improve docs

docs/src/catalyst_applications/structural_identifiability.md

@@ -3,16 +3,93 @@ During parameter fitting, parameter values are inferred from data. Identifiabili
 
 Structural identifiability can be illustrated in the following example network:
 ${dx \over dt} = p1*p2*x(t)$
-where, however much data I have on *x*, it is impossible to determine the values of *p1* and *p2* (these are non-identifiable).
+where, however much data is collected on *x*, it is impossible to determine the distinct values of *p1* and *p2* (these are non-identifiable).
 
-Catalyst contains a special extension for carrying out structural identifiability analysis using the [StructuralIdentifiability.jl](https://github.com/SciML/StructuralIdentifiability.jl) package. This enables StructuralIdentifiability's `assess_identifiability` and `assess_local_identifiability` functions to be called directly on Catalyst `ReactionSystems`. How to use these functions are described in the following tutorial, with [StructuralIdentifiability providing a more extensive documentation](https://docs.sciml.ai/StructuralIdentifiability/stable/). If you use this in your research, please [cite the StructuralIdentifiability.jl](@ref structural_identifiability_citation) and [Catalyst.jl](@ref catalyst_citation) publications.
+Catalyst contains a special extension for carrying out structural identifiability analysis using the [StructuralIdentifiability.jl](https://github.com/SciML/StructuralIdentifiability.jl) package. This enables StructuralIdentifiability's `assess_identifiability`, `assess_local_identifiability`, and `find_identifiable_functions` functions to be called directly on Catalyst `ReactionSystems`. It also implements specialised routines to make these more efficient when applied to reaction network models. How to use these functions are described in the following tutorial, with [StructuralIdentifiability providing a more extensive documentation](https://docs.sciml.ai/StructuralIdentifiability/stable/). If you use this in your research, please [cite the StructuralIdentifiability.jl](@ref structural_identifiability_citation) and [Catalyst.jl](@ref catalyst_citation) publications.
 
-## Global sensitivity analysis
+Structural identifiability can be divided into *local* and *global* identifiability. If a model quantity (which can either be a parameter or an initial condition) is locally identifiable, it means that its true value can be determined down to a finite-number of possible options. This also means that there is some limited region around its true value, where the true value is the only possible value. Globally identifiable quantities' values, on the other hand, can be uniquely determined. Again, while parameter (and initial condition) identifiability can be confirmed structurally for a model, it does not necessarily mean that they are practically* identifiable for some given data.
 
-## Local sensitivity analysis
+
+## Global identifiability analysis
+
+### Basic example
+
+Local identifiability can be assessed using the `assess_identifiability` function. For each model quantity (parameters and initial conditions), it will asses whether they are:
+- globally identifiable.
+- locally identifiable.
+- Unidentifiable.
+
+To it, we provide our `ReactionSystem` model and a list of quantities that we are able to measure. Here, we consider a ... model. Let us say that we are able to measure teh values of ... and ..., we provide these at the `measured_quantities` argument. We can now assess identifiability in the following way:
+```example si1
+```
+Here, ... are determined to be globally identifiable (and could theoretically be determined from data) and ... are locally identifiable (and for each, a finite number of candidate values can be determined from the data). Finally, ... are unidentifiable, and cannot be determined from data.
+
+### Indicating known parameters
+In the previous case we assumed that all parameters are unknown, however, this is not necessarily true. If there are parameters which value's are known, we can supply these using the `known_ps` argument. Indeed, this might turn other, previously unidentifiable, parameters identifiable. Let us consider this simple example:
+```example si2
+using Catalyst, StructuralIdentifiability # hide
+rn = @reaction_network begin
+    (p1+p2, d), 0 <--> X
+end
+```
+Typically, the two production parameters ($p1$ and $p2$) are unidentifiable. However, we we already know the value of $p1$, then $p2$#s value becomes identifiable:
+```example si2
+assess_identifiability(rn; measured_quantities=[:X], known_ps=[:p1])
+```
+
+### Providing non-trivial measured quantities
+Sometimes, we are not actually measuring species species, but rather some combinations of species (or possibly parameters). Here, any algebraic expression can be used in `measured_quantities`. If so, used species and parameters have to first be `@unpack`'ed from the model. Say that we have a model where an enzyme ($E$) is converted between an active and inactive form, which in turns activates the production of a product, $P$:
+```example si3
+using Catalyst, StructuralIdentifiability # hide
+enzyme_activation = @reaction_network begin
+    (kA,kD), Ei <--> Ea
+    (Ea, d), 0 <-->P
+end
+```
+and we can measure the total amount of $E$ ($=$Ei+Ea$), as well as the amount of $P$, we can use the following to assess identifiability:
+```example si3
+@unpack Ea, Ei = enzyme_activation
+assess_identifiability(enzyme_activation; measured_quantities=[Ei+Ea, :P])
+```
+
+### Checking identifiability of specific quantities
+By default, `asses_identifiability` assesses the identifiability of all parameters and species initial conditions. Sometimes, it is desirable to assess the identifiability of specific quantities. This can be done through the `funcs_to_check` argument. Let us consider our previous example, but say that we can only measure the amount of active enzyme ($Ea$), as well as the product ($P$). If we wish to determine whether the total amount of enzyme ($Ei+Ea$) is identifiable, we could use the following (again using `@unpack` to enable the formation of algebraic expression using the specific quantities):
+```example si3
+assess_identifiability(enzyme_activation; measured_quantities=[:Ei, :P], funcs_to_check=[Ei+Ea])
+```
+
+### Probability of correctness
+The identifiabiltiy methods used can, in theory, produce erroneous results. However, it is possible to adjust the lower bound for the probability of correctness using the argument `p` (by default set to `0.99`, that is, at least a $99%$ chance of correctness). We can e.g. increase the bound through:
+```example si2
+assess_identifiability(rn; measured_quantities=[:X], p=0.999)
+nothing # hide
+```
+giving a minimum bound of $99.9%$  chance of correctness. In practise, the bounds used by StructuralIdentifiability are very conservative, which means that while the minimum guaranteed probability of correctness in the default case is $99%$, in practise it is higher. While increasing teh value of `p` increases the certainty of correctness, it will also increase the time required to assess correctness.
+
+## Local identifiability analysis
+Local identifiability can be assessed using the `assess_local_identifiability` function. While this is already determined by the `assess_identifiability` function, local identifiability have the advantage that it is easier to compute. Hence, there might be models where global identifiability analysis fails (or takes prohibitively long time), where instead `assess_local_identifiability` can be used. This functions takes the same inputs as `assess_identifiability` and returns, for each quantity, `true` if iti is locally identifiable and `false` if it is not. Here we assesses local identifiability for the same model as used in the previous example:
+```example si1
+```
+We note that all parameters that `assess_identifiability` determined as either globally or locally identifiable are determined to be locally identifiable, while teh remaining are considered unidentifiable.
+
+
+## Finding identifiable functions
+Finally, StructuralIdentifiability provides the `find_identifiable_functions` function. Rather than determining the identifiability of each parameter and initial condition of the model, it finds a minimal set of identifiable functions, such as any other identifiable expression of the model can be generated by these. Here, let us consider the following model ...
+```example si5
+```
+
+The `find_identifiable_functions` functions tries to simplify its output functions to create nice expression. The degree to which it does this can be adjusted using the `simplify` keywords. Using the `:weak`, `:standard` (default), and `:strong` arguments, increased simplification can be forced (at the expense of longer runtimes).
 
 ## Creating StructuralIdentifiability compatible ODE models from Catalyst `ReactionSystem`s
+While the functionality described above covers the vast majority of analysis as user might want to perform, the StructuralIdentifiability package supports several additional features . While these does not have inherent Catalyst support, we do provide the `make_si_ode` function to simplify their use. Similarly to the previous functions, it takes a `ReactionSystem` and lists of measured quantities and known parameter values. The output is a [ode of the standard form supported by StructuralIdentifiability](https://docs.sciml.ai/StructuralIdentifiability/stable/tutorials/creating_ode/#Defining-the-model-using-@ODEmodel-macro). It can be created using the following syntax:
+```example si4
+using Catalyst, StructuralIdentifiability # hide
 
+```
+and then used as input to various StructuralIdentifiability functions. In the following example we use the produced ode to
+```example si4
+
+```
 
 ---
 ## [Citation](@id structural_identifiability_citation)

ext/CatalystStructuralIdentifiabilityExtension/structural_identifiability_extension.jl

@@ -23,7 +23,7 @@ si_ode(rs; measured_quantities = [:X], known_p = [:p])
 function Catalyst.make_si_ode(rs::ReactionSystem; measured_quantities=observed(rs), known_p = [], ignore_no_measured_warn=false)
     ignore_no_measured_warn || isempty(measured_quantities) && @warn "No measured quantity provided to the `measured_quantities` argument, any further identifiability analysis will likely fail."
     known_quantities = make_measured_quantities(rs, measured_quantities, known_p)
-    return StructuralIdentifiability.preprocess_ode(convert(ODESystem, rs), known_quantities)
+    return StructuralIdentifiability.preprocess_ode(convert(ODESystem, rs), known_quantities)[1]
 end
 
 # For input measured quantities, if this is not a vector of equations, convert it to a proper form.
@@ -47,3 +47,8 @@ function StructuralIdentifiability.assess_identifiability(rs::ReactionSystem, ar
     return StructuralIdentifiability.assess_identifiability(Catalyst.make_si_ode(rs; measured_quantities, known_p), args...; kwargs...)
 end
 
+# Identifiable functions.
+function StructuralIdentifiability.find_identifiable_functions(rs::ReactionSystem, args...; measured_quantities=observed(rs), known_p = Num[], kwargs...)
+    return StructuralIdentifiability.find_identifiable_functions(Catalyst.make_si_ode(rs; measured_quantities, known_p), args...; kwargs...)
+end
+