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Ridge Regression

Introduction to Ridge Regression

Coefficient estimates for the models described in Linear Regression rely on the independence of the model terms. When terms are correlated and the columns of the design matrix X have an approximate linear dependence, the matrix (XTX)–1 becomes close to singular. As a result, the least-squares estimate


becomes highly sensitive to random errors in the observed response y, producing a large variance. This situation of multicollinearity can arise, for example, when data are collected without an experimental design.

Ridge regression addresses the problem by estimating regression coefficients using


where k is the ridge parameter and I is the identity matrix. Small positive values of k improve the conditioning of the problem and reduce the variance of the estimates. While biased, the reduced variance of ridge estimates often result in a smaller mean square error when compared to least-squares estimates.

The Statistics and Machine Learning Toolbox™ function ridge carries out ridge regression.

Ridge Regression

This example shows how to perform ridge regression.

Load the data in acetylene.mat, with observations of the predictor variables x1, x2, x3, and the response variable y.

load acetylene

Plot the predictor variables against each other.

grid on
axis square

grid on
axis square

grid on
axis square

Figure contains 3 axes objects. axes object 1 with xlabel x1, ylabel x2 contains a line object which displays its values using only markers. axes object 2 with xlabel x1, ylabel x3 contains a line object which displays its values using only markers. axes object 3 with xlabel x2, ylabel x3 contains a line object which displays its values using only markers.

Note the correlation between x1 and the other two predictor variables.

Use ridge and x2fx to compute coefficient estimates for a multilinear model with interaction terms, for a range of ridge parameters.

X = [x1 x2 x3];
D = x2fx(X,'interaction');
D(:,1) = []; % No constant term
k = 0:1e-5:5e-3;
betahat = ridge(y,D,k);

Plot the ridge trace.

ylim([-100 100])
grid on 
xlabel('Ridge Parameter') 
ylabel('Standardized Coefficient') 
title('{\bf Ridge Trace}') 

Figure contains an axes object. The axes object with title blank Ridge blank Trace, xlabel Ridge Parameter, ylabel Standardized Coefficient contains 6 objects of type line. These objects represent x1, x2, x3, x1x2, x1x3, x2x3.

The estimates stabilize to the right of the plot. Note that the coefficient of the x2x3 interaction term changes sign at a value of the ridge parameter 5×10-4.

See Also

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