Dense case

Last updated: 2020-06-02

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Rmd	654b67a	fmorgante	2020-06-02	Add additional simulation to the dense scenario
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Rmd	ceda278	fmorgante	2020-06-01	Add case study with dense scenario

library(mr.mash.alpha)
library(glmnet)

Loading required package: Matrix

Loading required package: foreach

Loaded glmnet 2.0-16

###Set options
options(stringsAsFactors = FALSE)

###Function to compute accuracy
compute_accuracy <- function(Y, Yhat) {
  bias <- rep(as.numeric(NA), ncol(Y))
  r2 <- rep(as.numeric(NA), ncol(Y))
  mse <- rep(as.numeric(NA), ncol(Y))
  
  for(i in 1:ncol(Y)){
    fit  <- lm(Y[, i] ~ Yhat[, i])
    bias[i] <- coef(fit)[2] 
    r2[i] <- summary(fit)$r.squared
    mse[i] <- mean((Y[, i] - Yhat[, i])^2)
  }
  
  return(list(bias=bias, r2=r2, mse=mse))
}

Here we investigate why mr.mash does not perform as well in denser scenarios. Let’s simulate data with n=600, p=1,000, p_causal=500, r=5, r_causal=5, PVE=0.5, shared effects, independent predictors, and independent residuals. The models will be fitted to the training data (80% of the full data).

###Set seed
set.seed(123)

###Set parameters
n <- 600
p <- 1000
p_causal <- 500
r <- 5
r_causal <- r
pve <- 0.5
B_cor <- 1
X_cor <- 0
V_cor <- 0

###Simulate V, B, X and Y
out <- mr.mash.alpha:::simulate_mr_mash_data(n, p, p_causal, r, r_causal=r, intercepts = rep(1, r),
                                             pve=pve, B_cor=B_cor, B_scale=0.8, X_cor=X_cor, X_scale=0.8, V_cor=V_cor)

colnames(out$Y) <- paste0("Y", seq(1, r))
rownames(out$Y) <- paste0("N", seq(1, n))
colnames(out$X) <- paste0("X", seq(1, p))
rownames(out$X) <- paste0("N", seq(1, n))

###Split the data in training and test sets
test_set <- sort(sample(x=c(1:n), size=round(n*0.2), replace=FALSE))
Ytrain <- out$Y[-test_set, ]
Xtrain <- out$X[-test_set, ]
Ytest <- out$Y[test_set, ]
Xtest <- out$X[test_set, ]

We build the mixture prior as usual including zero matrix, identity matrix, rank-1 matrices, low, medium and high heterogeneity matrices, shared matrix, each scaled by a grid from 0.1 to 2.1 in steps of 0.2.

grid <- seq(0.1, 2.1, 0.2)
S0 <- mr.mash.alpha:::compute_cov_canonical(ncol(Ytrain), singletons=TRUE, hetgrid=c(0, 0.25, 0.5, 0.75, 1), grid, zeromat=TRUE)

We run glmnet with \(\alpha=1\) to obtain an inital estimate for the regression coefficients to provide to mr.mash, and for comparison.

###Fit group-lasso
cvfit_glmnet <- cv.glmnet(x=Xtrain, y=Ytrain, family="mgaussian", alpha=1)
coeff_glmnet <- coef(cvfit_glmnet, s="lambda.min")
Bhat_glmnet <- matrix(as.numeric(NA), nrow=p, ncol=r)
ahat_glmnet <- vector("numeric", r)
for(i in 1:length(coeff_glmnet)){
  Bhat_glmnet[, i] <- as.vector(coeff_glmnet[[i]])[-1]
  ahat_glmnet[i] <- as.vector(coeff_glmnet[[i]])[1]
}
Yhat_test_glmnet <- drop(predict(cvfit_glmnet, newx=Xtest, s="lambda.min"))

We run mr.mash with EM updates of the mixture weights, updating V, and initializing the regression coefficients with the estimates from glmnet.

###Fit mr.mash
fit_mrmash <- mr.mash(Xtrain, Ytrain, S0, tol=1e-2, convergence_criterion="ELBO", update_w0=TRUE, 
                     update_w0_method="EM", compute_ELBO=TRUE, standardize=TRUE, verbose=FALSE, update_V=TRUE,
                     mu1_init = Bhat_glmnet, w0_threshold=1e-8)

Processing the inputs... Done!
Fitting the optimization algorithm... Done!
Processing the outputs... Done!
mr.mash successfully executed in 2.71866 minutes!

Yhat_test_mrmash <- predict(fit_mrmash, Xtest)

Let’s now compare the results.

layout(matrix(c(1, 1, 2, 2,
                1, 1, 2, 2,
                0, 3, 3, 0,
                0, 3, 3, 0), 4, 4, byrow = TRUE))

###Plot estimated vs true coeffcients
##mr.mash
plot(out$B, fit_mrmash$mu1, main="mr.mash", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
##glmnet
plot(out$B, Bhat_glmnet, main="glmnet", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)

###Plot mr.mash vs glmnet estimated coeffcients
colorz <- matrix("black", nrow=p, ncol=r)
zeros <- apply(out$B, 2, function(x) x==0)
for(i in 1:ncol(colorz)){
  colorz[zeros[, i], i] <- "red"
}

plot(Bhat_glmnet, fit_mrmash$mu1, main="mr.mash vs glmnet", 
     xlab="glmnet estimated coefficients", ylab="mr.mash estimated coefficients",
     col=colorz, cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2)

Version	Author	Date
42ec51c	fmorgante	2020-06-01

print("Prediction performance with glmnet")

[1] "Prediction performance with glmnet"

compute_accuracy(Ytest, Yhat_test_glmnet)

$bias
[1] 0.7973441 0.7171240 1.1201318 0.9992656 0.9375002

$r2
[1] 0.12560024 0.10044653 0.14357016 0.13204939 0.09516285

$mse
[1] 446.4747 472.2412 509.8068 511.4644 642.4144

print("Prediction performance with mr.mash")

[1] "Prediction performance with mr.mash"

compute_accuracy(Ytest, Yhat_test_mrmash)

$bias
[1] 2.269604 2.196994 3.438804 2.832561 3.030053

$r2
[1] 0.1310534 0.1127552 0.2346756 0.1833728 0.1537069

$mse
[1] 460.6600 471.2208 525.7587 522.0806 649.4204

print("True V (full data)")

[1] "True V (full data)"

out$V

       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 294.59   0.00   0.00   0.00   0.00
[2,]   0.00 294.59   0.00   0.00   0.00
[3,]   0.00   0.00 294.59   0.00   0.00
[4,]   0.00   0.00   0.00 294.59   0.00
[5,]   0.00   0.00   0.00   0.00 294.59

print("Estimated V (training data)")

[1] "Estimated V (training data)"

fit_mrmash$V

         Y1       Y2       Y3       Y4       Y5
Y1 534.0861 241.6130 189.3974 175.6404 206.0833
Y2 241.6130 517.0130 217.6493 216.6763 227.4030
Y3 189.3974 217.6493 482.4770 224.0472 212.9069
Y4 175.6404 216.6763 224.0472 473.4047 190.4193
Y5 206.0833 227.4030 212.9069 190.4193 459.0911

It looks mr.mash shrinks the coeffcients too much and the residual covariance is absorbing part of the effects. Let’s now try to fix the residual covariance to the truth (although it is going to be a little different since we are fitting the model to the training set only).

###Fit mr.mash
fit_mrmash_fixV <- mr.mash(Xtrain, Ytrain, S0, tol=1e-2, convergence_criterion="ELBO", update_w0=TRUE, 
                     update_w0_method="EM", compute_ELBO=TRUE, standardize=TRUE, verbose=FALSE, update_V=FALSE,
                     mu1_init = Bhat_glmnet, w0_threshold=1e-8, V=out$V)

Processing the inputs... Done!
Fitting the optimization algorithm... Done!
Processing the outputs... Done!
mr.mash successfully executed in 3.421747 minutes!

Yhat_test_mrmash_fixV <- predict(fit_mrmash_fixV, Xtest)

Let’s compare the results again.

layout(matrix(c(1, 1, 2, 2,
                1, 1, 2, 2,
                0, 3, 3, 0,
                0, 3, 3, 0), 4, 4, byrow = TRUE))

###Plot estimated vs true coeffcients
##mr.mash
plot(out$B, fit_mrmash_fixV$mu1, main="mr.mash", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
##glmnet
plot(out$B, Bhat_glmnet, main="glmnet", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)

###Plot mr.mash vs glmnet estimated coeffcients
colorz <- matrix("black", nrow=p, ncol=r)
zeros <- apply(out$B, 2, function(x) x==0)
for(i in 1:ncol(colorz)){
  colorz[zeros[, i], i] <- "red"
}

plot(Bhat_glmnet, fit_mrmash_fixV$mu1, main="mr.mash vs glmnet", 
     xlab="glmnet estimated coefficients", ylab="mr.mash estimated coefficients",
     col=colorz, cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2)

Version	Author	Date
42ec51c	fmorgante	2020-06-01

print("Prediction performance with glmnet")

[1] "Prediction performance with glmnet"

compute_accuracy(Ytest, Yhat_test_glmnet)

$bias
[1] 0.7973441 0.7171240 1.1201318 0.9992656 0.9375002

$r2
[1] 0.12560024 0.10044653 0.14357016 0.13204939 0.09516285

$mse
[1] 446.4747 472.2412 509.8068 511.4644 642.4144

print("Prediction performance with mr.mash")

[1] "Prediction performance with mr.mash"

compute_accuracy(Ytest, Yhat_test_mrmash_fixV)

$bias
[1] 0.3667682 0.4788130 0.5380369 0.5509452 0.4881568

$r2
[1] 0.06928235 0.11834339 0.12732200 0.13692969 0.08741368

$mse
[1] 576.2796 535.1912 571.8264 567.9749 716.4623

Prediction performance looks worse in terms of MSE and \(r^2\) while bias, although still off, is closer to 1. However, looking at the plots of the coefficients, we can see that many coefficients are no longer overshrunk. Let’s see what happens if we initialize V to the truth but then we update it.

###Fit mr.mash
fit_mrmash_initV <- mr.mash(Xtrain, Ytrain, S0, tol=1e-2, convergence_criterion="ELBO", update_w0=TRUE, 
                     update_w0_method="EM", compute_ELBO=TRUE, standardize=TRUE, verbose=FALSE, update_V=TRUE,
                     mu1_init = Bhat_glmnet, w0_threshold=1e-8, V=out$V)

Processing the inputs... Done!
Fitting the optimization algorithm... Done!
Processing the outputs... Done!
mr.mash successfully executed in 2.818994 minutes!

Yhat_test_mrmash_initV <- predict(fit_mrmash_initV, Xtest)

Let’s look at the results.

layout(matrix(c(1, 1, 2, 2,
                1, 1, 2, 2,
                0, 3, 3, 0,
                0, 3, 3, 0), 4, 4, byrow = TRUE))

###Plot estimated vs true coeffcients
##mr.mash
plot(out$B, fit_mrmash_initV$mu1, main="mr.mash", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
##glmnet
plot(out$B, Bhat_glmnet, main="glmnet", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)

###Plot mr.mash vs glmnet estimated coeffcients
colorz <- matrix("black", nrow=p, ncol=r)
zeros <- apply(out$B, 2, function(x) x==0)
for(i in 1:ncol(colorz)){
  colorz[zeros[, i], i] <- "red"
}

plot(Bhat_glmnet, fit_mrmash_initV$mu1, main="mr.mash vs glmnet", 
     xlab="glmnet estimated coefficients", ylab="mr.mash estimated coefficients",
     col=colorz, cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2)

print("Prediction performance with glmnet")

[1] "Prediction performance with glmnet"

compute_accuracy(Ytest, Yhat_test_glmnet)

$bias
[1] 0.7973441 0.7171240 1.1201318 0.9992656 0.9375002

$r2
[1] 0.12560024 0.10044653 0.14357016 0.13204939 0.09516285

$mse
[1] 446.4747 472.2412 509.8068 511.4644 642.4144

print("Prediction performance with mr.mash")

[1] "Prediction performance with mr.mash"

compute_accuracy(Ytest, Yhat_test_mrmash_initV)

$bias
[1] 2.250388 2.175903 3.417905 2.814384 3.010519

$r2
[1] 0.1300321 0.1116303 0.2340150 0.1826561 0.1531588

$mse
[1] 460.7387 471.3525 525.6027 522.0123 649.3229

print("True V (full data)")

[1] "True V (full data)"

out$V

       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 294.59   0.00   0.00   0.00   0.00
[2,]   0.00 294.59   0.00   0.00   0.00
[3,]   0.00   0.00 294.59   0.00   0.00
[4,]   0.00   0.00   0.00 294.59   0.00
[5,]   0.00   0.00   0.00   0.00 294.59

print("Estimated V (training data)")

[1] "Estimated V (training data)"

fit_mrmash_initV$V

         Y1       Y2       Y3       Y4       Y5
Y1 534.0255 241.6133 189.3821 175.6086 206.0468
Y2 241.6133 517.0124 217.6608 216.6686 227.3917
Y3 189.3821 217.6608 482.4518 224.0279 212.8791
Y4 175.6086 216.6686 224.0279 473.3262 190.3769
Y5 206.0468 227.3917 212.8791 190.3769 458.9969

Here, we can see that that initilizing the residual covariance to the truth leads to essentially the same solutions as using the default initialization method.

For comparison let’s try to repeat the analysis above (updating V) on a sparser scenario, i.e., p_causal=50.

###Set parameters
n <- 600
p <- 1000
p_causal <- 50
r <- 5
r_causal <- r
pve <- 0.5
B_cor <- 1
X_cor <- 0
V_cor <- 0

###Simulate V, B, X and Y
out50 <- mr.mash.alpha:::simulate_mr_mash_data(n, p, p_causal, r, r_causal=r, intercepts = rep(1, r),
                                             pve=pve, B_cor=B_cor, B_scale=0.8, X_cor=X_cor, X_scale=0.8, V_cor=V_cor)

colnames(out50$Y) <- paste0("Y", seq(1, r))
rownames(out50$Y) <- paste0("N", seq(1, n))
colnames(out50$X) <- paste0("X", seq(1, p))
rownames(out50$X) <- paste0("N", seq(1, n))

###Split the data in training and test sets
test_set50 <- sort(sample(x=c(1:n), size=round(n*0.2), replace=FALSE))
Ytrain50 <- out50$Y[-test_set50, ]
Xtrain50 <- out50$X[-test_set50, ]
Ytest50 <- out50$Y[test_set50, ]
Xtest50 <- out50$X[test_set50, ]

###Fit group-lasso
cvfit_glmnet50 <- cv.glmnet(x=Xtrain50, y=Ytrain50, family="mgaussian", alpha=1)
coeff_glmnet50 <- coef(cvfit_glmnet50, s="lambda.min")
Bhat_glmnet50 <- matrix(as.numeric(NA), nrow=p, ncol=r)
ahat_glmnet50 <- vector("numeric", r)
for(i in 1:length(coeff_glmnet50)){
  Bhat_glmnet50[, i] <- as.vector(coeff_glmnet50[[i]])[-1]
  ahat_glmnet50[i] <- as.vector(coeff_glmnet50[[i]])[1]
}
Yhat_test_glmnet50 <- drop(predict(cvfit_glmnet50, newx=Xtest50, s="lambda.min"))

###Fit mr.mash
fit_mrmash50 <- mr.mash(Xtrain50, Ytrain50, S0, tol=1e-2, convergence_criterion="ELBO", update_w0=TRUE, 
                     update_w0_method="EM", compute_ELBO=TRUE, standardize=TRUE, verbose=FALSE, update_V=TRUE,
                     mu1_init = Bhat_glmnet50, w0_threshold=1e-8)

Processing the inputs... Done!
Fitting the optimization algorithm... Done!
Processing the outputs... Done!
mr.mash successfully executed in 1.083553 minutes!

Yhat_test_mrmash50 <- predict(fit_mrmash50, Xtest50)

Let’s look at the results.

layout(matrix(c(1, 1, 2, 2,
                1, 1, 2, 2,
                0, 3, 3, 0,
                0, 3, 3, 0), 4, 4, byrow = TRUE))

###Plot estimated vs true coeffcients
##mr.mash
plot(out50$B, fit_mrmash50$mu1, main="mr.mash", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
##glmnet
plot(out50$B, Bhat_glmnet50, main="glmnet", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)

###Plot mr.mash vs glmnet estimated coeffcients
colorz <- matrix("black", nrow=p, ncol=r)
zeros <- apply(out50$B, 2, function(x) x==0)
for(i in 1:ncol(colorz)){
  colorz[zeros[, i], i] <- "red"
}

plot(Bhat_glmnet50, fit_mrmash50$mu1, main="mr.mash vs glmnet", 
     xlab="glmnet estimated coefficients", ylab="mr.mash estimated coefficients",
     col=colorz, cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2)

Version	Author	Date
42ec51c	fmorgante	2020-06-01

print("Prediction performance with glmnet")

[1] "Prediction performance with glmnet"

compute_accuracy(Ytest50, Yhat_test_glmnet50)

$bias
[1] 1.1055781 1.2342390 0.8929952 1.2093933 1.1997154

$r2
[1] 0.3720343 0.4451268 0.2709590 0.3909355 0.4424055

$mse
[1] 50.11301 49.98023 54.17894 53.96508 46.36995

print("Prediction performance with mr.mash")

[1] "Prediction performance with mr.mash"

compute_accuracy(Ytest50, Yhat_test_mrmash50)

$bias
[1] 0.9295387 1.1205934 0.8382439 0.9741666 1.0034935

$r2
[1] 0.4227190 0.5544520 0.3641726 0.4142314 0.4747928

$mse
[1] 46.04314 39.62819 48.00426 51.00798 42.98684

print("True V (full data)")

[1] "True V (full data)"

out50$V

       [,1]   [,2]   [,3]   [,4]   [,5]
[1,] 43.262  0.000  0.000  0.000  0.000
[2,]  0.000 43.262  0.000  0.000  0.000
[3,]  0.000  0.000 43.262  0.000  0.000
[4,]  0.000  0.000  0.000 43.262  0.000
[5,]  0.000  0.000  0.000  0.000 43.262

print("Estimated V (training data)")

[1] "Estimated V (training data)"

fit_mrmash50$V

           Y1        Y2         Y3        Y4         Y5
Y1 37.0783688  2.693322  2.7229456  3.426152 -0.6935028
Y2  2.6933223 42.243137  2.0293575  1.966294  2.6486737
Y3  2.7229456  2.029357 37.9331012 -2.456907 -0.2269764
Y4  3.4261520  1.966294 -2.4569072 42.752140 -3.0717702
Y5 -0.6935028  2.648674 -0.2269764 -3.071770 43.5576891

The results of this simulation look very good in terms of prediction performance, estimation of the coefficients, and estimation of the residual covariance.

Just as an additional test let’s now try to maintain the same signal-to-noise ratio as the first simulation but with fewer total variables, i.e., p=500 and p_causal=250.

###Set parameters
n <- 600
p <- 500
p_causal <- 250
r <- 5
r_causal <- r
pve <- 0.5
B_cor <- 1
X_cor <- 0
V_cor <- 0

###Simulate V, B, X and Y
out500 <- mr.mash.alpha:::simulate_mr_mash_data(n, p, p_causal, r, r_causal=r, intercepts = rep(1, r),
                                             pve=pve, B_cor=B_cor, B_scale=0.8, X_cor=X_cor, X_scale=0.8, V_cor=V_cor)

colnames(out500$Y) <- paste0("Y", seq(1, r))
rownames(out500$Y) <- paste0("N", seq(1, n))
colnames(out500$X) <- paste0("X", seq(1, p))
rownames(out500$X) <- paste0("N", seq(1, n))

###Split the data in training and test sets
test_set500 <- sort(sample(x=c(1:n), size=round(n*0.2), replace=FALSE))
Ytrain500 <- out500$Y[-test_set500, ]
Xtrain500 <- out500$X[-test_set500, ]
Ytest500 <- out500$Y[test_set500, ]
Xtest500 <- out500$X[test_set500, ]

###Fit group-lasso
cvfit_glmnet500 <- cv.glmnet(x=Xtrain500, y=Ytrain500, family="mgaussian", alpha=1)
coeff_glmnet500 <- coef(cvfit_glmnet500, s="lambda.min")
Bhat_glmnet500 <- matrix(as.numeric(NA), nrow=p, ncol=r)
ahat_glmnet500 <- vector("numeric", r)
for(i in 1:length(coeff_glmnet500)){
  Bhat_glmnet500[, i] <- as.vector(coeff_glmnet500[[i]])[-1]
  ahat_glmnet500[i] <- as.vector(coeff_glmnet500[[i]])[1]
}
Yhat_test_glmnet500 <- drop(predict(cvfit_glmnet500, newx=Xtest500, s="lambda.min"))

###Fit mr.mash
fit_mrmash500 <- mr.mash(Xtrain500, Ytrain500, S0, tol=1e-2, convergence_criterion="ELBO", update_w0=TRUE, 
                     update_w0_method="EM", compute_ELBO=TRUE, standardize=TRUE, verbose=FALSE, update_V=TRUE,
                     mu1_init = Bhat_glmnet500, w0_threshold=1e-8)

Processing the inputs... Done!
Fitting the optimization algorithm... Done!
Processing the outputs... Done!
mr.mash successfully executed in 1.406657 minutes!

Yhat_test_mrmash500 <- predict(fit_mrmash500, Xtest500)

Let’s look at the results.

layout(matrix(c(1, 1, 2, 2,
                1, 1, 2, 2,
                0, 3, 3, 0,
                0, 3, 3, 0), 4, 4, byrow = TRUE))

###Plot estimated vs true coeffcients
##mr.mash
plot(out500$B, fit_mrmash500$mu1, main="mr.mash", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
##glmnet
plot(out500$B, Bhat_glmnet500, main="glmnet", xlab="True coefficients", ylab="Estimated coefficients",
     cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)

###Plot mr.mash vs glmnet estimated coeffcients
colorz <- matrix("black", nrow=p, ncol=r)
zeros <- apply(out500$B, 2, function(x) x==0)
for(i in 1:ncol(colorz)){
  colorz[zeros[, i], i] <- "red"
}

plot(Bhat_glmnet500, fit_mrmash500$mu1, main="mr.mash vs glmnet", 
     xlab="glmnet estimated coefficients", ylab="mr.mash estimated coefficients",
     col=colorz, cex=2, cex.lab=1.8, cex.main=2, cex.axis=1.8)
legend("topleft", 
       legend = c("Non-zero", "Zero"), 
       col = c("black", "red"), 
       pch = c(1, 1), 
       horiz = FALSE,
       cex=2)

Version	Author	Date
42ec51c	fmorgante	2020-06-01

print("Prediction performance with glmnet")

[1] "Prediction performance with glmnet"

compute_accuracy(Ytest500, Yhat_test_glmnet500)

$bias
[1] 0.7460558 0.8379627 0.8827708 1.0150154 0.8988217

$r2
[1] 0.1566151 0.1754140 0.2246133 0.2126951 0.2354353

$mse
[1] 262.5938 307.5913 280.3553 259.2769 265.2623

print("Prediction performance with mr.mash")

[1] "Prediction performance with mr.mash"

compute_accuracy(Ytest500, Yhat_test_mrmash500)

$bias
[1] 0.8007963 1.0543346 0.9437460 0.9812104 1.0269983

$r2
[1] 0.1917870 0.2740736 0.2402338 0.2626540 0.2816563

$mse
[1] 249.0113 269.0810 268.4814 242.8198 247.6815

print("True V (full data)")

[1] "True V (full data)"

out500$V

         [,1]     [,2]     [,3]     [,4]     [,5]
[1,] 168.3364   0.0000   0.0000   0.0000   0.0000
[2,]   0.0000 168.3364   0.0000   0.0000   0.0000
[3,]   0.0000   0.0000 168.3364   0.0000   0.0000
[4,]   0.0000   0.0000   0.0000 168.3364   0.0000
[5,]   0.0000   0.0000   0.0000   0.0000 168.3364

print("Estimated V (training data)")

[1] "Estimated V (training data)"

fit_mrmash500$V

           Y1         Y2        Y3        Y4         Y5
Y1 190.385218   4.826421  31.00999  22.18042   5.743413
Y2   4.826421 181.999035  16.65374   8.40475   9.590063
Y3  31.009989  16.653736 198.77957  39.23005  35.494926
Y4  22.180421   8.404750  39.23005 209.04480  25.167324
Y5   5.743413   9.590063  35.49493  25.16732 194.837000

In this last simulation, we can see that mr.mash performs pretty well, even in a denser scenario with fewer variables.

sessionInfo()

R version 3.5.1 (2018-07-02)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Scientific Linux 7.4 (Nitrogen)

Matrix products: default
BLAS/LAPACK: /software/openblas-0.2.19-el7-x86_64/lib/libopenblas_haswellp-r0.2.19.so

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] glmnet_2.0-16        foreach_1.4.4        Matrix_1.2-15       
[4] mr.mash.alpha_0.1-75

loaded via a namespace (and not attached):
 [1] MBSP_1.0           Rcpp_1.0.4.6       compiler_3.5.1    
 [4] later_0.7.5        git2r_0.26.1       workflowr_1.6.1   
 [7] iterators_1.0.10   tools_3.5.1        digest_0.6.25     
[10] evaluate_0.12      lattice_0.20-38    GIGrvg_0.5        
[13] yaml_2.2.1         SparseM_1.77       mvtnorm_1.0-12    
[16] coda_0.19-3        stringr_1.4.0      knitr_1.20        
[19] fs_1.3.1           MatrixModels_0.4-1 rprojroot_1.3-2   
[22] grid_3.5.1         glue_1.4.0         R6_2.4.1          
[25] rmarkdown_1.10     mixsqp_0.3-43      irlba_2.3.3       
[28] magrittr_1.5       whisker_0.3-2      codetools_0.2-15  
[31] backports_1.1.5    promises_1.0.1     htmltools_0.3.6   
[34] matrixStats_0.55.0 mcmc_0.9-6         MASS_7.3-51.1     
[37] httpuv_1.4.5       quantreg_5.36      stringi_1.4.3     
[40] MCMCpack_1.4-4

Dense case

Fabio Morgante

June 02, 2020