lassoblm

Bayesian linear regression model with lasso regularization

Description

The Bayesian linear regression model object lassoblm specifies the joint prior distribution of the regression coefficients and the disturbance variance (β, σ²) for implementing Bayesian lasso regression [1]. For j = 1,…,NumPredictors, the conditional prior distribution of β_j|σ² is the Laplace (double exponential) distribution with a mean of 0 and scale σ²/λ, where λ is the lasso regularization, or shrinkage, parameter. The prior distribution of σ² is inverse gamma with shape A and scale B.

The data likelihood is $${\displaystyle \prod _{t=1}^T\varphi \left(y_t;x_t\beta ,{\sigma }^2\right)},$$ where ϕ(y_t;x_tβ,σ²) is the Gaussian probability density evaluated at y_t with mean x_tβ and variance σ². The resulting posterior distribution is not analytically tractable. For details on the posterior distribution, see Analytically Tractable Posteriors.

In general, when you create a Bayesian linear regression model object, it specifies the joint prior distribution and characteristics of the linear regression model only. That is, the model object is a template intended for further use. Specifically, to incorporate data into the model for posterior distribution analysis and feature selection, pass the model object and data to the appropriate object function.

Creation

Syntax

PriorMdl = lassoblm(NumPredictors)

PriorMdl = lassoblm(NumPredictors,Name,Value)

Description

PriorMdl = lassoblm(NumPredictors) creates a Bayesian linear regression model object (PriorMdl) composed of NumPredictors predictors and an intercept, and sets the NumPredictors property. The joint prior distribution of (β, σ²) is appropriate for implementing Bayesian lasso regression [1]. PriorMdl is a template that defines the prior distributions and specifyies the values of the lasso regularization parameter λ and the dimensionality of β.

PriorMdl = lassoblm(NumPredictors,Name,Value) sets properties (except NumPredictors) using name-value pair arguments. Enclose each property name in quotes. For example, lassoblm(3,'Lambda',0.5) specifies a shrinkage of 0.5 for the three coefficients (not the intercept).

Properties

You can set writable property values when you create the model object by using name-value pair argument syntax, or after you create model object by using dot notation. For example, to set the shrinkage for all coefficients, except the intercept, to 0.5, enter

PriorMdl.Lambda = 0.5;

NumPredictors — Number of predictor variables

nonnegative integer

Number of predictor variables in the Bayesian multiple linear regression model, specified as a nonnegative integer.

NumPredictors must be the same as the number of columns in your predictor data, which you specify during model estimation or simulation.

When specifying NumPredictors, exclude any intercept term for the value.

After creating a model, if you change the of value NumPredictors using dot notation, then these parameters revert to the default values:

• Variable names (VarNames)

• Shrinkage parameter (Lambda)

Data Types: double

Intercept — Flag for including regression model intercept

true (default) | false

Flag for including a regression model intercept, specified as a value in this table.

Value	Description
false	Exclude an intercept from the regression model. Therefore, β is a p-dimensional vector, where p is the value of NumPredictors.
true	Include an intercept in the regression model. Therefore, β is a (p + 1)-dimensional vector. This specification causes a T-by-1 vector of ones to be prepended to the predictor data during estimation and simulation.

If you include a column of ones in the predictor data for an intercept term, then set Intercept to false.

Example: 'Intercept',false

Data Types: logical

VarNames — Predictor variable names

string vector | cell vector of character vectors

Predictor variable names for displays, specified as a string vector or cell vector of character vectors. VarNames must contain NumPredictors elements. VarNames(j) is the name of the variable in column j of the predictor data set, which you specify during estimation, simulation, or forecasting.

The default is {'Beta(1)','Beta(2),...,Beta(p)}, where p is the value of NumPredictors.

Example: 'VarNames',["UnemploymentRate"; "CPI"]

Data Types: string | cell | char

Lambda — Lasso regularization parameter

1 (default) | positive numeric scalar | positive numeric vector

Lasso regularization parameter for all regression coefficients, specified as a positive numeric scalar or (Intercept + NumPredictors)-by-1 positive numeric vector. Larger values of Lambda cause corresponding coefficients to shrink closer to zero.

Suppose X is a T-by-NumPredictors matrix of predictor data, which you specify during estimation, simulation, or forecasting.

• If Lambda is a vector and Intercept is true, Lambda(1) is the shrinkage for the intercept, Lambda(2) is the shrinkage for the coefficient of the first predictor X(:,1), Lambda(3) is the shrinkage for the coefficient of the second predictor X(:,2),…, and Lambda(NumPredictors + 1) is the shrinkage for the coefficient of the last predictor X(:,NumPredictors).

• If Lambda is a vector and Intercept is false, Lambda(1) is the shrinkage for the coefficient of the first predictor X(:,1),…, and Lambda(NumPredictors) is the shrinkage for the coefficient of the last predictor X(:,NumPredictors).

• If you supply the scalar s for Lambda, then all coefficients of the predictors in X have a shrinkage of s.

  • If Intercept is true, the intercept has a shrinkage of 0.01, and lassoblm stores [0.01; s*ones(NumPredictors,1)] in Lambda.

  • Otherwise, lassoblm stores s*ones(NumPredictors,1) in Lambda.

Example: 'Lambda',6

Data Types: double

A — Shape hyperparameter of inverse gamma prior on σ²

3 (default) | numeric scalar

Shape hyperparameter of the inverse gamma prior on σ², specified as a numeric scalar.

A must be at least –(Intercept + NumPredictors)/2.

With B held fixed, the inverse gamma distribution becomes taller and more concentrated as A increases. This characteristic weighs the prior model of σ² more heavily than the likelihood during posterior estimation.

For the functional form of the inverse gamma distribution, see Analytically Tractable Posteriors.

Example: 'A',0.1

Data Types: double

B — Scale hyperparameter of inverse gamma prior on σ²

1 (default) | positive scalar | Inf

Scale parameter of inverse gamma prior on σ², specified as a positive scalar or Inf.

With A held fixed, the inverse gamma distribution becomes taller and more concentrated as B increases. This characteristic weighs the prior model of σ² more heavily than the likelihood during posterior estimation.

Example: 'B',5

Data Types: double

Object Functions

estimate	Perform predictor variable selection for Bayesian linear regression models
simulate	Simulate regression coefficients and disturbance variance of Bayesian linear regression model
forecast	Forecast responses of Bayesian linear regression model
plot	Visualize prior and posterior densities of Bayesian linear regression model parameters
summarize	Distribution summary statistics of Bayesian linear regression model for predictor variable selection

Examples

Create Prior Model for Bayesian Lasso Regression

Consider the multiple linear regression model that predicts the US real gross national product (GNPR) using a linear combination of industrial production index (IPI), total employment (E), and real wages (WR).

$${\text{GNPR}}_t={\beta }_0+{\beta }_1{\text{IPI}}_t+{\beta }_2{\text{E}}_t+{\beta }_3{\text{WR}}_t+{\epsilon }_t.$$

For all $$t$$, $${\epsilon }_t$$ is a series of independent Gaussian disturbances with a mean of 0 and variance $${\sigma }^2$$.

Assume these prior distributions:

• For j = 0,...,3, $${\beta }_j|{\sigma }^2$$ has a Laplace distribution with a mean of 0 and a scale of ${\sigma }^2/\lambda$, where $\lambda$ is the shrinkage parameter. The coefficients are conditionally independent.

• $${\sigma }^2\sim IG(A,B)$$. $$A$$ and $$B$$ are the shape and scale, respectively, of an inverse gamma distribution.

Create a prior model for Bayesian linear regression. Specify the number of predictors p.

p = 3;
Mdl = lassoblm(p);

PriorMdl is a lassoblm Bayesian linear regression model object representing the prior distribution of the regression coefficients and disturbance variance. At the command window, lassoblm displays a summary of the prior distributions.

Alternatively, you can create a prior model for Bayesian lasso regression by passing the number of predictors to bayeslm and setting the ModelType name-value pair argument to 'lasso'.

MdlBayesLM = bayeslm(p,'ModelType','lasso')

MdlBayesLM = 
  lassoblm with properties:

    NumPredictors: 3
        Intercept: 1
         VarNames: {4x1 cell}
           Lambda: [4x1 double]
                A: 3
                B: 1

 
           |  Mean     Std           CI95         Positive   Distribution  
---------------------------------------------------------------------------
 Intercept |  0       100    [-200.000, 200.000]    0.500   Scale mixture  
 Beta(1)   |  0       1        [-2.000,  2.000]     0.500   Scale mixture  
 Beta(2)   |  0       1        [-2.000,  2.000]     0.500   Scale mixture  
 Beta(3)   |  0       1        [-2.000,  2.000]     0.500   Scale mixture  
 Sigma2    | 0.5000  0.5000    [ 0.138,  1.616]     1.000   IG(3.00,    1) 
 

Mdl and MdlBayesLM are equivalent model objects.

You can set writable property values of created models using dot notation. Set the regression coefficient names to the corresponding variable names.

Mdl.VarNames = ["IPI" "E" "WR"]

Mdl = 
  lassoblm with properties:

    NumPredictors: 3
        Intercept: 1
         VarNames: {4x1 cell}
           Lambda: [4x1 double]
                A: 3
                B: 1

 
           |  Mean     Std           CI95         Positive   Distribution  
---------------------------------------------------------------------------
 Intercept |  0       100    [-200.000, 200.000]    0.500   Scale mixture  
 IPI       |  0       1        [-2.000,  2.000]     0.500   Scale mixture  
 E         |  0       1        [-2.000,  2.000]     0.500   Scale mixture  
 WR        |  0       1        [-2.000,  2.000]     0.500   Scale mixture  
 Sigma2    | 0.5000  0.5000    [ 0.138,  1.616]     1.000   IG(3.00,    1) 
 

MATLAB® associates the variable names to the regression coefficients in displays.

Perform Variable Selection Using Default Lasso Shrinkage

Consider the linear regression model in Create Prior Model for Bayesian Lasso Regression.

Create a prior model for performing Bayesian lasso regression. Specify the number of predictors p and the names of the regression coefficients.

p = 3;
PriorMdl = bayeslm(p,'ModelType','lasso','VarNames',["IPI" "E" "WR"]);
shrinkage = PriorMdl.Lambda

shrinkage = 4×1

    0.0100
    1.0000
    1.0000
    1.0000

PriorMdl stores the shrinkage values for all predictors in its Lambda property. shrinkage(1) is the shrinkage for the intercept, and the elements of shrinkage(2:end) correspond to the coefficients of the predictors in Mdl.VarNames. The default shrinkage for the intercept is 0.01, and the default is 1 for all other coefficients.

Load the Nelson-Plosser data set. Create variables for the response and predictor series. Because lasso is sensitive to variable scales, standardize all variables.

load Data_NelsonPlosser
X = DataTable{:,PriorMdl.VarNames(2:end)};
y = DataTable{:,'GNPR'};

X = (X - nanmean(X))./nanstd(X);
y = (y - nanmean(y))/nanstd(y);

Although this example standardizes variables, you can specify different shrinkage values for each coefficient by setting the Lambda property of PriorMdl to a numeric vector of shrinkage values.

Implement Bayesian lasso regression by estimating the marginal posterior distributions of $$\beta$$ and $${\sigma }^2$$. Because Bayesian lasso regression uses Markov chain Monte Carlo (MCMC) for estimation, set a random number seed to reproduce the results.

rng(1);
PosteriorMdl = estimate(PriorMdl,X,y);

Method: lasso MCMC sampling with 10000 draws
Number of observations: 62
Number of predictors:   4
 
           |   Mean     Std         CI95        Positive  Distribution 
-----------------------------------------------------------------------
 Intercept | -0.4490  0.0527  [-0.548, -0.344]    0.000     Empirical  
 IPI       |  0.6679  0.1063  [ 0.456,  0.878]    1.000     Empirical  
 E         |  0.1114  0.1223  [-0.110,  0.365]    0.827     Empirical  
 WR        |  0.2215  0.1367  [-0.024,  0.494]    0.956     Empirical  
 Sigma2    |  0.0343  0.0062  [ 0.024,  0.048]    1.000     Empirical  
 

PosteriorMdl is an empiricalblm model object that stores draws from the posterior distributions of $$\beta$$ and $${\sigma }^2$$ given the data. estimate displays a summary of the marginal posterior distributions at the command line. Rows of the summary correspond to regression coefficients and the disturbance variance, and columns correspond to characteristics of the posterior distribution. The characteristics include:

• CI95, which contains the 95% Bayesian equitailed credible intervals for the parameters. For example, the posterior probability that the regression coefficient of E (standardized) is in [-0.110, 0.365] is 0.95.

• Positive, which contains the posterior probability that the parameter is greater than 0. For example, the probability that the intercept is greater than 0 is 0.

By default, estimate draws and discards a burn-in sample of size 5000. However, a good practice is to inspect a trace plot of the draws for adequate mixing and lack of transience. Plot a trace plot of the draws for each parameter. You can access the draws that compose the distribution (the properties BetaDraws and Sigma2Draws) using dot notation.

figure;
for j = 1:(p + 1)
    subplot(2,2,j);
    plot(PosteriorMdl.BetaDraws(j,:));
    title(sprintf('%s',PosteriorMdl.VarNames{j}));
end

figure;
plot(PosteriorMdl.Sigma2Draws);
title('Sigma2');

The trace plots indicate that the draws seem to mix well. The plot shows no detectable transience or serial correlation, and the draws do not jump between states.

Plot the posterior distributions of the coefficients and disturbance variance.

figure;
plot(PosteriorMdl)

E and WR might not be important predictors because 0 is within the region of high density in their posterior distributions.

Attribute Different Shrinkage Values to Coefficients

Consider the linear regression model in Create Prior Model for Bayesian Lasso Regression and its implementation in Perform Variable Selection Using Default Lasso Shrinkage.

When you implement lasso regression, a common practice is to standardize variables. However, if you want to preserve the interpretation of the coefficients, but the variables have different scales, then you can perform differential shrinkage by specifying a different shrinkage for each coefficient.

Create a prior model for performing Bayesian lasso regression. Specify the number of predictors p and the names of the regression coefficients.

p = 3;
PriorMdl = bayeslm(p,'ModelType','lasso','VarNames',["IPI" "E" "WR"]);

Load the Nelson-Plosser data set. Create variables for the response and predictor series. Determine whether the variables have exponential trends by plotting each in separate figures.

load Data_NelsonPlosser
X = DataTable{:,PriorMdl.VarNames(2:end)};
y = DataTable{:,'GNPR'};

figure;
plot(dates,y)
title('GNPR')

for j = 1:3
    figure;
    plot(dates,X(:,j));
    title(PriorMdl.VarNames(j + 1));
end

The variables GNPR, IPI, and WR appear to have an exponential trend.

Remove the exponential trend from the variables GNPR, IPI, and WR.

y = log(y);
X(:,[1 3]) = log(X(:,[1 3]));

All predictor variables have different scales (for more details, enter Description at the command line). Display the mean of each predictor. Because the variables contain leading missing values, use nanmean.

predmeans = nanmean(X)

predmeans = 1×3
104 ×

    0.0002    4.7700    0.0004

The values of the second predictor are much greater than those of the other two predictors and the response. Therefore, the regression coefficient of the second predictor can appear close to zero.

Using dot notation, attribute a very low shrinkage to the intercept, a shrinkage of 0.1 to the first and third predictors, and a shrinkage of 1000 to the second predictor.

PriorMdl.Lambda = [1e-5 0.1 1e4 0.1];

Implement Bayesian lasso regression by estimating the marginal posterior distributions of $$\beta$$ and $${\sigma }^2$$. Because Bayesian lasso regression uses MCMC for estimation, set a random number seed to reproduce the results.

rng(1);
PosteriorMdl = estimate(PriorMdl,X,y);

Method: lasso MCMC sampling with 10000 draws
Number of observations: 62
Number of predictors:   4
 
           |  Mean     Std         CI95        Positive  Distribution 
----------------------------------------------------------------------
 Intercept | 2.0281  0.6839  [ 0.679,  3.323]    0.999     Empirical  
 IPI       | 0.3534  0.2497  [-0.139,  0.839]    0.923     Empirical  
 E         | 0.0000  0.0000  [-0.000,  0.000]    0.762     Empirical  
 WR        | 0.5250  0.3482  [-0.126,  1.209]    0.937     Empirical  
 Sigma2    | 0.0315  0.0055  [ 0.023,  0.044]    1.000     Empirical  
 

Forecast Responses Using Posterior Predictive Distribution

Consider the linear regression model in Create Prior Model for Bayesian Lasso Regression.

Perform Bayesian lasso regression:

1. Create a Bayesian lasso prior model for the regression coefficients and disturbance variance. Use the default shrinkage.

2. Hold out the last 10 periods of data from estimation.

3. Estimate the marginal posterior distributions.

p = 3;
PriorMdl = bayeslm(p,'ModelType','lasso','VarNames',["IPI" "E" "WR"]);

load Data_NelsonPlosser
fhs = 10; % Forecast horizon size
X = DataTable{1:(end - fhs),PriorMdl.VarNames(2:end)};
y = DataTable{1:(end - fhs),'GNPR'};
XF = DataTable{(end - fhs + 1):end,PriorMdl.VarNames(2:end)}; % Future predictor data
yFT = DataTable{(end - fhs + 1):end,'GNPR'};                  % True future responses

rng(1); % For reproducibility
PosteriorMdl = estimate(PriorMdl,X,y,'Display',false);

Forecast responses using the posterior predictive distribution and the future predictor data XF. Plot the true values of the response and the forecasted values.

yF = forecast(PosteriorMdl,XF);

figure;
plot(dates,DataTable.GNPR);
hold on
plot(dates((end - fhs + 1):end),yF)
h = gca;
hp = patch([dates(end - fhs + 1) dates(end) dates(end) dates(end - fhs + 1)],...
    h.YLim([1,1,2,2]),[0.8 0.8 0.8]);
uistack(hp,'bottom');
legend('Forecast Horizon','True GNPR','Forecasted GNPR','Location','NW')
title('Real Gross National Product: 1909 - 1970');
ylabel('rGNP');
xlabel('Year');
hold off

yF is a 10-by-1 vector of future values of real GNP corresponding to the future predictor data.

Estimate the forecast root mean squared error (RMSE).

frmse = sqrt(mean((yF - yFT).^2))

frmse = 25.4831

The forecast RMSE is a relative measure of forecast accuracy. Specifically, you estimate several models using different assumptions. The model with the lowest forecast RMSE is the best-performing model of the ones being compared.

When you perform Bayesian lasso regression, a best practice is to search for appropriate shrinkage values. One way to do so is to estimate the forecast RMSE over a grid of shrinkage values, and choose the shrinkage that minimizes the forecast RMSE.

More About

Bayesian Linear Regression Model

A Bayesian linear regression model treats the parameters β and σ² in the multiple linear regression (MLR) model y_t = x_tβ + ε_t as random variables.

For times t = 1,...,T:

• y_t is the observed response.

• x_t is a 1-by-(p + 1) row vector of observed values of p predictors. To accommodate a model intercept, x_1t = 1 for all t.

• β is a (p + 1)-by-1 column vector of regression coefficients corresponding to the variables that compose the columns of x_t.

• ε_t is the random disturbance with a mean of zero and Cov(ε) = σ²I_T×T, while ε is a T-by-1 vector containing all disturbances. These assumptions imply that the data likelihood is

  $$\ell \left(\beta ,{\sigma }^2|y,x\right)={\displaystyle \prod _{t=1}^T\varphi \left(y_t;x_t\beta ,{\sigma }^2\right).}$$

  ϕ(y_t;x_tβ,σ²) is the Gaussian probability density with mean x_tβ and variance σ² evaluated at y_t;.

Before considering the data, you impose a joint prior distribution assumption on (β,σ²). In a Bayesian analysis, you update the distribution of the parameters by using information about the parameters obtained from the likelihood of the data. The result is the joint posterior distribution of (β,σ²) or the conditional posterior distributions of the parameters.

Tips

• Lambda is a tuning parameter. Therefore, perform Bayesian lasso regression using a grid of shrinkage values, and choose the model that best balances a fit criterion and model complexity.

• For estimation, simulation, and forecasting, MATLAB^® does not standardize predictor data. If the variables in the predictor data have different scales, then specify a shrinkage parameter for each predictor by supplying a numeric vector for Lambda.

Alternative Functionality

The bayeslm function can create any supported prior model object for Bayesian linear regression.

References

[1] Park, T., and G. Casella. "The Bayesian Lasso." Journal of the American Statistical Association. Vol. 103, No. 482, 2008, pp. 681–686.