Polynomial Regression.R

#MODELLING NON LINEARITIES in DATA using various Non linear functions-


#The most basic idea behind adding Non linear properties in the Model is by transforming the data 
#or the variables,alomst every trick transforms the variables to Model Non linearitites.
#Say Kernel Smoothing techniques , Splines or Step functions etc.


#Package containing the Dataset
require(ISLR)
attach(Wage)#Dataset

#Polynomial Regression
#First we will use polynomials , and focus on only one predictor age.


mod<-lm(wage~poly(age,4),data =Wage)
#Summary of the Model
summary(mod)

#Range of age variable
agelims<-range(age)
#Generating Test Data
age.grid<-seq(from=agelims[1], to = agelims[2])
#Making Predctions on Test data
pred<-predict(mod,newdata = list(age=age.grid),se=TRUE)
#Standard Error Bands- within 2 Standard Deviations
se.tab<-cbind(pred$fit+2*pred$se.fit,pred$fit -  2*pred$se.fit)
plot(age,wage,col="darkgrey")
#Plotting the age values vs Predicted Wage values for those Ages
lines(age.grid,pred$fit,col="blue",lwd=2)
#To plot the Error bands around the Regression Line
matlines(x=age.grid,y=se.tab,lty =2,col="blue")

#This time we will use different basis of polynomials
fit2<-lm(wage ~ age + I(age^2) + I(age^3) + I(age^4),data = Wage)
summary(fit2)

plot(fitted(mod),fitted(fit2),xlab="First Polynomial Model",
     ylab="Polynomial Model wrapped inside Identity function", 
     main="Fitted values of Both models are exactly same")


#We can notice that the coefficients and the summary is different though we have used 
#the same degree of polynomials and this is merely due to the different representations 
#of the polynomils using Identity I() function.

#Things we are interested in is the Fitted polynomial and we can notice that the fitted 
#values of both The model above and this Model has not changed.


#Now we will use anova() to test different Models with different Predictors

#Making Nested Models-i.e Each Next Model includes previous Model and is a special case for previous one
mod1<-lm(wage ~ education , data = Wage)
mod2<-lm(wage ~ education + age,data = Wage)
mod3<-lm(wage ~ education + age + poly(age,2),data = Wage)
mod4<-lm(wage ~ education + age + poly(age,3),data = Wage)
#using anova() function
anova(mod1,mod2,mod3,mod4)

BIC(mod1,mod2,mod3,mod4)

#Seeing the Above values,Model 4 which is the most Complex one is the most Insignificant Model 
#as the p-values indicate.Though the RSS value of Model 4 is least,and this is a expected as it
#fitting data too hard(Overfitting).

#Model2 and Model3 are the best ones and seem to balance the Bias-Variace Tradeoffs.


#Polynomial Logistic Regression....

#Logistic Regression Model the Binary Response variable;
logmod<-glm(I(wage > 250 ) ~ poly(age,3),data = Wage , family = "binomial")
summary(logmod)


#doing Predictions
pred2<-predict(logmod,newdata = list(age=age.grid),se=TRUE)
#Standard Error Bands
#a Matrix with 3 columns
#Confidence intervals
se.band<-pred2$fit + cbind(fit=0,lower=-2*pred2$se.fit , upper = 2*pred2$se.fit )
se.band[1:5,]

#p=e^n/(1+e^n)

#comuting the 95% confidence interval for the Fitted Probabilities value
prob.bands = exp(se.band)/ (1 + exp(se.band))
matplot(age.grid,prob.bands,col="blue",lwd = c(2,2,2),lty=c(1,2,2),
        type="l",ylim=c(0,.1),xlab="Age",ylab="Probability Values")

#jitter() function to uniformly add random noise to properly see the densities
points(jitter(age),I(wage > 250)/10 , pch="I",cex=0.5)

#The blue dotted lines represent the 95% Confidence Interval of the fitted Probabilities.

#The black dots are the actual Probability values for Binary Response Wage, i.e if wage > 250 
#is true then 1(TRUE) ,otherwise 0(FALSE).