save_figs = FALSE

source('../Chapter4/exercise_4_5.R')
m = 4
y_mean = mean(DF$MPG)
y_std = sd(DF$MPG)

# Part (a): PC regression:
#
CXY = cor(DF)
print(CXY) # notice no r_ij > 0.95 (two are very close however)

ES = eigen(CXY[ 1:m, 1:m ])
print('Eigenvalues:')
print(ES$values) # one eigenvalue is less than 0.05

# Compute the principal components (and a scatter plot of them):
#
DFs = scale(DF)

P = ES$vectors
Z = as.matrix(DFs[ , 1:m ]) %*% P
colnames(Z) = sprintf('Z%d', seq(1, m))
print(summary(Z))

if( save_figs ){ postscript("../../WriteUp/Graphics/Chapter5/ex_5_16_scatter_plot_of_Z.eps", onefile=FALSE, horizontal=FALSE) }
pairs(Z, xlim=c(-3, +3), ylim=c(-3, +3))
if( save_figs ){ dev.off() }

# Part (b):
#
# PC regression:
#
#source('../../../Hastie/Code/Chapter3/pcr_wwx.R')
source('http://www.waxworksmath.com/Authors/G_M/Hastie/Code/Chapter3/pcr_wwx.R')
res = pcr_wwx(DF, DF)
beta_pc_z = res[[1]][, m-1]

pt_1 = sprintf('PC model: MPG = %+.3f', y_mean)
c_names = colnames(DF)[1:m]
z_names = paste(c_names, 'Z', sep='')
pt_2 = paste(sprintf('%+.3f %s', beta_pc_z, z_names), sep='', collapse=' ')
print(sprintf('%s %s', pt_1, pt_2))

# Ridge Regression:
#
cv = 0.05

XTX = CXY[1:m, 1:m]
XTy = CXY[1:m, m+1]
beta_ridge_z = solve(XTX + cv*diag(m), XTy) # the coefficients of the standardized predictors

pt_1 = sprintf('Ridge model (c=%.2f): MPG = %+.3f', cv, y_mean)
c_names = names(beta_ridge_z)
z_names = paste(c_names, 'Z', sep='')
pt_2 = paste(sprintf('%+.3f %s', beta_ridge_z * y_std, z_names), sep='', collapse=' ')
print(sprintf('%s %s', pt_1, pt_2))