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Ridge regression

Syntax

b = ridge(y,X,k)
b = ridge(y,X,k,scaled)

Description

b = ridge(y,X,k) returns a vector b of coefficient estimates for a multilinear ridge regression of the responses in y on the predictors in X. X is an n-by-p matrix of p predictors at each of n observations. y is an n-by-1 vector of observed responses. k is a vector of ridge parameters. If k has m elements, b is p-by-m. By default, b is computed after centering and scaling the predictors to have mean 0 and standard deviation 1. The model does not include a constant term, and X should not contain a column of 1s.

b = ridge(y,X,k,scaled) uses the {0,1}-valued flag scaled to determine if the coefficient estimates in b are restored to the scale of the original data. ridge(y,X,k,0) performs this additional transformation. In this case, b contains p+1 coefficients for each value of k, with the first row corresponding to a constant term in the model. ridge(y,X,k,1) is the same as ridge(y,X,k). In this case, b contains p coefficients, without a coefficient for a constant term.

The relationship between b0 = ridge(y,X,k,0) and b1 = ridge(y,X,k,1) is given by

       m = mean(X);
       s = std(X,0,1)';
       b1_scaled = b1./s;
       b0 = [mean(y)-m*temp; b1_scaled]

This can be seen by replacing the x_i (i = ۱, …, n) in the multilinear model y = b₀⁰ + b₁⁰x₁ + … + b_n⁰x_n with the z-scores z_i = (x_i – μ_i)/σ_i , and replacing y with y – μ_y.

In general, b1 is more useful for producing plots in which the coefficients are to be displayed on the same scale, such as a ridge trace (a plot of the regression coefficients as a function of the ridge parameter). b0 is more useful for making predictions.

Coefficient estimates for multiple linear regression models 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 (X^TX)^–۱ 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.

Example

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:

subplot(1,3,1)
plot(x1,x2,'.')
xlabel('x1'); ylabel('x2'); grid on; axis square

subplot(1,3,2)
plot(x1,x3,'.')
xlabel('x1'); ylabel('x3'); grid on; axis square

subplot(1,3,3)
plot(x2,x3,'.')
xlabel('x2'); ylabel('x3'); grid on; axis square

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;
b = ridge(y,D,k);

Plot the ridge trace:

figure
plot(k,b,'LineWidth',2)
ylim([-100 100])
grid on 
xlabel('Ridge Parameter') 
ylabel('Standardized Coefficient') 
title('{\bf Ridge Trace}') 
legend('constant','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 ≈ ۵ × ۱۰^–۴.