lambda0: Penalty parameter for sparse LTS regression

Description

Use bivariate winsorization to estimate the smallest value of the penalty parameter for sparse least trimmed squares regression that sets all coefficients to zero.

Usage

lambda0(
  x,
  y,
  normalize = TRUE,
  intercept = TRUE,
  const = 2,
  prob = 0.95,
  tol = .Machine$double.eps^0.5,
  eps = .Machine$double.eps,
  ...
)

Value

A robust estimate of the smallest value of the penalty parameter for sparse LTS regression that sets all coefficients to zero.

Arguments

x: a numeric matrix containing the predictor variables.
y: a numeric vector containing the response variable.
normalize: a logical indicating whether the winsorized predictor variables should be normalized to have unit \(L_{2}\) norm (the default is TRUE).
intercept: a logical indicating whether a constant term should be included in the model (the default is TRUE).
const: numeric; tuning constant to be used in univariate winsorization (defaults to 2).
prob: numeric; probability for the quantile of the \(\chi^{2}\) distribution to be used in bivariate winsorization (defaults to 0.95).
tol: a small positive numeric value used to determine singularity issues in the computation of correlation estimates for bivariate winsorization (see corHuber).
eps: a small positive numeric value used to determine whether the robust scale estimate of a variable is too small (an effective zero).
...: additional arguments to be passed to robStandardize.

Author

Andreas Alfons

Details

The estimation procedure is inspired by the calculation of the respective penalty parameter in the first step of the classical LARS algorithm. First, two-dimensional data blocks consisting of the response with each predictor variable are cleaned via bivariate winsorization. For each block, the following computations are then performed. If an intercept is included in the model, the cleaned response is centered and the corresponding cleaned predictor is centered and scaled to have unit norm. Otherwise the variables are not centered, but the predictor is scaled to have unit norm. Finally, the dot product of the response and the corresponding predictor is computed. The largest absolute value of those dot products, rescaled to fit the parametrization of the sparse LTS definition, yields the estimate of the smallest penalty parameter that sets all coefficients to zero.

References

Alfons, A., Croux, C. and Gelper, S. (2013) Sparse least trimmed squares regression for analyzing high-dimensional large data sets. The Annals of Applied Statistics, 7(1), 226--248. tools:::Rd_expr_doi("10.1214/12-AOAS575")

Efron, B., Hastie, T., Johnstone, I. and Tibshirani, R. (2004) Least angle regression. The Annals of Statistics, 32(2), 407--499. tools:::Rd_expr_doi("10.1214/009053604000000067")

Khan, J.A., Van Aelst, S. and Zamar, R.H. (2007) Robust linear model selection based on least angle regression. Journal of the American Statistical Association, 102(480), 1289--1299. tools:::Rd_expr_doi("10.1198/016214507000000950")

Examples

Run this code

## generate data
# example is not high-dimensional to keep computation time low
library("mvtnorm")
set.seed(1234)  # for reproducibility
n <- 100  # number of observations
p <- 25   # number of variables
beta <- rep.int(c(1, 0), c(5, p-5))  # coefficients
sigma <- 0.5      # controls signal-to-noise ratio
epsilon <- 0.1    # contamination level
Sigma <- 0.5^t(sapply(1:p, function(i, j) abs(i-j), 1:p))
x <- rmvnorm(n, sigma=Sigma)    # predictor matrix
e <- rnorm(n)                   # error terms
i <- 1:ceiling(epsilon*n)       # observations to be contaminated
e[i] <- e[i] + 5                # vertical outliers
y <- c(x %*% beta + sigma * e)  # response
x[i,] <- x[i,] + 5              # bad leverage points

## estimate smallest value of the penalty parameter 
## that sets all coefficients to 0
lambda0(x, y)

Run the code above in your browser using DataCamp Workspace