3 Illustration
3.1 Spatial autocorrelation
We first create a toy dataset with spatial coordinates.
library(nlme)
n <- 75 #Sample size
p <- 100 #Number of features
g <- 10 #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
## Iteration 3Standard extraction functions like print(), coef() and predict() are defined for the new “pengls” object.
## pengls model with correlation structure: corGaus
## and 42 non-zero coefficients3.2 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:
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
## Iteration 3Show the output
## pengls model with correlation structure: corAR1
## and 44 non-zero coefficients3.3 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 :
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:
## Cross-validated pengls model with correlation structure: corGaus
## and 41 non-zero coefficients.
## 10 fold cross-validation yielded an estimated R2 of -1.31033 .By default, the 1 standard error is used to determine the optimal value of \(\lambda\) :
## [1] 0.2204881## [1] -1.31033Extract coefficients and fold IDs.
## [1] -0.14333032 0.13893419 0.00000000 0.00000000 -0.02448515 0.00000000## 73 59 87 43 49 20 28 63 48 100 89 78 88 30 5 22 69 11 9 23
## 5 1 9 8 1 10 10 8 4 9 3 9 3 10 10 1 9 2 10 2
## 81 79 6 67 46 93 74 42 10 14 27 58 31 82 50 4 75 54 91 16
## 6 3 4 1 1 6 5 2 4 2 4 10 5 5 10 2 6 2 5 4
## 24 64 44 62 36 21 65 77 8 71 80 26 52 13 1 47 37 41 33 53
## 2 5 7 5 1 6 1 9 4 6 9 10 7 2 8 7 4 6 8 2
## 2 55 35 51 96 85 95 34 76 94 7 56 99 25 29
## 2 6 2 5 1 6 9 1 5 5 7 6 3 8 1By 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(df), 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(df), pattern = "b", value =TRUE),
glsSt <- corStructTime, nfolds = nfolds, cvType = "blocked", transFun = function(x) x-mean(x))
penglsFitCVtimeCenter$cvOpt #Better performance## [1] 0.5621167