Control the number of \(\lambda\) parameters to test in the cross-validation processes

`n_lambda <- 20`

Control the number of cores to use in the cross-validation processes. If equals \(1\), then no parallel computation structure is deployed

`NCORES <- 7`

Load the **ddsPLS** package

`library(ddsPLS)`

The here used method can be used on mono and multi-block datasets with no missing values. In the case on mono-block datasets, there is no current developpment to deal with missing samples. Actually this would imply that for a given individual, no covariate is known. Which block any inference. ## Regression case The regression case has been treated through a toy example well know dataset in that section. ### Build a model We have worked on the Liver Toxicity dataset, see Bushel, Wolfinger, and Gibson (2007). This data set contains the expression measure of 3116 genes and 10 clinical measurements for 64 subjects (rats) that were exposed to non-toxic, moderately toxic or severely toxic doses of acetaminophen in a controlled experiment. Therefore the structure is : \[\mathbf{X}\in\mathbb{R}^{64\times3116},\mathbf{Y}\in\mathbb{R}^{64\times10}\]

```
library(doParallel)
library(RColorBrewer)
data("liver.toxicity")
X <- scale(liver.toxicity$gene)
Y <- scale(liver.toxicity$clinic)
mddsPLS_model_reg <- mddsPLS(Xs = X,Y = Y,lambda=0.9,R = 1,
mode = "reg",verbose = TRUE)
```

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))
```

The data set penicilliumYES has 36 rows and 3754 columns, see Clemmensen et al. (2007) The variables are 1st order statistics from multi-spectral images of three species of Penicillium fungi: Melanoconidium, Polonicum, and Venetum. These are the data used in the Clemmemsen et al “Sparse Discriminant Analysis” paper. Therefore the structure is, where \(\mathbf{Y}\) is the dummy matrix of the \(3\) classes : \[\mathbf{X}\in\mathbb{R}^{36\times3754},\mathbf{Y}\in\mathbb{R}^{36\times3}\]

```
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)
```

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")
```

A simulation process is proposed to have an idea of the behavior of the method in the multi-block context with missing values.

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.

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
```

**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
}
```

**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)
```

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 \(\lambda \approx 0.72\).

Bushel, Pierre R, Russell D Wolfinger, and Greg Gibson. 2007. “Simultaneous Clustering of Gene Expression Data with Clinical Chemistry and Pathological Evaluations Reveals Phenotypic Prototypes.” *BMC Systems Biology* 1 (1). BioMed Central: 15.

Clemmensen, Line H, Michael E Hansen, Jens C Frisvad, and Bjarne K Ersbøll. 2007. “A Method for Comparison of Growth Media in Objective Identification of Penicillium Based on Multi-Spectral Imaging.” *Journal of Microbiological Methods* 69 (2). Elsevier: 249–55.

Johnstone, Iain M, and Arthur Yu Lu. 2004. “Sparse Principal Components Analysis.” *Unpublished Manuscript* 7.