source('../../Data/data_loaders.R')

# Part (a):
#
# Do the computation "by hand":
# EPage 123
#
# Order of variables is: CLOT PROG ENZ LIV
#
data = c( 1, 0.09, -0.15, 0.502, 0.09, 1, -0.024, 0.369,
      -0.15, -0.024, 1, 0.416, 0.502, 0.369, 0.416, 1 )
XTX = matrix( data=data, nrow=4, ncol=4, byrow=FALSE )
colnames(XTX) = c("CLOT", "FROG", "ENZ", "LIV")
rownames(XTX) = c("CLOT", "PROG", "ENZ", "LIV")

XTy = c( 0.373, 0.554, 0.58, 0.722 ) # these are the correlations with ( y - y bar J ) and the standardized features

by_hand = solve( XTX, XTy )

print(XTX)
print(XTy)

# Check our results using the function lm:
#
DF = load_appendix_survival_data()

cm = colMeans(DF)
cs = apply(DF, 2, sd)
DF_standardized = sweep( sweep( DF, 2, cm, FUN='-' ), 2, cs, FUN='/' )

# Here all the variables are standardized (even time):
# Note that we don't strictly need to include the -l in the lm command
# below but if we don't we will get an "Intercept" term which is effectively zero.
#
ms = lm( TIME ~ CLOT + PROG + ENZ + LIV - 1, data=DF_standardized )

by_lm = coefficients( ms )

print( rbind( by_hand, by_lm ) )