# reproducibility - simulation

This script requires that the working directory includes the folders “data”, “results”, and “manuscript”. We obtained our results using R 4.3.0 (2023-04-21) with cornet 0.0.8 (2023-06-01) on a local machine (aarch64-apple-darwin20, macOS Ventura 13.4). Floating point errors might lead to slightly different results on other platforms.

# Graphical abstract

    grDevices::pdf("manuscript/figure_idea.pdf",width=5,height=2.5)

    box <- function(x,y,width=0.22,height=0.2,labels="",cex=1,col="black",...){
      xs <- x + 0.5*c(-1,-1,1,1)*width
      ys <- y + 0.5*c(-1,1,1,-1)*height
      graphics::polygon(x=xs,y=ys,border=col,lwd=2,...)
      graphics::text(x=x,y=y,labels=labels,col=col,cex=cex)
    }

    graphics::par(mar=c(0,0,0,0))
    graphics::plot.new()
    graphics::plot.window(xlim=c(0,1),ylim=c(0,1))

    v <- h <- 0.1

    box(x=0+h,y=0.5,labels="outcomes,\nfeatures")
    box(x=0.5,y=1-v,labels="initial binary\nclassification",col="red")
    box(x=0.5,y=0+v,labels="numerical\nprediction",col="blue")
    box(x=1-h,y=0.5,labels="final binary\nclassification",col="red")

    d <- 0.02
    graphics::arrows(x0=0.2+d,y0=0.5+c(-d,d),x1=0.4-d,y1=c(v,1-v),lwd=2,col=c("blue","red"))
    graphics::arrows(x0=0.6+d,y0=c(v,1-v),x1=0.8-d,y1=0.5+c(-d,d),lwd=2,col=c("blue","red"))

    graphics::text(x=0.4,y=0.55,labels="binary outcome:\nlogistic regression",col="red",cex=0.7,pos=3)
    graphics::text(x=0.4,y=0.45,labels="numerical outcome:\nlinear regression",col="blue",cex=0.7,pos=1)
    graphics::text(x=0.63,y=0.5,labels="combine\npredicted\nprobabilities",col="darkgrey",cex=0.7)
    graphics::text(x=0.8,y=0.3,labels="transform\npredicted values to\npredicted probabilities",col="darkgrey",cex=0.7,pos=1)

    grDevices::dev.off()

# Examples

    loss <- list()
    for(i in seq_len(4)){
      loss[[i]] <- list()
      cat("mode:",i,"\n")
      for(j in seq_len(100)){
        set.seed(j)
        cat("iteration:",j,"\n")
        n0 <- 100; n1 <- 10000; p <- 500
        n <- n0 + n1
        X <- matrix(data=stats::rnorm(n*p),nrow=n,ncol=p)
        beta <- stats::rbinom(n=p,size=1,prob=0.05)*stats::rnorm(n=p)
        eta <- X %*% beta
        epsilon <- stats::rnorm(n=n)
        if(i==1){
          y <- eta + epsilon
        } else if(i==2){
          y <- ifelse(eta<0,-2,+2)+epsilon
          table(y>=0,eta>=0)
        } else if(i==3){
          y <- ifelse(eta<0,-sqrt(abs(eta+epsilon)),(eta+epsilon)^2)
        } else if(i==4){
          y <- eta + epsilon + stats::rbinom(n=n,size=1,prob=0.05)*(2*stats::rbinom(n=n,size=1,prob=0.5)-1)*1.5*max(abs(eta))
        }
        foldid <- rep(c(0,1),times=c(n0,n1))
        loss[[i]][[j]] <- cornet::cv.cornet(y=y,cutoff=0,X=X,foldid.ext=foldid)
      }
    }

    save(loss,file="results/simulation.RData")
    writeLines(text=capture.output(utils::sessionInfo(),cat("\n"),
            sessioninfo::session_info()),con="results/info_sim.txt")

    load("results/simulation.RData")

    grDevices::pdf("manuscript/figure_EXA.pdf",width=5,height=5)

    graphics::par(mfrow=c(2,2),mar=c(2,2,1,1))
    pos <- c(binomial=1,combined=2,gaussian=3)
    col <- c(binomial="red",combined="grey",gaussian="blue")
    cex <- 0.7
    names <- c("binomial","combined","gaussian")
    for(i in seq_len(4)){
      frame <- as.data.frame(t(sapply(loss[[i]],function(x) x$deviance)))
      graphics::boxplot(x=frame[,names],at=pos[names],col=col[names],cex.axis=cex,main=paste0("example ",i),cex.main=cex,axes=FALSE)
      graphics::box()
      graphics::axis(side=1,at=pos[names],labels=names,cex.axis=cex,tick=FALSE,line=-1)
      graphics::axis(side=2,cex.axis=cex)
      for(j in c("binomial","combined","gaussian")){
        mean <- mean(frame[[j]])
        graphics::points(x=pos[j],y=mean,pch=21,col="white",bg="black")
        if(j=="combined"){next}
        pvalue <- stats::wilcox.test(x=frame$combined,y=frame[[j]],alternative="less")$p.value
        signif <- ifelse(pvalue<=0.05/8,"*","")
        graphics::text(x=mean(c(pos["combined"],pos[[j]])),y=min(frame),labels=paste0("p=",format(pvalue,digits=2,scientific=TRUE),signif),pos=3,cex=0.7)
      }
    }

    grDevices::dev.off()

# (not included)

    iter <- 1000
    set.seed(1)
    frame <- data.frame(cor=runif(n=iter,min=0,max=0.9),
                        n=round(runif(n=iter,min=100,max=200))+10000,
                        prob=runif(n=iter,min=0.01,max=0.1),
                        sd=runif(n=iter,min=1,max=2),
                        exp=runif(n=iter,min=0.1,max=2),
                        frac=runif(n=iter,min=0.5,max=0.9))

    ridge <- lasso <- list()
    pb <- utils::txtProgressBar(min=0,max=nrow(frame),width=20,style=3)
    for(i in seq_len(nrow(frame))){
        utils::setTxtProgressBar(pb=pb,value=i)
        set.seed(i)
        data <- do.call(what=cornet:::.simulate,args=cbind(frame[i,],p=500))
        foldid <- rep(c(0,1),times=c(frame$n[i],10000))
        set.seed(i)
        ridge[[i]] <- do.call(what=cornet:::cv.cornet,args=c(data,alpha=0,foldid=foldid))
        set.seed(i)
        lasso[[i]] <- do.call(what=cornet:::cv.cornet,args=c(data,alpha=1,foldid=foldid))
    }
    names(lasso) <- names(ridge) <- paste0("set",seq_len(nrow(frame)))
    save(lasso,ridge,frame,file="results/simulation.RData")

    writeLines(text=capture.output(utils::sessionInfo(),cat("\n"),
            sessioninfo::session_info()),con="results/info_sim.txt")

    #--- boxplot of different metrics ---
    load("results/simulation.RData",verbose=TRUE)

    fuse0 <- fuse1 <- list()
    for(i in c("deviance","class","mse","mae","auc")){
      fuse0[[i]] <- sapply(ridge,function(x) (x[[i]]["combined"]-x[[i]]["binomial"]))
      fuse1[[i]] <- sapply(lasso,function(x) (x[[i]]["combined"]-x[[i]]["binomial"]))
    }

    grDevices::pdf("manuscript/figure_BOX.pdf",width=6,height=4)
    graphics::par(mar=c(1.9,1.9,0.1,0.1))
    graphics::plot.new()
    ylim <- range(unlist(fuse0),unlist(fuse1))
    at <- seq(from=1,to=9,by=2)
    graphics::plot.window(xlim=c(min(at)-0.6,max(at)+0.6),ylim=ylim)
    graphics::axis(side=2)
    graphics::abline(h=0,col="grey",lty=2)
    graphics::abline(v=at+1,col="grey",lty=2)
    graphics::box()
    graphics::boxplot(fuse1,at=at-0.5,add=TRUE,axes=FALSE,col="white",border="black")
    graphics::boxplot(fuse0,at=at+0.5,add=TRUE,axes=FALSE,col="white",border="darkgrey")
    labels <- names(fuse1)
    labels <- ifelse(labels=="class","mcr",labels)
    labels <- ifelse(labels %in% c("mcr","mse","mae","auc"),toupper(labels),labels)
    for(i in seq_along(labels)){
      graphics::axis(side=1,at=at[i],labels=bquote(Delta ~ .(labels[i])))
    }
    grDevices::dev.off()

    # decrease
    sapply(fuse1,function(x) mean(x<0)) # lasso
    sapply(fuse0,function(x) mean(x<0)) # ridge

    # constant
    sapply(fuse1,function(x) mean(x==0)) # lasso
    sapply(fuse0,function(x) mean(x==0)) # ridge

    # increase
    sapply(fuse1,function(x) mean(x>0)) # lasso
    sapply(fuse0,function(x) mean(x>0)) # ridge

    #--- plot of percentage changes ---
    load("results/simulation.RData",verbose=TRUE)

    loss <- list()
    loss$ridge <- as.data.frame(t(sapply(ridge,function(x) x$deviance)))
    loss$lasso <- as.data.frame(t(sapply(lasso,function(x) x$deviance)))

    data <- list()
    for(i in c("ridge","lasso")){
      data[[i]] <- data.frame(row.names=rownames(frame))
      data[[i]]$"(1)" <- 100*(loss[[i]]$binomial-loss[[i]]$intercept)/loss[[i]]$intercept
      data[[i]]$"(2)" <- 100*(loss[[i]]$combined-loss[[i]]$intercept)/loss[[i]]$intercept
      data[[i]]$"(3)" <- 100*(loss[[i]]$combined-loss[[i]]$binomial)/loss[[i]]$binomial
    }

    row <- colnames(data$lasso)
    col <- colnames(frame)
    txt <- expression(rho,n,s,sigma,t,q)

    for(k in c("ridge","lasso")){
      grDevices::pdf(paste0("manuscript/figure_",k,".pdf"),width=6.5,height=4)
      graphics::par(mfrow=c(length(row),length(col)),
                  mar=c(0.2,0.2,0.2,0.2),oma=c(4,4,0,0))
      for(i in seq_along(row)){
        for(j in seq_along(col)){
          y <- data[[k]][[row[i]]]
          x <- frame[[col[j]]]
          graphics::plot.new()
          graphics::plot.window(xlim=range(x),ylim=range(y),xaxs="i")
          graphics::box()
          graphics::abline(h=0,lty=1,col="grey")
          graphics::points(y=y,x=x,cex=0.5,pch=16,col=ifelse(y>0,"black","grey"))
          line <- stats::loess.smooth(y=y,x=x,evaluation=200)
          graphics::lines(x=line$x,y=line$y,col="black",lty=2,lwd=1)
          if(j==1){
            graphics::mtext(text=row[i],side=2,line=2.5,las=2)
            graphics::axis(side=2)
          }
          if(i==length(row)){
            graphics::mtext(text=txt[j],side=1,line=2.5)
            graphics::axis(side=1)
          }
        }
      }
      grDevices::dev.off()
    }

    cbind(col,as.character(txt)) # verify