Univariate regression with some visuals to get the idea of what a transformation looks like

Faux PCA.R

x<-c(1:50)
y<-x+rnorm(50,sd=10)

plot(x,y,pch=20,main="Distance from Fitted Line")
abline(coef(lm(y~x)[1],coef(lm(y~x)[2])),col="blue",lwd=2)
segments(x,predict(lm(y~x)),y1=lm(y~x)$model[,1])

#This bit took longer to do than I care to admit.  Basically you want distance from the data point to the line 
#as an adjacent side to the triangle that would have the vertical distance as the hypotenuse.
#Because I don't know how to tell R to draw a segment with a given slope and starting point, I have to give
#the start and end points.  In order to get that I compute the rise and run via similar triangles.  
#x.shift and y.shift do most of the work in a vectorized computation.
x.shift<- sin(atan(-1/coef(lm(y~x))[2]))*cos(atan(-1/coef(lm(y~x))[2]))*sqrt((simple.y[,2]-simple.pred[,2])^2)
y.shift<- sin(atan(-1/coef(lm(y~x))[2]))*sin(atan(-1/coef(lm(y~x))[2]))*sqrt((simple.y[,2]-simple.pred[,2])^2)
#The for loop is here because we need to change the rise/run from positive to negative depending on whether or not 
#the data point is above or below the regression line.  
for (i in 1:50) {
		if (simple.y[i,2]>simple.pred[i,2]) {
			x.shift[i]<- -x.shift[i]
			y.shift[i]<- -y.shift[i]
			}
		}
		
plot(x,y,pch=20,main="Transformed Distance from Fitted Line")
abline(coef(lm(y~x)[1],coef(lm(y~x)[2])),col="blue",lwd=2)
segments(x,y,x0=x+x.shift,y0=y+y.shift)