Sparse least trimmed squares regression

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Description

Compute least trimmed squares regression with an L1 penalty on the regression coefficients, which allows for sparse model estimates.

Usage

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sparseLTS(x, ...)

## S3 method for class 'formula'
sparseLTS(formula, data, ...)

## Default S3 method:
sparseLTS(x, y, lambda, mode = c("lambda", "fraction"),
  alpha = 0.75, normalize = TRUE, intercept = TRUE, nsamp = c(500, 10),
  initial = c("sparse", "hyperplane", "random"), ncstep = 2,
  use.correction = TRUE, tol = .Machine$double.eps^0.5,
  eps = .Machine$double.eps, use.Gram, crit = c("BIC", "PE"),
  splits = foldControl(), cost = rtmspe, costArgs = list(),
  selectBest = c("hastie", "min"), seFactor = 1, ncores = 1, cl = NULL,
  seed = NULL, model = TRUE, ...)

Arguments

x

a numeric matrix containing the predictor variables.

formula

a formula describing the model.

data

an optional data frame, list or environment (or object coercible to a data frame by as.data.frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which sparseLTS is called.

y

a numeric vector containing the response variable.

lambda

a numeric vector of non-negative values to be used as penalty parameter.

mode

a character string specifying the type of penalty parameter. If "lambda", lambda gives the grid of values for the penalty parameter directly. If "fraction", the smallest value of the penalty parameter that sets all coefficients to 0 is first estimated based on bivariate winsorization, then lambda gives the fractions of that estimate to be used (hence all values of lambda should be in the interval [0,1] in that case).

alpha

a numeric value giving the percentage of the residuals for which the L1 penalized sum of squares should be minimized (the default is 0.75).

normalize

a logical indicating whether the predictor variables should be normalized to have unit L2 norm (the default is TRUE). Note that normalization is performed on the subsamples rather than the full data set.

intercept

a logical indicating whether a constant term should be included in the model (the default is TRUE).

nsamp

a numeric vector giving the number of subsamples to be used in the two phases of the algorithm. The first element gives the number of initial subsamples to be used. The second element gives the number of subsamples to keep after the first phase of ncstep C-steps. For those remaining subsets, additional C-steps are performed until convergence. The default is to first perform ncstep C-steps on 500 initial subsamples, and then to keep the 10 subsamples with the lowest value of the objective function for additional C-steps until convergence.

initial

a character string specifying the type of initial subsamples to be used. If "sparse", the lasso fit given by three randomly selected data points is first computed. The corresponding initial subsample is then formed by the fraction alpha of data points with the smallest squared residuals. Note that this is optimal from a robustness point of view, as the probability of including an outlier in the initial lasso fit is minimized. If "hyperplane", a hyperplane through p randomly selected data points is first computed, where p denotes the number of variables. The corresponding initial subsample is then again formed by the fraction alpha of data points with the smallest squared residuals. Note that this cannot be applied if p is larger than the number of observations. Nevertheless, the probability of including an outlier increases with increasing dimension p. If "random", the initial subsamples are given by a fraction alpha of randomly selected data points. Note that this leads to the largest probability of including an outlier.

ncstep

a positive integer giving the number of C-steps to perform on all subsamples in the first phase of the algorithm (the default is to perform two C-steps).

use.correction

currently ignored. Small sample correction factors may be added in the future.

tol

a small positive numeric value giving the tolerance for convergence.

eps

a small positive numeric value used to determine whether the variability within a variable is too small (an effective zero).

use.Gram

a logical indicating whether the Gram matrix of the explanatory variables should be precomputed in the lasso fits on the subsamples. If the number of variables is large, computation may be faster when this is set to FALSE. The default is to use TRUE if the number of variables is smaller than the number of observations in the subsamples and smaller than 100, and FALSE otherwise.

crit

a character string specifying the optimality criterion to be used for selecting the final model. Possible values are "BIC" for the Bayes information criterion and "PE" for resampling-based prediction error estimation.

splits

an object giving data splits to be used for prediction error estimation (see perryTuning).

cost

a cost function measuring prediction loss (see perryTuning for some requirements). The default is to use the root trimmed mean squared prediction error (see cost).

costArgs

a list of additional arguments to be passed to the prediction loss function cost.

selectBest,seFactor

arguments specifying a criterion for selecting the best model (see perryTuning). The default is to use a one-standard-error rule.

ncores

a positive integer giving the number of processor cores to be used for parallel computing (the default is 1 for no parallelization). If this is set to NA, all available processor cores are used. For prediction error estimation, parallel computing is implemented on the R level using package parallel. Otherwise parallel computing is implemented on the C++ level via OpenMP (http://openmp.org/).

cl

a parallel cluster for parallel computing as generated by makeCluster. This is preferred over ncores for prediction error estimation, in which case ncores is only used on the C++ level for computing the final model.

seed

optional initial seed for the random number generator (see .Random.seed). On parallel R worker processes for prediction error estimation, random number streams are used and the seed is set via clusterSetRNGStream.

model

a logical indicating whether the data x and y should be added to the return object. If intercept is TRUE, a column of ones is added to x to account for the intercept.

...

additional arguments to be passed down.

Value

If crit is "PE", an object of class "perrySparseLTS" (inheriting from class "perryTuning", see perryTuning). It contains information on the prediction error criterion, and includes the final model with the optimal tuning paramter as component finalModel.

Otherwise an object of class "sparseLTS" with the following components:

lambda

a numeric vector giving the values of the penalty parameter.

best

an integer vector or matrix containing the respective best subsets of h observations found and used for computing the raw estimates.

objective

a numeric vector giving the respective values of the sparse LTS objective function, i.e., the L1 penalized sums of the h smallest squared residuals from the raw fits.

coefficients

a numeric vector or matrix containing the respective coefficient estimates from the reweighted fits.

fitted.values

a numeric vector or matrix containing the respective fitted values of the response from the reweighted fits.

residuals

a numeric vector or matrix containing the respective residuals from the reweighted fits.

center

a numeric vector giving the robust center estimates of the corresponding reweighted residuals.

scale

a numeric vector giving the robust scale estimates of the corresponding reweighted residuals.

cnp2

a numeric vector giving the respective consistency factors applied to the scale estimates of the reweighted residuals.

wt

an integer vector or matrix containing binary weights that indicate outliers from the respective reweighted fits, i.e., the weights are 1 for observations with reasonably small reweighted residuals and 0 for observations with large reweighted residuals.

df

an integer vector giving the respective degrees of freedom of the obtained reweighted model fits, i.e., the number of nonzero coefficient estimates.

intercept

a logical indicating whether the model includes a constant term.

alpha

a numeric value giving the percentage of the residuals for which the L1 penalized sum of squares was minimized.

quan

the number h of observations used to compute the raw estimates.

raw.coefficients

a numeric vector or matrix containing the respective coefficient estimates from the raw fits.

raw.fitted.values

a numeric vector or matrix containing the respective fitted values of the response from the raw fits.

raw.residuals

a numeric vector or matrix containing the respective residuals from the raw fits.

raw.center

a numeric vector giving the robust center estimates of the corresponding raw residuals.

raw.scale

a numeric vector giving the robust scale estimates of the corresponding raw residuals.

raw.cnp2

a numeric value giving the consistency factor applied to the scale estimate of the raw residuals.

raw.wt

an integer vector or matrix containing binary weights that indicate outliers from the respective raw fits, i.e., the weights used for the reweighted fits.

crit

an object of class "bicSelect" containing the BIC values and indicating the final model (only returned if argument crit is "BIC" and argument lambda contains more than one value for the penalty parameter).

x

the predictor matrix (if model is TRUE).

y

the response variable (if model is TRUE).

call

the matched function call.

Note

Package robustHD has a built-in back end for sparse least trimmed squares using the C++ library Armadillo. Another back end is available through package sparseLTSEigen, which uses the C++ library Eigen. The latter is faster, currently does not work on 32-bit R for Windows.

For both C++ back ends, parallel computing is implemented via OpenMP (http://openmp.org/).

Author(s)

Andreas Alfons

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.

See Also

coef, fitted, plot, predict, residuals, wt, ltsReg

Examples

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## 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

## fit sparse LTS model for one value of lambda
sparseLTS(x, y, lambda = 0.05, mode = "fraction")

## fit sparse LTS models over a grid of values for lambda
frac <- seq(0.2, 0.05, by = -0.05)
sparseLTS(x, y, lambda = frac, mode = "fraction")

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