Lasso or elastic net regularization for generalized linear models
returns penalized, maximum-likelihood fitted coefficients for generalized linear
models of the predictor data B
= lassoglm(X
,y
)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.
fits regularized generalized linear regressions with additional options specified by
one or more name-value pair arguments. For example, B
= lassoglm(X
,y
,distr
,Name,Value
)'Alpha',0.5
sets elastic net as the regularization method, with the parameter
Alpha
equal to 0.5.
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.
load examgrades
X = grades(:,1:4);
y = grades(:,5);
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.
X
— Predictor dataPredictor data, specified as a numeric matrix. Each row represents one observation, and each column represents one predictor variable.
Data Types: single
| double
y
— Response dataResponse 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
distr
— Distribution of response data'normal'
| 'binomial'
| 'poisson'
| 'gamma'
| 'inverse gaussian'
Distribution of response data, specified as one of the following:
'normal'
'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.
Specify optional
comma-separated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
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.Alpha
— Weight of lasso versus ridge optimization1
(default) | positive scalarWeight of lasso (L^{1})
versus ridge (L^{2})
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
CV
— Cross-validation specification for estimating deviance'resubstitution'
(default) | positive integer scalar | cvpartition
objectCross-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 K
—
lassoglm
uses K
-fold
cross-validation.
cvpartition
object
cvp
—
lassoglm
uses the cross-validation
method expressed in cvp
. You cannot use a
'leaveout'
partition with
lassoglm
.
Example: 'CV',10
DFmax
— Maximum number of nonzero coefficientsInf
(default) | positive integer scalarMaximum 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
Lambda
— Regularization coefficientsRegularization 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
LambdaRatio
— Ratio of smallest to largest Lambda
values1e–4
(default) | positive scalarRatio 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
Link
— Mapping between mean of response and linear predictor'comploglog'
| 'identity'
| 'log'
| 'logit'
| 'loglog'
| ...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.
Value | Description |
---|---|
'comploglog' | log(–log((1 – µ))) = Xb |
| µ = Xb |
| log(µ) = Xb |
| log(µ/(1 – µ)) = Xb |
| 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 |
| µ^{p} = Xb |
A cell array of the form | User-specified link function (see Custom Link Function) |
Example: 'Link','probit'
Data Types: char
| string
| single
| double
| cell
MaxIter
— Maximum number of iterations allowed1e4
(default) | positive integer scalarMaximum 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
MCReps
— Number of Monte Carlo repetitions for cross-validation1
(default) | positive integer scalarNumber 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
NumLambda
— Number of Lambda
values100
(default) | positive integer scalarNumber 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
Offset
— Additional predictor variableAdditional 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
Options
— Option to cross-validate in parallel and specify random streamsOption to cross-validate in parallel and specify the random streams,
specified as the comma-separated pair consisting of
'Options'
and a structure. This option 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
PredictorNames
— Names of predictor variables{}
(default) | string array | cell array of character vectorsNames 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
RelTol
— Convergence threshold for coordinate descent algorithm1e–4
(default) | positive scalarConvergence 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 L^{2} norm by a
relative amount less than RelTol
.
Example: 'RelTol',2e–3
Data Types: single
| double
Standardize
— Flag for standardizing predictor data before fitting modelstrue
(default) | false
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
Weights
— Observation weights1/n*ones(n,1)
(default) | nonnegative vectorObservation 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
B
— Fitted coefficientsFitted 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
FitInfo
— Fit information of modelsFit information of the generalized linear models, returned as a structure with the fields described in this table.
Field in
FitInfo | Description |
---|---|
Intercept | Intercept term
β_{0} for each
linear model, a 1 -by-L
vector |
Lambda | Lambda parameters in ascending order, a
1 -by-L
vector |
Alpha | Value of the Alpha parameter, a
scalar |
DF | Number 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
If the model is cross-validated, then
the values for |
PredictorNames | Value of the PredictorNames parameter,
stored as a cell array of character vectors |
If you set the CV
name-value pair argument to
cross-validate, the FitInfo
structure contains these
additional fields.
Field in
FitInfo | Description |
---|---|
SE | Standard error of Deviance for each
Lambda , as calculated during
cross-validation, a
1 -by-L
vector |
LambdaMinDeviance | Lambda value with minimum expected
deviance, as calculated by cross-validation, a
scalar |
Lambda1SE | Largest Lambda value such that
Deviance is within one standard error
of the minimum, a scalar |
IndexMinDeviance | Index of Lambda with the value
LambdaMinDeviance , a scalar |
Index1SE | Index of Lambda with the value
Lambda1SE , a scalar |
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 Family | Default Link Function | Other Typical Link Functions |
---|---|---|
'normal' | 'identity' | |
'binomial' | 'logit' | 'comploglog' , 'loglog' ,
'probit' |
'poisson' | 'log' | |
'gamma' | 'reciprocal' | |
'inverse gaussian' | –2 |
For a nonnegative value of λ, lassoglm
solves the
problem
$$\underset{{\beta}_{0},\beta}{\mathrm{min}}\left(\frac{1}{N}\text{Deviance}\left({\beta}_{0},\beta \right)+\lambda {\displaystyle \sum _{j=1}^{p}\left|{\beta}_{j}\right|}\right).$$
The function Deviance in this equation is the deviance of the model fit to the
responses using the intercept β_{0} 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 β_{0} 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 L^{1} norm of β, as contrasted with the elastic net algorithm.
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
$${P}_{\alpha}\left(\beta \right)=\frac{(1-\alpha )}{2}{\Vert \beta \Vert}_{2}^{2}+\alpha {\Vert \beta \Vert}_{1}={\displaystyle \sum _{j=1}^{p}\left(\frac{(1-\alpha )}{2}{\beta}_{j}^{2}+\alpha \left|{\beta}_{j}\right|\right)}.$$
Elastic net is the same as lasso when α = 1. For other values of α,
the penalty term P_{α}(β)
interpolates between the L^{1} norm
of β and the squared L^{2} norm
of β. As α shrinks
toward 0, elastic net approaches ridge
regression.
[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.
To run in parallel, specify the 'Options'
name-value argument in the call
to this function and set the 'UseParallel'
field of the options
structure to true
using statset
.
For example: 'Options',statset('UseParallel',true)
For more information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).
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