#=================================================================
# Figure 15.2.R.
# R script to calculate least squares estimates for pronghorn data.
# Variable for prediction is y, spring fawn count.
# Predictor variables are size of adult pronghorn population (u),
# annual inches precipitation (v), winter severity index (w;  scale
# of 1:mild-5:severe. 
# Script produces a time plot of the data along with overlayed
# predictions.
#================================================================= 

#-----------------------------------------------------
# Input the data.
#-----------------------------------------------------
Pronghorn=read.table("pronghorn.txt",header=TRUE) # Change file name
                                                  #  if necessary.
attach(Pronghorn)

#-----------------------------------------------------
# Calculate least squares intercept and slope.
#-----------------------------------------------------
n=length(y)               # Number of observations is n.
X=matrix(1,n,4)           # Form the X matrix:  col 1 has 1’s,
X[,2]=u                   #  cols 2-4 have predictor variables.
X[,3]=v                   #       ---
X[,4]=w                   #       ---
b=solve(t(X)%*%X,t(X)%*%y)   # Least squares estimates in b;
                              #  t( ) is transpose function.
                              # Alternatively can use
                              #  b=solve((t(X)%*%X)%*%t(X)%*%y .

#-----------------------------------------------------
# Draw a time plot of data, with superimposed least
# squares predictions.
#-----------------------------------------------------
plot((1:8),y,type="o",xlab="time (yr)",
  ylab="spring fawn count")   # Time plot of data.


ypredict=X%*%b  # Calculate predicted y values at values of predictors. 


points((1:8),ypredict,type="p",pch=2)  # Plot predicted values
                                       #  with points.

#-----------------------------------------------------
# Print the least squares estimates to the console.
#-----------------------------------------------------
"least squares intercept and coefficients: "
b