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)

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## Warning in pengls(data = df, outVar = "a", xNames = grep(names(df), pattern = "b", : No convergence achieved in pengls!

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

penglsFit

## pengls model with correlation structure: corGaus 
##  and 18 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 8 non-zero coefficients.
##  3 fold cross-validation yielded an estimated R2 of -1.024022 .

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

penglsFitCV$lambda.1se #Lambda for 1 standard error rule

## [1] 0.9441916

penglsFitCV$cvOpt #Corresponding R2

## [1] -1.024022

Extract coefficients and fold IDs.

head(coef(penglsFitCV))

## [1] 1.0552207 0.6493971 0.0000000 0.0000000 0.0000000 0.0000000

penglsFitCV$foldid #The folds used

##  67 120  22  36 121 107 178   3 157   8 140  30 167  90  53 126  51 221  72  60 
##   1   3   1   1   2   2   3   1   2   3   2   3   2   3   1   2   1   2   1   3 
##  87  77  99  81 199 
##   3   1   3   1   2

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

## [1] 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")