robustHD: Robust Methods for High-Dimensional Data

To cite package robustHD in publications, please use:

A. Alfons (2021). robustHD: An R package for robust regression with high-dimensional data. Journal of Open Source Software, 6(67), 3786. DOI 10.21105/joss.03786.

Summary

In regression analysis with high-dimensional data, variable selection is an important step to (i) overcome computational problems, (ii) improve prediction performance by variance reduction, and (iii) increase interpretability of the resulting models due to the smaller number of variables. However, robust methods are necessary to prevent outlying data points from distorting the results. The add-on package robustHD for the statistical computing environment R provides functionality for robust linear regression and model selection with high-dimensional data. More specifically, the implemented functionality includes robust least angle regression (Khan et al., 2007), robust groupwise least angle regression (Alfons et al., 2016), as well as sparse least trimmed squares regression (Alfons et al., 2013). The latter can be seen as a trimmed version of the popular lasso regression estimator (Tibshirani, 1996). Selecting the optimal model can be done via cross-validation or an information criterion, and various plots are available to illustrate model selection and to evaluate the final model estimates. Furthermore, the package includes functionality for pre-processing such as robust standardization and winsorization. Finally, robustHD follows a clear object-oriented design and takes advantage of C++ code and parallel computing to reduce computing time.

Main functionality

• sparseLTS(): Sparse least trimmed squares regression.

• rlars(): Robust least angle regression.

• grplars() and rgrplars(): (Robust) groupwise least angle regression.

• tslars() and rtslars(): (Robust) least angle regression for time series data.

• corHuber(): Robust correlation based on winsorization.

• winsorize(): Winsorization of the data.

• robStandardize(): Robust standardization of the data with given functions for computing center and scale. By default, the median and MAD are used.

Installation

Package robustHD is on CRAN (The Comprehensive R Archive Network), hence the latest release can be easily installed from the R command line via

install.packages("robustHD")

Building from source

To install the latest (possibly unstable) development version from GitHub, you can pull this repository and install it from the R command line via

install.packages("devtools")
devtools::install_github("aalfons/robustHD")

If you already have package devtools installed, you can skip the first line. Moreover, package robustHD contains C++ code that needs to be compiled, so you may need to download and install the necessary tools for MacOS or the necessary tools for Windows.

Example: Sparse least trimmed squares regression

The well-known NCI-60 cancer cell panel is used to illustrate the functionality for sparse least trimmed squares regression. The protein expressions for a specific protein are selected as the response variable, and the gene expressions of the 100 genes that have the highest (robustly estimated) correlations with the response variable are screened as candidate predictors.

# load package and data
library("robustHD")
data("nci60")  # contains matrices 'protein' and 'gene'

# define response variable
y <- protein[, 92]
# screen most correlated predictor variables
correlations <- apply(gene, 2, corHuber, y)
keep <- partialOrder(abs(correlations), 100, decreasing = TRUE)
X <- gene[, keep]

Sparse least trimmed squares is a regularized estimator of the linear regression model, whose results depend on a non-negative regularization parameter [see Alfons et al., 2013]. In general, a larger value of this regularization parameter yields more regression coefficients being set to zero, which can be seen as a form of variable selection.

For convenience, sparseLTS() can internally estimate the smallest value of the regularization parameter that sets all coefficients to zero. With mode = "fraction", the values supplied via the argument lambda are then taken as fractions of this estimated value (i.e., they are multiplied with the internally estimated value). In this example, the optimal value of the the regularization parameter is selected by estimating the prediction error (crit = "PE") via 5-fold cross-validation with one replication (splits = foldControl(K = 5, R = 1)). The default prediction loss function is the root trimmed mean squared prediction error. Finally, the seed of the random number generator is supplied for reproducibility.

# fit sparse least trimmed squares regression and print results
lambda <- seq(0.01, 0.5, length.out = 10)
fit <- sparseLTS(X, y, lambda = lambda, mode = "fraction", crit = "PE",
                 splits = foldControl(K = 5, R = 1), seed = 20210507)
fit

## 
## 5-fold CV results:
##         lambda reweighted       raw
## 1  0.347703682  1.4572964 1.6215821
## 2  0.309842614  1.3157592 1.5232998
## 3  0.271981547  1.1783524 1.4194223
## 4  0.234120479  1.0837333 1.3122837
## 5  0.196259412  1.0137302 1.2284962
## 6  0.158398344  0.9490259 1.1709616
## 7  0.120537276  0.8450414 1.0669884
## 8  0.082676209  0.7736669 0.9138857
## 9  0.044815141  0.7119135 0.7244448
## 10 0.006954074  0.7733818 0.7733818
## 
## Optimal lambda:
## reweighted        raw 
## 0.04481514 0.04481514 
## 
## Final model:
## 
## Call:
## sparseLTS(x = X, y = y, lambda = 0.0448151412422958)
## 
## Coefficients:
##  (Intercept)         8502        21786          134         4454         1106 
## -3.709498642  0.593549132  0.033366829  0.115955965  0.015899659  0.020447909 
##        20125         8510        14785        17400         8460         8120 
## -0.091451556  0.111369625 -0.014556471  0.002262256 -0.003669024  0.112165149 
##        18447        15622         7696         5550        16784        13547 
## -0.229292900 -0.008785651  0.020915212  0.005880150  0.015398316  0.037262275 
##                                    
## Penalty parameter:       0.04481514
## Residual scale estimate: 0.62751742

Among other information, the output prints the results of the final model fit, which here consists of 17 genes with non-zero coefficients.

When selecting the optimal model fit by estimating the prediction error, the final model estimate on the full data is computed only with the optimal value of the regularization parameter instead of the full grid. For visual inspection of the results, function critPlot() plots the values of the optimality criterion (in this example, the root trimmed mean squared error) against the values of the regularization parameter. Moreover, function diagnosticPlot() allows to produce various diagnostic plots for the optimal model fit.

<img src="./inst/doc/paper/figure_sparseLTS-1.svg" width="67%" style="display: block; margin: auto;" />

Examples of the optimality criterion plot (left) and the regression diagnostic plot (right) for output of function sparseLTS().

Example: Robust groupwise least angle regression

Robust least angle regression (Khan et al., 2007) and robust groupwise least angle regression (Alfons et al., 2016) follow a hybrid model selection strategy: first obtain a sequence of important candidate predictors, then fit submodels along that sequence via robust regressions. Here, data on cars featured in the popular television show Top Gear are used to illustrate this functionality.

The response variable is fuel consumption in miles per gallon (MPG), with all remaining variables used as candidate predictors. Information on the car model is first removed from the data set, and the car price is log-transformed. In addition, only observations with complete information are used in this illustrative example.

# load package and data
library("robustHD")
data("TopGear")

# keep complete observations and remove information on car model
keep <- complete.cases(TopGear)
TopGear <- TopGear[keep, -(1:3)]
# log-transform price
TopGear$Price <- log(TopGear$Price)

As the Top Gear data set contains several categorical variables, robust groupwise least angle regression is used. Through the formula interface, function rgrplars() by default takes each categorical variable (factor) as a group of dummy variables while all remaining variables are taken individually. However, the group assignment can be defined by the user through argument assign. The maximum number of candidate predictor groups to be sequenced is determined by argument sMax. Furthermore, with crit = "BIC", the optimal submodel along the sequence is selected via the Bayesian information criterion (BIC). Note that each submodel along the sequence is fitted using a robust regression estimator with a non-deterministic algorithm, hence the seed of the random number generator is supplied for reproducibility.

# fit robust groupwise least angle regression and print results
fit <- rgrplars(MPG ~ ., d