#This is another linear regression example
source( "../gnrnd4.R")
#  generate and inspect the values that we will use
gnrnd4(6271641306,4161000704, 30000100)
L1
L2
plot( L1, L2 )  # make a first, rough, scatter plot
#refine that plot a bit
plot( L1, L2, main="Our Linear Regression Example",
      xlim=c(-15,25), ylim=c(-15,45),
      xaxp=c(-15,25,8), yaxp=c(-15,45,12), las=1,
      cex.axis=0.75, xlab="L1 -- the x value",
      ylab="L2 -- the y value", pch=20, col="darkgreen")
abline( h=seq(-15,45,5), v=seq(-15,25,5), lty="dotted",
        col="red")
abline( h=0, v=0, col="darkred")

#  look at the correlation coefficient
cor( L1, L2 )

# generate the linear regression model
lm_hold <- lm( L2 ~ L1 )
lm_hold  # look at the linear regression coefficients

abline( lm_hold, col="blue") #plot the regression line

# find all of the expected values
expected <- 26.224 + (-1.636)*L1
expected #look at those values

# now add the expected values to the plot
points( L1, expected, pch=6, col="darkorange")

# now, compute all of the residual values
resid_vals <- L2 - expected
resid_vals
#  we did not have to do that because we could 
#  have just pulled the 'true' residual values 
#  from the model
residuals( lm_hold)
#  the difference is due to the fact that we used 
#  the output from our linear model as the coefficients
#  for the regression equation.  A more precise pair of
#  values could be pulled from the linear model
coeffs <- coefficients( lm_hold)
coeffs
#  and we could separate those values, and use 
#   as.numeric() to remove the label
intercept <- as.numeric(coeffs[1])
slope <- as.numeric(coeffs[2])
intercept
slope

#  then, just as an extra display, if we want to sort
#  the values, according to the x-values we could
#  put them into a data frame
comb <- data.frame(L1,L2,expected)
comborder<-comb[order(L1),]
L1sort<-comborder$L1
L2sort<-comborder$L2
expectedsort <- comborder$expected
L1sort
L2sort
expectedsort