# 1 Introduction

This vignette demonstrates the use of the pengls package for high-dimensional data with spatial or temporal autocorrelation. It consists of an iterative loop around the nlme and glmnet packages. Currently, only continuous outcomes and $$R^2$$ and MSE as performance measure are implemented.

# 2 Installation instuctions

The pengls package is available from BioConductor, and can be installed as follows:

library(BiocManager)
install("pengls")

Once installed, it can be loaded and version info printed.

suppressPackageStartupMessages(library(pengls))
cat("pengls package version", as.character(packageVersion("pengls")), "\n")
## pengls package version 1.6.0

# 3 Illustration

## 3.1 Spatial autocorrelation

We first create a toy dataset with spatial coordinates.

library(nlme)
n <- 25 #Sample size
p <- 50 #Number of features
g <- 15 #Size of the grid
#Generate grid
Grid <- expand.grid("x" = seq_len(g), "y" = seq_len(g))
# Sample points from grid without replacement
GridSample <- Grid[sample(nrow(Grid), n, replace = FALSE),]
#Generate outcome and regressors
b <- matrix(rnorm(p*n), n , p)
a <- rnorm(n, mean = b %*% rbinom(p, size = 1, p = 0.25), sd = 0.1) #25% signal
#Compile to a matrix
df <- data.frame("a" = a, "b" = b, GridSample)

The pengls method requires prespecification of a functional form for the autocorrelation. This is done through the corStruct objects defined by the nlme package. We specify a correlation decaying as a Gaussian curve with distance, and with a nugget parameter. The nugget parameter is a proportion that indicates how much of the correlation structure explained by independent errors; the rest is attributed to spatial autocorrelation. The starting values are chosen as reasonable guesses; they will be overwritten in the fitting process.

# Define the correlation structure (see ?nlme::gls), with initial nugget 0.5 and range 5
corStruct <- corGaus(form = ~ x + y, nugget = TRUE, value = c("range" = 5, "nugget" = 0.5))

Finally the model is fitted with a single outcome variable and large number of regressors, with the chosen covariance structure and for a prespecified penalty parameter $$\lambda=0.2$$.

#Fit the pengls model, for simplicity for a simple lambda
penglsFit <- pengls(data = df, outVar = "a", xNames = grep(names(df), pattern = "b", value =TRUE), glsSt = corStruct, lambda = 0.2, verbose = TRUE)
## Starting iterations...
## Iteration 1
## Iteration 2

Standard extraction functions like print(), coef() and predict() are defined for the new “pengls” object.

penglsFit
## pengls model with correlation structure: corGaus
##  and 13 non-zero coefficients
penglsCoef <- coef(penglsFit)
penglsPred <- predict(penglsFit)

## 3.2 Temporal autocorrelation

The method can also account for temporal autocorrelation by defining another correlation structure from the nlme package, e.g. autocorrelation structure of order 1:

set.seed(354509)
n <- 100 #Sample size
p <- 10 #Number of features
#Generate outcome and regressors
b <- matrix(rnorm(p*n), n , p)
a <- rnorm(n, mean = b %*% rbinom(p, size = 1, p = 0.25), sd = 0.1) #25% signal
#Compile to a matrix
dfTime <- data.frame("a" = a, "b" = b, "t" = seq_len(n))
corStructTime <- corAR1(form = ~ t, value = 0.5)

The fitting command is similar, this time the $$\lambda$$ parameter is found through cross-validation of the naive glmnet (for full cross-validation , see below). We choose $$\alpha=0.5$$ this time, fitting an elastic net model.

penglsFitTime <- pengls(data = dfTime, outVar = "a", verbose = TRUE,
xNames = grep(names(dfTime), pattern = "b", value =TRUE),
glsSt = corStructTime, nfolds = 5, alpha = 0.5)
## Fitting naieve model...
## Starting iterations...
## Iteration 1
## Iteration 2

Show the output

penglsFitTime
## pengls model with correlation structure: corAR1
##  and 2 non-zero coefficients

## 3.3 Penalty parameter and cross-validation

The pengls package also provides cross-validation for finding the optimal $$\lambda$$ value. If the tuning parameter $$\lambda$$ is not supplied, the optimal $$\lambda$$ according to cross-validation with the naive glmnet function (the one that ignores dependence) is used. Hence we recommend to use the following function to use cross-validation. Multithreading is supported through the BiocParallel package :

library(BiocParallel)
register(MulticoreParam(3)) #Prepare multithereading
nfolds <- 3 #Number of cross-validation folds

The function is called similarly to cv.glmnet:

penglsFitCV <- cv.pengls(data = df, outVar = "a", xNames = grep(names(df), pattern = "b", value =TRUE), glsSt = corStruct, nfolds = nfolds)

Check the result:

penglsFitCV
## Cross-validated pengls model with correlation structure: corGaus
##  and 13 non-zero coefficients.
##  3 fold cross-validation yielded an estimated R2 of 0.1584249 .

By default, the 1 standard error is used to determine the optimal value of $$\lambda$$ :

penglsFitCV$lambda.1se #Lambda for 1 standard error rule ##  0.2340147 penglsFitCV$cvOpt #Corresponding R2
##  0.1584249

Extract coefficients and fold IDs.

head(coef(penglsFitCV))
##  -0.06120535  0.03269066  0.00000000  0.00000000 -0.35250614  0.00000000
penglsFitCV$foldid #The folds used ## 217 27 7 176 136 139 185 201 18 15 61 103 32 122 58 170 113 93 187 120 ## 1 3 3 1 1 1 1 1 2 3 1 3 2 1 3 1 1 1 1 3 ## 140 60 134 214 205 ## 1 3 3 1 1 By default, blocked cross-validation is used, but random cross-validation is also available (but not recommended for timecourse or spatial data). First we illustrate the different ways graphically, again using the timecourse example: set.seed(5657) randomFolds <- makeFolds(nfolds = nfolds, dfTime, "random", "t") blockedFolds <- makeFolds(nfolds = nfolds, dfTime, "blocked", "t") plot(dfTime$t, randomFolds, xlab ="Time", ylab ="Fold")
points(dfTime$t, blockedFolds, col = "red") legend("topleft", legend = c("random", "blocked"), pch = 1, col = c("black", "red")) To perform random cross-validation penglsFitCVtime <- cv.pengls(data = dfTime, outVar = "a", xNames = grep(names(dfTime), pattern = "b", value =TRUE), glsSt = corStructTime, nfolds = nfolds, cvType = "random") To negate baseline differences at different timepoints, it may be useful to center or scale the outcomes in the cross validation. For instance for centering only: penglsFitCVtimeCenter <- cv.pengls(data = dfTime, outVar = "a", xNames = grep(names(dfTime), pattern = "b", value =TRUE), glsSt = corStructTime, nfolds = nfolds, cvType = "blocked", transFun = function(x) x-mean(x)) penglsFitCVtimeCenter$cvOpt #Better performance
##  0.9949127

Alternatively, the mean squared error (MSE) can be used as loss function, rather than the default $$R^2$$:

penglsFitCVtime <- cv.pengls(data = dfTime, outVar = "a", xNames = grep(names(dfTime), pattern = "b", value =TRUE), glsSt = corStructTime, nfolds = nfolds, loss =  "MSE")

# 4 Session info

sessionInfo()
## R version 4.3.0 RC (2023-04-13 r84269)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 22.04.2 LTS
##
## Matrix products: default
## BLAS:   /home/biocbuild/bbs-3.17-bioc/R/lib/libRblas.so
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.10.0
##
## locale:
##   LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
##   LC_TIME=en_GB              LC_COLLATE=C
##   LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
##   LC_PAPER=en_US.UTF-8       LC_NAME=C
##  LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
##
## time zone: America/New_York
## tzcode source: system (glibc)
##
## attached base packages:
##  stats     graphics  grDevices utils     datasets  methods   base
##
## other attached packages:
##  BiocParallel_1.34.0 nlme_3.1-162        pengls_1.6.0
##
## loaded via a namespace (and not attached):
##   cli_3.6.1        knitr_1.42       rlang_1.1.0      xfun_0.39
##   highr_0.10       jsonlite_1.8.4   htmltools_0.5.5  sass_0.4.5
##   glmnet_4.1-7     rmarkdown_2.21   grid_4.3.0       evaluate_0.20
##  jquerylib_0.1.4  fastmap_1.1.1    yaml_2.3.7       foreach_1.5.2
##  compiler_4.3.0   codetools_0.2-19 Rcpp_1.0.10      lattice_0.21-8
##  digest_0.6.31    R6_2.5.1         parallel_4.3.0   splines_4.3.0
##  shape_1.4.6      bslib_0.4.2      Matrix_1.5-4     tools_4.3.0
##  iterators_1.0.14 survival_3.5-5   cachem_1.0.7