2 PLSR Training
In Partial Least Square Regression (PLSR), we want to estimate a linear combination of \(\mathbf{z}_1, \mathbf{z}_2, \ldots, \mathbf{z}_k\) that are good predictors for both the input \(\mathbf{X}\in\mathbb{R}^{n\times p}\) and the response \(\mathbf{y}\in\mathbb{R}^{n}\). The PLSR linear relationships can be written as
\[ \begin{split} \mathbf{X} &= \mathbf{Z}\mathbf{V}^T + \mathbf{E} \\ \mathbf{y} &= \mathbf{Z}\mathbf{b} + \mathbf{e} \end{split} \]
where \(\mathbf{Z}\in\mathbb{R}^{n\times k}\) are the PLS scores, \(\mathbf{V}\in\mathbb{R}^{k\times p}\) are the PLS loadings, and \(\mathbf{b}\in\mathbb{R}^{k}\) are the PLS coefficients. The terms \(\mathbf{E}\) and \(\mathbf{e}\) are the residual matrices for both input and response, respectively. The scalars \(n\), \(p\), and \(k\) denote the number of samples, input predictors, and PLS components.
Hence, to build a PLSR model, we need the number of PLS components, or \(k\), and that’s what we do the PLSR training.
In this paper,
- The input predictor \(\mathbf{X}\) contains
age,sex, and the first \(m\) principal components from the LV shape atlas. The number of \(m\) depends on the cohort: MESA or UKBB. - The response \(\mathbf{y}\) is a binary variable to denote the presence of a risk (=0) or not (=1). Five cardiovacular risk factors were used: hypertension, diabetes, obesity, hypercholesterolemia, and smoking.
2.1 k-Folds Cross Validation
We used five-fold cross validation to determine the optimal number of PLSR components. We did this for each cohort and for each risk factor. The general function to perform the k-fold cross validation for PLSR training is given below:
train_pls <- function(form, dt, n_folds=5, n_comps=30,
prep=c("center"), probMethod="softmax")
{
# create frequency table to calculate the weights
response <- model.frame(form, data=dt)[[form[[2]]]]
# create cross-validation folds
cvIndex <- createFolds(factor(response), n_folds, returnTrain = T)
# create caret's training controller
ctrl <- trainControl(method = "cv",
index = cvIndex,
classProbs = TRUE,
verboseIter=TRUE,
summaryFunction = twoClassSummary,
savePredictions = TRUE,
allowParallel = TRUE)
# train using PLS, metric is ROC.
# Note that the number of PLS modes is given in the tuneLength argument.
model <- train(form=form,
data=dt,
method="pls",
probMethod=probMethod,
metric="ROC",
tuneLength = n_comps,
preProc = prep,
trControl = ctrl)
return(model)
}