## Lect1, 30 Aug 2016 


# Decompose a two-way table into an additive pattern plus residual 
## Vectors need not look like columns

y <- c(5,6,8,4,5.1,7.2)
Y <- matrix(y,
			byrow=T,nrow=2)

## What is the difference?			
Y1 <- as.vector(Y)
Y2 <- t(Y1)
Y3 <- t(Y2)

# Which are vectors (row or column?), which are matrices?
Y; is.matrix(Y); dim(Y)
Y1; is.matrix(Y1); dim(Y1)
Y2; is.matrix(Y2); dim(Y2)
Y3; is.matrix(Y3); dim(Y3)

# least squares fit

ROW <- factor( c("R1","R1","R1","R2","R2","R2") )
COL <- factor( c("C1","C2","C3","C1","C2","C3") )

mydata <- data.frame(y, ROW,COL)
rowm <- matrix(ROW,byrow=T,nrow=2)
colm <- matrix(COL,byrow=T,nrow=2)

# different ways to think about the ``data''
mydata; Y; rowm; colm

# least squares fit
syd <- lm(y ~ ROW + COL,data=mydata)

# How R sees the exercise:
names(syd)
syd$model
mm <- model.matrix(syd)

matrix(mm[,1],byrow=T,nrow=2)
matrix(mm[,2],byrow=T,nrow=2)
matrix(mm[,3],byrow=T,nrow=2);
matrix(mm[,4],byrow=T,nrow=2)

thefit <- matrix(syd$fit,byrow=T,nrow=2) # an additive pattern
theresid <- round(matrix(syd$resid,byrow=T,nrow=2),3)
thefit + theresid

# Come back to this stuff later.
make.onb <- function(myqr=syd$qr,as.tables=F){
	Q <- qr.Q(myqr)
	if ( !as.tables){
		return( Q )
	}else{
		qr.tables <- list()
		thedim <- myqr$rank
		for (j in seq(thedim) )
		{
			qr.tables[[j]] <- matrix(Q[,j],byrow=T,nrow=2)
		}
		return(qr.tables)		
	}
}


# might come in handy:
ss <- make.onb()
round( t(ss) %*% ss,3)

ss <- make.onb(as.tables = T) 
lapply(ss,round,2)