Linear Regression
Turing is powerful when applied to complex hierarchical models, but it can also be put to task at common statistical procedures, like linear regression. This tutorial covers how to implement a linear regression model in Turing.
Set Up
We begin by importing all the necessary libraries.
# Import Turing and Distributions.
using Turing, Distributions
# Import RDatasets.
using RDatasets
# Import MCMCChains, Plots, and StatPlots for visualizations and diagnostics.
using MCMCChains, Plots, StatsPlots
# Functionality for splitting and normalizing the data.
using MLDataUtils: shuffleobs, splitobs, rescale!
# Functionality for evaluating the model predictions.
using Distances
# Functionality for constructing arrays with identical elements efficiently.
using FillArrays
# Functionality for working with scaled identity matrices.
using LinearAlgebra
# Set a seed for reproducibility.
using Random
Random.seed!(0)
# Hide the progress prompt while sampling.
Turing.setprogress!(false);
We will use the mtcars dataset from the RDatasets package. mtcars contains a variety of statistics on different car models, including their miles per gallon, number of cylinders, and horsepower, among others.
We want to know if we can construct a Bayesian linear regression model to predict the miles per gallon of a car, given the other statistics it has. Lets take a look at the data we have.
# Import the "Default" dataset.
data = RDatasets.dataset("datasets", "mtcars");
# Show the first six rows of the dataset.
first(data, 6)
6×12 DataFrame
Row │ Model MPG Cyl Disp HP DRat WT
QS ⋯
│ String31 Float64 Int64 Float64 Int64 Float64 Float64
Fl ⋯
─────┼─────────────────────────────────────────────────────────────────────
─────
1 │ Mazda RX4 21.0 6 160.0 110 3.9 2.62
⋯
2 │ Mazda RX4 Wag 21.0 6 160.0 110 3.9 2.875
3 │ Datsun 710 22.8 4 108.0 93 3.85 2.32
4 │ Hornet 4 Drive 21.4 6 258.0 110 3.08 3.215
5 │ Hornet Sportabout 18.7 8 360.0 175 3.15 3.44
⋯
6 │ Valiant 18.1 6 225.0 105 2.76 3.46
5 columns om
itted
size(data)
(32, 12)
The next step is to get our data ready for testing. We'll split the mtcars dataset into two subsets, one for training our model and one for evaluating our model. Then, we separate the targets we want to learn (MPG, in this case) and standardize the datasets by subtracting each column's means and dividing by the standard deviation of that column. The resulting data is not very familiar looking, but this standardization process helps the sampler converge far easier.
# Remove the model column.
select!(data, Not(:Model))
# Split our dataset 70%/30% into training/test sets.
trainset, testset = splitobs(shuffleobs(data), 0.7)
# Turing requires data in matrix form.
target = :MPG
train = Matrix(select(trainset, Not(target)))
test = Matrix(select(testset, Not(target)))
train_target = trainset[:, target]
test_target = testset[:, target]
# Standardize the features.
μ, σ = rescale!(train; obsdim=1)
rescale!(test, μ, σ; obsdim=1)
# Standardize the targets.
μtarget, σtarget = rescale!(train_target; obsdim=1)
rescale!(test_target, μtarget, σtarget; obsdim=1);
Model Specification
In a traditional frequentist model using OLS, our model might look like:
$$ \mathrm{MPG}_i = \alpha + \boldsymbol{\beta}^\mathsf{T}\boldsymbol{X_i} $$
where $\boldsymbol{\beta}$ is a vector of coefficients and $\boldsymbol{X}$ is a vector of inputs for observation $i$. The Bayesian model we are more concerned with is the following:
$$ \mathrm{MPG}_i \sim \mathcal{N}(\alpha + \boldsymbol{\beta}^\mathsf{T}\boldsymbol{X_i}, \sigma^2) $$
where $\alpha$ is an intercept term common to all observations, $\boldsymbol{\beta}$ is a coefficient vector, $\boldsymbol{X_i}$ is the observed data for car $i$, and $\sigma^2$ is a common variance term.
For $\sigma^2$, we assign a prior of truncated(Normal(0, 100); lower=0).
This is consistent with Andrew Gelman's recommendations on noninformative priors for variance.
The intercept term ($\alpha$) is assumed to be normally distributed with a mean of zero and a variance of three.
This represents our assumptions that miles per gallon can be explained mostly by our assorted variables, but a high variance term indicates our uncertainty about that.
Each coefficient is assumed to be normally distributed with a mean of zero and a variance of 10.
We do not know that our coefficients are different from zero, and we don't know which ones are likely to be the most important, so the variance term is quite high.
Lastly, each observation $y_i$ is distributed according to the calculated mu term given by $\alpha + \boldsymbol{\beta}^\mathsf{T}\boldsymbol{X_i}$.
# Bayesian linear regression.
@model function linear_regression(x, y)
# Set variance prior.
σ² ~ truncated(Normal(0, 100); lower=0)
# Set intercept prior.
intercept ~ Normal(0, sqrt(3))
# Set the priors on our coefficients.
nfeatures = size(x, 2)
coefficients ~ MvNormal(Zeros(nfeatures), 10.0 * I)
# Calculate all the mu terms.
mu = intercept .+ x * coefficients
return y ~ MvNormal(mu, σ² * I)
end
linear_regression (generic function with 2 methods)
With our model specified, we can call the sampler. We will use the No U-Turn Sampler (NUTS) here.
model = linear_regression(train, train_target)
chain = sample(model, NUTS(0.65), 3_000)
Chains MCMC chain (3000×24×1 Array{Float64, 3}):
Iterations = 1001:1:4000
Number of chains = 1
Samples per chain = 3000
Wall duration = 5.19 seconds
Compute duration = 5.19 seconds
parameters = σ², intercept, coefficients[1], coefficients[2], coeffi
cients[3], coefficients[4], coefficients[5], coefficients[6], coefficients[
7], coefficients[8], coefficients[9], coefficients[10]
internals = lp, n_steps, is_accept, acceptance_rate, log_density, h
amiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error,
tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std naive_se mcse ess
r ⋯
Symbol Float64 Float64 Float64 Float64 Float64 F
loa ⋯
σ² 0.3117 0.1834 0.0033 0.0063 783.2997
1.0 ⋯
intercept -0.0016 0.1131 0.0021 0.0018 3502.6971
0.9 ⋯
coefficients[1] -0.0450 0.5486 0.0100 0.0145 1435.1582
1.0 ⋯
coefficients[2] 0.3194 0.7314 0.0134 0.0216 1006.0850
1.0 ⋯
coefficients[3] -0.3857 0.3972 0.0073 0.0099 1616.0794
1.0 ⋯
coefficients[4] 0.1694 0.2927 0.0053 0.0084 1266.8139
1.0 ⋯
coefficients[5] -0.3347 0.7274 0.0133 0.0236 843.4465
0.9 ⋯
coefficients[6] 0.0764 0.3670 0.0067 0.0097 1230.1250
0.9 ⋯
coefficients[7] 0.0123 0.3945 0.0072 0.0116 1415.7843
1.0 ⋯
coefficients[8] 0.1782 0.3127 0.0057 0.0092 1075.5653
1.0 ⋯
coefficients[9] 0.1111 0.2862 0.0052 0.0085 1007.7045
1.0 ⋯
coefficients[10] -0.2593 0.4142 0.0076 0.0145 795.1212
1.0 ⋯
2 columns om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
σ² 0.1182 0.1931 0.2658 0.3720 0.7957
intercept -0.2377 -0.0717 0.0011 0.0687 0.2204
coefficients[1] -1.1493 -0.3834 -0.0425 0.2987 1.0444
coefficients[2] -1.1663 -0.1402 0.3273 0.7817 1.7194
coefficients[3] -1.1710 -0.6219 -0.3953 -0.1528 0.4414
coefficients[4] -0.4003 -0.0161 0.1712 0.3494 0.7630
coefficients[5] -1.7905 -0.8004 -0.3388 0.1256 1.0998
coefficients[6] -0.6517 -0.1630 0.0670 0.3126 0.8036
coefficients[7] -0.8004 -0.2296 0.0193 0.2623 0.8055
coefficients[8] -0.4758 -0.0140 0.1837 0.3747 0.8049
coefficients[9] -0.4626 -0.0696 0.1075 0.2929 0.6702
coefficients[10] -1.1081 -0.5109 -0.2541 -0.0036 0.5771
We can also check the densities and traces of the parameters visually using the plot functionality.
plot(chain)
It looks like all parameters have converged.
Comparing to OLS
A satisfactory test of our model is to evaluate how well it predicts. Importantly, we want to compare our model to existing tools like OLS. The code below uses the GLM.jl package to generate a traditional OLS multiple regression model on the same data as our probabilistic model.
# Import the GLM package.
using GLM
# Perform multiple regression OLS.
train_with_intercept = hcat(ones(size(train, 1)), train)
ols = lm(train_with_intercept, train_target)
# Compute predictions on the training data set
# and unstandardize them.
p = GLM.predict(ols)
train_prediction_ols = μtarget .+ σtarget .* p
# Compute predictions on the test data set
# and unstandardize them.
test_with_intercept = hcat(ones(size(test, 1)), test)
p = GLM.predict(ols, test_with_intercept)
test_prediction_ols = μtarget .+ σtarget .* p;
The function below accepts a chain and an input matrix and calculates predictions. We use the samples of the model parameters in the chain starting with sample 200, which is where the warm-up period for the NUTS sampler ended.
# Make a prediction given an input vector.
function prediction(chain, x)
p = get_params(chain[200:end, :, :])
targets = p.intercept' .+ x * reduce(hcat, p.coefficients)'
return vec(mean(targets; dims=2))
end
prediction (generic function with 1 method)
When we make predictions, we unstandardize them so they are more understandable.
# Calculate the predictions for the training and testing sets
# and unstandardize them.
p = prediction(chain, train)
train_prediction_bayes = μtarget .+ σtarget .* p
p = prediction(chain, test)
test_prediction_bayes = μtarget .+ σtarget .* p
# Show the predictions on the test data set.
DataFrame(; MPG=testset[!, target], Bayes=test_prediction_bayes, OLS=test_prediction_ols)
10×3 DataFrame
Row │ MPG Bayes OLS
│ Float64 Float64 Float64
─────┼─────────────────────────────
1 │ 19.2 17.9818 18.1265
2 │ 15.0 6.67636 6.37891
3 │ 16.4 13.8408 13.883
4 │ 14.3 11.8707 11.7337
5 │ 21.4 25.1902 25.1916
6 │ 18.1 20.6603 20.672
7 │ 19.7 15.9947 15.8408
8 │ 15.2 18.292 18.3391
9 │ 26.0 28.4566 28.4865
10 │ 17.3 14.5441 14.534
Now let's evaluate the loss for each method, and each prediction set. We will use the mean squared error to evaluate loss, given by $$ \mathrm{MSE} = \frac{1}{n} \sum_{i=1}^n {(y_i - \hat{y_i})^2} $$ where $y_i$ is the actual value (true MPG) and $\hat{y_i}$ is the predicted value using either OLS or Bayesian linear regression. A lower SSE indicates a closer fit to the data.
println(
"Training set:",
"\n\tBayes loss: ",
msd(train_prediction_bayes, trainset[!, target]),
"\n\tOLS loss: ",
msd(train_prediction_ols, trainset[!, target]),
)
println(
"Test set:",
"\n\tBayes loss: ",
msd(test_prediction_bayes, testset[!, target]),
"\n\tOLS loss: ",
msd(test_prediction_ols, testset[!, target]),
)
Training set:
Bayes loss: 4.651083751584832
OLS loss: 4.648142085690521
Test set:
Bayes loss: 14.10580325710221
OLS loss: 14.796847779051523
As we can see above, OLS and our Bayesian model fit our training and test data set about the same.
Appendix
These tutorials are a part of the TuringTutorials repository, found at: https://github.com/TuringLang/TuringTutorials.
To locally run this tutorial, do the following commands:
using TuringTutorials
TuringTutorials.weave("05-linear-regression", "05_linear-regression.jmd")
Computer Information:
Julia Version 1.6.7
Commit 3b76b25b64 (2022-07-19 15:11 UTC)
Platform Info:
OS: Linux (x86_64-pc-linux-gnu)
CPU: AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-11.0.1 (ORCJIT, znver2)
Environment:
JULIA_CPU_THREADS = 16
BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/7aa0085e-79a4-45f3-a5bd-9743c91cf3da
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[efe28fd5] OpenSpecFun_jll v0.5.5+0
[91d4177d] Opus_jll v1.3.2+0
[2f80f16e] PCRE_jll v8.44.0+0
[30392449] Pixman_jll v0.40.1+0
[ea2cea3b] Qt5Base_jll v5.15.3+0
[f50d1b31] Rmath_jll v0.3.0+0
[a2964d1f] Wayland_jll v1.19.0+0
[2381bf8a] Wayland_protocols_jll v1.23.0+0
[02c8fc9c] XML2_jll v2.9.12+0
[aed1982a] XSLT_jll v1.1.34+0
[4f6342f7] Xorg_libX11_jll v1.6.9+4
[0c0b7dd1] Xorg_libXau_jll v1.0.9+4
[935fb764] Xorg_libXcursor_jll v1.2.0+4
[a3789734] Xorg_libXdmcp_jll v1.1.3+4
[1082639a] Xorg_libXext_jll v1.3.4+4
[d091e8ba] Xorg_libXfixes_jll v5.0.3+4
[a51aa0fd] Xorg_libXi_jll v1.7.10+4
[d1454406] Xorg_libXinerama_jll v1.1.4+4
[ec84b674] Xorg_libXrandr_jll v1.5.2+4
[ea2f1a96] Xorg_libXrender_jll v0.9.10+4
[14d82f49] Xorg_libpthread_stubs_jll v0.1.0+3
[c7cfdc94] Xorg_libxcb_jll v1.13.0+3
[cc61e674] Xorg_libxkbfile_jll v1.1.0+4
[12413925] Xorg_xcb_util_image_jll v0.4.0+1
[2def613f] Xorg_xcb_util_jll v0.4.0+1
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
[35661453] Xorg_xkbcomp_jll v1.4.2+4
[33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
[c5fb5394] Xorg_xtrans_jll v1.4.0+3
[3161d3a3] Zstd_jll v1.5.2+0
[0ac62f75] libass_jll v0.15.1+0
[f638f0a6] libfdk_aac_jll v2.0.2+0
[b53b4c65] libpng_jll v1.6.38+0
[f27f6e37] libvorbis_jll v1.3.7+1
[1270edf5] x264_jll v2021.5.5+0
[dfaa095f] x265_jll v3.5.0+0
[d8fb68d0] xkbcommon_jll v0.9.1+5
[0dad84c5] ArgTools
[56f22d72] Artifacts
[2a0f44e3] Base64
[ade2ca70] Dates
[8bb1440f] DelimitedFiles
[8ba89e20] Distributed
[f43a241f] Downloads
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[4af54fe1] LazyArtifacts
[b27032c2] LibCURL
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[a63ad114] Mmap
[ca575930] NetworkOptions
[44cfe95a] Pkg
[de0858da] Printf
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays
[10745b16] Statistics
[4607b0f0] SuiteSparse
[fa267f1f] TOML
[a4e569a6] Tar
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll
[deac9b47] LibCURL_jll
[29816b5a] LibSSH2_jll
[c8ffd9c3] MbedTLS_jll
[14a3606d] MozillaCACerts_jll
[4536629a] OpenBLAS_jll
[05823500] OpenLibm_jll
[83775a58] Zlib_jll
[8e850ede] nghttp2_jll
[3f19e933] p7zip_jll