Introduction

The package provides a tool to select variables in a nonlinear multivariate model. More precisely, it consists in providing a variable selection tool from \(n\) observations satisfying the following nonparametric regression model: \[\begin{equation} \label{eq:model} Y_i = f(x_i) + \varepsilon_i, \quad x_i = \left(x_i^{(1)}, \ldots, x_i^{(p)}\right), \quad 1\leq i \leq n, \end{equation}\] where \(f\) is an unknown real-valued function and where the \(\varepsilon_i\)’s are i.i.d centered random variables of variance \(\sigma^2\). The \(x_i\)’s are observation points which belong to a compact set \(S\) of \(\mathbb{R}^p\). We will also assume that \(f\) actually depends on only \(d\) variables instead of \(p\), with \(d<p\), which means that there exists a real-valued function \(\widetilde{f}\) such that \(f(x)=\widetilde{f}(\widetilde{x})\), where \(x\in\mathbb{R}^p\) and \(\widetilde{x}\in\mathbb{R}^d\). Variable selection consists in identifying the components of \(\widetilde{x}\). This variable selection approach is described in [1]. We refer the reader to this paper for further details and references.

Installing

You can install the released version of from CRAN with:

install.packages("absorber")

Variable selection

We first propose to apply our method to \(n=700\) observations satisfying Model \(\eqref{eq:model}\) with \(f=f_1\) where \(p=5\), defined in [1]. These observations are obtained with a Gaussian noise of \(\sigma = 0.25\). In the following, the \(d=2\) relevant variables to select are \(\{3,5\}\) and the irrelevant ones to discard are \(\{1,2,4\}\):

true.dimensions = c(3,5) ; false.dimensions = c(1,2,4)

Description of the dataset

The observation set is loaded from files which are provided within the package, as follows:

# --- Loading the values of the observation sets --- ##
data('x_obs') ;
head(x_obs)

##           [,1]       [,2]      [,3]      [,4]      [,5]
## [1,] 0.3687684 0.16895845 0.7114856 0.1493075 0.2300115
## [2,] 0.7162858 0.47407370 0.2271114 0.8187909 0.3845692
## [3,] 0.5543277 0.63473174 0.9341467 0.4209710 0.1551578
## [4,] 0.2551628 0.55242762 0.8940447 0.8587429 0.6602330
## [5,] 0.1468073 0.21261063 0.8249912 0.7159358 0.6177809
## [6,] 0.3917696 0.01350068 0.6862343 0.8377919 0.6143807

## --- Loading the values of  corresponding noisy values of the response variable --- ##
data('y_obs') ;
head(y_obs)

## [1] -0.09049367 -1.56817050  0.02365417  0.32580069  1.07158399  1.21354888

Application of \(\texttt{absorber}\) to select the relevant variables

The \(\texttt{absorber}\) function of the \(\texttt{absorber}\) package is applied by using the following arguments:

the input values \((x_i)_{1 \leq i \leq n}\) (\(\texttt{x}\)) where \(x_i\) belongs to \([0,1]^p\), \(1\leq i \leq n\),
the corresponding \((Y_i)_{1 \leq i \leq n}\) (\(\texttt{y}\)),
the order of the B-spline basis used in the regression model (\(\texttt{M}\)). The default value is \(3\) (quadratic B-splines).

res = absorber(x = x_obs, y = y_obs, M = 3)

Additional arguments can also be provided in this function:

\(\texttt{K}\): Integer, number \(K\) of evenly spaced knots to use in the B-spline basis. The default value is \(1\).
\(\texttt{all.variables}\): List of characters or integers, labels of the variables. The default value is \(\texttt{NULL}\).
\(\texttt{parallel}\): Logical, if set to \(\texttt{TRUE}\) then a parallelized version of the code is used. The default value is \(\texttt{FALSE}\).
\(\texttt{nbCore}\): Numerical, it represents the number of cores used for parallelization, if parallel is set to \(\texttt{TRUE}\).

The resulting outputs are the following:

\(\texttt{lambdas}\): sequence of the used penalization parameters \(\lambda\).
\(\texttt{selec.var}\): list of sequences of the selected variables, one sequence for each penalization parameter.
\(\texttt{aic.var}\): sequence of variables selected using the AIC.

First, we can print the sequence of penalization parameters \(\lambda\) used in our method:

head(res$lambdas)

## [1] 0.01563831 0.01492752 0.01424904 0.01360140 0.01298320 0.01239309

We can then print the corresponding sequences of selected variables for each penalization parameter:

head(res$selec.var)

## [[1]]
## NULL
## 
## [[2]]
## [1] 3
## 
## [[3]]
## [1] 3
## 
## [[4]]
## [1] 3
## 
## [[5]]
## [1] 3
## 
## [[6]]
## [1] 3

and finally the variables selected with AIC:

res$aic.var

## [1] 3 5

Visualization of the percentage of selection for each variable with \(\texttt{plot\_selection}\)

The \(\texttt{plot\_selection}\) function of the \(\texttt{absorber}\) package produces a histogram of the variable selection percentage for each variable on which \(f\) depends. It also displays in red the results obtained with the AIC.

plot_selection(res)

We can compare this visualization to the one indicating the relevant and the irrelevant variables in red and green, respectively, as in Figure 6 of [1]. To do so, we gather the results into a data.frame as follows:

nlam = length(res$lambdas)
occurrence = data.frame(table(unlist(res$selec.var))) ; 
colnames(occurrence) = c("Covariable", "Percentage") ;
occurrence$Percentage =occurrence$Percentage*100/nlam ;
occurrence = occurrence[order(-occurrence$Percentage),,drop=FALSE] ;
occurrence$Covariable = factor(occurrence$Covariable,
                                       levels = unique(occurrence$Covariable)) ;

occurrence$Category = as.factor(ifelse(occurrence$Covariable %in% true.dimensions, 
                                   'real features', 'fake features')) ;
str(occurrence) ;

## 'data.frame':    5 obs. of  3 variables:
##  $ Covariable: Factor w/ 5 levels "3","5","4","2",..: 1 2 3 4 5
##  $ Percentage: num  99 65 45 37 36
##  $ Category  : Factor w/ 2 levels "fake features",..: 2 2 1 1 1

We can then plot the results as a histogram of variable selection percentage:

color.order = c('firebrick', 'forestgreen')[which( c('fake features', 'real features') 
                                                   %in% levels(occurrence$Category))]

plt_occ = ggplot(data = occurrence, aes(x = Covariable, y = Percentage, fill = Category)) +
  geom_bar(stat = 'identity') +
  scale_fill_manual(values = color.order) +
  ylab('Percentage of selection') +
  theme_bw() +
  theme(legend.title = element_blank(),
        axis.text.x = element_text(size = 16, face = 'bold'),
        axis.text.y = element_text(size = 14),
        axis.title.x = element_blank(),
        axis.title.y = element_text(size = 15),
        legend.text =  element_text(size = 14),
        legend.position = 'bottom',
        legend.key.size = unit(1, "cm"), 
        panel.grid.major = element_line(size = 0.6, linetype = 'solid',
                                           colour = "darkgrey"), 
           panel.grid.minor = element_line(size = 0.2, linetype = 'solid',
                                           colour = "darkgrey"))

print(plt_occ)

The results obtained with the AIC allows us to retrieve the correct relevant variables since it selects \(\{3,5\}\) while discarding the irrelevant ones.

References

[1] Savino, M. E. and Lévy-Leduc, C. (2024) A novel variable selection method in nonlinear multivariate models using B-splines with an application to geoscience. ⟨hal-04434820⟩.