# This sample program shows how that simple linear regression may not be good enough
# when the underlying model is complex. Nonetheless, it can be use as an approximation
# locally. Be careful. Extrapolation might lead to unexpected and even ridiculous conclusion.
#  We will illustrate these points by simulation studies.
#
# The program is divided into the following steps
#
# 1. Generate the random "errors" e_1, ..., e_20 iid ~ N(0, sgm2)
# 2. Create X=1, 2, ...,20 with   a)  Y_i = 1 +2 X_i^2 + e_i
#                                                b) Y_i = 4*sin(X_i) + e_i
#                                                c) Y_i = exp(X_i /4) + e_i
#                                                d) Y_i = 1 + 2 X_i + e_i, i=1, ..., 20.
# 3. Sketch the scatterplots and the fitted lines, respectively.
#
# Question: 1. Based on the simulated examples, when is it appropriate to use simple linear regression?
#                    2.  When the underlying relation between Y and X is not linear, do you find the fitted model
#                          based on  simple linear model satisfactory?


par(mfrow=c(2,2));    # Setup the graphical device
sgm<-2;                     # sgm = sqrt(sgm^2)
x<-as.vector(seq(1,20,1));
y1<-1+ 2*x^2;  y2<-4*sin(x);
y3<-exp(x/4); y4<-1+2*x;

# Generate the random errors and add them to y

e<-as.vector(rnorm(mean=0,sd=sgm,n=20)) ;
y1<-as.vector(y1+e); y2<-as.vector(y2+e);
y3<-as.vector(y3+e); y4<-as.vector(y4+e);

plot(x, y1); m1<-lm(y1~x); summary(m1); abline(coef(m1))
plot(x, y2); m2<-lm(y2~x); summary(m2); abline(coef(m2))
plot(x, y3); m3<-lm(y3~x); summary(m3); abline(coef(m3))
plot(x, y4); m4<-lm(y4~x); summary(m4); abline(coef(m4))

# You are welcom to try something other relations nd see what happens!
quit()
# Then Exit R without saving workpace image.
# Created 030930 by YesKno&Mhouse