04-curve-fitting.jl

### A Pluto.jl notebook ###
# v0.20.4

using Markdown
using InteractiveUtils

# ╔═╡ 23daae04-ab90-4ce9-8058-864cab13972d
using Random;Random.seed!(1234)

# ╔═╡ 8c821f17-da8c-4288-a960-f042b3b8b973
using Plots

# ╔═╡ 80c6de8f-6d91-49dc-9941-7ff2451231b6
using Optimization

# ╔═╡ 5fff1493-e74f-4df6-9b21-38c86562b913
using OptimizationOptimJL


# ╔═╡ 17cb3c66-4e77-4308-96e2-b83d30e3204f
using ForwardDiff

# ╔═╡ 3e04f697-b12a-44aa-b7de-25295e1ea364
using Turing

# ╔═╡ f1b6fd4f-2f0e-4d7a-a5d8-7f88f0fd9f4c
using PairPlots

# ╔═╡ 0e5f8dfb-9890-448c-847e-a8cf9f63dc12
using CairoMakie: Makie

# ╔═╡ 0f6e13ab-575f-4011-ba0d-43dacc5edfb7
md"""
# Curve Fitting


This tutorial will demonstrate fitting data with a straight line (linear regression), an abitrary non-linear model, and finally a Bayesian model.

## Packages

* [`LinearAlgebra`](https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/) we'll use this built-in Julia standard library to perform a linear regression
* [`Optimization`](http://optimization.sciml.ai/stable/): we'll use this package to display coordinates along the image and add the scalebar
* `OptimizationOptimJL`: the specific optimizer backend we will use.  For your own problems, select the best backend from the Optimization.jl documentation page.
* [`Turing`](https://turing.ml/stable/): we'll use this package for Bayesian modelling.
* [`PairPlots`](https://github.com/sefffal/PairPlots.jl): we'll use this for creating a corner plot of the posterior from our Bayesian models.

In your own code, you most likely won't need all of these packages. Pick and choose the one that best fits your problem.

## Generating the data

We'll generate synthetic data for this problem. We'll make a weak parabola with some noise. For consistency, we'll seed the Julia random number generator so that we see the same noise each time the tutorial is run.

"""

# ╔═╡ 5dd3e860-590c-4c80-9b06-e3f499810835
md"""
By calling `seed!`, the pattern of random numbers generated by `rand` and `randn` will be the same each time.

Now we'll generate the data:
"""

# ╔═╡ a5680cc0-17b9-4205-ace8-19ec773e4502
x = 0:5:100 # Or equivalently: range(0, 100, step=5)

# ╔═╡ 922a86df-155a-4a5c-b4d5-c5d859883d39
y = (x ./ 20 .- 0.2).^2 .+ 2 .+ randn(length(x))

# ╔═╡ 8061efe2-f7f0-4ad3-bfa3-6e261e515af0
md"""
The `randn` function generates a random value normally distributed around `0` with a standard deviation of `1`. `rand` on the other hand creates uniformly distributed random values distributed between `0` and `1`.

Let's plot the data to see what it looks like:
"""

# ╔═╡ b9a6dbb2-2ed5-4d66-9906-956db1bb455d
scatter(x, y; xlabel="x", ylabel="y", label="data")

# ╔═╡ 3a70d376-acfa-4b40-b676-6776efd0a67a
md"""
## Linear regression

Before using any packages, let's perform a linear fit from scratch using some linear algebra.

The equation of a line can be written in matrix form as 
```math
\quad
\begin{pmatrix} 
N & \sum y_i \\
\sum y_i & \sum y_{i}^2
\end{pmatrix}
\begin{pmatrix}
c_1 \\
c_2 \\
\end{pmatrix}=
\begin{pmatrix}
\sum y_i \\
\sum y_i x_i
\end{pmatrix}
```

where $c_1$ and $c_2$ are the intercept and slope.

Multiplying both sides by the inverse of the first matrix gives

```math
\quad
\begin{pmatrix}
c_1 \\
c_2 \\
\end{pmatrix}=
\begin{pmatrix} 
N & \sum y_i \\
\sum y_i & \sum y_{i}^2
\end{pmatrix}^{-1}
\begin{pmatrix}
\sum y_i \\
\sum y_i x_i
\end{pmatrix}
```

We can write the right-hand side matrix and vector (let's call them `A` and `b`) in Julia notation like so:

"""

# ╔═╡ 7a1c1019-9d0e-442b-bb76-e5e8494a9b4d
A = [
  length(x) sum(x)
  sum(x)    sum(x.^2)
]

# ╔═╡ 29fcd76d-4e1b-42c9-9282-2186b5ba42e8
b = [
   sum(y)
   sum(y .* x)                                          
]

# ╔═╡ 6f70d9a9-d3c6-43e3-92be-3bc01e8ec7ba
md"""
We can now perform the linear fit by solving the system of equations with the `\` operator:
"""

# ╔═╡ caeb0ecf-0d5c-4913-972b-e5fe38f588c2
c = A\b

# ╔═╡ fcdeb663-5215-4dc0-a265-b90a5ce62f41
md"""
Let's make a helper function `linfunc` that takes an x value, a slope, and an intercept and calculates the corresponding y value:
"""

# ╔═╡ 4b32d879-e3e0-40ad-a777-d81b3beb91df
linfunc(x; slope, intercept) = slope*x + intercept

# ╔═╡ 52e4201f-b23d-4a20-aae5-94219094da17
md"""

Finally, we can plot the solution:

"""

# ╔═╡ cbd9f8cf-ad8c-40d9-a515-eb9172e5ba29
yfit = linfunc.(x; slope=c[2], intercept=c[1])

# ╔═╡ d9dcc518-9a27-4e92-8a30-5f2ad3401514
let
	scatter(x, y, xlabel="x", ylabel="y", label="data")
	plot!(x, yfit, label="best fit")
end

# ╔═╡ b212fc99-fe7a-4f6c-991c-405186177902
md"""

The packages [LsqFit](https://julianlsolvers.github.io/LsqFit.jl/latest/) and [GLM](https://juliastats.org/GLM.jl/v0.11/#Minimal-examples-1) (for generalized linear models) contain functions for performing and evaluating these types of linear fits.

## (Non-)linear curve fit

The packages above can be used to fit different polynomial models, but if we have a truly arbitrary Julia function we would like to fit to some data we can use the [Optimization.jl](http://optimization.sciml.ai/stable/) package. Through its various backends, Optimization.jl supports a very wide range of algorithms for local, global, convex, and non-convex optimization. 

The first step is to define our objective function. We'll reuse our simple `linfunc` linear function from above:
"""

# ╔═╡ 5ed7e9a1-aa5f-4bc1-8904-36c09a2e90a3

# We must supply an objective function that will be minimized
# The u argument is a vector of parameters from the optimizer.
# data is a vector of static parameters passed through below.
function objective_linear(u, data)
    # Get our fit parameters from u
    slope, intercept = u
    # equivalent to:
    # slope = u[1]
    # intercept = u[2]

    # Get the x and y vectors from data
    x, y = data
    
    # Calculate the residuals between our model and the data
    residuals = linfunc.(x; slope, intercept) .- y

    # Return the sum of squares of the residuals to minimize
    return sum(residuals.^2)
end

# Define the initial parameter values for slope and intercept

# ╔═╡ a9960993-249f-4cf3-ac94-1151fd15bbbb
u0_linear = [1.0, 1.0]
# Pass through the data we want to fit

# ╔═╡ 5c382a4f-c263-4991-8c38-69402b1a4077
data = [x,y]


# ╔═╡ 1878366e-3dda-4734-ae07-84f4d6c79a58
# Create an OptimizationProblem object to hold the function, initial
# values, and data.
prob = OptimizationProblem(objective_linear,u0_linear,data)

# Import the optimization backend we want to use

# ╔═╡ c43d8e58-62cb-4deb-a256-4453a19068e4

# Minimize the function. Optimization.jl uses the SciML common solver
# interface. Pass the problem you want to solve (optimization problem
# here) and a solver to use.
# NelderMead() is a derivative-free method for finding a function's
# local minimum.
sol_linear = solve(prob,NelderMead())



# ╔═╡ f464a85f-f2bc-4006-ab28-180a639dbf5e
# Exctract the best-fitting parameters
slope_opt, intercept_opt = sol_linear.u

# ╔═╡ 55fca124-23db-4577-bb65-76d627789358
md"""
Note: the `NelderMead()` algorithm behaves nearly identically to MATLAB's `fminsearch`. 


We can now plot the solution:
"""

# ╔═╡ a71591a5-b2b8-4b5a-8b57-4900dcdea818
yfit_linear = linfunc.(
	x; 
	slope=sol_linear.u[1],
	intercept=sol_linear.u[2]
)

# ╔═╡ 0ac4b717-7c54-4c19-923c-9c49840295b3
begin
	scatter(x, y, xlabel="x", ylabel="y", label="data")
	plot!(x, yfit_linear, label="best fit")
end

# ╔═╡ 099fce63-9f9a-47c0-9454-6780c609692e
md"""

We can now test out a quadratic fit using the same package:
"""

# ╔═╡ ef40cf71-9b09-4f4d-8d43-9de003490717
function objective_quad(u, data)
    x, y = data
    
    # Define an equation of a quadratic, e.g.:
    # 3x^2 + 2x + 1
    model = u[1] .* x.^2 .+ u[2] .* x .+ u[3]

    # Calculate the residuals between our model and the data
    residuals = model .- y

    # Return the sum of squares of the residuals to minimize
    return sum(residuals.^2)
end

# ╔═╡ 528ce1ed-78a6-4381-a4bc-28a11004a839
u0_quad = [1.0, 1.0, 1.0]

# ╔═╡ d9fa281f-ef11-480e-bc4c-b8e00b8c826f
prob_quad = OptimizationProblem(objective_quad,u0_quad,data)

# ╔═╡ cc3984e5-586b-4d3a-bf3c-ef92568f455a
sol_quad = solve(prob_quad,NelderMead())

# ╔═╡ 07897a8d-457c-42c8-aa41-d6ef6754f69a
sol_quad.u

# ╔═╡ ad38861a-3e3a-4805-ac26-aa919f0291d7
yfit_quad = sol_quad.u[1] .* x.^2 .+ sol_quad.u[2] .* x .+ sol_quad.u[3]

# ╔═╡ b5dbf92b-040c-477e-8061-20a6ee0aedb4
let
	scatter(x, y, xlabel="x", ylabel="y", label="data")
	plot!(x,  yfit_quad, label="quadratic fit")
end

# ╔═╡ 0613407d-98e7-4dba-8f96-04115f73b198
md"""


This is already very fast; however, as the scale of your problem grows, there are several routes you can take to improve the optimization performance.
First, you can use automatic differentiation and a higher order optimization algorithm:
"""

# ╔═╡ 3a9af7c6-31ff-41dc-92bb-bbda172eea71
optf = OptimizationFunction(objective_quad, Optimization.AutoForwardDiff())

# ╔═╡ a45bc312-6c1b-4872-a492-9d2a84b5a4e2
prob_quad_autodiff = OptimizationProblem(optf,u0_quad,data)

# ╔═╡ 5e838203-9bcd-4ff4-ae2f-13611042e0e2
@time sol = solve(prob_quad_autodiff,BFGS())  # another good algorithm is Newton()

# ╔═╡ eab5634f-de29-4702-9224-7d5170c5dad1
md"""
You can also write an "in-place" version of `objective` that doesn't allocate new arrays with each iteration.


## Bayesian models
Let's shift gears and now create a fully Bayesian model using the [Turing.jl](https://turing.ml/stable/) package.

Instead of defining an arbitrary Julia function, this package requires us to use a macro called `@model`.

Let's start with a linear model once more, now with the Turing `@model` syntax:

"""


# ╔═╡ b272d287-b6b4-4acd-9092-3f1fd8860c35
# Bayesian linear regression.
@model function linear_regression(x, y)
    # Set variance prior.
    σ₂ ~ truncated(Normal(0, 100), 0, Inf)
    # Typed as \sigma <tab> \_2 <tab>

    # Set intercept prior.
    intercept ~ Normal(0, 5)

    # Set the prior on our slope coefficient.
    slope ~ Normal(0, 10)

    # Each point is drawn from a gaussian (Normal) distribution
    # with mean calculated form our linear model, and standard
    # deviation as the square root of the variance variable
    for i in eachindex(x,y)
        y[i] ~ Normal(x[i] * slope + intercept, sqrt(σ₂))
    end
end

# ╔═╡ f0264420-be19-4250-86c9-3e4c490f4a91
md"""

We can now draw posterior samples from this model using one of many available samplers, `NUTS`, the No U-Turn Sampler.

"""

# ╔═╡ 055298cd-1bbc-4d37-a2a9-595e8523b41b
model = linear_regression(x, y)

# ╔═╡ 0e30378a-1b6e-47b7-a70a-873ea61e9aa7
chain = sample(model, NUTS(0.65), 500)

# ╔═╡ 846c91dc-bdf8-48f5-aae8-e7e11f2394cf
md"""
We now plot all draws from the posterior 
"""

# ╔═╡ b5fc7474-13b8-4f8f-a802-cf2cffb5b96c
let 
	intercept = chain["intercept"]
	slope = chain["slope"]
	σ₂ = chain["σ₂"]
	y_fit =  x .* slope' .+ intercept'
	plot(x, y_fit;
	    label="",
	    color=:gray,
	    alpha=0.05
	)
	scatter!(x, y, xlabel="x", ylabel="y", label="data", color=1)
end

# ╔═╡ c65205db-35fb-48b4-9a29-4200bc9a9a78
md"""

Each gray curve is a sample from the posterior distribution of this model. To examine the model parameters and their covariance in greater detail, we can make a corner plot using the PairPlots.jl package. 

Exercise: consider increasing the number of iterations to make a smooth plot.
"""

# ╔═╡ 2509c7ea-1eff-4d4e-800c-8cfc2a9fcedc
pairplot(chain)

# ╔═╡ 4f3b74d5-8a7a-4115-a7c2-1dd2a1e3b341
md"""
Let's now repeat this proceedure with a Bayesian quadratic model.

"""

# ╔═╡ 8a5ff894-4d56-42e3-9c1e-eeaa134b58e0
@model function quad_regression(x, y)
    # Prior on the variance of the data around the best-fit line
    σ₂ ~ truncated(Normal(0, 10), 0, Inf)

    # Priors on the three quadratic parameters
    u1 ~ Normal(0,0.01)
    u2 ~ Normal(0,0.1)
    u3 ~ Normal(0,5)

    for i in eachindex(x,y)
        model = u1 * x[i]^2 + u2*x[i] + u3
        y[i] ~ Normal(model, sqrt(σ₂))
    end
end

# ╔═╡ eec6af16-9a10-4289-a0a8-55d35936fea6
md"""

We can now draw posterior samples from this model using one of many available samplers, `NUTS`, or the No U-Turn Sampler.

"""

# ╔═╡ 4d611e81-e88d-42c4-acbc-c4016b2ce15b
model_quad = quad_regression(x, y)

# ╔═╡ df1b7e09-a88e-4486-b8d3-c1f0f8382c37
chain_quad = sample(model_quad, NUTS(0.65), 500)

# ╔═╡ 8f391130-062b-46c0-998e-bb1ed4ff4238
posterior = let
	u1 = chain_quad["u1"]
	u2 = chain_quad["u2"]
	u3 = chain_quad["u3"]
	u1' .* x.^2 .+ u2' .* x .+ u3'
end

# ╔═╡ 16181717-4e30-4c9c-a963-f6ea24e067d2
let
	plot(x, posterior;
	    label="",
	    color=:gray,
	    alpha=0.1
	)
	scatter!(x, y, xlabel="x", ylabel="y", label="data", color=1)
end

# ╔═╡ 83c89249-c67b-4308-b5fa-c6aa195326b4
pairplot(chain_quad)

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[[deps.gperf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3516a5630f741c9eecb3720b1ec9d8edc3ecc033"
uuid = "1a1c6b14-54f6-533d-8383-74cd7377aa70"
version = "3.1.1+0"

[[deps.isoband_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51b5eeb3f98367157a7a12a1fb0aa5328946c03c"
uuid = "9a68df92-36a6-505f-a73e-abb412b6bfb4"
version = "0.2.3+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "522c1df09d05a71785765d19c9524661234738e9"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.11.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "e17c115d55c5fbb7e52ebedb427a0dca79d4484e"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.2+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.11.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "141fe65dc3efabb0b1d5ba74e91f6ad26f84cc22"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.11.0+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "8a22cf860a7d27e4f3498a0fe0811a7957badb38"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.3+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "ad50e5b90f222cfe78aa3d5183a20a12de1322ce"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.18.0+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "b7bfd3ab9d2c58c3829684142f5804e4c6499abc"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.45+0"

[[deps.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "libpng_jll"]
git-tree-sha1 = "1e53ffe8941ee486739f3c0cf11208c26637becd"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.10.4+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "490376214c4721cdaca654041f635213c6165cb3"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+2"

[[deps.libwebp_jll]]
deps = ["Artifacts", "Giflib_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libglvnd_jll", "Libtiff_jll", "libpng_jll"]
git-tree-sha1 = "ccbb625a89ec6195856a50aa2b668a5c08712c94"
uuid = "c5f90fcd-3b7e-5836-afba-fc50a0988cb2"
version = "1.4.0+0"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "814e154bdb7be91d78b6802843f76b6ece642f11"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.6+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.59.0+0"

[[deps.oneTBB_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "7d0ea0f4895ef2f5cb83645fa689e52cb55cf493"
uuid = "1317d2d5-d96f-522e-a858-c73665f53c3e"
version = "2021.12.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+2"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "9c304562909ab2bab0262639bd4f444d7bc2be37"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+1"
"""

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