Cross-validation

The cross-validation process is started in a leave-one-out design along \(1\) dimension. NCORES fixes the number of cores in the paralellized process, n_lambda fixes the number of regularization terms to be tested.

res_cv_reg <- perf_mddsPLS(Xs = X,Y = Y,
                           R = 1,lambda_min=0.7,n_lambda=n_lambda,
                           mode = "reg",NCORES = NCORES,kfolds = "loo")
plot(res_cv_reg,legend_names=colnames(Y))

Result of cross-validation funciton perf_mddspls.

Classification case

data("penicilliumYES")
X <- penicilliumYES$X
X <- scale(X[,which(apply(X,2,sd)>0)])
classes <- c("Melanoconidium","Polonicum","Venetum")
Y <- as.factor(unlist(lapply(classes,
                             function(tt){rep(tt,12)})))
mddsPLS_model_class <- mddsPLS(Xs = X,Y = Y,lambda = 0.956,R = 2,
                               mode = "clas",verbose = TRUE)

## At most 4 variable(s) can be selected
##     For each block of X, are selected in order of component:
##         @ (2,2) variable(s)
##     For the Y block, are selected in order of component:
##         @ (1,1) variable(s)

plot(mddsPLS_model_class$mod$t,col=Y,pch=as.numeric(Y)+15,cex=2,
     xlab="1st X component, 2 var. selected",
     ylab="2nd X component, 2 var. selected")
legend(-2,0,legend=classes,col=1:3,pch=15+(1:3),box.lty=0,y.intersp=2)

Cross-validation

The cross-validation process is started in a fold-fixed design, because each sample is repeated \(3\) times. In that sense this is a leave-one-out process. \(R=2\) fixes the number of dimensions to 2. NCORES fixes the number of cores in the paralellized process, n_lambda fixes the number of regularization terms to be tested.

res_cv_class <- perf_mddsPLS(X,Y,R = 2,lambda_min=0.92,n_lambda=n_lambda,
                             mode = "clas",NCORES = NCORES,
                             fold_fixed = rep(1:12,3))
plot(res_cv_class,legend_names = levels(Y),pos_legend="bottomleft")

Result of cross-validation funciton perf_mddspls.

Simulation model

According to what have been proposed in (Johnstone and Lu 2004), the following simulations follow the spike covariance models \[\left\{\begin{aligned} {\bf X}_1&= {\bf L}{\bf \Omega}_1^{1/2}{\bf U}^T_{1,mod}+{\bf E}_1\\ &\vdots\\ {\bf X}_T&= {\bf L}{\bf \Omega}_T^{1/2}{\bf U}^T_{T,mod}+{\bf E}_T\\ {\bf Y}&= {\bf L}{\bf \Omega}_y^{1/2}{\bf V}^T_{mod}+{\bf E}_y\\ \end{aligned} \right.,\] where \(({\bf \Omega}_t)_{t=1\cdots T}\) and \({\bf \Omega}_y\) are \(R\)-dimensional diagonal matrices with strictly positive diagonal elements. \(({\bf U}_{t,mod}\in\mathbb{R}^{p_t\times R})_{t=1\cdots T}\) and \({\bf V}_{mod}\in\mathbb{R}^{q\times R}\) are matrices with orthonormal columns. \({\bf L}\in\mathbb{R}^{n\times R}\) is a matrix where elements are i.i.d. standard Gaussian random effects, \(({\bf E}_t\in\mathbb{R}^{n\times p_t})_{t=1\cdots T}\) (respectively \({\bf E}_y\in\mathbb{R}^{n\times q}\)) are matrices such that each row follows the standard multivariate normal distribution \((N_{p_t}(0,\mathbb{I}_{p_t}))_{t=1\cdots T}\) (respectively \(N_q(0,\mathbb{I}_q)\)) and the \(n\) rows are independent and mutually independent noise vectors. Let us mention that the matrix \({\bf L}\) does not depend of \(t\) and thus introduces a common structure between the \({\bf X}_t\)’s and \(\bf Y\) models.

Simulation parameters

Usual parameters used to simulate datasets

n <- 20 # number of individuals
R <- 5 # number of created dimensions in __L__
T_ <- 10 # number of blocks
sd_error <- 0.1 # Standard-deviation of the spike-covariance model element matrices of $E_t$ and $E_y$
p_s <- sample(x = 100:200,size = T_,replace = T) # number of variables per block $X_t$
q <- 10  # number of variable in $Y$
R_real <- 3 # number of components of __L__ described in __Y__
p_missing <- 0.3 # the proportion of missing values

Possible values for \(({\bf \Omega}_t^{1/2})_{t=1\cdots T}\) and \({\bf \Omega}_y^{1/2}\) diagonal elements are then chosen. It has been chosen to consider low elements, close to \(0\), and high elements, \(\approx 1\).

o_x <- seq(0,1,length.out = 1000)
o_y <- (o_x-0.5)^2
o_y[which(o_y<0.2)] <- 0 # keep only low or high potential diagonal elements
all_omegas <- sample(o_x,prob = o_y,size = R*T_) # Select R*T_ elements

all_omegas_y <- sample(o_x,prob = o_y,size = R_real) # Select R_real elements
Omegas_y <- diag(c(all_omegas_y,rep(0,R-R_real))) # Create the Omega_y diagonal matrix

Generate covariate dataset

Xs is a list of matrices corresponding to the defined spike-covariance model.

Xs <- list()
L <- matrix(rnorm(n*R),nrow = n)
for(k in 1:T_){
    Omegas <- diag(all_omegas[1:R+(k-1)*R])
    Us <- svd(matrix(rnorm(p_s[k]*n),nrow = n))$v[,1:R]
    E_k <- matrix(rnorm(n*p_s[k],sd = sd_error),nrow = n)
    Xs[[k]]<- scale(E_k + tcrossprod(L%*%Omegas,Us))
}

A proportion \(p_{missing}\) of the data is missing, the following script permits to remove that proportion of samplestaking into account that a given participant must not be missing for all blocks.

values <- expand.grid(1:n,1:T_)
values_id <- 1:(n*T_)
probas <- rep(1,n*T_)/(n*T_)
number_miss_samp <- floor(n*T_*p_missing)
missin_samp <- matrix(NA,nrow = number_miss_samp,ncol = 2)
for(sam in 1:number_miss_samp){
  curr_id <- values_id[sample(values_id,size = 1,prob = probas)]
  missin_samp[sam,1] <- values[curr_id,1]
  missin_samp[sam,2] <- values[curr_id,2]
  probas[curr_id] <- 0
  if(length(which(na.omit(missin_samp[,1])==missin_samp[sam,1]))==n){
    probas[which(values[,1]==missin_samp[sam,1])] <- 0
  }
  Xs[[missin_samp[sam,2]]][missin_samp[sam,1],] <- NA ## Remove individual value
}

Generate response matrix

Y is a matrix also corresponding to the defined spike-covariance model.

V <- svd(matrix(rnorm(q*n),nrow = n))$v[,1:R]
E_y <- matrix(rnorm(q*n,sd = sd_error),nrow = n)
Y <- tcrossprod(L%*%Omegas_y,V)
Y <- scale(E_y + Y)

Cross-validation

The cross-validation process is started in a leave-one-out design along \(3\) dimensions. NCORES fixes the number of cores in the paralellized process, n_lambda fixes the number of regularization terms to be tested.

cross_valid <- perf_mddsPLS(Xs,Y,n_lambda = n_lambda,
                            R = 3,kfolds = "loo",NCORES = NCORES)
plot(cross_valid,plot_mean = T)

In that simulation, \(5\) Y variables are very well predicted for \(\lambda <0.8\) and a minimum of error is reached for \(\lambda \approx 0.72\).