Recursive Estimation and Time Series analysis

# Recursive Estimation and Time Series analysis

# RLS(Recursive Linear Least Square) p.33

library(dplyr)

data=data.frame(anscombe)

char=c("x1","x2","x3","x4")

Y=data$y2

X=as.matrix(data[,colnames(data) %in% char])

# times=5

# for(j in 1:times){
  
# X=rbind(X,X)

# Y=c(Y,Y)
  
# }

a=as.matrix(rep(1,ncol(X)),dim=c(ncol(X),1))

P=diag(0,ncol(X))

for(k in 1:nrow(data)){
  
P=P+t(t(c(X[k,])))%*%t(c(X[k,])) 

P_mat=eigen(P)$vectors%*%diag(1/eigen(P)$values)%*%t(eigen(P)$vectors)

sig=sum((Y[1:k]-X[1:k,]%*%a)^2)/(nrow(data)-ncol(data)-1)

P_star=sig*P_mat
  
e=Y[k]-t(c(X[k,]))%*%a  

print(P_mat%*%t(t(c(X[k,]))))

a=a+P_mat%*%t(t(c(X[k,])))*as.numeric(e)

}

a_hat=eigen(t(X)%*%X)$vectors%*%diag(1/eigen(t(X)%*%X)$values)%*%t(eigen(t(X)%*%X)$vectors)%*%t(X)%*%Y

cbind(Y,X%*%a,X%*%a_hat)

cor(cbind(Y,X%*%a,X%*%a_hat))