enetLTS: Robust and Sparse Methods for High Dimensional Linear and Binary and Multinomial Regression

Summary

enetLTS is an R package that provides a fully robust version of elastic net estimator for high dimensional linear, binary, and multinomial regression. The elastic net penalization provides intrinsic variable selection and coefficient estimates for highly correlated variables, in particular for high-dimensional low sample size data sets, and it has been extended to generalized linear regression models (Friedman et al., 2010). Combining these advantages with trimming idea yields the robust solutions. The main idea of the algorithm is to search for outlier-free subsets on which the classical elastic net estimator can be applied. Outlier-free subsets are determined by trimming the penalized log-likelihood function for considered regression model. The algorithm starts with 500 elemental subsets only for one combination of elastic net parameters $\alpha$ and $\lambda$, and takes the warm start strategy for subsequent combinations in order to save the computation time. The final reweighting step is added to improve the statistical efficiency of the proposed methods. From this point of view, the enet-LTS estimator can be seen as trimmed version of the elastic net regression estimator for linear, binary and multinomial regression (Friedman et al., 2010). Selecting model with optimal tuning parameters is done via cross-validation, and various plots are available to illustrate model and to evaluate the final model estimates.

Implemented Methods

• enetLTS(): elastic net trimmed squared regression for families:

1- gaussian

2- binomial

3- multinomial

Installation

Package enetLTS is on CRAN (The Comprehensive R Archive Network), hence the latest release can be easily installed from the R command as follows

{R, eval = TRUE} install.packages("enetLTS")

Building from source

The package enetLTS can also installed from Github using following the R command line

{R, eval = TRUE} install.packages("devtools") devtools::install_github("fatmasevinck/enetLTS")

If you already have package devtools installed, the first line can be skipped.

Example: Robust and Sparse Linear Regression (family="gaussian")

We have considered the NCI-60 cancer cell panel data in order to provide an example for the enetLTS model. NCI-60 data includes 60 human cancer cell lines with nine cancer types, which are breast, central nervous system, colon, leukemia, lung, melanoma,ovearian, prostate and renal cancers. In this example, we regress the protein espression on gene expression data. Using the Affymetrix HG-U133A chip and normalizing with the GCRMA method, the number of predictors is obtained as 22,283. Since one observation has all missing values, it is omitted. The number of observations is decreased to 59. This data set is available in package robustHD.

As in Alfons (Alfons, 2021), we determine the response variable with one of the protein expressions, which is 92th protein. Out of the gene expressions of the 22,283 genes for predictors, we have considered the gene expressions of the 100 genes that have the highest (robustly estimated) correlations with the response variable. The code lines for loading and re-organizing response variable and predictors are follows:

```{R, eval = TRUE}

load 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] ```

The package enetLTS can either be installed from CRAN or directly from Github. The main function is enetLTS, and the default family option is gaussian, which corresponds to linear regression.

```{R}

install and load package

install.packages("enetLTS") library(enetLTS)

fit the model for family="gaussian"

set.seed(1) fit.gaussian <- enetLTS(X,y)

[1] "optimal model: lambda = 0.1043 alpha = 0.8"

fit.gaussian ```

```{R}

enetLTS estimator

Call: enetLTS(xx = X, yy = y, crit.plot = TRUE)

number of the nonzero coefficients:

[1] 23

alpha: 0.8

lambda: 0.1043

lambdaw: 0.1824974

```

The main idea to obtain an outlier-free subset is to carry out concentration steps (C-steps). This means that in each iteration of the algorithm, the value of the objective function improves. Thus, one has to start with several initial subsets, and the C-steps will lead at least to a local optimum.

For the argument hsize one needs to provide a numeric value with the trimming percentage used in the penalized objective function. The default value is 0.75. The argument nsamp is a numeric vector: The first element gives the number of initial subsamples to be used. The second element gives the number of subsamples to keep after a number of C-steps has been performed. For those remaining subsets, additional C-steps are performed until convergence. The default is to start the C-steps with 500 initial subsamples for a first combination of tuning parameters $\alpha$ and $\lambda$, and then to keep the 10 subsamples with the lowest value of the objective function for additional C-steps until convergence. For the next combination of tuning parameters $\alpha$ and $\lambda$, the algorithm makes use of the $warm start$ idea, which means that the best subset of the neighboring grid value is taken, and C-steps are started from this best subset until convergence. The nsamp entries can also be supplied by the user. These arguments are the same for the other family options.

The main function enetLTS() allows the user to specify a sequence of values for $\alpha$ for the elastic net penalty. If this is not provided, a default sequence of 41 equally spaced values between 0 and 1 is taken. For the other tuning parameter $\lambda$ that keeps the strength of the elastic net penalty, a user supplied sequence is available. If not provided, the default for family="gaussian" is chosen with steps of size -0.025 lambda0 with $0 \le \lambda \le$ lambda0, where lambda0 is determined as in Alfons, 2021.

After computing all candidates based on the best subsets for certain combinations of $\alpha$ and $\lambda$, the combination of the optimal tuning parameters is defined by 5-fold cross-validation. The evaluation criterion for 5-fold cross-validation is summarized by a heatmap, see Figure \ref{fig:hatmapGauss}, if the argument crit.plot is assigned to "TRUE".

To determine updated parameter $\lambda$ (lambdaw) in a reweighting step, 5-fold cross-validation based on the cv.glmnet() function from glmnet (Friedman et al., 2021) is used.

Several plots are available for the results. plotCoef.enetLTS() visualizes the coefficients where the coefficients which are set to zeros are shown, plotResid.enetLTS() plots the values of residuals vs fitted values, and plotDiagnostic.enetLTS() allows to produce various diagnostic plots for the final model fit. Some examples of these plots are shown in Figure \ref{fig:plotexamplesGuas}.

Examples of the residuals plot (left) and the diagnostic plot (right) for output of function enetLTS() with the arguman family="gaussian".

Example: Robust and Sparse Binary Regression (family="binomial")

For binary regression, we have considered the same NCI-60 data with some modifications. In order to provide an example for binary regression, the response variable is re-organized as follows. If mean(y) is smaller than 0.5, the response will be assigned to 0, otherwise, the response will be assigned to 1. The predictors are the same as in the previous section.

```{R, eval = TRUE} y <- protein[, 92]

for binary class

y.binom <- ifelse(y <= mean(y),0,1) ```

For the binary regression, the family arguman of enetLTS() function should be set to "binomial".

```{R}

determine alpha and lambda sequences

alphas=seq(0,1,length=41) l0 <- lambda00(X, y.binom, normalize = TRUE, intercept = TRUE) lambdas <- seq(l0,0.001,by=-0.025*l0)

fit the model for family="binomial"

set.seed(12) fit.binomial <- enetLTS(X, y.binom, alphas=alphas, lambdas=lambdas, family="binomial") fit.binomial

enetLTS estimator

```

```{R}

Call: enetLTS(xx = X, yy = y.binom, family = "binomial", alphas = alphas, lambdas = lambdas)

number of the nonzero coefficients:

[1] 48

alpha: 0.325

lambda: 0.0011

lambdaw: 0.01456879

```

The main function enetLTS() provides similar options for the values of the elastic net penalty. For the tuning parameter $\lambda$, a user supplied sequence option is available. If this is not provided, the default is chosen with steps of size -0.025 lambda00 with $0 \le \lambda \le$ lambda00, where lambda00 is determined based on the robustified point-biserial correlation, see Kurnaz et al., 2018.

The evaluation criterion results for to the candidates of tuning parameters is avaliable in a heatmap if the argument crit.plot is assigned to "TRUE" (which is omitted here). To determine the updated parameter $\lambda$ (lambdaw) for the reweighting step, 5-fold cross-validation based on the cv.glmnet() function is used from the glmnet package for the current family option.

Similarly, plotCoef.enetLTS() visualizes the coefficients. The other plot functions are re-organized for binary regression. In plotResid.enetLTS(), residuals are turned into the deviances, and this plot function produces two plots which are deviances vs index, and deviances vs fitted values (link function). plotDiagnostic.enetLTS() shows the response variable vs fitted values (link function). Some of these plots are presented in Figure \ref{fig:ResidDiagbinom}.

Examples of plot functions of deviances (left); diagnostic (right) for binary regression

Example: Robust and Sparse Multinomial Regression (family="multinomial")

The fruit data set has been well-known in the context of robust discrimination studies. Therefore, we have considered the fruit data set in order to illustrate the multinomial regression. It contains spectral information with 256 wavelengths for observations from 3 different cultivars of the same fruit, named D, M, and HA, with group sizes 490, 106, and 500. This data set is available in R package rrcov and it is taken into consideration to illustrate the enetLTS model for multinomial regression.

```{R, eval = TRUE}

load data

library(rrcov) data(fruit)

d <- fruit[,-1] # first column includes the fruid names X <- as.matrix(d)

define response variable

grp <- c(rep(1,490),rep(2,106),rep(3,500)) y <- factor(grp-1) ```

With family="multinomial", the model enetLTS() produces the results of multinomial regression. Here user supplied values of lambdas are considered.

```{R, eval = TRUE} alphas=seq(from=0.01,to=0.1,by=0.01) set.seed(4) fit.multinom <- enetLTS(X, y, family="multinomial", alphas=alphas, lambdas=seq(from=0.01,to=0.1,by=0.01), crit.plot=FALSE)

[1] "optimal model: lambda = 0.01 alpha = 0.01"

fit.mutinom ```

```{R}

enetLTS estimator

Call: enetLTS(xx = X, yy = y, family = "multinomial", alphas = alphas, lambdas = lambdas, seq(from=0.01,to=0.1,by=0.01), crit.plot=FALSE)

number of the nonzero coefficients:

[1] 745

alpha: 0.01

lambda: 0.01

lambdaw: 0.004347225

```

```R

load data

library(rrcov) data(fruit)

d <- fruit[,-1] # first column includes the fruit names X <- as.matrix(d)

define response variable

grp <- c(rep(1,490),rep(2,106),rep(3,500)) y <- factor(grp-1) `` Withfamily="multinomial", the modelenetLTS()produces the results of multinomial regression. Here user supplied values oflambdas` are considered.

```R lambdas=seq(from=0.01,to=0.1,by=0.01) set.seed(4) fit.multinom <- enetLTS(X, y, family="multinomial", lambdas=lambdas, crit.plot=FALSE)

[1] "optimal model: lambda = 0.01 alpha = 0.02"

fit.mutinom ```

```R

enetLTS estimator

Call: enetLTS(xx = X, yy = y, family = "multinomial", lambdas=lambdas, crit.plot = FALSE)

number of the nonzero coefficients:

[1] 704

alpha: 0.02

lambda: 0.01

lambdaw: 0.003971358

```

The main function enetLTS() provides similar options for the $\alpha$ sequence of the elastic net penalty. The default for the tuning parameters $\lambda$ are values from 0.95 to 0.05 with steps of size -0.05, see Kurnaz and Filzmoser, 2022.

The combination of the optimal tuning parameters is evaluated by 5-fold cross-validation. A heatmap is available if the argument crit.plot is assigned to "TRUE". As for the other models, an updated tuning parameter $\lambda$ (lambdaw) for the reweighting step is obtained by the cv.glmnet() function from the package glmnet (Friedman et al., 2021).

The plot functions are adjusted for multinomial regression. plotCoef.enetLTS() gives the coefficients plots which includes group information. In plotResid.enetLTS(), residuals are turned into deviances, as in the binary regression case, with group information. plotDiagnostic.enetLTS() shows the scores of all groups in the space of the first two principal components.

Examples of plot functions of deviances (left); diagnostic (right) for multinomial regression

Especially for 'family="multinomial"', runtime is long because the algorithm is based on repeated C-steps.

References

Friedman J., Hastie T. and Tibshirani R. (2010). Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33(1), 1-22. DOI 10.1163/ej.9789004178922.i-328.7

Reinhold, W. C., Sunshine, M., Liu, H., Varma, S., Kohn, K. W., Morris, J., Doroshow, J., & Pommier, Y. (2012). CellMiner: A web-based suite of genomic and pharmacologic tools to explore transcript and drug patterns in the NCI-60 cell line set. Cancer Research, 72(14), 3499–3511. DOI 10.1158/0008-5472.can-12-1370

F. S. Kurnaz and I. Hoffmann and P. Filzmoser (2018). Robust and sparse estimation methods for high-dimensional linear and logistic regression. Chemometrics and Intelligent Laboratory Systems, 172, 211-222. DOI 10.1016/j.chemolab.2017.11.017"

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

J. Friedman and T. Hastie and R. Tibshirani and B. Narasimhan and K. Tay and N. Simon and J. Qian and J. Yang (2021). glmnet: Lasso and Elastic-Net Regularized Generalized Linear Models. R Foundation for Statistical Computing, Vienna, Austria. R package version 4.1--3 (https://CRAN.R-project.org/package=glmnet)

F. S. Kurnaz and P. Filzmoser (2022). Robust and Sparse Multinomial Regression in High Dimensions. DOI 10.48550/arXiv.2205.11835