# 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

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(), 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     = 6.37 seconds
Compute duration  = 6.37 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    0.3084    0.1750     0.0032    0.0056    937.3860
1.0 ⋯
intercept    0.0011    0.1210     0.0022    0.0024   2862.8287
1.0 ⋯
coefficients[1]   -0.0237    0.5481     0.0100    0.0137   1914.3621
0.9 ⋯
coefficients[2]    0.2714    0.7075     0.0129    0.0209   1183.2423
0.9 ⋯
coefficients[3]   -0.3893    0.3975     0.0073    0.0084   2214.6240
1.0 ⋯
coefficients[4]    0.1784    0.2921     0.0053    0.0094   1452.3292
0.9 ⋯
coefficients[5]   -0.2860    0.7084     0.0129    0.0247    966.8302
0.9 ⋯
coefficients[6]    0.0646    0.3607     0.0066    0.0107   1335.7522
0.9 ⋯
coefficients[7]    0.0068    0.3856     0.0070    0.0091   1374.8672
0.9 ⋯
coefficients[8]    0.1726    0.3034     0.0055    0.0082   1488.1037
1.0 ⋯
coefficients[9]    0.1200    0.2795     0.0051    0.0070   1413.5057
0.9 ⋯
coefficients[10]   -0.2817    0.4036     0.0074    0.0141    881.2970
0.9 ⋯
2 columns om
itted

Quantiles
parameters      2.5%     25.0%     50.0%     75.0%     97.5%
Symbol   Float64   Float64   Float64   Float64   Float64

σ²    0.1095    0.1949    0.2625    0.3677    0.7854
intercept   -0.2377   -0.0722    0.0021    0.0739    0.2433
coefficients[1]   -1.1069   -0.3709   -0.0245    0.3130    1.0966
coefficients[2]   -1.1437   -0.1643    0.2667    0.7141    1.6685
coefficients[3]   -1.1653   -0.6434   -0.3810   -0.1447    0.4003
coefficients[4]   -0.4171    0.0025    0.1815    0.3652    0.7569
coefficients[5]   -1.6252   -0.7339   -0.2902    0.1601    1.1150
coefficients[6]   -0.6262   -0.1639    0.0666    0.2854    0.7863
coefficients[7]   -0.7833   -0.2342    0.0076    0.2481    0.7665
coefficients[8]   -0.4022   -0.0276    0.1627    0.3602    0.8012
coefficients[9]   -0.4292   -0.0568    0.1225    0.2935    0.6880
coefficients[10]   -1.0792   -0.5369   -0.2774   -0.0299    0.5264


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, 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.
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 │    19.2  18.1121  18.1265
2 │    15.0   6.4727   6.37891
3 │    16.4  14.034   13.883
4 │    14.3  11.7826  11.7337
5 │    21.4  25.2335  25.1916
6 │    18.1  20.7122  20.672
7 │    19.7  15.9436  15.8408
8 │    15.2  18.3863  18.3391
9 │    26.0  28.5231  28.4865
10 │    17.3  14.6567  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: 4.65052654134636
OLS loss: 4.648142085690516
Test set:
Bayes loss: 14.496894318145872
OLS loss: 14.796847779051504


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:
BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/7aa0085e-79a4-45f3-a5bd-9743c91cf3da



Package Information:

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[aacddb02] JpegTurbo_jll v2.1.91+0
[c1c5ebd0] LAME_jll v3.100.1+0
[88015f11] LERC_jll v3.0.0+1
[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
[94ce4f54] Libiconv_jll v1.16.1+2
[4b2f31a3] Libmount_jll v2.35.0+0
[89763e89] Libtiff_jll v4.4.0+0
[38a345b3] Libuuid_jll v2.36.0+0
[856f044c] MKL_jll v2022.2.0+0
[e7412a2a] Ogg_jll v1.3.5+1
[458c3c95] OpenSSL_jll v1.1.20+0
[efe28fd5] OpenSpecFun_jll v0.5.5+0
[91d4177d] Opus_jll v1.3.2+0
[30392449] Pixman_jll v0.40.1+0
[ea2cea3b] Qt5Base_jll v5.15.3+2
[f50d1b31] Rmath_jll v0.4.0+0
[a2964d1f] Wayland_jll v1.21.0+0
[2381bf8a] Wayland_protocols_jll v1.25.0+0
[02c8fc9c] XML2_jll v2.10.3+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
[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 v1.5.4+0
[214eeab7] fzf_jll v0.29.0+0
[a4ae2306] libaom_jll v3.4.0+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 v1.4.1+0
[56f22d72] Artifacts
[2a0f44e3] Base64
[8bb1440f] DelimitedFiles
[8ba89e20] Distributed
[9fa8497b] Future
[b77e0a4c] InteractiveUtils
[4af54fe1] LazyArtifacts
[b27032c2] LibCURL
[76f85450] LibGit2
[8f399da3] Libdl
[37e2e46d] LinearAlgebra
[56ddb016] Logging
[d6f4376e] Markdown
[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
[efcefdf7] PCRE2_jll
[83775a58] Zlib_jll
[8e850ede] nghttp2_jll
[3f19e933] p7zip_jll