LassoBT: Fit linear models with interactions using the Lasso.

Description

Computes a number of Lasso solution paths with increasing numbers of interactions present in the design matrices corresponding to each path. Previous paths are used to speed up computation of subsequent paths so the process is very fast.

Usage

LassoBT(
  x,
  y,
  nlambda = 100L,
  iter_max = 1L,
  lambda.min.ratio = ifelse(nobs < nvars, 0.01, 1e-04),
  lambda = NULL,
  thresh = 1e-07,
  verbose = FALSE,
  inter_orig
)

Value

An object with S3 class "BT".

call: the call that produced the object
a0: list of intercept vectors
beta: list of matrices of coefficients stored in sparse column format (CsparseMatrix)
fitted: list of fitted values
lambda: the sequence of lambda values used
nobs: the number of observations
nvars: the number of variables
var_indices: the indices of the non-constant columns of the design matrix
interactions: a 2-row matrix with columns giving the interactions that were added to the design matrix
path_lookup: a matrix with columns corresponding to iterations and rows to lambda values. Entry \(ij\) gives the component of the a0 and beta lists that gives the coefficients for the \(i\)th lambda value and \(j\)th iteration
l_start: a vector with component entries giving the minimimum lambda index in the corresponding copmonents of beta and a0

Arguments

x: input matrix of dimension nobs by nvars; each row is an observation vector.
y: response variable; shoud be a numeric vector.
nlambda: the number of lambda values. Must be at least 3.
iter_max: the number of iterations of the Backtracking algorithm to run. iter_max=1 corresponds to a single lasso or elasticnet fit. Values greater than 1 will fit interactions.
lambda.min.ratio: smallest value in lambda as a fraction of the largest value at which all main effects coefficients are 0.
lambda: user supplied lambda sequence of decreasing penalty parameters. Typical usage is to allow the function to compute its own lambda sequence. Inappropriate sequences may cause convergence problems.
thresh: convergence threshold for coordinate descent. Each inner coordinate descent loop continues until either the maximum change in the objective after any coefficient update is less than thresh or 1E5 iterations have been performed.
verbose: if TRUE will print iteration numbers.
inter_orig: an optional 2-row matrix with each column giving interactions that are to be added to the design matrix before the algorithm begins.

Details

The Lasso optimisations are performed using coordinate descent similarly to the glmnet package. An intercept term is always included. Variables are centred and scaled to have equal empirical variance. Interactions are constructed from these centred and scaled variables, and the interactions themselves are also centred and scaled. Note the coefficients are returned on the original scale of the variables. Coefficients returned for interactions are for simple pointwise products of the original variables with no scaling.

References

Shah, R. D. (2016) Shah, R. D. (2016) Modelling interactions in high-dimensional data with Backtracking. JMLR, 17, 1-31 https://www.jmlr.org/papers/volume17/13-515/13-515.pdf

Examples

Run this code

x <- matrix(rnorm(100*250), 100, 250)
y <- x[, 1] + x[, 2] - x[, 1]*x[, 2] + x[, 3] + rnorm(100)
out <- LassoBT(x, y, iter_max=10)

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