Over a million developers have joined DZone.
{{announcement.body}}
{{announcement.title}}

Some Intuition About the Theory of Statistical Learning

DZone's Guide to

Some Intuition About the Theory of Statistical Learning

· Big Data Zone
Free Resource

Intelligently automate your Big Data operations to lower your costs, make your team more productive, scale more efficiently, and lower the risk of failure. Learn how >>

While I was working on the Theory of Statistical Learning, and the concept of consistency, I found the following popular graph (e.g. from  thoses slides, here in French)

The curve below is the error on the training sample, as a function of the size of the training sample. Above, it is the error on a validation sample. Our learning process is consistent if the two converge.

I was wondering if it was possible to generate such a graph, with some data, and some statistical model. And indeed, it is rather simple, and it gives nice intuition about possible interpretations. Consider some (simple) classification problem. Here, we consider a logistic regression. We generate a sample of size , we fit our model, we compute the misclassification rate, then we generate another sample of size , we use our previous model to make some prediction, and we compute the misclassifiation rate. And we play with .

missclassification <- function(n){
  U=data.frame(X1=runif(n),X2=runif(n))
  p=(U[,1]+U[,2])/2
  U$Y=rbinom(n,size=1,prob=p)
  reg=glm(Y~X1+X2,data=U,family=binomial)
  pd=function(x1,x2) predict(reg,newdata=data.frame(X1=x1,X2=x2),type="response")>.5
  x=seq(0,1,length=101)
  z=outer(x,x,pd)
  cl2=c(rgb(1,0,0,.4),rgb(0,0,1,.4))
  cl1=c("red","blue")
  image(x,x,z,col=cl2,xlab="",ylab="",main="Training Sample")
  points(U$X1,U$X2,pch=19,col=cl1[1+U$Y])

  V=data.frame(X1=runif(n),X2=runif(n))
  p=(V[,1]+V[,2])/2
  V$Y=rbinom(n,size=1,prob=p)
  screen(4)
  image(x,x,z,col=cl2,xlab="",ylab="",main="Validation Sample")
  points(V$X1,V$X2,pch=19,col=cl1[1+V$Y])

  MissClassU=mean(abs(pd(U$X1,U$X2)-U$Y))
  MissClassV=mean(abs(pd(V$X1,V$X2)-V$Y))
return(c(MissClassU,MissClassV))
}


If we plot it, we get (in purple, it is the training sample, and in black, the validation sample)

The graph is not exactly the same as above, but it is probably due to the randomness of our samples. If we generate hundreds of samples, it should be just fine.

MCU=rep(NA,500)
MCV=rep(NA,500)
n=250
  for(i in 1:500){
    U=data.frame(X1=runif(n),X2=runif(n))
    p=(U[,1]+U[,2])/2
    U$Y=rbinom(n,size=1,prob=p)
    reg=glm(Y~X1+X2,data=U,family=binomial)
    pd=function(x1,x2) predict(reg,newdata=data.frame(X1=x1,X2=x2),type="response")>.5
    MCU[i]=mean(abs(pd(U$X1,U$X2)-U$Y))

    V=data.frame(X1=runif(n),X2=runif(n))
    p=(V[,1]+V[,2])/2
    V$Y=rbinom(n,size=1,prob=p)
    MCV[i]=mean(abs(pd(V$X1,V$X2)-V$Y))
  }
  MissClassV=mean(MCU)
  MissClassU=mean(MCV)


Find the perfect platform for a scalable self-service model to manage Big Data workloads in the Cloud. Download the free O'Reilly eBook to learn more.

Topics:
bigdata ,big data ,statistical learning

Published at DZone with permission of Arthur Charpentier, DZone MVB. See the original article here.

Opinions expressed by DZone contributors are their own.

{{ parent.title || parent.header.title}}

{{ parent.tldr }}

{{ parent.urlSource.name }}