A Two-Stage Regularization Method for Variable Selection and Forecasting in High-Order Interaction Model

Algorithm 1

The TSRRHR algorithm.

Inputs: the dataset

: maximum number of iterations used in the SRHR algorithm of the first stage;

: maximum number of iterations used in the SRHR algorithm of the second stage;

: the error tolerance used in the SRHR algorithm of the first stage;

: the error tolerance used in the SRHR algorithm of the second stage;

: scale parameter used in the SRHR algorithm of the first stage;

: scale parameter used in the SRHR algorithm of the second stage;

Output: The forecasting test error and selected pattern;

Randomly divide the original data into the training dataset and

test dataset .

The first stage using SRHR algorithm:

Generate grid values of and .

forto

forto

Initialization: ,

Scaling:

while or do

Step 1.

Step 2.

Step 3.

Step 4:

end while

end for

end for

Obtain the solution path and the corresponding sparsity patterns based on

EBIC criterion.

The second stage using SRHR algorithm:

Generate the high-order interaction model and based

on the sparsity pattern .

Generate grid values of and .

forto

forto

Initialization:

Scaling:

while or do

Step 1.

Step 2.

Step 3.

Step 4:

end while

end for

end for

Obtain the solution path and update the sparsity patterns using

HDBIC criterion.

Calculate the test error using test dataset