Monthly Archives: April 2015

Regression

I Fought the (distribution) Law (and the Law did not win)

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A few days ago, I was asked if we should spend a lot of time to choose the distribution we use, in GLMs, for (actuarial) ratemaking. On that topic, I usually claim that the family is not the most important parameter in the regression model. Consider the following dataset

> db <- data.frame(x=c(1,2,3,4,5),y=c(1,2,4,2,6))
> plot(db,xlim=c(0,6),ylim=c(-1,8),pch=19)

To visualize a regression model, use the following code

> nd=data.frame(x=seq(0,6,by=.1))
> add_predict = function(reg){
+ prd1=predict(reg,newdata=nd,se.fit = TRUE,type="response")
+ y1=prd1$fit
+ y1_upp=prd1$fit+prd1$residual.scale*1.96*
prd1$se.fit   
+ y1_low=prd1$fit-prd1$residual.scale*1.96*
prd1$se.fit 
+ polygon(c(nd$x,rev(nd$x)),c(y1_upp,
rev(y1_low)),col="light green",angle=90,
density=40,border=NA)
+ lines(nd$x,y1,col="red",lwd=2)
+ }

For instance, with a Poisson regression (with a log link function) we get

> plot(db)
> reg1=glm(y~x,family=poisson(link="log"),
+ data=db)
> add_predict(reg1)

while, with a Gaussian regresion (but still with a log link function), we get

> plot(db)
> reg2=glm(y~x,family=gaussian(link="log"),
+ data=db)
> add_predict(reg2)

If we just care about the expected value of our prediction, the output is more or less the same

> plot(db)
> lines(nd$x,predict(reg1,newdata=nd,
+ type="response"),col="red",lwd=1.5)
> lines(nd$x,predict(reg2,newdata=nd,
+ type="response"),col="blue",lwd=1.5)

So, indeed, forget about the (distribution) law when running a GLM. Not convinced? Consider – on the same dataset – a Poisson regression (with an identity link function this time)

> plot(db)
> reg1=glm(y~x,family=poisson(link="identity"),
+ data=db)
> add_predict(reg1)

while, with a Gaussian regresion (but still with an identity link function), we get

> plot(db)
> reg2=glm(y~x,family=gaussian(link="identity"),
+ data=db)
> add_predict(reg2)

Again, if we just plot the expected value of our prediction, the output is more or less the same

> plot(db)
> lines(nd$x,predict(reg1,newdata=nd,
+ type="response"),col="red",lwd=1.5)
> lines(nd$x,predict(reg2,newdata=nd,
+ type="response"),col="blue",lwd=1.5)

So clearly, the simplistic message you should not care too much about the (distribution) law seems to be valid…

Continue reading I Fought the (distribution) Law (and the Law did not win) →

M1-Data Science

Visualising a Classification in High Dimension, part 2

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A few weeks ago, I published a post on Visualising a Classification in High Dimension, based on the use of a principal component analysis, to get a projection on the first two components. Following that post, I was wondering what could be done in the context of a classification on categorical covariates. A natural idea would be to consider a correspondance analysis, and to run a similar code.

Consider here the dataset used in a recent post,

> source("http://freakonometrics.free.fr/import_data_credit.R")

If we consider a correspondance analysis, we get

> library(FactoMineR)
> acm=MCA(train.db,quali.sup = 
+ which(names(train.db,)=="class"),ncp=10)

For the covariates (including also the variable we want to model, considered here as some supplementary variable), the visualisation – on the first two components – is

and for the individuals

Continue reading Visualising a Classification in High Dimension, part 2 →

M1-Data Science, Regression

Classification with Categorical Variables (the fuzzy side)

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The Gaussian and the (log) Poisson regressions share a very interesting property,

i.e. the average predicted value is the empirical mean of our sample.

> mean(predict(lm(dist~speed,data=cars)))
[1] 42.98
> mean(cars$dist)
[1] 42.98

One can prove that it is also the prediction for the average individual in our sample

> predict(lm(dist~speed,data=cars),
+ newdata=data.frame(speed=mean(cars$speed))) 
42.98

The geometric interpretation is that the regression line passes through the centroid,

> plot(cars)
> abline(lm(dist~speed,data=cars),col="red")
> abline(h=mean(cars$dist),col="blue")
> abline(v=mean(cars$speed),col="blue")
> points(mean(cars$speed),mean(cars$dist))

But in all other cases, it is no longer the case. Consider for instance the case of a logistic regression. And to ask for something even more complicated, consider the case where we have only categorical explanatory variables. In that context, it is more difficult to get a prediction for the “average individual”. Unless we consider some fuzzy interpretation of the regression.

Continue reading Classification with Categorical Variables (the fuzzy side) →