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Copy pathL6 Selection bias and shrinkage (HWK Version).R
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L6 Selection bias and shrinkage (HWK Version).R
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# Simulation to show bias & MSPE in variable selection and shrinkage
#- Generate data from lin reg model with 1 important var and 9 zeroes
#- Fit MLR and estimate all parameters --- show properties
#- Use stepwise to select best model
#- Estimate parameters only if chosen, otherwise record 0
#- show properties
#- Use LASSO to select and estimate
#- Show properties
library(stringr)
library(glmnet)
library(MASS)
set.seed(3014312)
n = 50 # PLAY WITH THIS NUMBER
p = 10 # PLAY WITH THIS NUMBER
beta1 = 1
sigma = 3 # PLAY WITH THIS NUMBER
iter = 100
coefs.lm = matrix(NA, nrow=iter, ncol=p+1)
coefs.sw = matrix(NA, nrow=iter, ncol=p+1)
coefs.ri = matrix(NA, nrow=iter, ncol=p+1)
coefs.la = matrix(NA, nrow=iter, ncol=p+1)
MSPEs = matrix(NA, nrow=iter, ncol=4)
colnames(MSPEs) = c("LM", "STEP", "RIDGE", "LASSO")
testx1 = rnorm(n=1000)
testx = cbind(testx1, matrix(0, ncol=(p-1), nrow=length(testx1)))
testx.all = cbind(testx1, matrix(rnorm(n=(p-1)*length(testx1)), ncol=(p-1), nrow=length(testx1)))
noms = c("X1")
for(q in 2:p){
noms = c(noms,paste0("X",q))
}
colnames(testx) = noms
colnames(testx.all) = colnames(testx)
testy = testx1*beta1
#Example Data Set
x = matrix(rnorm(n=n*p), nrow=n)
eps = rnorm(n, 0, sigma)
y = beta1*x[,1] + eps
x11()
curve(expr=beta1*x, from=-3, to=3,
ylim=c(-8,8),
col="red", lwd=3, xlab="X", ylab="Y",
main=paste("One data set with n=",n, "\n Population R-squared=",
round(1/(1+sigma^2),2)))
points(x=x[,1], y=y, pch="X", col="blue")
for(i in 1:iter){
x = matrix(rnorm(n=n*p), nrow=n)
eps = rnorm(n, 0, sigma)
y = beta1*x[,1] + eps
xydat = data.frame(y,x)
mod.lm = lm(y~., data=xydat)
coefs.lm[i,] = coef(mod.lm)
step1 <- step(object=lm(y~1, data=xydat), scope=list(upper=mod.lm), direction="both",
k = log(nrow(xydat)), trace=0)
coef.locs = c(1, 1+as.numeric(str_remove_all(names(coef(step1))[-1], "X")))
coefs.sw[i,] = 0
coefs.sw[i,coef.locs] = coef(step1)
ridgec <- lm.ridge(y ~., lambda = seq(0, 100, .05), data=xydat)
coefs.ri[i,] = coef(ridgec)[which.min(ridgec$GCV),]
cv.lasso.1 <- cv.glmnet(y=y, x= x, family="gaussian")
coefs.la[i,] = coef(cv.lasso.1)[,1]
pred.lm = predict(mod.lm, as.data.frame(testx.all))
pred.sw = predict(step1, as.data.frame(testx.all))
pred.ri = as.matrix(cbind(1,testx.all)) %*% coef(ridgec)[which.min(ridgec$GCV),]
pred.la = predict(cv.lasso.1, testx.all, s="lambda.min")
MSPEs[i,] = c(mean((testy-pred.lm)^2),
mean((testy-pred.sw)^2),
mean((testy-pred.ri)^2),
mean((testy-pred.la)^2))
}
#x11()
#boxplot(MSPEs[,1:2],
# main=paste0("Comparison of MSPEs\n R-squared=",
# round(1/(1+sigma^2),2), ", n=",n,", p=",p),
# names=c("lm","step"))
#min.t = apply(MSPEs[,1:2],1,min)
#x11()
#boxplot(MSPEs[,1:2]/min.t,
# main=paste0("Comparison of Relative MSPEs\n R-squared=",
# round(1/(1+sigma^2),2), ", n=",n,", p=",p),
# names=c("lm","step"))
#x11()
#boxplot(MSPEs[,1:3],
# main=paste0("Comparison of MSPEs\n R-squared=",
# round(1/(1+sigma^2),2), ", n=",n,", p=",p),
# names=c("lm","step", "ridge"))
#min.t = apply(MSPEs[,1:3],1,min)
#x11()
#boxplot(MSPEs[,1:3]/min.t,
# main=paste0("Comparison of Relative MSPEs\n R-squared=",
# round(1/(1+sigma^2),2), ", n=",n,", p=",p),
# names=c("lm","step", "ridge"))
x11()
boxplot(MSPEs,
main=paste0("Comparison of MSPEs\n R-squared=",
round(1/(1+sigma^2),2), ", n=",n,", p=",p),
names=c("lm","step", "ridge","LASSO"))
#min.t = apply(MSPEs,1,min)
#x11()
#boxplot(MSPEs/min.t,
# main=paste0("Comparison of Relative MSPEs\n R-squared=",
# round(1/(1+sigma^2),2), ", n=",n,", p=",p),
# names=c("lm","step", "ridge","LASSO"))
MSE = matrix(NA, nrow=p+1, ncol=4)
for(j in 2:3){
truec = ifelse(j-1==1, 1, 0)
x11(height=7, width=7)
boxplot(cbind(coefs.lm[,j], coefs.sw[,j], coefs.ri[,j], coefs.la[,j]),
main=paste0("Comparison of coefs for variable ",j-1,
"\n R-squared=",round(1/(1+sigma^2),2), ", n=",n,", p=",p),
names=c("lm","step","Ridge", "LASSO"))
abline(h=truec, col="red")
points(x=1, y=mean(coefs.lm[,j]), col="blue")
points(x=2, y=mean(coefs.sw[,j]), col="blue")
points(x=3, y=mean(coefs.ri[,j]), col="blue")
points(x=4, y=mean(coefs.la[,j]), col="blue")
# x11(height=7, width=7)
# boxplot(cbind(coefs.lm[,j], coefs.sw[,j], coefs.ri[,j]),
# main=paste0("Comparison of coefs for variable ",j-1,
# "\n R-squared=",round(1/(1+sigma^2),2), ", n=",n,", p=",p),
# names=c("lm","step","Ridge"))
# abline(h=truec, col="red")
# points(x=1, y=mean(coefs.lm[,j]), col="blue")
# points(x=2, y=mean(coefs.sw[,j]), col="blue")
# points(x=3, y=mean(coefs.ri[,j]), col="blue")
MSE[j,]=round(c(mean((coefs.lm[,j]-truec)^2),
mean((coefs.sw[,j]-truec)^2),
mean((coefs.ri[,j]-truec)^2),
mean((coefs.la[,j]-truec)^2)), 3)
# x11(height=7, width=7)
# boxplot(cbind(coefs.lm[,j], coefs.sw[,j]),
# main=paste0("Comparison of coefs for variable ",j-1,
# "\n R-squared=",round(1/(1+sigma^2),2), ", n=",n,", p=",p),
# names=c("lm","step"))
# abline(h=truec, col="red")
# points(x=1, y=mean(coefs.lm[,j]), col="blue")
# points(x=2, y=mean(coefs.sw[,j]), col="blue")
# x11(h=5, w=10)
# par(mfrow = c(1,2))
# hist(coefs.lm[,j], breaks = seq(-2.25, 2.25, 0.5)+truec, ylim=c(0,100),
# main=paste0("Histogram of beta-hat", j, " from LS"))
# abline(v=truec, col="red")
# abline(v=mean(coefs.lm[,j]), col="blue")
# hist(coefs.sw[,j], breaks = seq(-2.25, 2.25, 0.5)+truec, ylim=c(0,100),
# main=paste0("Histogram of beta-hat", j, " from Step"))
# abline(v=truec, col="red")
# abline(v=mean(coefs.sw[,j]), col="blue")
# x11(h=5, w=10)
# par(mfrow = c(1,2))
# hist(coefs.ri[,j], breaks = seq(-2.25, 2.25, 0.5)+truec, ylim=c(0,100),
# main=paste0("Histogram of beta-hat", j, " from Ridge"))
# abline(v=truec, col="red")
# abline(v=mean(coefs.ri[,j]), col="blue")
# hist(coefs.la[,j], breaks = seq(-2.25, 2.25, 0.5)+truec, ylim=c(0,100),
# main=paste0("Histogram of beta-hat", j, " from LASSO"))
# abline(v=truec, col="red")
# abline(v=mean(coefs.la[,j]), col="blue")
}
MSE
nonz = function(x){mean(x!=0)}
# Here are the proportion of nonzero estimates for Stepwise and LASSO
round(rbind(
apply(X=coefs.lm, MARGIN=2, FUN=nonz),
apply(X=coefs.sw, MARGIN=2, FUN=nonz),
apply(X=coefs.la, MARGIN=2, FUN=nonz)), 2)
###############################################################
preds.ls.all = coefs.lm %*% t(cbind(1,testx.all))
preds.sw.all = coefs.sw %*% t(cbind(1,testx.all))
preds.ri.all = coefs.ri %*% t(cbind(1,testx.all))
preds.la.all = coefs.la %*% t(cbind(1,testx.all))
x11(h=7, w=12)
par(mfrow=c(1,2))
# Plot this line
curve(expr=beta1*x, from=-3, to=3,
ylim=c(-8,8),
col="red", lwd=3, xlab="X", ylab="Y",
main=paste("Predictions from LS when p=",p, " n=",n))
for(i in 1:iter){
points(x=testx1, y=preds.ls.all[i,], cex=0.2, col="lightblue")
}
abline(a=0, b=1, col="red")
meanp.ls.all = apply(preds.ls.all, 2, mean)
points(x=testx1, y=meanp.ls.all, cex=0.3, pch=20, col="blue")
# Plot this line
curve(expr=beta1*x, from=-3, to=3,
ylim=c(-8,8),
col="red", lwd=3, xlab="X", ylab="Y",
main=paste("Predictions from step() when p=",p, " n=",n))
for(i in 1:iter){
points(x=testx1, y=preds.sw.all[i,], cex=0.2, col="lightblue")
}
abline(a=0, b=1, col="red")
meanp.sw.all = apply(preds.sw.all, 2, mean)
points(x=testx1, y=meanp.sw.all, cex=0.3, pch=20, col="blue")
x11(h=7, w=12)
par(mfrow=c(1,2))
# Plot this line
curve(expr=beta1*x, from=-3, to=3,
ylim=c(-8,8),
col="red", lwd=3, xlab="X", ylab="Y",
main=paste("Predictions from Ridge when p=",p, " n=",n))
for(i in 1:iter){
points(x=testx1, y=preds.ri.all[i,], cex=0.2, col="lightblue")
}
abline(a=0, b=1, col="red")
meanp.ri.all = apply(preds.ri.all, 2, mean)
points(x=testx1, y=meanp.ri.all, cex=0.3, pch=20, col="blue")
# Plot this line
curve(expr=beta1*x, from=-3, to=3,
ylim=c(-8,8),
col="red", lwd=3, xlab="X", ylab="Y",
main=paste("Predictions from LASSO when p=",p, " n=",n))
for(i in 1:iter){
points(x=testx1, y=preds.la.all[i,], cex=0.2, col="lightblue")
}
abline(a=0, b=1, col="red")
meanp.la.all = apply(preds.la.all, 2, mean)
points(x=testx1, y=meanp.la.all, cex=0.3, pch=20, col="blue")