"Contributing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to StructuralIdentifiability.\nThere are a few community forums:\nThe #diffeq-bridged channel in the Julia Slack\nJuliaDiffEq on Gitter\nOn the Julia Discourse forums\nSee also SciML Community page","category":"page"},{"location":"#Reproducibility","page":"Home","title":"Reproducibility","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"
The documentation of this SciML package was built using these direct dependencies,","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg # hide\nPkg.status() # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"
and using this machine and Julia version.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using InteractiveUtils # hide\nversioninfo() # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"
A more complete overview of all dependencies and their versions is also provided.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg # hide\nPkg.status(; mode = PKGMODE_MANIFEST) # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"You can also download the \nmanifest file and the\nproject file.","category":"page"},{"location":"input/input/#Parsing-input-ODE-system","page":"Parsing input ODE system","title":"Parsing input ODE system","text":"","category":"section"},{"location":"input/input/","page":"Parsing input ODE system","title":"Parsing input ODE system","text":"@ODEmodel(ex::Expr...)\nODE\nset_parameter_values","category":"page"},{"location":"input/input/#StructuralIdentifiability.@ODEmodel-Tuple{Vararg{Expr}}","page":"Parsing input ODE system","title":"StructuralIdentifiability.@ODEmodel","text":"macro ODEmodel\n\nMacro for creating an ODE from a list of equations. It also injects all variables into the global scope.\n\nExample\n\nCreating a simple ODE:\n\nusing StructuralIdentifiability\n\node = @ODEmodel(\n x1'(t) = a * x1(t) + u(t),\n x2'(t) = b * x2(t) + c*x1(t)*x2(t),\n y(t) = x1(t)\n)\n\nHere,\n\nx1, x2 are state variables\ny is an output variable\nu is an input variable\na, b, c are time-indepdendent parameters\n\n\n\n\n\n","category":"macro"},{"location":"input/input/#StructuralIdentifiability.ODE","page":"Parsing input ODE system","title":"StructuralIdentifiability.ODE","text":"The main structure that represents input ODE system.\n\nStores information about states (x_vars), outputs (y_vars), inputs (u_vars), parameters (parameters) and the equations.\n\nThis structure is constructed via @ODEmodel macro.\n\n\n\n\n\n","category":"type"},{"location":"input/input/#StructuralIdentifiability.set_parameter_values","page":"Parsing input ODE system","title":"StructuralIdentifiability.set_parameter_values","text":"set_parameter_values(ode, param_values)\n\nInput:\n\node - an ODE as above\nparam_values - values for (possibly, some of) the parameters as dictionary parameter => value\n\nOutput:\n\nnew ode with the parameters in param_values plugged with the given numbers\n\n\n\n\n\n","category":"function"},{"location":"input/input/#Create-Compartmental-Model","page":"Parsing input ODE system","title":"Create Compartmental Model","text":"","category":"section"},{"location":"input/input/","page":"Parsing input ODE system","title":"Parsing input ODE system","text":"linear_compartment_model","category":"page"},{"location":"input/input/#StructuralIdentifiability.linear_compartment_model","page":"Parsing input ODE system","title":"StructuralIdentifiability.linear_compartment_model","text":"linear_compartment_model(graph, inputs, outputs, leaks)\n\nInput: defines a linear compartment model with nodes numbered from 1 to n by\n\ngraph - and array of integer arrays representing the adjacency lists of the graph\ninputs - array of input nodes\noutputs - array of output nodes\nleaks - array of sink nodes\n\nOutput:\n\nthe corresponding ODE system in the notation of https://doi.org/10.1007/s11538-015-0098-0\n\n\n\n\n\n","category":"function"}] +[{"location":"utils/ode/#Functions-to-work-with-the-ODE-structure","page":"ODE Tools","title":"Functions to work with the ODE structure","text":"","category":"section"},{"location":"utils/ode/","page":"ODE Tools","title":"ODE Tools","text":"Modules = [StructuralIdentifiability]\nPages = [\"ODE.jl\", \"submodels.jl\"]\nOrder = [:function]","category":"page"},{"location":"utils/ode/#StructuralIdentifiability.__preprocess_ode-Tuple{ModelingToolkit.AbstractTimeDependentSystem, Array{<:Tuple{String, var\"#s6\"} where var\"#s6\"<:SymbolicUtils.BasicSymbolic}}","page":"ODE Tools","title":"StructuralIdentifiability.__preprocess_ode","text":"function __preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{Tuple{String, SymbolicUtils.BasicSymbolic}})\n\nInput:\n\nde - ModelingToolkit.AbstractTimeDependentSystem, a system for identifiability query\nmeasured_quantities - array of input function in the form (name, expression)\n\nOutput:\n\nODE object containing required data for identifiability assessment\nconversion dictionary from the symbols in the input MTK model to the variable involved in the produced ODE object\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability._get_var-Tuple{Any}","page":"ODE Tools","title":"StructuralIdentifiability._get_var","text":"For an expression of the form f'(t) or f(t) returns (f, true) and (f, false), resp\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability._reduce_mod_p-Tuple{Nemo.QQMPolyRingElem, Int64}","page":"ODE Tools","title":"StructuralIdentifiability._reduce_mod_p","text":"_reduce_mod_p(f, p)\n\nReduces a polynomial/rational function over Q modulo p\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability.power_series_solution-Union{Tuple{P}, Tuple{T}, Tuple{ODE{P}, Dict{P, T}, Dict{P, T}, Dict{P, Vector{T}}, Int64}} where {T<:AbstractAlgebra.FieldElem, P<:AbstractAlgebra.MPolyRingElem{T}}","page":"ODE Tools","title":"StructuralIdentifiability.power_series_solution","text":"power_series_solution(ode, param_values, initial_conditions, input_values, prec)\n\nInput:\n\node - an ode to solve\nparam_values - parameter values, must be a dictionary mapping parameter to a value\ninitial_conditions - initial conditions of ode, must be a dictionary mapping state variable to a value\ninput_values - power series for the inputs presented as a dictionary variable => list of coefficients\nprec - the precision of solutions\n\nOutput:\n\ncomputes a power series solution with precision prec presented as a dictionary variable => corresponding coordinate of the solution\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability.preprocess_ode-Tuple{ModelingToolkit.AbstractTimeDependentSystem, Array{Symbolics.Equation}}","page":"ODE Tools","title":"StructuralIdentifiability.preprocess_ode","text":"function preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{ModelingToolkit.Equation})\nfunction preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{SymbolicUtils.BasicSymbolic})\n\nInput:\n\nde - ModelingToolkit.AbstractTimeDependentSystem, a system for identifiability query\nmeasured_quantities - array of output functions (as equations of just functions)\n\nOutput:\n\nODE object containing required data for identifiability assessment\nconversion dictionary from the symbols in the input MTK model to the variable involved in the produced ODE object\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability.reduce_ode_mod_p-Tuple{ODE{<:AbstractAlgebra.MPolyRingElem{Nemo.QQFieldElem}}, Int64}","page":"ODE Tools","title":"StructuralIdentifiability.reduce_ode_mod_p","text":"reduce_ode_mod_p(ode, p)\n\nInput: ode is an ODE over QQ, p is a prime number Output: the reduction mod p, throws an exception if p divides one of the denominators\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability.set_parameter_values-Union{Tuple{P}, Tuple{T}, Tuple{ODE{P}, Dict{P, T}}} where {T<:AbstractAlgebra.FieldElem, P<:AbstractAlgebra.MPolyRingElem{T}}","page":"ODE Tools","title":"StructuralIdentifiability.set_parameter_values","text":"set_parameter_values(ode, param_values)\n\nInput:\n\node - an ODE as above\nparam_values - values for (possibly, some of) the parameters as dictionary parameter => value\n\nOutput:\n\nnew ode with the parameters in param_values plugged with the given numbers\n\n\n\n\n\n","category":"method"},{"location":"utils/ode/#StructuralIdentifiability.find_submodels-Union{Tuple{ODE{P}}, Tuple{P}} where P<:AbstractAlgebra.MPolyRingElem","page":"ODE Tools","title":"StructuralIdentifiability.find_submodels","text":"find_submodels(ode)\n\nThe function calculates and returns all valid submodels given a system of ODEs.\n\nInput:\n\node - an ODEs system to be studied\n\nOutput: \n\nA list of submodels represented as ode objects\n\nExample:\n\n>ode = @ODEmodel(x1'(t) = x1(t)^2, \n x2'(t) = x1(t) * x2(t), \n y1(t) = x1(t), \n y2(t) = x2(t))\n>find_submodels(ode)\n ODE{fmpq_mpoly}[\n \n x1'(t) = a(t)*x2(t)^2 + x1(t)\n y1(t) = x1(t)\n ]\n\n\n\n\n\n","category":"method"},{"location":"utils/primality/#Primality-Checks","page":"Primality Checks","title":"Primality Checks","text":"","category":"section"},{"location":"utils/primality/","page":"Primality Checks","title":"Primality Checks","text":"Pages=[\"primality.md\"]","category":"page"},{"location":"utils/primality/","page":"Primality Checks","title":"Primality Checks","text":"StructuralIdentifiability.check_primality","category":"page"},{"location":"utils/primality/#StructuralIdentifiability.check_primality","page":"Primality Checks","title":"StructuralIdentifiability.check_primality","text":"check_primality(polys::Dict{fmpq_mpoly, fmpq_mpoly}, extra_relations::Array{fmpq_mpoly, 1})\n\nThe function checks if the ideal generated by the polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals.\n\nThe extra_relations allows adding more polynomials to the generators (not affecting the saturation).\n\n\n\n\n\ncheck_primality(polys::Dict{fmpq_mpoly, fmpq_mpoly})\n\nThe function checks if the ideal generated by the polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals.\n\n\n\n\n\n","category":"function"},{"location":"tutorials/identifiable_functions/#Globally-Identifiable-Functions","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"","category":"section"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"In addition to assessing identifiabuility of given functions of parameters or states, StructuralIdentifiability.jl provides the function find_identifiable_functions which finds a set of identifiable functions such that any other identifiable function can be expressed using them. This allows to find out what actually is identifiable and what does non-identifiability in the model at hand looks like.","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"For example, consider the following model[1].","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"using StructuralIdentifiability # hide\nLLW1987 = @ODEmodel(\n x1'(t) = -p1 * x1(t) + p2 * u(t),\n x2'(t) = -p3 * x2(t) + p4 * u(t),\n x3'(t) = -(p1 + p3) * x3(t) + (p4 * x1(t) + p2 * x2(t)) * u(t),\n y1(t) = x3(t)\n)","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"Several decades ago, this model was introduced to demonstrate the presence of nontrivial unidentifiability in nonlinear systems of ODEs. Nowadays, we can automatically find the identifiable combinations of parameters:","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"using StructuralIdentifiability # hide\nfind_identifiable_functions(LLW1987)","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"From these expressions, we see that the values of p1 and p3 are not identifiable but an unordered pair of numbers {p1, p3} is uniquely determined since p1 + p3 and p1 * p3 are known. Furthermore, we see that, for fixed input and output, p2 and p4 can take infinitely many values but knowing one of them, we would also be able to determine the other.","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"Moreover, we can find generators of all identifiable functions in parameters and states:","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"find_identifiable_functions(LLW1987, with_states = true)","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"By default, find_identifiable_functions tries to simplify the output functions as much as possible, and it has simplify keyword responsible for the degree of simplification. The default value is :standard but one could use :strong to try to simplify further (at the expense of heavier computation) or use :weak to simplify less (but compute faster).","category":"page"},{"location":"tutorials/identifiable_functions/","page":"Globally Identifiable Functions","title":"Globally Identifiable Functions","text":"[1]: Y. Lecourtier, F. Lamnabhi-Lagarrigue, and E. Walter, A method to prove that nonlinear models can be unidentifiable, Proceedings of the 26th Conference on Decision and Control, December 1987, 2144-2145;","category":"page"},{"location":"tutorials/identifiability/#Identifiability-of-Differential-Models-(Local-and-Global)","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"","category":"section"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Recall that we consider ODE models in the state-space form","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"begincases\nmathbfx(t) = mathbff(mathbfx(t) mathbfp mathbfu(t))\nmathbfy(t) = mathbfg(mathbfx(t) mathbfp mathbfu(t))\nendcases","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"where mathbfx(t) mathbfy(t), and mathbfu(t) are time-dependent states, outputs, and inputs, respectively, and mathbfp are scalar parameters. We will call that a parameter or a states (or a function of them) is identifiable if its value can be recovered from time series for inputs and outputs. Typically, two types of identifiability are distinguished","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"local identifiability: the value can be recovered up to finitely many options;\nglobal identifiability: the value can be recovered uniquely.","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Note that in the case of states, term observability it typically used. We will use identifiability for both states and parameters for brevity and uniformity. While the notion of global identifiability is much more precise, assessing local identifiability is typically much faster, and can be performed for the models whose global identifiability analysis is out of reach.","category":"page"},{"location":"tutorials/identifiability/#Local-identifiability","page":"Identifiability of Differential Models (Local and Global)","title":"Local identifiability","text":"","category":"section"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"We consider the Lotka-Volterra model:","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"begincases\nx_1(t) = a x_1(t) - b x_1(t) x_2(t) + u(t)\nx_2(t) = -c x_2(t) + d x_1(t) x_2(t)\ny(t) = x_1(t)\nendcases","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"The local identifiability of all parameters and states in this model can be assessed as follows","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"using StructuralIdentifiability\n\node = @ODEmodel(\n x1'(t) = a * x1(t) - b * x1(t) * x2(t) + u(t),\n x2'(t) = -c * x2(t) + d * x1(t) * x2(t),\n y(t) = x1(t)\n)\n\nassess_local_identifiability(ode)","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"We see that x_1(t) is locally identifiable (no surprises, this is an output), a c and d are identifiable as well. On the other hand, x_2(t) and b are nonidentifiable. This can be explained by the following scaling symmetry","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"x_2(t) to lambda x_2(t) quad b to fracblambda","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"which preserves input and output for every nonzero lambda. The algorithm behind this call is the one due to Sedoglavic[1].","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Function assess_local_identifiability has several optional parameters","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"funcs_to_check a list of specific functions of parameters and states to check identifiability for (see an example below). If not provided, the identifiability is assessed for all parameters and states.\np (default 099) is the probability of correctness. The algorithm can, in theory, produce wrong result, but the probability that it is correct is guaranteed to be at least p. However, the probability bounds we use are quite conservative, so the actual probability of correctness is likely to be much higher.\ntype (default :SE). By default, the algorithm checks the standard single-experiment identifiability. If one sets type = :ME, then the algorithm checks multi-experiment identifiability, that is, identifiability from several experiments with independent initial conditions (the algorithm from [2] is used).","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Note that the scaling symmetry given above suggests that b x_2(t) may in fact be identifiable. This can be checked using funcs_to_check parameter:","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"assess_local_identifiability(ode, funcs_to_check = [b * x2])","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Indeed!","category":"page"},{"location":"tutorials/identifiability/#Global-identifiability","page":"Identifiability of Differential Models (Local and Global)","title":"Global identifiability","text":"","category":"section"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"One can obtain more refined information about a model using assess_identifiability function. We will showcase it using the Goodwin oscillator model [3].","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"using StructuralIdentifiability\n\node = @ODEmodel(\n x1'(t) = -b * x1(t) + 1 / (c + x4(t)),\n x2'(t) = alpha * x1(t) - beta * x2(t),\n x3'(t) = gama * x2(t) - delta * x3(t),\n x4'(t) = sigma * x4(t) * (gama * x2(t) - delta * x3(t)) / x3(t),\n y(t) = x1(t)\n)\n\nassess_identifiability(ode)","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"As a result, each parameter/state is assigned one of the labels :globally (globally identifiable), :locally (locally but not globally identifiable), or :nonidentifiable (not identifiable, even locally). The algorithm behind this computation follows [4].","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Similarly to assess_local_identifiability, this function has optional parameters:","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"funcs_to_check a list of specific functions of parameters and states to check identifiability for (see an example below). If not provided, the identifiability is assessed for all parameters and states. Note that the computations for states may be more involved than for the parameters, so one may want to call the function with funcs_to_check = ode.parameters if the call assess_identifiability(ode) takes too long.\np (default 099) is the probability of correctness. Same story as above: the probability estimates are very conservative, so the actual error probability is much lower than 1%. Also, currently, the probability of correctness does not include the probability of correctness of the modular reconstruction for Groebner bases. This probability is ensured by an additional check modulo a large prime, and can be neglected for practical purposes.","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"Using funcs_to_check parameter, one can further inverstigate the nature of the lack of identifiability in the model at hand. For example, for the Goodwin oscillator, we can check if beta + delta and beta * delta are identifiable:","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"assess_identifiability(ode, funcs_to_check = [beta + delta, beta * delta])","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"And we see that they indeed are. This means, in particular, that the reason why beta and delta are not identifiable is because their values can be exchanged. One may wonder how could we guess these functions beta + delta, beta * delta. In fact, they can be just computed using find_identifiable_functions function as we will explain in the next tutorial. Stay tuned!","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"[1]: A. Sedoglavic, A probabilistic algorithm to test local algebraic observability in polynomial time, Journal of Symbolic Computation, 2002.","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"[2]: A. Ovchinnikov, A. Pillay, G. Pogudin, T. Scanlon, Multi-experiment Parameter Identifiability of ODEs and Model Theory, SIAM Journal on Applied Algebra and Geometry, 2022.","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"[3]: D. Gonze, P. Ruoff, The Goodwin Oscillator and its Legacy, Acta Biotheoretica, 2020.","category":"page"},{"location":"tutorials/identifiability/","page":"Identifiability of Differential Models (Local and Global)","title":"Identifiability of Differential Models (Local and Global)","text":"[4]: R. Dong, C. Goodbrake, H. Harrington, G. Pogudin, Differential elimination for dynamical models via projections with applications to structural identifiability, SIAM Journal on Applied Algebra and Geometry, 2023.","category":"page"},{"location":"export/export/#Exporting-to-Other-Systems","page":"Exporting to Other Systems","title":"Exporting to Other Systems","text":"","category":"section"},{"location":"export/export/","page":"Exporting to Other Systems","title":"Exporting to Other Systems","text":"Here we put some helpful utilities to export your code to other identifiability software.","category":"page"},{"location":"export/export/","page":"Exporting to Other Systems","title":"Exporting to Other Systems","text":"print_for_maple\nprint_for_DAISY\nprint_for_GenSSI\nprint_for_COMBOS","category":"page"},{"location":"export/export/#StructuralIdentifiability.print_for_maple","page":"Exporting to Other Systems","title":"StructuralIdentifiability.print_for_maple","text":"print_for_maple(ode, package)\n\nPrints the ODE in the format accepted by maple packages\n\nSIAN (https://github.com/pogudingleb/SIAN) if package=:SIAN\nDifferentialAlgebra if package=:DifferentialAlgebra\nDifferentialThomas if package=:DifferentialThomas\n\n\n\n\n\n","category":"function"},{"location":"export/export/#StructuralIdentifiability.print_for_DAISY","page":"Exporting to Other Systems","title":"StructuralIdentifiability.print_for_DAISY","text":"print_for_DAISY(ode)\n\nPrints the ODE in the format accepted by DAISY (https://daisy.dei.unipd.it/)\n\n\n\n\n\n","category":"function"},{"location":"export/export/#StructuralIdentifiability.print_for_GenSSI","page":"Exporting to Other Systems","title":"StructuralIdentifiability.print_for_GenSSI","text":"print_for_GenSSI(ode)\n\nPrints the ODE in the format accepted by GenSSI 2.0 (https://github.com/genssi-developer/GenSSI)\n\n\n\n\n\n","category":"function"},{"location":"export/export/#StructuralIdentifiability.print_for_COMBOS","page":"Exporting to Other Systems","title":"StructuralIdentifiability.print_for_COMBOS","text":"print_for_COMBOS(ode)\n\nPrints the ODE in the format accepted by COMBOS (http://biocyb1.cs.ucla.edu/combos/)\n\n\n\n\n\n","category":"function"},{"location":"utils/util/#Other-Helpful-Functions","page":"Other Utilities","title":"Other Helpful Functions","text":"","category":"section"},{"location":"utils/util/","page":"Other Utilities","title":"Other Utilities","text":"Pages=[\"util.md\"]","category":"page"},{"location":"utils/util/","page":"Other Utilities","title":"Other Utilities","text":"Modules = [StructuralIdentifiability]\nPages = [\"util.jl\"]","category":"page"},{"location":"utils/util/#StructuralIdentifiability.compare_rational_func_by-Union{Tuple{By}, Tuple{Any, Any, By}, Tuple{Any, Any, By, Any}} where By","page":"Other Utilities","title":"StructuralIdentifiability.compare_rational_func_by","text":"compare_rational_func_by(f, g, by)\n\nReturns \n\n-1 if f < g,\n0 if f = g, and \n1 if f > g.\n\nFunctions' numerators and denominators are compared using by.\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.decompose_derivative-Tuple{String, Array{String}}","page":"Other Utilities","title":"StructuralIdentifiability.decompose_derivative","text":"decompose_derivative(varname, prefixes)\n\nDetermines if it is possible to represent the varname as a_number where a is an element of prefixes If yes, returns a pair (a, number), otherwise nothing\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.dennums_to_fractions-Union{Tuple{Array{Vector{T}, 1}}, Tuple{T}} where T","page":"Other Utilities","title":"StructuralIdentifiability.dennums_to_fractions","text":"dennums_to_fractions(dennums)\n\nReturns the field generators represented by fractions.\n\nInput: an array of arrays of polynomials, as in [[f1, f2, f3, ...], [g1, g2, g3, ...], ...]\n\nOutput: an array of fractions [f2/f1, f3/f1, ..., g2/g1, g3/g1, ...]\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.eval_at_dict-Union{Tuple{P}, Tuple{P, Dict{P, <:AbstractAlgebra.RingElem}}} where P<:AbstractAlgebra.MPolyRingElem","page":"Other Utilities","title":"StructuralIdentifiability.eval_at_dict","text":"eval_at_dict(f, d)\n\nEvaluates a polynomial/rational function on a dictionary of type var => val and missing values are replaced with zeroes\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.extract_coefficients-Union{Tuple{P}, Tuple{P, Vector{P}}} where P<:AbstractAlgebra.MPolyRingElem","page":"Other Utilities","title":"StructuralIdentifiability.extract_coefficients","text":"extract_coefficients(poly, variables)\n\nInput:\n\npoly - multivariate polynomial\nvariables - a list of variables from the generators of the ring of p\n\nOutput:\n\ndictionary with keys being tuples of length lenght(variables) and values being polynomials in the variables other than those which are the coefficients at the corresponding monomials (in a smaller polynomial ring)\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.fractions_to_dennums-Tuple{Any}","page":"Other Utilities","title":"StructuralIdentifiability.fractions_to_dennums","text":"fractions_to_dennums(fractions)\n\nReturns the field generators represented by lists of denominators and numerators.\n\nInput: an array of fractions, as in [f2/f1, f3/f1, ..., g2/g1, g3/g1, ...]\n\nOutput: an array of arrays of polynomials, [[f1, f2, f3, ...], [g1, g2, g3, ...], ...]\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.gen_tag_name","page":"Other Utilities","title":"StructuralIdentifiability.gen_tag_name","text":"gen_tag_name(base; stop_words)\ngen_tag_names(n, base; stop_words)\n\nGenerates a string which will not collide with the words in stop_words.\n\nArguments\n\nn: Generates a sequence of unique strings of length n\nbase: A string or a vector of strings, the base for the generated sequence\nstop_words: A vector of strings, stop words\n\n\n\n\n\n","category":"function"},{"location":"utils/util/#StructuralIdentifiability.make_substitution-Union{Tuple{P}, NTuple{4, P}} where P<:AbstractAlgebra.MPolyRingElem","page":"Other Utilities","title":"StructuralIdentifiability.make_substitution","text":"make_substitution(f, var_sub, val_numer, val_denom)\n\nSubstitute a variable in a polynomial with an expression\n\nInput:\n\nf - the polynomial\nvar_sub - the variable to be substituted\nvar_numer - numerator of the substitution expression\nvar_denom - denominator of the substitution expression\n\nOutput:\n\npolynomial - result of substitution\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.parent_ring_change-Tuple{AbstractAlgebra.MPolyRingElem, AbstractAlgebra.MPolyRing}","page":"Other Utilities","title":"StructuralIdentifiability.parent_ring_change","text":"parent_ring_change(poly, new_ring)\n\nConverts a polynomial to a different polynomial ring Input\n\npoly - a polynomial to be converted\nnew_ring - a polynomial ring such that every variable name appearing in poly appears among the generators\n\nOutput:\n\na polynomial in new_ring “equal” to poly\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.switch_ring-Tuple{AbstractAlgebra.MPolyRingElem, AbstractAlgebra.MPolyRing}","page":"Other Utilities","title":"StructuralIdentifiability.switch_ring","text":"switch_ring(v, ring)\n\nFor a variable v, returns a variable in ring with the same name\n\n\n\n\n\n","category":"method"},{"location":"utils/util/#StructuralIdentifiability.uncertain_factorization-Tuple{AbstractAlgebra.MPolyRingElem{Nemo.QQFieldElem}}","page":"Other Utilities","title":"StructuralIdentifiability.uncertain_factorization","text":"uncertain_factorization(f)\n\nInput:\n\nf - polynomial with rational coefficients\n\nOutput:\n\nlist of pairs (div, certainty) where\ndiv's are divisors of f such that f is their product with certain powers\nif certainty is true, div is Q-irreducible\n\n\n\n\n\n","category":"method"},{"location":"utils/wronskian/#Wronskian-Tools","page":"Wronskian Tools","title":"Wronskian Tools","text":"","category":"section"},{"location":"utils/wronskian/","page":"Wronskian Tools","title":"Wronskian Tools","text":"Modules = [StructuralIdentifiability]\nPages = [\"wronskian.jl\"]","category":"page"},{"location":"utils/wronskian/#StructuralIdentifiability.get_max_below-Tuple{StructuralIdentifiability.ExpVectTrie, Vector{Int64}}","page":"Wronskian Tools","title":"StructuralIdentifiability.get_max_below","text":"get_max_below(t, vect)\n\nInput:\n\nt - a trie with exponent vectors\nvect - yet another exponent vector\n\nOutput:\n\na pair (d, v) where v is a vector in the trie which is componentwise ≤ vect and the difference d is as small as possible\n\n\n\n\n\n","category":"method"},{"location":"utils/wronskian/#StructuralIdentifiability.massive_eval-Tuple{Any, Any}","page":"Wronskian Tools","title":"StructuralIdentifiability.massive_eval","text":"massive_eval(polys, eval_dict)\n\nInput:\n\npolys - a list of polynomials\neval_dict - dictionary from variables to the values. Missing values are treated as zeroes\n\nOutput:\n\na list of values of the polynomials\n\nEvaluates a list of polynomials at a point. Assumes that multiplications are relatively expensive (like in truncated power series) so all the monomials are precomputed first and the values of monomials of lower degree are cached and used to compute the values of the monomials of higher degree\n\n\n\n\n\n","category":"method"},{"location":"utils/wronskian/#StructuralIdentifiability.monomial_compress-Tuple{Any, ODE}","page":"Wronskian Tools","title":"StructuralIdentifiability.monomial_compress","text":"monomial_compress(io_equation, ode)\n\nCompresses an input-output equation for the rank computation Input:\n\nio_equation - input-output equation\node - the corresponding ODE model\n\nOutput:\n\npair (coeffs, terms) such that:\nsum of coeffs[i] * terms[i] = io_equation\ncoeffs involve only parameters, terms involve only inputs and outputs\nlength of the representation is the smallest possible\n\n\n\n\n\n","category":"method"},{"location":"utils/wronskian/#StructuralIdentifiability.wronskian-Union{Tuple{P}, Tuple{Dict{P, P}, ODE{P}}} where P<:AbstractAlgebra.MPolyRingElem","page":"Wronskian Tools","title":"StructuralIdentifiability.wronskian","text":"wronskian(io_equations, ode)\n\nInput:\n\nio_equations - a set of io-equations in the form of the Dict as returned by find_ioequations\node - the ODE object\n\nOutput:\n\na list of Wronskians evaluated at a point modulo prime\n\nComputes the Wronskians of io_equations\n\n\n\n\n\n","category":"method"},{"location":"utils/global_identifiability/#Global-Identifiability-Tools","page":"Global Identifiability Tools","title":"Global Identifiability Tools","text":"","category":"section"},{"location":"utils/global_identifiability/","page":"Global Identifiability Tools","title":"Global Identifiability Tools","text":"Pages=[\"global_identifiability.md\"]","category":"page"},{"location":"utils/global_identifiability/","page":"Global Identifiability Tools","title":"Global Identifiability Tools","text":"CurrentModule=StructuralIdentifiability","category":"page"},{"location":"utils/global_identifiability/","page":"Global Identifiability Tools","title":"Global Identifiability Tools","text":"StructuralIdentifiability.RationalFunctionField\nStructuralIdentifiability.field_contains\nStructuralIdentifiability.get_degree_and_coeffsize","category":"page"},{"location":"utils/global_identifiability/#StructuralIdentifiability.RationalFunctionField","page":"Global Identifiability Tools","title":"StructuralIdentifiability.RationalFunctionField","text":"RationalFunctionField\n\nA subfield of the field of rational functions over the rationals.\n\nExample\n\nusing Nemo\nusing StructuralIdentifiability: RationalFunctionField\n\nR, (x, y, z) = QQ[\"x\", \"y\", \"z\"]\n\n# Constructs a subfield generated by x / y, y / z\nrff = RationalFunctionField([x // y, y // z])\n\n# Constructs a subfield generated by y / x, 1 / x, z / y\nrff = RationalFunctionField([[x, y, R(1)], [y, z]])\n\n\n\n\n\n","category":"type"},{"location":"utils/global_identifiability/#StructuralIdentifiability.field_contains","page":"Global Identifiability Tools","title":"StructuralIdentifiability.field_contains","text":"field_contains(field, ratfuncs, p)\n\nChecks whether given rational function field field contains given rational functions ratfuncs (represented as a list of lists). The result is correct with probability at least p\n\nInputs:\n\nfield - a rational function field\nratfuncs - a list of lists of polynomials. Each of the lists, say, [f1, ..., fn], defines generators f2/f1, ..., fn/f1.\np real number from (0, 1)\n\nOutput:\n\na list L[i] of bools of length length(rat_funcs) such that L[i] is true iff the i-th function belongs to field\n\n\n\n\n\n","category":"function"},{"location":"utils/global_identifiability/#StructuralIdentifiability.get_degree_and_coeffsize","page":"Global Identifiability Tools","title":"StructuralIdentifiability.get_degree_and_coeffsize","text":"get_degree_and_coeffsize(f)\n\nfor f being a polynomial/rational function over rationals (QQ) returns a tuple (degree, max_coef_size)\n\n\n\n\n\n","category":"function"},{"location":"utils/reparametrization/#Model-reparametrization","page":"Model reparametrization","title":"Model reparametrization","text":"","category":"section"},{"location":"utils/reparametrization/","page":"Model reparametrization","title":"Model reparametrization","text":"StructuralIdentifiability.reparametrize_global","category":"page"},{"location":"utils/reparametrization/#StructuralIdentifiability.reparametrize_global","page":"Model reparametrization","title":"StructuralIdentifiability.reparametrize_global","text":"reparametrize_global(ode, options...)\n\nFinds a reparametrization of ode in terms of globally identifiabile functions.\n\nReturns a tuple (new_ode, new_vars, implicit_relations), such that:\n\nnew_ode is the reparametrized ODE system\nnew_vars is a mapping from the new variables to the original ones\nrelations is the array of implicit algebraic relations among new_vars. Usually, relations is empty.\n\nOptions\n\nThe function accepts the following optional arguments.\n\nseed: A float in the range from 0 to 1, random seed (default is seed = 42). \np: The probability of correctness (default is p = 0.99).\n\nExample\n\nusing StructuralIdentifiability\n\node = @ODEmodel(\n x1'(t) = a * x1(t) - b*x1(t)*x2(t),\n x2'(t) = -c * x2(t) + d*x1(t)*x2(t),\n y(t) = x1(t)\n)\n\nnew_ode, new_vars, relations = reparametrize_global(ode)\n\nThen, we have the following:\n\n# new_ode\nX2'(t) = X1(t)*X2(t)*a2 - X2(t)*a1\nX1'(t) = -X1(t)*X2(t) + X1(t)*a3\ny1(t) = X1(t)\n\n# new_vars\nDict{Nemo.QQMPolyRingElem, AbstractAlgebra.Generic.Frac{Nemo.QQMPolyRingElem}} with 6 entries:\n X2 => b*x2\n y1 => y\n X1 => x1\n a2 => d\n a3 => a\n a1 => c\n\nNotice that the new_ode is fully identifiabile, and has 1 less parameter compared to the original one.\n\n\n\n\n\n","category":"function"},{"location":"utils/power_series_utils/#Power-Series-Utilities","page":"Power Series Tools","title":"Power Series Utilities","text":"","category":"section"},{"location":"utils/power_series_utils/","page":"Power Series Tools","title":"Power Series Tools","text":"Pages =[\"power_series_utils.md\"]","category":"page"},{"location":"utils/power_series_utils/","page":"Power Series Tools","title":"Power Series Tools","text":"Modules = [StructuralIdentifiability]\nPages = [\"power_series_utils.jl\"]","category":"page"},{"location":"utils/power_series_utils/#StructuralIdentifiability._matrix_inv_newton_iteration-Union{Tuple{T}, Tuple{AbstractAlgebra.MatElem{T}, AbstractAlgebra.MatElem{T}}} where T<:(AbstractAlgebra.AbsPowerSeriesRingElem{<:AbstractAlgebra.FieldElem})","page":"Power Series Tools","title":"StructuralIdentifiability._matrix_inv_newton_iteration","text":"_matrix_inv_newton_iteration(M, Minv)\n\nPerforms a single step of Newton iteration for inverting M with Minv being a partial result\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_diff-Tuple{AbstractAlgebra.AbsPowerSeriesRingElem{<:AbstractAlgebra.RingElem}}","page":"Power Series Tools","title":"StructuralIdentifiability.ps_diff","text":"ps_diff(ps)\n\nInput:\n\nps - (absolute capped) univariate power series\n\nOutput:\n\nthe derivative of ps\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_integrate-Tuple{AbstractAlgebra.AbsPowerSeriesRingElem{<:AbstractAlgebra.FieldElem}}","page":"Power Series Tools","title":"StructuralIdentifiability.ps_integrate","text":"ps_integrate(ps)\n\nInput:\n\nps - (absolute capped) univariate power series\n\nOutput:\n\nthe integral of ps without constant term\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_matrix_homlinear_de-Union{Tuple{T}, Tuple{AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:T}}, Tuple{AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:T}, Int64}} where T<:AbstractAlgebra.FieldElem","page":"Power Series Tools","title":"StructuralIdentifiability.ps_matrix_homlinear_de","text":"ps_matrix_homlinear_de(A, Y0, prec)\n\nInput:\n\nA - a square matrix with entries in a univariate power series ring\nY0 - a square invertible matrix over the base field\n\nOutput:\n\nmatrix Y such that Y' = AY up to precision of A - 1 and Y(0) = Y0\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_matrix_inv","page":"Power Series Tools","title":"StructuralIdentifiability.ps_matrix_inv","text":"ps_matrix_inv(M, prec)\n\nInput:\n\nM - a square matrix with entries in a univariate power series ring it is assumed that M(0) is invertible and all entries having the same precision\nprec - an integer, precision, if -1 then defaults to precision of M\n\nOutput:\n\nthe inverse of M computed up to prec\n\n\n\n\n\n","category":"function"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_matrix_linear_de-Union{Tuple{T}, Tuple{AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:T}}, Tuple{AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{T}}, AbstractAlgebra.MatElem{<:T}, Int64}} where T<:AbstractAlgebra.FieldElem","page":"Power Series Tools","title":"StructuralIdentifiability.ps_matrix_linear_de","text":"ps_matrix_linear_de(A, B, Y0, prec)\n\nInput:\n\nA, B - square matrices with entries in a univariate power series ring\nY0 - a matrix over the base field with the rows number the same as A\n\nOutput:\n\nmatrix Y such that Y' = AY + B up to precision of A - 1 and Y(0) = Y0\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_matrix_log-Tuple{AbstractAlgebra.MatElem{<:AbstractAlgebra.AbsPowerSeriesRingElem{<:AbstractAlgebra.FieldElem}}}","page":"Power Series Tools","title":"StructuralIdentifiability.ps_matrix_log","text":"ps_matrix_log(M)\n\nInput:\n\nM - a square matrix with entries in a univariate power series ring it is assumed that M(0) is the identity\n\nOutput:\n\nthe natural log of M\n\n\n\n\n\n","category":"method"},{"location":"utils/power_series_utils/#StructuralIdentifiability.ps_ode_solution-Union{Tuple{P}, Tuple{T}, Tuple{Vector{P}, Dict{P, T}, Dict{P, Vector{T}}, Int64}} where {T<:AbstractAlgebra.FieldElem, P<:AbstractAlgebra.MPolyRingElem{T}}","page":"Power Series Tools","title":"StructuralIdentifiability.ps_ode_solution","text":"ps_ode_solution(equations, ic, inputs, prec)\n\nInput:\n\nequations - a system of the form A(x u mu)x - B(x u mu) = 0, where A is a generically nonsingular square matrix. Assumption: A is nonzero at zero\nic - initial conditions for x's (dictionary)\ninputs - power series for inputs represented as arrays (dictionary)\nprec - precision of the solution\n\nOutput:\n\npower series solution of the system\n\n\n\n\n\n","category":"method"},{"location":"identifiability/identifiability/#Functions-to-Assess-Identifiability","page":"Functions to Assess Identifiability","title":"Functions to Assess Identifiability","text":"","category":"section"},{"location":"identifiability/identifiability/#Assessing-All-Types-of-Identifiability","page":"Functions to Assess Identifiability","title":"Assessing All Types of Identifiability","text":"","category":"section"},{"location":"identifiability/identifiability/","page":"Functions to Assess Identifiability","title":"Functions to Assess Identifiability","text":"assess_identifiability","category":"page"},{"location":"identifiability/identifiability/#StructuralIdentifiability.assess_identifiability","page":"Functions to Assess Identifiability","title":"StructuralIdentifiability.assess_identifiability","text":"assess_identifiability(ode; funcs_to_check = [], p=0.99)\n\nInput:\n\node - the ODE model\nfuncs_to_check - list of functions to check identifiability for; if empty, all parameters and states are taken\np - probability of correctness.\n\nAssesses identifiability of a given ODE model. The result is guaranteed to be correct with the probability at least p. The function returns a dictionary from the functions to check to their identifiability properties (one of :nonidentifiable, :locally, :globally).\n\n\n\n\n\nassess_identifiability(ode::ModelingToolkit.ODESystem; measured_quantities=Array{ModelingToolkit.Equation}[], funcs_to_check=[], p = 0.99)\n\nInput:\n\node - the ModelingToolkit.ODESystem object that defines the model\nmeasured_quantities - the output functions of the model\nfuncs_to_check - functions of parameters for which to check the identifiability\np - probability of correctness.\n\nAssesses identifiability (both local and global) of a given ODE model (parameters detected automatically). The result is guaranteed to be correct with the probability at least p.\n\n\n\n\n\n","category":"function"},{"location":"identifiability/identifiability/#Assessing-Local-Identifiability","page":"Functions to Assess Identifiability","title":"Assessing Local Identifiability","text":"","category":"section"},{"location":"identifiability/identifiability/","page":"Functions to Assess Identifiability","title":"Functions to Assess Identifiability","text":"assess_local_identifiability","category":"page"},{"location":"identifiability/identifiability/#StructuralIdentifiability.assess_local_identifiability","page":"Functions to Assess Identifiability","title":"StructuralIdentifiability.assess_local_identifiability","text":"function assess_local_identifiability(ode::ModelingToolkit.ODESystem; measured_quantities=Array{ModelingToolkit.Equation}[], funcs_to_check=Array{}[], p::Float64=0.99, type=:SE)\n\nInput:\n\node - the ODESystem object from ModelingToolkit\nmeasured_quantities - the measurable outputs of the model\nfuncs_to_check - functions of parameters for which to check identifiability\np - probability of correctness\ntype - identifiability type (:SE for single-experiment, :ME for multi-experiment)\n\nOutput:\n\nfor type=:SE, the result is a dictionary from each parameter to boolean;\nfor type=:ME, the result is a tuple with the dictionary as in :SE case and array of number of experiments.\n\nThe function determines local identifiability of parameters in funcs_to_check or all possible parameters if funcs_to_check is empty\n\nThe result is correct with probability at least p.\n\ntype can be either :SE (single-experiment identifiability) or :ME (multi-experiment identifiability). The return value is a tuple consisting of the array of bools and the number of experiments to be performed.\n\n\n\n\n\nassess_local_identifiability(ode::ODE{P}; funcs_to_check::Array{<: Any, 1}, p::Float64=0.99, type=:SE) where P <: MPolyElem{Nemo.fmpq}\n\nChecks the local identifiability/observability of the functions in funcs_to_check. The result is correct with probability at least p.\n\nCall this function if you have a specific collection of parameters of which you would like to check local identifiability.\n\ntype can be either :SE (single-experiment identifiability) or :ME (multi-experiment identifiability). If the type is :ME, states are not allowed to appear in the funcs_to_check.\n\n\n\n\n\nfunction assess_local_identifiability(\n dds::ModelingToolkit.DiscreteSystem; \n measured_quantities=Array{ModelingToolkit.Equation}[], \n funcs_to_check=Array{}[], \n known_ic=Array{}[],\n p::Float64=0.99)\n\nInput:\n\ndds - the DiscreteSystem object from ModelingToolkit\nmeasured_quantities - the measurable outputs of the model\nfuncs_to_check - functions of parameters for which to check identifiability (all parameters and states if not specified)\nknown_ic - functions (of states and parameter) whose initial conditions are assumed to be known\np - probability of correctness\n\nOutput:\n\nthe result is a dictionary from each function to to boolean;\n\nThe result is correct with probability at least p.\n\n\n\n\n\n","category":"function"},{"location":"identifiability/identifiability/#Assessing-Global-Identifiability","page":"Functions to Assess Identifiability","title":"Assessing Global Identifiability","text":"","category":"section"},{"location":"identifiability/identifiability/","page":"Functions to Assess Identifiability","title":"Functions to Assess Identifiability","text":"assess_global_identifiability","category":"page"},{"location":"identifiability/identifiability/#StructuralIdentifiability.assess_global_identifiability","page":"Functions to Assess Identifiability","title":"StructuralIdentifiability.assess_global_identifiability","text":"assess_global_identifiability(ode::ODE{P}, p::Float64=0.99; var_change=:default) where P <: MPolyElem{fmpq}\n\nInput:\n\node - the ODE model\nknown - a list of functions in states which are assumed to be known and generic\np - probability of correctness\nvar_change - a policy for variable change (:default, :yes, :no), affects only the runtime\n\nOutput:\n\na dictionary mapping each parameter to a boolean.\n\nChecks global identifiability for parameters of the model provided in ode. Call this function to check global identifiability of all parameters automatically.\n\n\n\n\n\nassess_global_identifiability(ode, [funcs_to_check, p=0.99, var_change=:default])\n\nInput:\n\node - the ODE model\nfuncs_to_check - rational functions in parameters\nknown - function in parameters that are assumed to be known and generic\np - probability of correctness\nvar_change - a policy for variable change (:default, :yes, :no), affects only the runtime\n\nOutput:\n\narray of length length(funcs_to_check) with true/false values for global identifiability or dictionary param => Bool if funcs_to_check are not given\n\nChecks global identifiability of functions of parameters specified in funcs_to_check.\n\n\n\n\n\n","category":"function"},{"location":"identifiability/identifiability/#Finding-Identifiable-Functions","page":"Functions to Assess Identifiability","title":"Finding Identifiable Functions","text":"","category":"section"},{"location":"identifiability/identifiability/","page":"Functions to Assess Identifiability","title":"Functions to Assess Identifiability","text":"find_identifiable_functions","category":"page"},{"location":"identifiability/identifiability/#StructuralIdentifiability.find_identifiable_functions","page":"Functions to Assess Identifiability","title":"StructuralIdentifiability.find_identifiable_functions","text":"find_identifiable_functions(ode::ODE; options...)\n\nFinds all functions of parameters/states that are identifiable in the given ODE system.\n\nOptions\n\nThis functions takes the following optional arguments:\n\nwith_states: When true, also reports the identifiabile functions in the ODE states. Default is false.\nsimplify: The extent to which the output functions are simplified. Stronger simplification may require more time. Possible options are:\n:standard: Default simplification.\n:weak: Weak simplification. This option is the fastest, but the output functions can be quite complex.\n:strong: Strong simplification. This option is the slowest, but the output\nfunctions are nice and simple.\n:absent: No simplification.\np: A float in the range from 0 to 1, the probability of correctness. Default is 0.99.\nseed: The rng seed. Default value is 42.\n\nExample\n\nusing StructuralIdentifiability\n\node = @ODEmodel(\n x0'(t) = -(a01 + a21) * x0(t) + a12 * x1(t),\n x1'(t) = a21 * x0(t) - a12 * x1(t),\n y(t) = x0(t)\n)\n\nfind_identifiable_functions(ode)\n\n# prints\n3-element Vector{AbstractAlgebra.Generic.Frac{Nemo.fmpq_mpoly}}:\n a12 + a01 + a21\n a12*a01\n\n\n\n\n\nfind_identifiable_functions(ode::ModelingToolkit.ODESystem; measured_quantities=[], options...)\n\nFinds all functions of parameters/states that are identifiable in the given ODE system.\n\nOptions\n\nThis functions takes the following optional arguments:\n\nmeasured_quantities - the output functions of the model.\n\nExample\n\nusing StructuralIdentifiability\nusing ModelingToolkit\n\n@parameters a01 a21 a12\n@variables t x0(t) x1(t) y1(t) [output = true]\nD = Differential(t)\n\neqs = [\n D(x0) ~ -(a01 + a21) * x0 + a12 * x1, \n D(x1) ~ a21 * x0 - a12 * x1, y1 ~ x0\n]\nde = ODESystem(eqs, t, name = :Test)\n\nfind_identifiable_functions(de, measured_quantities = [y1 ~ x0])\n\n# prints\n2-element Vector{Num}:\n a01*a12\n a01 + a12 + a21\n\n\n\n\n\n","category":"function"},{"location":"utils/elimination/#Elimination","page":"Elimination","title":"Elimination","text":"","category":"section"},{"location":"utils/elimination/","page":"Elimination","title":"Elimination","text":"Pages=[\"elimination.md\"]","category":"page"},{"location":"utils/elimination/","page":"Elimination","title":"Elimination","text":"Modules = [StructuralIdentifiability]\nPages = [\"elimination.jl\"]","category":"page"},{"location":"utils/elimination/#StructuralIdentifiability.Bezout_matrix-Union{Tuple{P}, Tuple{P, P, P}} where P<:AbstractAlgebra.MPolyRingElem","page":"Elimination","title":"StructuralIdentifiability.Bezout_matrix","text":"Bezout_matrix(f, g, var_elim)\n\nCompute the Bezout matrix of two polynomials f, g with respect to var_elim\n\nInputs:\n\nf - first polynomial\ng - second polynomial\nvar_elim - variable, of which f and g are considered as polynomials\n\nOutput:\n\nM::MatrixElem - The Bezout matrix\n\n\n\n\n\n","category":"method"},{"location":"utils/elimination/#StructuralIdentifiability.Sylvester_matrix-Union{Tuple{P}, Tuple{P, P, P}} where P<:AbstractAlgebra.MPolyRingElem","page":"Elimination","title":"StructuralIdentifiability.Sylvester_matrix","text":"Sylvester_matrix(f, g, var_elim)\n\nCompute the Sylvester matrix of two polynomials f, g with respect to var_elim Inputs:\n\nf - first polynomial\ng - second polynomial\nvar_elim - variable, of which f and g are considered as polynomials\n\nOutput:\n\nM::MatrixElem - The Sylvester matrix\n\n\n\n\n\n","category":"method"},{"location":"utils/elimination/#StructuralIdentifiability.choose-Union{Tuple{P}, Tuple{Vector{P}, Any}} where P<:(AbstractAlgebra.MPolyRingElem{<:AbstractAlgebra.FieldElem})","page":"Elimination","title":"StructuralIdentifiability.choose","text":"choose(polys, generic_point_generator)\n\nInput:\n\npolys - an array of distinct irreducible polynomials in the same ring\ngeneric_point_generator - a generic point generator as described above for one of polys\n\nOutput:\n\nthe polynomial that vanishes at the generic_point_generator\n\n\n\n\n\n","category":"method"},{"location":"utils/elimination/#StructuralIdentifiability.eliminate_var-Union{Tuple{P}, Tuple{P, P, P, Any}} where P<:(AbstractAlgebra.MPolyRingElem{<:AbstractAlgebra.FieldElem})","page":"Elimination","title":"StructuralIdentifiability.eliminate_var","text":"eliminate_var(f, g, var_elim, generic_point_generator)\n\nEliminate a variable from a pair of polynomials\n\nInput:\n\nf and g - polynomials\nvar_elim - variable to be eliminated\ngeneric_point_generator - a generic point generator object for the factor of the resultant of f and g of interest\n\nOutput:\n\npolynomial - the desired factor of the resultant of f and g\n\n\n\n\n\n","category":"method"},{"location":"utils/elimination/#StructuralIdentifiability.simplify_matrix-Union{Tuple{AbstractAlgebra.MatElem{P}}, Tuple{P}} where P<:AbstractAlgebra.MPolyRingElem","page":"Elimination","title":"StructuralIdentifiability.simplify_matrix","text":"simplify_matrix(M)\n\nEliminate GCD of entries of every row and column\n\nInput:\n\nM::MatrixElem - matrix to be simplified\n\nOutput:\n\nM::MatrixElem - Simplified matrix\nextra_factors::Vector{AbstractAlgebra.MPolyElem} - array of GCDs eliminated from M.\n\n\n\n\n\n","category":"method"},{"location":"tutorials/discrete_time/#Identifiability-of-Discrete-Time-Models-(Local)","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"","category":"section"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"Now we consider a discrete-time model in the state-space form","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"begincases\nmathbfx(t + 1) = mathbff(mathbfx(t) mathbfp mathbfu(t))\nmathbfy(t) = mathbfg(mathbfx(t) mathbfp mathbfu(t))\nendcases","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"where mathbfx(t) mathbfy(t), and mathbfu(t) are time-dependent states, outputs, and inputs, respectively, and mathbfp are scalar parameters. As in the ODE case, we will call that a parameter or a states (or a function of them) is identifiable if its value can be recovered from time series for inputs and outputs (in the generic case, see Definition 3 in [1] for details). Again, we will distinguish two types of identifiability","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"local identifiability: the value can be recovered up to finitely many options;\nglobal identifiability: the value can be recovered uniquely.","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"Currently, StructuralIdentifiability.jl allows to assess only local identifiability for discrete-time models, and below we will describe how this can be done. As a running example, we will use the following discrete version of the SIR model:","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"begincases\nS(t + 1) = S(t) - beta S(t) I(t)\nI(t + 1) = I(t) + beta S(t) I(t) - alpha I(t)\nR(t + 1) = R(t) + alpha I(t)\ny(t) = I(t)\nendcases","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"where the observable is I, the number of infected people. We start with creating a system as a DiscreteSystem from ModelingToolkit:","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"using ModelingToolkit\nusing StructuralIdentifiability\n\n@parameters α β\n@variables t S(t) I(t) R(t) y(t)\nD = Difference(t; dt = 1.0)\n\neqs = [D(S) ~ S - β * S * I, D(I) ~ I + β * S * I - α * I, D(R) ~ R + α * I]\n@named sir = DiscreteSystem(eqs)","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"Once the model is defined, we can assess identifiability by providing the formula for the observable:","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"assess_local_identifiability(sir; measured_quantities = [y ~ I])","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"For each parameter or state, the value in the returned dictionary is true (1) if the parameter is locally identifiable and false (0) otherwise. We see that R(t) is not identifiable, which makes sense: it does not affect the dynamics of the observable in any way.","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"In principle, it is not required to give a name to the observable, so one can write this shorter","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"assess_local_identifiability(sir; measured_quantities = [I])","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"The assess_local_identifiability function has two important keyword arguments:","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"funcs_to_check is a list of functions for which one want to assess identifiability, for example, the following code will check if β * S is locally identifiable.","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"assess_local_identifiability(sir; measured_quantities = [I], funcs_to_check = [β * S])","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"p is the probability of correctness (default value 0.99, i.e., 99%). The underlying algorithm is a Monte-Carlo algorithm, so in principle it may produce incorrect result but the probability of correctness of the returned result is guaranteed to be at least p (in fact, the employed bounds are quite conservative, so in practice incorrect result is almost never produced).","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"The implementation is based on a version of the observability rank criterion and will be described in a forthcoming paper.","category":"page"},{"location":"tutorials/discrete_time/","page":"Identifiability of Discrete-Time Models (Local)","title":"Identifiability of Discrete-Time Models (Local)","text":"[1]: S. Nõmm, C. Moog, Identifiability of discrete-time nonlinear systems, IFAC Proceedings Volumes, 2004.","category":"page"},{"location":"tutorials/creating_ode/#Creating-ODE-System","page":"Creating ODE System","title":"Creating ODE System","text":"","category":"section"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"Most of the algorithms in StructuralIdentifiability.jl take as input system of ordinary differential equations (ODEs) in the state space form, that is:","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"begincases\nmathbfx(t) = mathbff(mathbfx(t) mathbfp mathbfu(t))\nmathbfy(t) = mathbfg(mathbfx(t) mathbfp mathbfu(t))\nendcases","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"which involves","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"a vector mathbfx(t) of the state variables of the system,\na vector mathbfu(t) of extermal inputs,\na vector mathbfp of scalar parameters,\na vector mathbfy(t) of outputs (i.e., observations),\nand vectors of rational functions mathbff and mathbfg (for discussion of the non-rational case, see this issue).","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"In the standard setup, inputs and outputs are assumed to be known, and the goal is to assess identifiability of parameters and/or states from the input-output data. In the case of states, this property is also called observability.","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"There are two ways to define such a system to be processed using StructuralIdentifiability.jl. We will demonstrate them using the following example system (Wright's population model of two mutualist species with control[1]):","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"begincases\nx_1(t) = r_1 x_1(t)(1 - c_1 x_1(t)) + fracbeta_1 x_1(t)x_2(t)chi_1 + x_2(t) + u(t)\nx_2(t) = r_2 x_2(t)(1 - c_2 x_2(t)) + fracbeta_2 x_1(t)x_2(t)chi_2 + x_1(t)\ny(t) = x_1(t)\nendcases","category":"page"},{"location":"tutorials/creating_ode/#Defining-the-model-using-@ODEmodel-macro","page":"Creating ODE System","title":"Defining the model using @ODEmodel macro","text":"","category":"section"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"One way to define the model is to use the @ODEmodel macro provided by the StructuralIdentifiability.jl package.","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"using StructuralIdentifiability\n\node = @ODEmodel(\n x1'(t) =\n r1 * x1(t) * (1 - c1 * x1(t)) + beta1 * x1(t) * x2(t) / (chi1 + x2(t)) + u(t),\n x2'(t) = r2 * x2(t) * (1 - c2 * x2(t)) + beta2 * x1(t) * x2(t) / (chi2 + x1(t)),\n y(t) = x1(t)\n)","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"Then one can, for example, assess identifiability of the parameters and states by","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"assess_identifiability(ode)","category":"page"},{"location":"tutorials/creating_ode/#Defining-using-ModelingToolkit","page":"Creating ODE System","title":"Defining using ModelingToolkit","text":"","category":"section"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"Alternatively, one can use ModelingToolkit: encode the equations for the states as ODESystem and specify the outputs separately. In order to do this, we first introduce all functions and scalars:","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"using StructuralIdentifiability, ModelingToolkit\n\n@parameters r1, r2, c1, c2, beta1, beta2, chi1, chi2\n@variables t, x1(t), x2(t), y(t), u(t)\n\nD = Differential(t)","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"And then defined the system and separately the outputs:","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"eqs = [\n D(x1) ~ r1 * x1 * (1 - c1 * x1) + beta1 * x1 * x2 / (chi1 + x2) + u,\n D(x2) ~ r2 * x2 * (1 - c2 * x2) + beta2 * x1 * x2 / (chi2 + x1),\n]\n\nmeasured_quantities = [y ~ x1]\n\node_mtk = ODESystem(eqs, t, name = :mutualist)","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"Then, for example, the identifiability of parameters and states can be assessed as follows:","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"assess_identifiability(ode_mtk, measured_quantities = measured_quantities)","category":"page"},{"location":"tutorials/creating_ode/","page":"Creating ODE System","title":"Creating ODE System","text":"[1]: D. H. Wright, A Simple, Stable Model of Mutualism Incorporating Handling Time, The American Naturalist, 1989, 134(4).","category":"page"},{"location":"ioequations/ioequations/#Finding-Input-Output-Equations","page":"Input-Output Equation tools","title":"Finding Input-Output Equations","text":"","category":"section"},{"location":"ioequations/ioequations/","page":"Input-Output Equation tools","title":"Input-Output Equation tools","text":"find_ioequations","category":"page"},{"location":"ioequations/ioequations/#StructuralIdentifiability.find_ioequations","page":"Input-Output Equation tools","title":"StructuralIdentifiability.find_ioequations","text":"find_ioequations(ode, [var_change_policy=:default])\n\nFinds the input-output equations of an ODE system Input:\n\node - the ODE system\nvar_change_policy - whether to perform automatic variable change, can be one of :default, :yes, :no\n\nOutput:\n\na dictionary from “leaders” to the corresponding input-output equations; if an extra projection is needed, it will be the value corresponding to rand_proj_var\n\n\n\n\n\n","category":"function"},{"location":"ioequations/ioequations/#Reducing-with-respect-to-Input-Output-Equations","page":"Input-Output Equation tools","title":"Reducing with respect to Input-Output Equations","text":"","category":"section"},{"location":"ioequations/ioequations/","page":"Input-Output Equation tools","title":"Input-Output Equation tools","text":"PBRepresentation\npseudodivision\ndiffreduce\nio_switch!","category":"page"},{"location":"ioequations/ioequations/#StructuralIdentifiability.PBRepresentation","page":"Input-Output Equation tools","title":"StructuralIdentifiability.PBRepresentation","text":"The structure for storing a projection-based representation of differential ideal (see Section 2.3 https://arxiv.org/abs/2111.00991). Contains the following fields:\n\ny_names - the names of the variables with finite order in the profile (typically, outputs)\nu_names - the names of the variables with infinite order in the profile (typically, inputs)\nparam_names - the names of the parameters\nprofile - the profile of the PB-representation (see Definition 2.13) as a dict from y_names with finite orders to the orders\nprojections - the corresponding projections (see Definition 2.15) as a dict from y_names to the projections\n\n\n\n\n\n","category":"type"},{"location":"ioequations/ioequations/#StructuralIdentifiability.pseudodivision","page":"Input-Output Equation tools","title":"StructuralIdentifiability.pseudodivision","text":"pseudodivision(f, g, x)\n\nComputes the result of pseudodivision of f by g as univariate polynomials in x Input:\n\nf - the polynomial to be divided\ng - the polynomial to divide by\nx - the variable for the division\n\nOutput: the pseudoremainder of f divided by g w.r.t. x\n\n\n\n\n\n","category":"function"},{"location":"ioequations/ioequations/#StructuralIdentifiability.diffreduce","page":"Input-Output Equation tools","title":"StructuralIdentifiability.diffreduce","text":"diffreduce(diffpoly, pbr)\n\nComputes the result of differential reduction of a differential polynomial diffpoly with respect to the charset defined by a PB-representation pbr Input:\n\ndiffpoly - a polynomial representing a differential polynomial to be reduced\npbr - a projection-based representation\n\nOutput: the result of differential reduction of diffpoly by pbr considered as a characteristic set (see Remark 2.20 in the paper)\n\n\n\n\n\n","category":"function"},{"location":"ioequations/ioequations/#StructuralIdentifiability.io_switch!","page":"Input-Output Equation tools","title":"StructuralIdentifiability.io_switch!","text":"io_switch(pbr)\n\nIn a single-output pb-representation pbr makes the leading variable to be the first of the inputs\n\n\n\n\n\n","category":"function"},{"location":"utils/local_identifiability/#Local-Identifiability-Tools","page":"Local Identifiability Tools","title":"Local Identifiability Tools","text":"","category":"section"},{"location":"utils/local_identifiability/","page":"Local Identifiability Tools","title":"Local Identifiability Tools","text":"Pages=[\"local_identifiability.md\"]","category":"page"},{"location":"utils/local_identifiability/","page":"Local Identifiability Tools","title":"Local Identifiability Tools","text":"CurrentModule=StructuralIdentifiability","category":"page"},{"location":"utils/local_identifiability/","page":"Local Identifiability Tools","title":"Local Identifiability Tools","text":"StructuralIdentifiability.differentiate_solution\nStructuralIdentifiability.differentiate_output","category":"page"},{"location":"utils/local_identifiability/#StructuralIdentifiability.differentiate_solution","page":"Local Identifiability Tools","title":"StructuralIdentifiability.differentiate_solution","text":"differentiate_solution(ode, params, ic, inputs, prec)\n\nInput:\n\nthe same as for power_series_solutions\n\nOutput:\n\na tuple consisting of the power series solution and a dictionary of the form (u, v) => power series, where u is a state variable v is a state or parameter, and the power series is the partial derivative of the function u w.r.t. v evaluated at the solution\n\n\n\n\n\n","category":"function"},{"location":"utils/local_identifiability/#StructuralIdentifiability.differentiate_output","page":"Local Identifiability Tools","title":"StructuralIdentifiability.differentiate_output","text":"differentiate_output(ode, params, ic, inputs, prec)\n\nSimilar to differentiate_solution but computes partial derivatives of prescribed outputs returns a dictionary of the form y_function => Dict(var => dy/dvar) where dy/dvar is the derivative of y_function with respect to var.\n\n\n\n\n\n","category":"function"},{"location":"#StructuralIdentifiability.jl","page":"Home","title":"StructuralIdentifiability.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"StructuralIdentifiability.jl is a comprehensive toolbox for assessing identifiability parameters.","category":"page"},{"location":"#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To install StructuralIdentifiability.jl, use the Julia package manager:","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg\nPkg.add(\"StructuralIdentifiability\")","category":"page"},{"location":"#Citation","page":"Home","title":"Citation","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"@article{structidjl,\n author = {Dong, R. and Goodbrake, C. and Harrington, H. and Pogudin G.},\n title = {Differential Elimination for Dynamical Models via Projections with Applications to Structural Identifiability},\n journal = {SIAM Journal on Applied Algebra and Geometry},\n url = {https://doi.org/10.1137/22M1469067},\n year = {2023}\n volume = {7},\n number = {1},\n pages = {194-235}\n}","category":"page"},{"location":"#Feature-Summary","page":"Home","title":"Feature Summary","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"StructuralIdentifiability.jl can assess local and global identifiability of ODE models. In addition to these straightforward identifiability queries on individual parameters, the package can distinguish between single- and multi-experiment identifiability.","category":"page"},{"location":"#Feature-List","page":"Home","title":"Feature List","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Local identifiability checks\nGlobal identifiability checks\nAssessment of identifiable functions of parameters and states\nModel reparametrization (experimental)","category":"page"},{"location":"#Contributing","page":"Home","title":"Contributing","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to StructuralIdentifiability.\nThere are a few community forums:\nThe #diffeq-bridged channel in the Julia Slack\nJuliaDiffEq on Gitter\nOn the Julia Discourse forums\nSee also SciML Community page","category":"page"},{"location":"#Reproducibility","page":"Home","title":"Reproducibility","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"
The documentation of this SciML package was built using these direct dependencies,","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg # hide\nPkg.status() # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"
and using this machine and Julia version.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using InteractiveUtils # hide\nversioninfo() # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"
A more complete overview of all dependencies and their versions is also provided.","category":"page"},{"location":"","page":"Home","title":"Home","text":"using Pkg # hide\nPkg.status(; mode = PKGMODE_MANIFEST) # hide","category":"page"},{"location":"","page":"Home","title":"Home","text":"
","category":"page"},{"location":"","page":"Home","title":"Home","text":"You can also download the \nmanifest file and the\nproject file.","category":"page"},{"location":"input/input/#Parsing-input-ODE-system","page":"Parsing input ODE system","title":"Parsing input ODE system","text":"","category":"section"},{"location":"input/input/","page":"Parsing input ODE system","title":"Parsing input ODE system","text":"@ODEmodel(ex::Expr...)\nODE\nset_parameter_values","category":"page"},{"location":"input/input/#StructuralIdentifiability.@ODEmodel-Tuple{Vararg{Expr}}","page":"Parsing input ODE system","title":"StructuralIdentifiability.@ODEmodel","text":"macro ODEmodel\n\nMacro for creating an ODE from a list of equations. It also injects all variables into the global scope.\n\nExample\n\nCreating a simple ODE:\n\nusing StructuralIdentifiability\n\node = @ODEmodel(\n x1'(t) = a * x1(t) + u(t),\n x2'(t) = b * x2(t) + c*x1(t)*x2(t),\n y(t) = x1(t)\n)\n\nHere,\n\nx1, x2 are state variables\ny is an output variable\nu is an input variable\na, b, c are time-indepdendent parameters\n\n\n\n\n\n","category":"macro"},{"location":"input/input/#StructuralIdentifiability.ODE","page":"Parsing input ODE system","title":"StructuralIdentifiability.ODE","text":"The main structure that represents input ODE system.\n\nStores information about states (x_vars), outputs (y_vars), inputs (u_vars), parameters (parameters) and the equations.\n\nThis structure is constructed via @ODEmodel macro.\n\n\n\n\n\n","category":"type"},{"location":"input/input/#StructuralIdentifiability.set_parameter_values","page":"Parsing input ODE system","title":"StructuralIdentifiability.set_parameter_values","text":"set_parameter_values(ode, param_values)\n\nInput:\n\node - an ODE as above\nparam_values - values for (possibly, some of) the parameters as dictionary parameter => value\n\nOutput:\n\nnew ode with the parameters in param_values plugged with the given numbers\n\n\n\n\n\n","category":"function"},{"location":"input/input/#Create-Compartmental-Model","page":"Parsing input ODE system","title":"Create Compartmental Model","text":"","category":"section"},{"location":"input/input/","page":"Parsing input ODE system","title":"Parsing input ODE system","text":"linear_compartment_model","category":"page"},{"location":"input/input/#StructuralIdentifiability.linear_compartment_model","page":"Parsing input ODE system","title":"StructuralIdentifiability.linear_compartment_model","text":"linear_compartment_model(graph, inputs, outputs, leaks)\n\nInput: defines a linear compartment model with nodes numbered from 1 to n by\n\ngraph - and array of integer arrays representing the adjacency lists of the graph\ninputs - array of input nodes\noutputs - array of output nodes\nleaks - array of sink nodes\n\nOutput:\n\nthe corresponding ODE system in the notation of https://doi.org/10.1007/s11538-015-0098-0\n\n\n\n\n\n","category":"function"}] } diff --git a/previews/PR238/tutorials/discrete_time/index.html b/previews/PR238/tutorials/discrete_time/index.html index d2cf1f478..0729c71e5 100644 --- a/previews/PR238/tutorials/discrete_time/index.html +++ b/previews/PR238/tutorials/discrete_time/index.html @@ -6,10 +6,12 @@
GitHub

Identifiability of Discrete-Time Models (Local)

Now we consider a discrete-time model in the state-space form

\[\begin{cases} \mathbf{x}(t + 1) = \mathbf{f}(\mathbf{x}(t), \mathbf{p}, \mathbf{u}(t)),\\ \mathbf{y}(t) = \mathbf{g}(\mathbf{x}(t), \mathbf{p}, \mathbf{u(t)}), -\end{cases}\]

where $\mathbf{x}(t), \mathbf{y}(t)$, and $\mathbf{u}(t)$ are time-dependent states, outputs, and inputs, respectively, and $\mathbf{p}$ are scalar parameters. As in the ODE case, we will call that a parameter or a states (or a function of them) is identifiable if its value can be recovered from time series for inputs and outputs (in the generic case, see Definition 3 in [1] for details). Again, we will distinguish two types of identifiability

  • local identifiability: the value can be recovered up to finitely many options;

  • global identifiability: the value can be recovered uniquely.

Currently, StructuralIdentifiability.jl allows to assess only local identifiability for discrete-time models, and below we will describe how this can be done. As a running example, we will use the following discrete version of the SIR model: $\begin{cases} S(t + 1) = S(t) - \beta S(t) I(t),\\ +\end{cases}\]

where $\mathbf{x}(t), \mathbf{y}(t)$, and $\mathbf{u}(t)$ are time-dependent states, outputs, and inputs, respectively, and $\mathbf{p}$ are scalar parameters. As in the ODE case, we will call that a parameter or a states (or a function of them) is identifiable if its value can be recovered from time series for inputs and outputs (in the generic case, see Definition 3 in [1] for details). Again, we will distinguish two types of identifiability

  • local identifiability: the value can be recovered up to finitely many options;

  • global identifiability: the value can be recovered uniquely.

Currently, StructuralIdentifiability.jl allows to assess only local identifiability for discrete-time models, and below we will describe how this can be done. As a running example, we will use the following discrete version of the SIR model:

\[\begin{cases} +S(t + 1) = S(t) - \beta S(t) I(t),\\ I(t + 1) = I(t) + \beta S(t) I(t) - \alpha I(t),\\ R(t + 1) = R(t) + \alpha I(t),\\ -y(t) = I(t), \end{cases}$

where the observable is I, the number of infected people. We start with creating a system as a DiscreteSystem from ModelingToolkit:

using ModelingToolkit
+y(t) = I(t),
+\end{cases}\]
where the observable is I, the number of infected people. We start with creating a system as a DiscreteSystem from ModelingToolkit:
using ModelingToolkit
 using StructuralIdentifiability
 
 @parameters α β
diff --git a/previews/PR238/utils/elimination/index.html b/previews/PR238/utils/elimination/index.html
index 243d4f96f..51b225757 100644
--- a/previews/PR238/utils/elimination/index.html
+++ b/previews/PR238/utils/elimination/index.html
@@ -3,4 +3,4 @@
   function gtag(){dataLayer.push(arguments);}
   gtag('js', new Date());
   gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash});
-
GitHub
Elimination
StructuralIdentifiability.Bezout_matrix — Method
Bezout_matrix(f, g, var_elim)
Compute the Bezout matrix of two polynomials f, g with respect to var_elim
Inputs:
  • f - first polynomial
  • g - second polynomial
  • var_elim - variable, of which f and g are considered as polynomials
Output:
  • M::MatrixElem - The Bezout matrix
source
StructuralIdentifiability.Sylvester_matrix — Method
Sylvester_matrix(f, g, var_elim)
Compute the Sylvester matrix of two polynomials f, g with respect to var_elim Inputs:
  • f - first polynomial
  • g - second polynomial
  • var_elim - variable, of which f and g are considered as polynomials
Output:
  • M::MatrixElem - The Sylvester matrix
source
StructuralIdentifiability.choose — Method
choose(polys, generic_point_generator)
Input:
  • polys - an array of distinct irreducible polynomials in the same ring
  • generic_point_generator - a generic point generator as described above for one of polys
Output:
  • the polynomial that vanishes at the generic_point_generator
source
StructuralIdentifiability.eliminate_var — Method
eliminate_var(f, g, var_elim, generic_point_generator)
Eliminate a variable from a pair of polynomials
Input:
  • f and g - polynomials
  • var_elim - variable to be eliminated
  • generic_point_generator - a generic point generator object for the factor     of the resultant of f and g of interest
Output:
  • polynomial - the desired factor of the resultant of f and g
source
StructuralIdentifiability.simplify_matrix — Method
simplify_matrix(M)
Eliminate GCD of entries of every row and column
Input:
  • M::MatrixElem - matrix to be simplified
Output:
  • M::MatrixElem - Simplified matrix
  • extra_factors::Vector{AbstractAlgebra.MPolyElem} - array of GCDs eliminated from M.
source

+
GitHub
Elimination
StructuralIdentifiability.Bezout_matrix — Method
Bezout_matrix(f, g, var_elim)
Compute the Bezout matrix of two polynomials f, g with respect to var_elim
Inputs:
  • f - first polynomial
  • g - second polynomial
  • var_elim - variable, of which f and g are considered as polynomials
Output:
  • M::MatrixElem - The Bezout matrix
source
StructuralIdentifiability.Sylvester_matrix — Method
Sylvester_matrix(f, g, var_elim)
Compute the Sylvester matrix of two polynomials f, g with respect to var_elim Inputs:
  • f - first polynomial
  • g - second polynomial
  • var_elim - variable, of which f and g are considered as polynomials
Output:
  • M::MatrixElem - The Sylvester matrix
source
StructuralIdentifiability.choose — Method
choose(polys, generic_point_generator)
Input:
  • polys - an array of distinct irreducible polynomials in the same ring
  • generic_point_generator - a generic point generator as described above for one of polys
Output:
  • the polynomial that vanishes at the generic_point_generator
source
StructuralIdentifiability.eliminate_var — Method
eliminate_var(f, g, var_elim, generic_point_generator)
Eliminate a variable from a pair of polynomials
Input:
  • f and g - polynomials
  • var_elim - variable to be eliminated
  • generic_point_generator - a generic point generator object for the factor     of the resultant of f and g of interest
Output:
  • polynomial - the desired factor of the resultant of f and g
source
StructuralIdentifiability.simplify_matrix — Method
simplify_matrix(M)
Eliminate GCD of entries of every row and column
Input:
  • M::MatrixElem - matrix to be simplified
Output:
  • M::MatrixElem - Simplified matrix
  • extra_factors::Vector{AbstractAlgebra.MPolyElem} - array of GCDs eliminated from M.
source

diff --git a/previews/PR238/utils/global_identifiability/index.html b/previews/PR238/utils/global_identifiability/index.html
index b9f753963..80997b7e3 100644
--- a/previews/PR238/utils/global_identifiability/index.html
+++ b/previews/PR238/utils/global_identifiability/index.html
@@ -12,4 +12,4 @@
 rff = RationalFunctionField([x // y, y // z])
 
 # Constructs a subfield generated by y / x, 1 / x, z / y
-rff = RationalFunctionField([[x, y, R(1)], [y, z]])
source
StructuralIdentifiability.field_contains — Function
field_contains(field, ratfuncs, p)

Checks whether given rational function field field contains given rational functions ratfuncs (represented as a list of lists). The result is correct with probability at least p

Inputs:

  • field - a rational function field
  • ratfuncs - a list of lists of polynomials. Each of the lists, say, [f1, ..., fn], defines generators f2/f1, ..., fn/f1.
  • p real number from (0, 1)

Output:

  • a list L[i] of bools of length length(rat_funcs) such that L[i] is true iff the i-th function belongs to field
source
+rff = RationalFunctionField([[x, y, R(1)], [y, z]])source
StructuralIdentifiability.field_contains — Function
field_contains(field, ratfuncs, p)

Checks whether given rational function field field contains given rational functions ratfuncs (represented as a list of lists). The result is correct with probability at least p

Inputs:

  • field - a rational function field
  • ratfuncs - a list of lists of polynomials. Each of the lists, say, [f1, ..., fn], defines generators f2/f1, ..., fn/f1.
  • p real number from (0, 1)

Output:

  • a list L[i] of bools of length length(rat_funcs) such that L[i] is true iff the i-th function belongs to field
source
diff --git a/previews/PR238/utils/local_identifiability/index.html b/previews/PR238/utils/local_identifiability/index.html index b7c5baba4..c38eb29d6 100644 --- a/previews/PR238/utils/local_identifiability/index.html +++ b/previews/PR238/utils/local_identifiability/index.html @@ -3,4 +3,4 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub

Local Identifiability Tools

StructuralIdentifiability.differentiate_solution — Function
differentiate_solution(ode, params, ic, inputs, prec)

Input:

  • the same as for power_series_solutions

Output:

  • a tuple consisting of the power series solution and a dictionary of the form (u, v) => power series, where u is a state variable v is a state or parameter, and the power series is the partial derivative of the function u w.r.t. v evaluated at the solution
source
StructuralIdentifiability.differentiate_output — Function
differentiate_output(ode, params, ic, inputs, prec)

Similar to differentiate_solution but computes partial derivatives of prescribed outputs returns a dictionary of the form y_function => Dict(var => dy/dvar) where dy/dvar is the derivative of y_function with respect to var.

source
+
GitHub

Local Identifiability Tools

StructuralIdentifiability.differentiate_solution — Function
differentiate_solution(ode, params, ic, inputs, prec)

Input:

  • the same as for power_series_solutions

Output:

  • a tuple consisting of the power series solution and a dictionary of the form (u, v) => power series, where u is a state variable v is a state or parameter, and the power series is the partial derivative of the function u w.r.t. v evaluated at the solution
source
StructuralIdentifiability.differentiate_output — Function
differentiate_output(ode, params, ic, inputs, prec)

Similar to differentiate_solution but computes partial derivatives of prescribed outputs returns a dictionary of the form y_function => Dict(var => dy/dvar) where dy/dvar is the derivative of y_function with respect to var.

source
diff --git a/previews/PR238/utils/ode/index.html b/previews/PR238/utils/ode/index.html index 81fb28087..eba4c4762 100644 --- a/previews/PR238/utils/ode/index.html +++ b/previews/PR238/utils/ode/index.html @@ -3,8 +3,8 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub

Functions to work with the ODE structure

StructuralIdentifiability.__preprocess_ode — Method
function __preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{Tuple{String, SymbolicUtils.BasicSymbolic}})

Input:

  • de - ModelingToolkit.AbstractTimeDependentSystem, a system for identifiability query
  • measured_quantities - array of input function in the form (name, expression)

Output:

  • ODE object containing required data for identifiability assessment
  • conversion dictionary from the symbols in the input MTK model to the variable involved in the produced ODE object
source
StructuralIdentifiability.power_series_solution — Method
power_series_solution(ode, param_values, initial_conditions, input_values, prec)

Input:

  • ode - an ode to solve
  • param_values - parameter values, must be a dictionary mapping parameter to a value
  • initial_conditions - initial conditions of ode, must be a dictionary mapping state variable to a value
  • input_values - power series for the inputs presented as a dictionary variable => list of coefficients
  • prec - the precision of solutions

Output:

  • computes a power series solution with precision prec presented as a dictionary variable => corresponding coordinate of the solution
source
StructuralIdentifiability.preprocess_ode — Method
function preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{ModelingToolkit.Equation})
-function preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{SymbolicUtils.BasicSymbolic})

Input:

  • de - ModelingToolkit.AbstractTimeDependentSystem, a system for identifiability query
  • measured_quantities - array of output functions (as equations of just functions)

Output:

  • ODE object containing required data for identifiability assessment
  • conversion dictionary from the symbols in the input MTK model to the variable involved in the produced ODE object
source
StructuralIdentifiability.set_parameter_values — Method
set_parameter_values(ode, param_values)

Input:

  • ode - an ODE as above
  • param_values - values for (possibly, some of) the parameters as dictionary parameter => value

Output:

  • new ode with the parameters in param_values plugged with the given numbers
source
StructuralIdentifiability.find_submodels — Method
find_submodels(ode)

The function calculates and returns all valid submodels given a system of ODEs.

Input:

  • ode - an ODEs system to be studied

Output:

  • A list of submodels represented as ode objects

Example:

>ode = @ODEmodel(x1'(t) = x1(t)^2, 
+
GitHub
Functions to work with the ODE structure
StructuralIdentifiability.__preprocess_ode — Method
function __preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{Tuple{String, SymbolicUtils.BasicSymbolic}})
Input:
  • de - ModelingToolkit.AbstractTimeDependentSystem, a system for identifiability query
  • measured_quantities - array of input function in the form (name, expression)
Output:
  • ODE object containing required data for identifiability assessment
  • conversion dictionary from the symbols in the input MTK model to the variable involved in the produced ODE object
source
StructuralIdentifiability.power_series_solution — Method
power_series_solution(ode, param_values, initial_conditions, input_values, prec)
Input:
  • ode - an ode to solve
  • param_values - parameter values, must be a dictionary mapping parameter to a value
  • initial_conditions - initial conditions of ode, must be a dictionary mapping state variable to a value
  • input_values - power series for the inputs presented as a dictionary variable => list of coefficients
  • prec - the precision of solutions
Output:
  • computes a power series solution with precision prec presented as a dictionary variable => corresponding coordinate of the solution
source
StructuralIdentifiability.preprocess_ode — Method
function preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{ModelingToolkit.Equation})
+function preprocess_ode(de::ModelingToolkit.AbstractTimeDependentSystem, measured_quantities::Array{SymbolicUtils.BasicSymbolic})
Input:
  • de - ModelingToolkit.AbstractTimeDependentSystem, a system for identifiability query
  • measured_quantities - array of output functions (as equations of just functions)
Output:
  • ODE object containing required data for identifiability assessment
  • conversion dictionary from the symbols in the input MTK model to the variable involved in the produced ODE object
source
StructuralIdentifiability.set_parameter_values — Method
set_parameter_values(ode, param_values)
Input:
  • ode - an ODE as above
  • param_values - values for (possibly, some of) the parameters as dictionary parameter => value
Output:
  • new ode with the parameters in param_values plugged with the given numbers
source
StructuralIdentifiability.find_submodels — Method
find_submodels(ode)
The function calculates and returns all valid submodels given a system of ODEs.
Input:
  • ode - an ODEs system to be studied
Output: 
  • A list of submodels represented as ode objects
Example:
>ode = @ODEmodel(x1'(t) = x1(t)^2, 
                  x2'(t) = x1(t) * x2(t), 
                  y1(t) = x1(t), 
                  y2(t) = x2(t))
@@ -13,4 +13,4 @@
         
         x1'(t) = a(t)*x2(t)^2 + x1(t)
         y1(t) = x1(t)
-    ]
source

+    ]
source
diff --git a/previews/PR238/utils/power_series_utils/index.html b/previews/PR238/utils/power_series_utils/index.html index 75c127606..383a0cf22 100644 --- a/previews/PR238/utils/power_series_utils/index.html +++ b/previews/PR238/utils/power_series_utils/index.html @@ -3,4 +3,4 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub

Power Series Utilities

StructuralIdentifiability.ps_matrix_homlinear_de — Method
ps_matrix_homlinear_de(A, Y0, prec)

Input:

  • A - a square matrix with entries in a univariate power series ring
  • Y0 - a square invertible matrix over the base field

Output:

  • matrix Y such that Y' = AY up to precision of A - 1 and Y(0) = Y0
source
StructuralIdentifiability.ps_matrix_inv — Function
ps_matrix_inv(M, prec)

Input:

  • M - a square matrix with entries in a univariate power series ring it is assumed that M(0) is invertible and all entries having the same precision
  • prec - an integer, precision, if -1 then defaults to precision of M

Output:

  • the inverse of M computed up to prec
source
StructuralIdentifiability.ps_matrix_linear_de — Method
ps_matrix_linear_de(A, B, Y0, prec)

Input:

  • A, B - square matrices with entries in a univariate power series ring
  • Y0 - a matrix over the base field with the rows number the same as A

Output:

  • matrix Y such that Y' = AY + B up to precision of A - 1 and Y(0) = Y0
source
StructuralIdentifiability.ps_ode_solution — Method
ps_ode_solution(equations, ic, inputs, prec)

Input:

  • equations - a system of the form $A(x, u, mu)x' - B(x, u, mu) = 0$, where A is a generically nonsingular square matrix. Assumption: A is nonzero at zero
  • ic - initial conditions for x's (dictionary)
  • inputs - power series for inputs represented as arrays (dictionary)
  • prec - precision of the solution

Output:

  • power series solution of the system
source
+
GitHub

Power Series Utilities

StructuralIdentifiability.ps_matrix_homlinear_de — Method
ps_matrix_homlinear_de(A, Y0, prec)

Input:

  • A - a square matrix with entries in a univariate power series ring
  • Y0 - a square invertible matrix over the base field

Output:

  • matrix Y such that Y' = AY up to precision of A - 1 and Y(0) = Y0
source
StructuralIdentifiability.ps_matrix_inv — Function
ps_matrix_inv(M, prec)

Input:

  • M - a square matrix with entries in a univariate power series ring it is assumed that M(0) is invertible and all entries having the same precision
  • prec - an integer, precision, if -1 then defaults to precision of M

Output:

  • the inverse of M computed up to prec
source
StructuralIdentifiability.ps_matrix_linear_de — Method
ps_matrix_linear_de(A, B, Y0, prec)

Input:

  • A, B - square matrices with entries in a univariate power series ring
  • Y0 - a matrix over the base field with the rows number the same as A

Output:

  • matrix Y such that Y' = AY + B up to precision of A - 1 and Y(0) = Y0
source
StructuralIdentifiability.ps_ode_solution — Method
ps_ode_solution(equations, ic, inputs, prec)

Input:

  • equations - a system of the form $A(x, u, mu)x' - B(x, u, mu) = 0$, where A is a generically nonsingular square matrix. Assumption: A is nonzero at zero
  • ic - initial conditions for x's (dictionary)
  • inputs - power series for inputs represented as arrays (dictionary)
  • prec - precision of the solution

Output:

  • power series solution of the system
source
diff --git a/previews/PR238/utils/primality/index.html b/previews/PR238/utils/primality/index.html index 0f9c8737a..39b3dc47f 100644 --- a/previews/PR238/utils/primality/index.html +++ b/previews/PR238/utils/primality/index.html @@ -3,4 +3,4 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub

Primality Checks

StructuralIdentifiability.check_primality — Function
check_primality(polys::Dict{fmpq_mpoly, fmpq_mpoly}, extra_relations::Array{fmpq_mpoly, 1})

The function checks if the ideal generated by the polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals.

The extra_relations allows adding more polynomials to the generators (not affecting the saturation).

source
check_primality(polys::Dict{fmpq_mpoly, fmpq_mpoly})

The function checks if the ideal generated by the polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals.

source
+
GitHub

Primality Checks

StructuralIdentifiability.check_primality — Function
check_primality(polys::Dict{fmpq_mpoly, fmpq_mpoly}, extra_relations::Array{fmpq_mpoly, 1})

The function checks if the ideal generated by the polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals.

The extra_relations allows adding more polynomials to the generators (not affecting the saturation).

source
check_primality(polys::Dict{fmpq_mpoly, fmpq_mpoly})

The function checks if the ideal generated by the polynomials and saturated at the leading coefficient with respect to the corresponding variables is prime over rationals.

source
diff --git a/previews/PR238/utils/reparametrization/index.html b/previews/PR238/utils/reparametrization/index.html index b675b6901..09c9ba2c3 100644 --- a/previews/PR238/utils/reparametrization/index.html +++ b/previews/PR238/utils/reparametrization/index.html @@ -23,4 +23,4 @@ X1 => x1 a2 => d a3 => a - a1 => c

Notice that the new_ode is fully identifiabile, and has 1 less parameter compared to the original one.

source + a1 => c

Notice that the new_ode is fully identifiabile, and has 1 less parameter compared to the original one.

source diff --git a/previews/PR238/utils/util/index.html b/previews/PR238/utils/util/index.html index ef4b383ba..93a756d51 100644 --- a/previews/PR238/utils/util/index.html +++ b/previews/PR238/utils/util/index.html @@ -3,5 +3,5 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub

Other Helpful Functions

StructuralIdentifiability.dennums_to_fractions — Method
dennums_to_fractions(dennums)

Returns the field generators represented by fractions.

Input: an array of arrays of polynomials, as in [[f1, f2, f3, ...], [g1, g2, g3, ...], ...]

Output: an array of fractions [f2/f1, f3/f1, ..., g2/g1, g3/g1, ...]

source
StructuralIdentifiability.extract_coefficients — Method
extract_coefficients(poly, variables)

Input:

  • poly - multivariate polynomial
  • variables - a list of variables from the generators of the ring of p

Output:

  • dictionary with keys being tuples of length lenght(variables) and values being polynomials in the variables other than those which are the coefficients at the corresponding monomials (in a smaller polynomial ring)
source
StructuralIdentifiability.fractions_to_dennums — Method
fractions_to_dennums(fractions)

Returns the field generators represented by lists of denominators and numerators.

Input: an array of fractions, as in [f2/f1, f3/f1, ..., g2/g1, g3/g1, ...]

Output: an array of arrays of polynomials, [[f1, f2, f3, ...], [g1, g2, g3, ...], ...]

source
StructuralIdentifiability.gen_tag_name — Function
gen_tag_name(base; stop_words)
-gen_tag_names(n, base; stop_words)

Generates a string which will not collide with the words in stop_words.

Arguments

  • n: Generates a sequence of unique strings of length n
  • base: A string or a vector of strings, the base for the generated sequence
  • stop_words: A vector of strings, stop words
source
StructuralIdentifiability.make_substitution — Method
make_substitution(f, var_sub, val_numer, val_denom)

Substitute a variable in a polynomial with an expression

Input:

  • f - the polynomial
  • var_sub - the variable to be substituted
  • var_numer - numerator of the substitution expression
  • var_denom - denominator of the substitution expression

Output:

  • polynomial - result of substitution
source
StructuralIdentifiability.parent_ring_change — Method
parent_ring_change(poly, new_ring)

Converts a polynomial to a different polynomial ring Input

  • poly - a polynomial to be converted
  • new_ring - a polynomial ring such that every variable name appearing in poly appears among the generators

Output:

  • a polynomial in new_ring “equal” to poly
source
StructuralIdentifiability.uncertain_factorization — Method
uncertain_factorization(f)

Input:

  • f - polynomial with rational coefficients

Output:

  • list of pairs (div, certainty) where
    • div's are divisors of f such that f is their product with certain powers
    • if certainty is true, div is $Q$-irreducible
source
+
GitHub

Other Helpful Functions

StructuralIdentifiability.dennums_to_fractions — Method
dennums_to_fractions(dennums)

Returns the field generators represented by fractions.

Input: an array of arrays of polynomials, as in [[f1, f2, f3, ...], [g1, g2, g3, ...], ...]

Output: an array of fractions [f2/f1, f3/f1, ..., g2/g1, g3/g1, ...]

source
StructuralIdentifiability.extract_coefficients — Method
extract_coefficients(poly, variables)

Input:

  • poly - multivariate polynomial
  • variables - a list of variables from the generators of the ring of p

Output:

  • dictionary with keys being tuples of length lenght(variables) and values being polynomials in the variables other than those which are the coefficients at the corresponding monomials (in a smaller polynomial ring)
source
StructuralIdentifiability.fractions_to_dennums — Method
fractions_to_dennums(fractions)

Returns the field generators represented by lists of denominators and numerators.

Input: an array of fractions, as in [f2/f1, f3/f1, ..., g2/g1, g3/g1, ...]

Output: an array of arrays of polynomials, [[f1, f2, f3, ...], [g1, g2, g3, ...], ...]

source
StructuralIdentifiability.gen_tag_name — Function
gen_tag_name(base; stop_words)
+gen_tag_names(n, base; stop_words)

Generates a string which will not collide with the words in stop_words.

Arguments

  • n: Generates a sequence of unique strings of length n
  • base: A string or a vector of strings, the base for the generated sequence
  • stop_words: A vector of strings, stop words
source
StructuralIdentifiability.make_substitution — Method
make_substitution(f, var_sub, val_numer, val_denom)

Substitute a variable in a polynomial with an expression

Input:

  • f - the polynomial
  • var_sub - the variable to be substituted
  • var_numer - numerator of the substitution expression
  • var_denom - denominator of the substitution expression

Output:

  • polynomial - result of substitution
source
StructuralIdentifiability.parent_ring_change — Method
parent_ring_change(poly, new_ring)

Converts a polynomial to a different polynomial ring Input

  • poly - a polynomial to be converted
  • new_ring - a polynomial ring such that every variable name appearing in poly appears among the generators

Output:

  • a polynomial in new_ring “equal” to poly
source
StructuralIdentifiability.uncertain_factorization — Method
uncertain_factorization(f)

Input:

  • f - polynomial with rational coefficients

Output:

  • list of pairs (div, certainty) where
    • div's are divisors of f such that f is their product with certain powers
    • if certainty is true, div is $Q$-irreducible
source
diff --git a/previews/PR238/utils/wronskian/index.html b/previews/PR238/utils/wronskian/index.html index ccff7847b..59204c139 100644 --- a/previews/PR238/utils/wronskian/index.html +++ b/previews/PR238/utils/wronskian/index.html @@ -3,4 +3,4 @@ function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-90474609-3', {'page_path': location.pathname + location.search + location.hash}); -
GitHub

Wronskian Tools

StructuralIdentifiability.get_max_below — Method
get_max_below(t, vect)

Input:

  • t - a trie with exponent vectors
  • vect - yet another exponent vector

Output:

  • a pair (d, v) where v is a vector in the trie which is componentwise ≤ vect and the difference d is as small as possible
source
StructuralIdentifiability.massive_eval — Method
massive_eval(polys, eval_dict)

Input:

  • polys - a list of polynomials
  • eval_dict - dictionary from variables to the values. Missing values are treated as zeroes

Output:

  • a list of values of the polynomials

Evaluates a list of polynomials at a point. Assumes that multiplications are relatively expensive (like in truncated power series) so all the monomials are precomputed first and the values of monomials of lower degree are cached and used to compute the values of the monomials of higher degree

source
StructuralIdentifiability.monomial_compress — Method
monomial_compress(io_equation, ode)

Compresses an input-output equation for the rank computation Input:

  • io_equation - input-output equation
  • ode - the corresponding ODE model

Output:

  • pair (coeffs, terms) such that:
    • sum of coeffs[i] * terms[i] = io_equation
    • coeffs involve only parameters, terms involve only inputs and outputs
    • length of the representation is the smallest possible
source
StructuralIdentifiability.wronskian — Method
wronskian(io_equations, ode)

Input:

  • io_equations - a set of io-equations in the form of the Dict as returned by find_ioequations
  • ode - the ODE object

Output:

  • a list of Wronskians evaluated at a point modulo prime

Computes the Wronskians of io_equations

source
+
GitHub

Wronskian Tools

StructuralIdentifiability.get_max_below — Method
get_max_below(t, vect)

Input:

  • t - a trie with exponent vectors
  • vect - yet another exponent vector

Output:

  • a pair (d, v) where v is a vector in the trie which is componentwise ≤ vect and the difference d is as small as possible
source
StructuralIdentifiability.massive_eval — Method
massive_eval(polys, eval_dict)

Input:

  • polys - a list of polynomials
  • eval_dict - dictionary from variables to the values. Missing values are treated as zeroes

Output:

  • a list of values of the polynomials

Evaluates a list of polynomials at a point. Assumes that multiplications are relatively expensive (like in truncated power series) so all the monomials are precomputed first and the values of monomials of lower degree are cached and used to compute the values of the monomials of higher degree

source
StructuralIdentifiability.monomial_compress — Method
monomial_compress(io_equation, ode)

Compresses an input-output equation for the rank computation Input:

  • io_equation - input-output equation
  • ode - the corresponding ODE model

Output:

  • pair (coeffs, terms) such that:
    • sum of coeffs[i] * terms[i] = io_equation
    • coeffs involve only parameters, terms involve only inputs and outputs
    • length of the representation is the smallest possible
source
StructuralIdentifiability.wronskian — Method
wronskian(io_equations, ode)

Input:

  • io_equations - a set of io-equations in the form of the Dict as returned by find_ioequations
  • ode - the ODE object

Output:

  • a list of Wronskians evaluated at a point modulo prime

Computes the Wronskians of io_equations

source