# lassoglm

Lasso or elastic net regularization for generalized linear models

## Description

example

B = lassoglm(X,y) returns penalized, maximum-likelihood fitted coefficients for generalized linear models of the predictor data X and the response y, where the values in y are assumed to have a normal probability distribution. Each column of B corresponds to a particular regularization coefficient in Lambda. By default, lassoglm performs lasso regularization using a geometric sequence of Lambda values.

B = lassoglm(X,y,distr) performs lasso regularization to fit the models using the probability distribution distr for y.

B = lassoglm(X,y,distr,Name,Value) fits regularized generalized linear regressions with additional options specified by one or more name-value pair arguments. For example, 'Alpha',0.5 sets elastic net as the regularization method, with the parameter Alpha equal to 0.5.

example

[B,FitInfo] = lassoglm(___) also returns the structure FitInfo, which contains information about the fit of the models, using any of the input arguments in the previous syntaxes.

## Examples

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Construct a data set with redundant predictors and identify those predictors by using lassoglm.

Create a random matrix X with 100 observations and 10 predictors. Create the normally distributed response y using only four of the predictors and a small amount of noise.

rng default
X = randn(100,10);
weights = [0.6;0.5;0.7;0.4];
y = X(:,[2 4 5 7])*weights + randn(100,1)*0.1; % Small added noise

Perform lasso regularization.

B = lassoglm(X,y);

Find the coefficient vector for the 75th Lambda value in B.

B(:,75)
ans = 10×1

0
0.5431
0
0.3944
0.6173
0
0.3473
0
0
0

lassoglm identifies and removes the redundant predictors.

Construct data from a Poisson model, and identify the important predictors by using lassoglm.

Create data with 20 predictors. Create a Poisson response variable using only three of the predictors plus a constant.

rng default % For reproducibility
X = randn(100,20);
weights = [.4;.2;.3];
mu = exp(X(:,[5 10 15])*weights + 1);
y = poissrnd(mu);

Construct a cross-validated lasso regularization of a Poisson regression model of the data.

[B,FitInfo] = lassoglm(X,y,'poisson','CV',10);

Examine the cross-validation plot to see the effect of the Lambda regularization parameter.

lassoPlot(B,FitInfo,'plottype','CV');
legend('show') % Show legend

The green circle and dotted line locate the Lambda with minimum cross-validation error. The blue circle and dotted line locate the point with minimum cross-validation error plus one standard deviation.

Find the nonzero model coefficients corresponding to the two identified points.

idxLambdaMinDeviance = FitInfo.IndexMinDeviance;
mincoefs = find(B(:,idxLambdaMinDeviance))
mincoefs = 7×1

3
5
6
10
11
15
16

idxLambda1SE = FitInfo.Index1SE;
min1coefs = find(B(:,idxLambda1SE))
min1coefs = 3×1

5
10
15

The coefficients from the minimum-plus-one standard error point are exactly those coefficients used to create the data.

Predict whether students got a B or above on their last exam by using lassoglm.

Load the examgrades data set. Convert the last exam grades to a logical vector, where 1 represents a grade of 80 or above and 0 represents a grade below 80.

yBinom = (y>=80);

Partition the data into training and test sets.

rng default    % Set the seed for reproducibility
c = cvpartition(yBinom,'HoldOut',0.3);
idxTrain = training(c,1);
idxTest = ~idxTrain;
XTrain = X(idxTrain,:);
yTrain = yBinom(idxTrain);
XTest = X(idxTest,:);
yTest = yBinom(idxTest);

Perform lasso regularization for generalized linear model regression with 3-fold cross-validation on the training data. Assume the values in y are binomially distributed. Choose model coefficients corresponding to the Lambda with minimum expected deviance.

[B,FitInfo] = lassoglm(XTrain,yTrain,'binomial','CV',3);
idxLambdaMinDeviance = FitInfo.IndexMinDeviance;
B0 = FitInfo.Intercept(idxLambdaMinDeviance);
coef = [B0; B(:,idxLambdaMinDeviance)]
coef = 5×1

-21.1911
0.0235
0.0670
0.0693
0.0949

Predict exam grades for the test data using the model coefficients found in the previous step. Specify the link function for a binomial response using 'logit'. Convert the prediction values to a logical vector.

yhat = glmval(coef,XTest,'logit');
yhatBinom = (yhat>=0.5);

Determine the accuracy of the predictions using a confusion matrix.

c = confusionchart(yTest,yhatBinom);

The function correctly predicts 31 exam grades. However, the function incorrectly predicts that 1 student receives a B or above and 4 students receive a grade below a B.

Create a matrix X of N p-dimensional normal variables, where N is large and p = 1000. Create a response vector y from the model y = X*beta + noise, where beta is a vector of coefficients with 50% nonzero values.

rng default % For reproducibility
N = 1e4; % Number of samples
p = 1e3; % Number of features
X = randn(N,p);
beta = 1 + 3*rand(p,1); % Multiplicative coefficients
activep = randperm(p,p/2); % 50% nonzero coefficients
y = X(:,activep)*beta(activep) + randn(N,1)*0.1; % Add noise

Construct the lasso fit without using the covariance matrix. Time the creation.

B = lassoglm(X,y,"normal",UseCovariance=false); % Warm up lasso for reliable timing data
tic
B = lassoglm(X,y,"normal",UseCovariance=false);
timefalse = toc
timefalse = 2.9323

Construct the lasso fit using the covariance matrix. Time the creation.

B2 = lassoglm(X,y,"normal",UseCovariance=true); % Warm up lasso for reliable timing data
tic
B2 = lassoglm(X,y,"normal",UseCovariance=true);
timetrue = toc
timetrue = 0.7596

The fitting time with the covariance matrix is less than the time without it. View the speedup factor that results from using the covariance matrix.

speedup = timefalse/timetrue
speedup = 3.8603

Check that the returned coefficients B and B2 are similar.

norm(B-B2)/norm(B)
ans = 3.6556e-15

The results are virtually identical.

## Input Arguments

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Predictor data, specified as a numeric matrix. Each row represents one observation, and each column represents one predictor variable.

Data Types: single | double

Response data, specified as a numeric vector, logical vector, categorical array, or two-column numeric matrix.

• When distr is not 'binomial', y is a numeric vector or categorical array of length n, where n is the number of rows in X. The response y(i) corresponds to row i in X.

• When distr is 'binomial', y is one of the following:

• Numeric vector of length n, where each entry represents success (1) or failure (0)

• Logical vector of length n, where each entry represents success or failure

• Categorical array of length n, where each entry represents success or failure

• Two-column numeric matrix, where the first column contains the number of successes for each observation and the second column contains the total number of trials

Data Types: single | double | logical | categorical

Distribution of response data, specified as one of the following:

• 'normal' (default)

• 'binomial'

• 'poisson'

• 'gamma'

• 'inverse gaussian'

lassoglm uses the default link function corresponding to distr. Specify another link function using the Link name-value pair argument.

### Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: lassoglm(X,y,'poisson','Alpha',0.5) performs elastic net regularization assuming that the response values are Poisson distributed. The 'Alpha',0.5 name-value pair argument sets the parameter used in the elastic net optimization.

Weight of lasso (L1) versus ridge (L2) optimization, specified as the comma-separated pair consisting of 'Alpha' and a positive scalar value in the interval (0,1]. The value Alpha = 1 represents lasso regression, Alpha close to 0 approaches ridge regression, and other values represent elastic net optimization. See Elastic Net.

Example: 'Alpha',0.75

Data Types: single | double

Size of the covariance matrix in megabytes, specified as a positive scalar or 'maximal'. The lassoglm function can use a covariance matrix for fitting when the UseCovariance argument is true or 'auto'.

If UseCovariance is true or 'auto' and CacheSize is 'maximal', lassoglm can attempt to allocate a covariance matrix that exceeds the available memory. In this case, MATLAB® issues an error.

Example: 'CacheSize','maximal'

Data Types: double | char | string

Cross-validation specification for estimating the deviance, specified as the comma-separated pair consisting of 'CV' and one of the following:

• 'resubstitution'lassoglm uses X and y to fit the model and to estimate the deviance without cross-validation.

• Positive scalar integer Klassoglm uses K-fold cross-validation.

• cvpartition object cvplassoglm uses the cross-validation method expressed in cvp. You cannot use a 'leaveout' partition with lassoglm.

Example: 'CV',10

Maximum number of nonzero coefficients in the model, specified as the comma-separated pair consisting of 'DFmax' and a positive integer scalar. lassoglm returns results only for Lambda values that satisfy this criterion.

Example: 'DFmax',25

Data Types: single | double

Regularization coefficients, specified as the comma-separated pair consisting of 'Lambda' and a vector of nonnegative values. See Lasso.

• If you do not supply Lambda, then lassoglm estimates the largest value of Lambda that gives a nonnull model. In this case, LambdaRatio gives the ratio of the smallest to the largest value of the sequence, and NumLambda gives the length of the vector.

• If you supply Lambda, then lassoglm ignores LambdaRatio and NumLambda.

• If Standardize is true, then Lambda is the set of values used to fit the models with the X data standardized to have zero mean and a variance of one.

The default is a geometric sequence of NumLambda values, with only the largest value able to produce B = 0.

Data Types: single | double

Ratio of the smallest to the largest Lambda values when you do not supply Lambda, specified as the comma-separated pair consisting of 'LambdaRatio' and a positive scalar.

If you set LambdaRatio = 0, then lassoglm generates a default sequence of Lambda values and replaces the smallest one with 0.

Example: 'LambdaRatio',1e–2

Data Types: single | double

Mapping between the mean µ of the response and the linear predictor Xb, specified as the comma-separated pair consisting of 'Link' and one of the values in this table.

ValueDescription
'comploglog'

log(–log((1 – µ))) = Xb

'identity', default for the distribution 'normal'

µ = Xb

'log', default for the distribution 'poisson'

log(µ) = Xb

'logit', default for the distribution 'binomial'

log(µ/(1 – µ)) = Xb

'loglog'

log(–log(µ)) = Xb

'probit'

Φ–1(µ) = Xb, where Φ is the normal (Gaussian) cumulative distribution function

'reciprocal', default for the distribution 'gamma'

µ–1 = Xb

p (a number), default for the distribution 'inverse gaussian' (with p = –2)

µp = Xb

A cell array of the form {FL FD FI}, containing three function handles created using @, which define the link (FL), the derivative of the link (FD), and the inverse link (FI). Or, a structure of function handles with the field Link containing FL, the field Derivative containing FD, and the field Inverse containing FI.

User-specified link function (see Custom Link Function)

Data Types: char | string | single | double | cell

Maximum number of iterations allowed, specified as the comma-separated pair consisting of 'MaxIter' and a positive integer scalar.

If the algorithm executes MaxIter iterations before reaching the convergence tolerance RelTol, then the function stops iterating and returns a warning message.

The function can return more than one warning when NumLambda is greater than 1.

Example: 'MaxIter',1e3

Data Types: single | double

Number of Monte Carlo repetitions for cross-validation, specified as the comma-separated pair consisting of 'MCReps' and a positive integer scalar.

• If CV is 'resubstitution' or a cvpartition of type 'resubstitution', then MCReps must be 1.

• If CV is a cvpartition of type 'holdout', then MCReps must be greater than 1.

Example: 'MCReps',2

Data Types: single | double

Number of Lambda values lassoglm uses when you do not supply Lambda, specified as the comma-separated pair consisting of 'NumLambda' and a positive integer scalar. lassoglm can return fewer than NumLambda fits if the deviance of the fits drops below a threshold fraction of the null deviance (deviance of the fit without any predictors X).

Example: 'NumLambda',150

Data Types: single | double

Additional predictor variable, specified as the comma-separated pair consisting of 'Offset' and a numeric vector with the same number of rows as X. The lassoglm function keeps the coefficient value of Offset fixed at 1.0.

Data Types: single | double

Option to cross-validate in parallel and specify the random streams, specified as the comma-separated pair consisting of 'Options' and a structure. Parallel computation requires Parallel Computing Toolbox™.

Create the Options structure with statset. The option fields are:

• UseParallel — Set to true to compute in parallel. The default is false.

• UseSubstreams — Set to true to compute in parallel in a reproducible fashion. For reproducibility, set Streams to a type allowing substreams: 'mlfg6331_64' or 'mrg32k3a'. The default is false.

• Streams — A RandStream object or cell array consisting of one such object. If you do not specify Streams, then lassoglm uses the default stream.

Example: 'Options',statset('UseParallel',true)

Data Types: struct

Names of the predictor variables, in the order in which they appear in X, specified as the comma-separated pair consisting of 'PredictorNames' and a string array or cell array of character vectors.

Example: 'PredictorNames',{'Height','Weight','Age'}

Data Types: string | cell

Convergence threshold for the coordinate descent algorithm [3], specified as the comma-separated pair consisting of 'RelTol' and a positive scalar. The algorithm terminates when successive estimates of the coefficient vector differ in the L2 norm by a relative amount less than RelTol.

Example: 'RelTol',2e–3

Data Types: single | double

Flag for standardizing the predictor data X before fitting the models, specified as the comma-separated pair consisting of 'Standardize' and either true or false. If Standardize is true, then the X data is scaled to have zero mean and a variance of one. Standardize affects whether the regularization is applied to the coefficients on the standardized scale or the original scale. The results are always presented on the original data scale.

Example: 'Standardize',false

Data Types: logical

Indication to use a covariance matrix for fitting, specified as 'auto' or a logical scalar.

• 'auto' causes lassoglm to attempt to use a covariance matrix for fitting when the number of observations is greater than the number of problem variables, Link = 'identity', and distr = 'normal'. This attempt can fail when memory is insufficient. To find out whether lassoglm used a covariance matrix for fitting, examine the UseCovariance field of the FitInfo output.

• true causes lassoglm to use a covariance matrix for fitting as long as the required size does not exceed CacheSize. If the required covariance matrix size exceeds CacheSize, lassoglm issues a warning and does not use a covariance matrix for fitting.

• false causes lassoglm not to use a covariance matrix for fitting.

Using a covariance matrix for fitting can be faster than not using one, especially for a normally-distributed response, but can require more memory. See Use Correlation Matrix for Fitting lassoglm. The speed increase can negatively affect numerical stability. For details, see Coordinate Descent Algorithm.

Example: 'UseCovariance',true

Data Types: logical | char | string

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a nonnegative vector. Weights has length n, where n is the number of rows of X. At least two values must be positive.

Data Types: single | double

## Output Arguments

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Fitted coefficients, returned as a numeric matrix. B is a p-by-L matrix, where p is the number of predictors (columns) in X, and L is the number of Lambda values. You can specify the number of Lambda values using the NumLambda name-value pair argument.

The coefficient corresponding to the intercept term is a field in FitInfo.

Data Types: single | double

Fit information of the generalized linear models, returned as a structure with the fields described in this table.

Field in FitInfoDescription
InterceptIntercept term β0 for each linear model, a 1-by-L vector
LambdaLambda parameters in ascending order, a 1-by-L vector
AlphaValue of the Alpha parameter, a scalar
DFNumber of nonzero coefficients in B for each value of Lambda, a 1-by-L vector
Deviance

Deviance of the fitted model for each value of Lambda, a 1-by-L vector

If the model is cross-validated, then the values for Deviance represent the estimated expected deviance of the model applied to new data, as calculated by cross-validation. Otherwise, Deviance is the deviance of the fitted model applied to the data used to perform the fit.

PredictorNamesValue of the PredictorNames parameter, stored as a cell array of character vectors
UseCovarianceLogical value indicating whether the covariance matrix was used in fitting. If the covariance was computed and used, this field is true. Otherwise, this field is false.

If you set the CV name-value pair argument to cross-validate, the FitInfo structure contains these additional fields.

Field in FitInfoDescription
SEStandard error of Deviance for each Lambda, as calculated during cross-validation, a 1-by-L vector
LambdaMinDevianceLambda value with minimum expected deviance, as calculated by cross-validation, a scalar
Lambda1SELargest Lambda value such that Deviance is within one standard error of the minimum, a scalar
IndexMinDevianceIndex of Lambda with the value LambdaMinDeviance, a scalar
Index1SEIndex of Lambda with the value Lambda1SE, a scalar

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A link function f(μ) maps a distribution with mean μ to a linear model with data X and coefficient vector b using the formula

f(μ) = Xb.

You can find the formulas for the link functions in the Link name-value pair argument description. This table lists the link functions that are typically used for each distribution.

Distribution FamilyDefault Link FunctionOther Typical Link Functions
'normal''identity'
'binomial''logit''comploglog', 'loglog', 'probit'
'poisson''log'
'gamma''reciprocal'
'inverse gaussian'–2

### Lasso

For a nonnegative value of λ, lassoglm solves the problem

$\underset{{\beta }_{0},\beta }{\mathrm{min}}\left(\frac{1}{N}\text{Deviance}\left({\beta }_{0},\beta \right)+\lambda \sum _{j=1}^{p}|{\beta }_{j}|\right).$

• The function Deviance in this equation is the deviance of the model fit to the responses using the intercept β0 and the predictor coefficients β. The formula for Deviance depends on the distr parameter you supply to lassoglm. Minimizing the λ-penalized deviance is equivalent to maximizing the λ-penalized loglikelihood.

• N is the number of observations.

• λ is a nonnegative regularization parameter corresponding to one value of Lambda.

• The parameters β0 and β are a scalar and a vector of length p, respectively.

As λ increases, the number of nonzero components of β decreases.

The lasso problem involves the L1 norm of β, as contrasted with the elastic net algorithm.

### Elastic Net

For α strictly between 0 and 1, and nonnegative λ, elastic net solves the problem

$\underset{{\beta }_{0},\beta }{\mathrm{min}}\left(\frac{1}{N}\text{Deviance}\left({\beta }_{0},\beta \right)+\lambda {P}_{\alpha }\left(\beta \right)\right),$

where

${P}_{\alpha }\left(\beta \right)=\frac{\left(1-\alpha \right)}{2}{‖\beta ‖}_{2}^{2}+\alpha {‖\beta ‖}_{1}=\sum _{j=1}^{p}\left(\frac{\left(1-\alpha \right)}{2}{\beta }_{j}^{2}+\alpha |{\beta }_{j}|\right).$

Elastic net is the same as lasso when α = 1. For other values of α, the penalty term Pα(β) interpolates between the L1 norm of β and the squared L2 norm of β. As α shrinks toward 0, elastic net approaches ridge regression.

## Algorithms

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### Coordinate Descent Algorithm

lassoglm fits many values of λ simultaneously by an efficient procedure named coordinate descent, based on Friedman, Tibshirani, and Hastie [3]. The procedure has two main code paths depending on whether the fitting uses a covariance matrix. You can affect this choice with the UseCovariance name-value argument.

When lassoglm uses a covariance matrix to fit N data points and D predictors, the fitting has a rough computational complexity of D*D. Without a covariance matrix, the computational complexity is roughly N*D. So, typically, using a covariance matrix can be faster when N > D, and the default 'auto' setting of the UseCovariance argument makes this choice. Using a covariance matrix causes lassoglm to subtract larger numbers than otherwise, which can be less numerically stable. For details of the algorithmic differences, see [3]. For one comparison of timing and accuracy differences, see Use Correlation Matrix for Fitting lassoglm.

## References

[1] Tibshirani, R. “Regression Shrinkage and Selection via the Lasso.” Journal of the Royal Statistical Society. Series B, Vol. 58, No. 1, 1996, pp. 267–288.

[2] Zou, H., and T. Hastie. “Regularization and Variable Selection via the Elastic Net.” Journal of the Royal Statistical Society. Series B, Vol. 67, No. 2, 2005, pp. 301–320.

[3] Friedman, J., R. Tibshirani, and T. Hastie. “Regularization Paths for Generalized Linear Models via Coordinate Descent.” Journal of Statistical Software. Vol. 33, No. 1, 2010. https://www.jstatsoft.org/v33/i01

[4] Hastie, T., R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. 2nd edition. New York: Springer, 2008.

[5] Dobson, A. J. An Introduction to Generalized Linear Models. 2nd edition. New York: Chapman & Hall/CRC Press, 2002.

[6] McCullagh, P., and J. A. Nelder. Generalized Linear Models. 2nd edition. New York: Chapman & Hall/CRC Press, 1989.

[7] Collett, D. Modelling Binary Data. 2nd edition. New York: Chapman & Hall/CRC Press, 2003.

## Version History

Introduced in R2012a