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.
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.
-
sparseLTS(): Sparse least trimmed squares regression. -
rlars(): Robust least angle regression. -
grplars()andrgrplars(): (Robust) groupwise least angle regression. -
tslars()andrtslars(): (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.
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")
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.
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.
Examples of the optimality criterion plot (left) and the regression
diagnostic plot (right) for output of function sparseLTS().
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 ~ ., data = TopGear, sMax = 15,
crit = "BIC", seed = 20210507)
fit
##
## Call:
## rgrplars(formula = MPG ~ ., data = TopGear, sMax = 15, crit = "BIC",
## seed = 20210507)
##
## Sequence of moves:
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
## Group 6 4 8 1 10 5 12 13 9 18 27 16 28 17 26
##
## Coefficients of optimal submodel:
## (Intercept) FuelPetrol Displacement DriveWheelFront DriveWheelRear
## 149.512783142 -12.905795146 -0.003968404 5.374692058 0.612416948
## BHP Acceleration TopSpeed Weight Width
## 0.015265327 0.530554736 -0.191667161 -0.001817037 -0.013641306
## Height
## -0.029560052
##
## Optimal step: 9
The output prints information on the sequence of predictor groups, as well as the results of the final model fit. Here, 9 predictor groups consisting of 10 individual covariates are selected into the final model.
When the optimal model fit is selected via BIC, each submodel along the
sequence is estimated on the full data set. In this case, a plot of the
coefficient path along the sequence can be produced via the function
coefPlot(). Functions critPlot() and diagnosticPlot() are again
available to produce similar plots as in the previous example.
Examples of the coefficient plot (left), the optimality criterion plot
(center), and the regression diagnostic plot (right) for output of
function rgrplars().
If you experience any bugs or issues or if you have any suggestions for additional features, please submit an issue via the Issues tab of this repository. Please have a look at existing issues first to see if your problem or feature request has already been discussed.
If you want to contribute to the package, you can fork this repository and create a pull request after implementing the desired functionality.
If you need help using the package, or if you are interested in collaborations related to this project, please get in touch with the package maintainer.
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. DOI 10.1214/12-AOAS575.
Alfons, A., Croux, C. and Gelper, S. (2016) Robust groupwise least angle regression. Computational Statistics & Data Analysis, 93, 421–435. DOI 10.1016/j.csda.2015.02.007.
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. DOI 10.1198/016214507000000950.
Tibshirani, R. (1996) Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58(1), 267–288. DOI 10.1111/j.2517-6161.1996.tb02080.x.