ridgeFit¶

Purpose¶

Fit a linear model with an L1 penalty.

Format¶

mdl = ridgeFit(y, X, lambda)¶

Parameters:

y (Nx1 vector) – The target, or dependent variable.
X (NxP matrix) – The model features, or independent variables.
lambda (Scalar, or Kx1 vector) – The L1 penalty parameter(s).

Returns:

mdl (struct) –

An instance of a ridgeModel structure. An instance named mdl will have the following members:

mdl.alpha_hat	(1 x nlambdas vector) The estimated value for the intercept for each provided lambda.
mdl.beta_hat	(P x nlambdas matrix) The estimated parameter values for each provided lambda.
mdl.mse_train	(nlambdas x 1 vector) The mean squared error for each set of parameters, computed on the training set.
mdl.lambda	(nlambdas x 1 vector) The lambda values used in the estimation.

Examples¶

Example 1: Basic Estimation and Prediction¶

new;

library gml;

// Specify dataset with full path
dataset = getGAUSSHome() $+ "pkgs/gml/examples/qsar_fish_toxicity.csv";

// Load dependent and independent variable
y = loadd(dataset, "LC50");
X = loadd(dataset, ". -LC50");

// Split data into training sets
y_test = y[1:636];
X_test = X[1:636,.];
y_train = y[637:rows(y)];
X_train = X[637:rows(X),.];

// Declare 'mdl' to be an instance of a
// ridgeModel structure to hold the estimation results
struct ridgeModel mdl;

lambda = seqm(90, 0.8, 60);

// Estimate the model with default settings
mdl = ridgeFit(y_train, X_train, lambda);

After the above code, mdl.beta_hat will be a \(6 \times 60\) matrix, where each column contains the estimates for a different lambda value. The graph below shows the path of the parameter values as the value of lambda changes.

_images/ridge-fit-example-1-beta-path.jpg

Continuing with our example, we can make test predictions like this:

// Make predictions on the test set
y_hat = X_test * mdl.beta_hat + mdl.alpha_hat;

After the above code, y_hat will be a matrix with the same number of observations as y_test. However, it will have one column for each value of lambda used in the estimation. We can compute the mean-squared error (MSE) for each of our predictions with the following code:

// Compute MSE for each prediction
mse_test = meanc((y_test - y_hat).^2);

Below is a plot of the change in MSE with the changes in lambda.

_images/ridge-fit-example-1-mse-path.jpg

Remarks¶

Each variable (column of X) is centered to have a mean of 0 and scaled to have unit length, (i.e. the vector 2-norm of each column of X is equal to 1).