Chapter_5.Rmd

### Statistical Learning with Math and R, Springer ###

# Chapter 5 Information Criteria

## 5.1 Information Criteria 

```{r}
RSS.min=function(X,y,T){
  m=ncol(T); S.min=Inf
  for(j in 1:m){
    q=T[,j]
	S=sum((lm(y~X[,q])$fitted.values-y)^2)
  if(S<S.min){S.min=S; set.q=q}
  }
return(list(value=S.min,set=set.q))
}
library(MASS)
df=Boston
X=as.matrix(df[,c(1,3,5,6,7,8,10,11,12,13)])
y=df[[14]]
p=ncol(X)
n=length(y)
AIC.min=Inf
for(k in 1:p){
  T=combn(1:p,k)
  res=RSS.min(X,y,T)
  AIC= n*log(res$value/n)+2*k ##
  if(AIC<AIC.min){
    AIC.min=AIC
	set.min= res$set
  }
}
AIC.min
set.min

y.bar=mean(y)
TSS=sum((y-y.bar)^2)
D.max=-Inf
for(k in 1:p){
  T=combn(1:p,k)
  res=RSS.min(X,y,T)
  D= 1-res$value/(n-k-1)/TSS/(n-1)
  if(D>D.max){
    D.max=D
	set.max= res$set
  }
}
D.max
set.max

library(MASS)
df=Boston
X=as.matrix(df[,c(1,3,5,6,7,8,10,11,12,13)])
y=df[[14]]
n=nrow(X)
p=ncol(X)
IC=function(k){
  T=combn(1:p,k)
  res=RSS.min(X,y,T)
  AIC= n*log(res$value/n)+2*k 
  BIC= n*log(res$value/n)+k*log(n)
  return(list(AIC=AIC,BIC=BIC))
}
AIC.seq=NULL
BIC.seq=NULL
for(k in 1:p){
  AIC.seq=c(AIC.seq, IC(k)$AIC)
  BIC.seq=c(BIC.seq, IC(k)$BIC)
}
plot(1:p, ylim=c(min(AIC.seq),max(BIC.seq)), type="n",xlab="# of variables", ylab="IC values")
lines(AIC.seq,col="red"); lines(BIC.seq,col="blue")
legend("topright",legend=c("AIC","BIC"), col=c("red","blue"), lwd=1, cex =.8)
```
# 4.2 Efficient Estimator and Fisher Information Matrix
# 4.3 Kullback-Leibler divergence
# 4.4 Derivation of Akaike's Information Criterion