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.
using Turing
# Package for loading the data set.
using RDatasets
# Package for visualization.
using StatsPlots
# Functionality for splitting the data.
using MLUtils: splitobs
# Functionality for constructing arrays with identical elements efficiently.
using FillArrays
# Functionality for normalizing the data and evaluating the model predictions.
using StatsBase
# Functionality for working with scaled identity matrices.
using LinearAlgebra
# Set a seed for reproducibility.
using Random
Random.seed!(0);
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. Let us take a look at the data we have.
# Load the 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 = map(DataFrame, splitobs(data; at=0.7, shuffle=true))
# 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.
dt_features = fit(ZScoreTransform, train; dims=1)
StatsBase.transform!(dt_features, train)
StatsBase.transform!(dt_features, test)
# Standardize the targets.
dt_targets = fit(ZScoreTransform, train_target)
StatsBase.transform!(dt_targets, train_target)
StatsBase.transform!(dt_targets, test_target);
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(), 5_000)
Chains MCMC chain (5000×24×1 Array{Float64, 3}):
Iterations = 1001:1:6000
Number of chains = 1
Samples per chain = 5000
Wall duration = 6.46 seconds
Compute duration = 6.46 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 mcse ess_bulk ess_tail
⋯
Symbol Float64 Float64 Float64 Float64 Float64
Flo ⋯
σ² 0.1505 0.0890 0.0024 1348.6424 1784.3448
1. ⋯
intercept -0.0005 0.0810 0.0011 5399.4530 3137.0985
1. ⋯
coefficients[1] 0.0551 0.3597 0.0058 3743.8676 2832.6811
1. ⋯
coefficients[2] -0.0455 0.3956 0.0084 2205.6502 2724.5070
1. ⋯
coefficients[3] 0.0406 0.3344 0.0067 2480.1402 2533.1857
1. ⋯
coefficients[4] 0.2035 0.1831 0.0034 2820.2254 3271.2229
1. ⋯
coefficients[5] -0.2007 0.4318 0.0101 1800.7685 2411.5459
1. ⋯
coefficients[6] 0.0921 0.2450 0.0048 2532.5912 2735.8429
1. ⋯
coefficients[7] -0.0129 0.2377 0.0035 4621.8658 3164.4100
0. ⋯
coefficients[8] 0.2324 0.1964 0.0032 3751.8621 3211.4205
0. ⋯
coefficients[9] 0.3408 0.2934 0.0058 2582.9846 2703.6005
1. ⋯
coefficients[10] -0.5718 0.4346 0.0103 1750.2600 2387.8880
1. ⋯
2 columns om
itted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
σ² 0.0569 0.0951 0.1279 0.1779 0.3819
intercept -0.1563 -0.0510 0.0004 0.0485 0.1630
coefficients[1] -0.6486 -0.1742 0.0481 0.2840 0.7832
coefficients[2] -0.8506 -0.2957 -0.0445 0.2034 0.7190
coefficients[3] -0.6301 -0.1616 0.0369 0.2503 0.7048
coefficients[4] -0.1541 0.0846 0.2012 0.3189 0.5755
coefficients[5] -1.0306 -0.4835 -0.2019 0.0783 0.6721
coefficients[6] -0.4118 -0.0655 0.0938 0.2467 0.5885
coefficients[7] -0.4723 -0.1666 -0.0121 0.1393 0.4568
coefficients[8] -0.1561 0.1095 0.2346 0.3575 0.6181
coefficients[9] -0.2221 0.1522 0.3416 0.5289 0.9271
coefficients[10] -1.4430 -0.8598 -0.5693 -0.2886 0.2827
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.
train_prediction_ols = GLM.predict(ols)
StatsBase.reconstruct!(dt_targets, train_prediction_ols)
# Compute predictions on the test data set and unstandardize them.
test_with_intercept = hcat(ones(size(test, 1)), test)
test_prediction_ols = GLM.predict(ols, test_with_intercept)
StatsBase.reconstruct!(dt_targets, test_prediction_ols);
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.
# 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.
train_prediction_bayes = prediction(chain, train)
StatsBase.reconstruct!(dt_targets, train_prediction_bayes)
test_prediction_bayes = prediction(chain, test)
StatsBase.reconstruct!(dt_targets, test_prediction_bayes)
# 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 │ 15.8 25.142 25.3071
2 │ 26.0 30.9893 31.1095
3 │ 21.4 26.9163 26.9598
4 │ 19.2 16.8195 16.8422
5 │ 32.4 29.895 29.9475
6 │ 22.8 28.9494 29.0082
7 │ 21.0 22.0858 22.0527
8 │ 21.0 22.2453 22.1979
9 │ 19.2 18.3686 18.3084
10 │ 14.7 11.279 11.2979
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: 2.0235329724536175
OLS loss: 2.022491012568111
Test set:
Bayes loss: 20.747901318693184
OLS loss: 21.243154363900427
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 1.9.3
Commit bed2cd540a1 (2023-08-24 14:43 UTC)
Build Info:
Official https://julialang.org/ release
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CPU: 128 × AMD EPYC 7502 32-Core Processor
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[700de1a5] ZygoteRules v0.2.3
⌅ [68821587] Arpack_jll v3.5.1+1
[6e34b625] Bzip2_jll v1.0.8+0
[83423d85] Cairo_jll v1.16.1+1
[2702e6a9] EpollShim_jll v0.0.20230411+0
[2e619515] Expat_jll v2.5.0+0
⌃ [b22a6f82] FFMPEG_jll v4.4.2+2
[f5851436] FFTW_jll v3.3.10+0
[a3f928ae] Fontconfig_jll v2.13.93+0
[d7e528f0] FreeType2_jll v2.13.1+0
[559328eb] FriBidi_jll v1.0.10+0
[0656b61e] GLFW_jll v3.3.8+0
[d2c73de3] GR_jll v0.72.10+0
[78b55507] Gettext_jll v0.21.0+0
[7746bdde] Glib_jll v2.76.5+0
[3b182d85] Graphite2_jll v1.3.14+0
[2e76f6c2] HarfBuzz_jll v2.8.1+1
[1d5cc7b8] IntelOpenMP_jll v2023.2.0+0
[aacddb02] JpegTurbo_jll v2.1.91+0
[c1c5ebd0] LAME_jll v3.100.1+0
[88015f11] LERC_jll v3.0.0+1
[dad2f222] LLVMExtra_jll v0.0.26+0
[1d63c593] LLVMOpenMP_jll v15.0.4+0
[dd4b983a] LZO_jll v2.10.1+0
⌅ [e9f186c6] Libffi_jll v3.2.2+1
[d4300ac3] Libgcrypt_jll v1.8.7+0
[7e76a0d4] Libglvnd_jll v1.6.0+0
[7add5ba3] Libgpg_error_jll v1.42.0+0
[94ce4f54] Libiconv_jll v1.17.0+0
[4b2f31a3] Libmount_jll v2.35.0+0
[89763e89] Libtiff_jll v4.5.1+1
[38a345b3] Libuuid_jll v2.36.0+0
[856f044c] MKL_jll v2023.2.0+0
[e7412a2a] Ogg_jll v1.3.5+1
⌅ [458c3c95] OpenSSL_jll v1.1.23+0
[efe28fd5] OpenSpecFun_jll v0.5.5+0
[91d4177d] Opus_jll v1.3.2+0
[30392449] Pixman_jll v0.42.2+0
[c0090381] Qt6Base_jll v6.5.2+2
[f50d1b31] Rmath_jll v0.4.0+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.21.0+1
[2381bf8a] Wayland_protocols_jll v1.25.0+0
[02c8fc9c] XML2_jll v2.11.5+0
[aed1982a] XSLT_jll v1.1.34+0
[ffd25f8a] XZ_jll v5.4.4+0
[f67eecfb] Xorg_libICE_jll v1.0.10+1
[c834827a] Xorg_libSM_jll v1.2.3+0
[4f6342f7] Xorg_libX11_jll v1.8.6+0
[0c0b7dd1] Xorg_libXau_jll v1.0.11+0
[935fb764] Xorg_libXcursor_jll v1.2.0+4
[a3789734] Xorg_libXdmcp_jll v1.1.4+0
[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 v0.1.1+0
[c7cfdc94] Xorg_libxcb_jll v1.15.0+0
[cc61e674] Xorg_libxkbfile_jll v1.1.2+0
[e920d4aa] Xorg_xcb_util_cursor_jll v0.1.4+0
[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 v1.4.6+0
[33bec58e] Xorg_xkeyboard_config_jll v2.39.0+0
[c5fb5394] Xorg_xtrans_jll v1.5.0+0
[3161d3a3] Zstd_jll v1.5.5+0
[35ca27e7] eudev_jll v3.2.9+0
⌅ [214eeab7] fzf_jll v0.29.0+0
[1a1c6b14] gperf_jll v3.1.1+0
[a4ae2306] libaom_jll v3.4.0+0
[0ac62f75] libass_jll v0.15.1+0
[2db6ffa8] libevdev_jll v1.11.0+0
[f638f0a6] libfdk_aac_jll v2.0.2+0
[36db933b] libinput_jll v1.18.0+0
[b53b4c65] libpng_jll v1.6.38+0
[f27f6e37] libvorbis_jll v1.3.7+1
[009596ad] mtdev_jll v1.1.6+0
[1270edf5] x264_jll v2021.5.5+0
[dfaa095f] x265_jll v3.5.0+0
[d8fb68d0] xkbcommon_jll v1.4.1+1
[0dad84c5] ArgTools v1.1.1
[56f22d72] Artifacts
[2a0f44e3] Base64
[ade2ca70] Dates
[8ba89e20] Distributed
[f43a241f] Downloads v1.6.0
[7b1f6079] FileWatching
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[4af54fe1] LazyArtifacts
[b27032c2] LibCURL v0.6.3
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[a63ad114] Mmap
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.9.2
[de0858da] Printf
[3fa0cd96] REPL
[9a3f8284] Random
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization
[1a1011a3] SharedArrays
[6462fe0b] Sockets
[2f01184e] SparseArrays
[10745b16] Statistics v1.9.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test
[cf7118a7] UUIDs
[4ec0a83e] Unicode
[e66e0078] CompilerSupportLibraries_jll v1.0.5+0
[deac9b47] LibCURL_jll v7.84.0+0
[29816b5a] LibSSH2_jll v1.10.2+0
[c8ffd9c3] MbedTLS_jll v2.28.2+0
[14a3606d] MozillaCACerts_jll v2022.10.11
[4536629a] OpenBLAS_jll v0.3.21+4
[05823500] OpenLibm_jll v0.8.1+0
[efcefdf7] PCRE2_jll v10.42.0+0
[bea87d4a] SuiteSparse_jll v5.10.1+6
[83775a58] Zlib_jll v1.2.13+0
[8e850b90] libblastrampoline_jll v5.8.0+0
[8e850ede] nghttp2_jll v1.48.0+0
[3f19e933] p7zip_jll v17.4.0+0
Info Packages marked with ⌃ and ⌅ have new versions available, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`