Supervised Classification, beyond the logistic

In our data-science class, after discussing limitations of the logistic regression, e.g. the fact that the decision boundary line was a straight line, we’ve mentioned possible natural extensions. Let us consider our (now) standard dataset

 clr1 <- c(rgb(1,0,0,1),rgb(0,0,1,1))
 clr2 <- c(rgb(1,0,0,.2),rgb(0,0,1,.2))
 x <- c(.4,.55,.65,.9,.1,.35,.5,.15,.2,.85)
 y <- c(.85,.95,.8,.87,.5,.55,.5,.2,.1,.3)
 z <- c(1,1,1,1,1,0,0,1,0,0)
 df <- data.frame(x,y,z)
 plot(x,y,pch=19,cex=2,col=clr1[z+1])

One can consider a quadratic function of the covariates (instead of a linear one)

 reg=glm(z~x+y+I(x^2)+I(y^2)+I(x*y),
     data=df,family=binomial)
 summary(reg)
 
 pred_1 <- function(x,y){
 predict(reg,newdata=data.frame(x=x,
 y=y),type="response")>.5 }
 
 x_grid<-seq(0,1,length=101)
 y_grid<-seq(0,1,length=101)
 z_grid <- outer(x_grid,y_grid,pred_1)
 image(x_grid,y_grid,z_grid,col=clr2)
 points(x,y,pch=19,cex=2,col=clr1[z+1])

But one can also consider some additive model, with splines

 library(splines)
 reg=glm(z~bs(x)+bs(y),data=df,family=binomial)

or even more general, a model with some bivariate splines,

 library(mgcv)
 reg=gam(z~s(x,y,k=3),data=df,family=binomial)

With a (generalized) linear model, with nonlinear transformation, we can get very general classifier.

We did also mention connexions between the multinomial regression model, and multiple logistic. Here we consider three classes, sayhttps://latex.codecogs.com/gif.latex?\{A,B,C\},

 clr1=c(rgb(1,0,0,1),rgb(1,1,0,1),rgb(0,0,1,1))
 clr2=c(rgb(1,0,0,.2),rgb(1,1,0,.2),
 rgb(0,0,1,.2))
 x=c(.4,.55,.65,.9,.1,.35,.5,.15,.2,.85)
 y=c(.85,.95,.8,.87,.5,.55,.5,.2,.1,.3)
 z=c(1,2,2,2,1,0,0,1,0,0)
 df=data.frame(x,y,z)
 plot(x,y,pch=19,cex=2,col=clr1[z+1])

Can’t we consider three (binomial) logistic regression, with https://latex.codecogs.com/gif.latex?\{A,A^C\}, https://latex.codecogs.com/gif.latex?\{B,B^C\} and https://latex.codecogs.com/gif.latex?\{C,C^C\}

 reg1=glm((z==1)~x+y,data=df,family=binomial)
 summary(reg1)
 reg0=glm((z==0)~x+y,data=df,family=binomial)
 summary(reg0)
 reg2=glm((z==2)~x+y,data=df,family=binomial)
 summary(reg2)

If we look at seperation lines

 pred_0 <- function(x,y){
   predict(reg0,newdata=data.frame(x=x,
   y=y),type="response")>.5
 }
 z_grid0 <- outer(x_grid,y_grid,pred_0)

 pred_1 <- function(x,y){
   predict(reg1,newdata=data.frame(x=x,
   y=y),type="response")>.5
 }
 z_grid1 <- outer(x_grid,y_grid,pred_1)
 
 pred_2 <- function(x,y){
   predict(reg2,newdata=data.frame(x=x,
   y=y),type="response")>.5
 }
 z_grid2 <- outer(x_grid,y_grid,pred_2)

and if we consider a multinomial regression

 library(nnet)
 reg=multinom(z~x+y,data=df)
 
 plot(x,y,pch=19,cex=2,col=clr1[z+1])

we get

 pred_3class <- function(x,y){
 which.max(predict(reg,
 newdata=data.frame(x=x,y=y),type="probs"))
 }

  Outer <- function(x,y,fun) {
   mat <- matrix(NA, length(x), length(y))
   for (i in seq_along(x)) {
   for (j in seq_along(y))    
         mat[i,j]=fun(x[i],y[j])}
   return(mat)}
 z_grid <- Outer(x_grid,y_grid,pred_3class)
 image(x_grid,y_grid,z_grid,col=clr2)
 points(x,y,pch=19,cex=2,col=clr1[z+1])
 
contour(x_grid,y_grid,z_grid0,levels=.5,add=TRUE)
contour(x_grid,y_grid,z_grid1,levels=.5,add=TRUE)
contour(x_grid,y_grid,z_grid2,levels=.5,add=TRUE)

which is slightly different. Since all the isoprobability curves are parallel with a logistic regression, we should focus on the slope of the lines, here. But except of the one on the left, the logistics and the multinomial regression generate different classifiers.

 


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