On the poor performance of classifiers in insurance models
(This article was first published on Renglish – Freakonometrics , and kindly contributed toRbloggers)
Each time we have a case study in my actuarial courses (with real data), students are surprised to have hard time getting a “good” model, and they are always surprised to have a low AUC , when trying to model the probability to claim a loss, to die, to fraud, etc. And each time, I keep saying, “yes, I know, and that’s what we expect because there a lot of ‘randomness’ in insurance”. To be more specific, I decided to run some simulations, and to compute AUCs to see what’s going on. And because I don’t want to waste time fitting models, we will assume that we have each time a perfect model. So I want to show that the upper bound of the AUC is actually quite low ! So it’s not a modeling issue, it is a fondamental issue in insurance !
By ‘perfect model’ I mean the following :
denotes the heterogeneity factor, because people are different. We would love to get . Unfortunately, is unobservable ! So we use covariates (like the age of the driver of the car in motor insurance, or of the policyholder in life insurance, etc). Thus, we have data ‘s and we use them to train a model, in order to approximate . And then, we check if our model is good (or not) using the ROC curve, obtained from confusion matrices, comparing ‘s and ‘s where when exceeds a given threshold. Here, I will not try to construct models. I will predict each time the true underlying probability exceeds a threshold ! The point is that it’s possible to claim a loss ( ) even if the probability is 3% (and most of the time ), and to not claim one ( ) even if the probability is 97% (and most of the time). That’s the idea with randomness, right ?
So, here
denote the probabilities to claim a loss, to die, to fraud, etc. There is heterogeneity here, and this heterogenity can be small, or large. Consider the graph below, to illustrate,
In both cases, there is, on average, 25% chance to claim a loss. But on the left, there is more heterogeneity, more dispersion. To illustrate, I used the arrow, which is a classical 90% interval : 90% of the individuals have a probability to claim a loss in that interval. (here 10%40%), 5% are below 10% (low risk), and 5% are above 40% (high risk). Later on, we will say that we have 25% on average, with a dispersion of 30% (40% minus 10%). On the right, it’s more 25% on average, with a dispersion of of 15%. What I call dispersion is the difference between the 95% and the 5% quantiles.
Consider now some dataset, with Bernoulli variables
, drawn with those probabilities. Then, let us assume that we are able to get a perfect model : I do not estimate a model based on some covariates, here, I assume that I know perfectly the probability (which is true, because I did generate those data). More specifically, to generate a vector of probabilities, here I use a Beta distribution with a given mean, and a given variance (to capture the heterogeneity I mentioned above)
a=m*(m*(1m)/v1) b=(1m)*(m*(1m)/v1) p=rbeta(n,a,b)
from those probabilities, I generate occurences of claims, or deaths,
Y=rbinom(n,size = 1,prob = p)
Then, I compute the AUC of my “perfect” model,
auc.tmp=performance(prediction(p,Y),"auc")
And then, I will generate many samples, to compute the average value of the AUC. And actually, we can do that for many values of the mean and the variance of the Beta distribution. Here is the code
library(ROCR) n=1000 ns=200 ab_beta = function(m,inter){ a=uniroot(function(a) qbeta(.95,a,a/ma)qbeta(.05,a,a/ma)inter, interval=c(.0000001,1000000))$root b=a/ma return(c(a,b)) } Sim_AUC_mean_inter=function(m=.5,i=.05){ V_auc=rep(NA,ns) b=1 essai = try(ab<ab_beta(m,i),TRUE) if(inherits(essai,what="tryerror")) a=1 if(!inherits(essai,what="tryerror")){ a=ab[1] b=ab[2] } if((a>=0)&(b>=0)){ for(s in 1:ns){ p=rbeta(n,a,b) Y=rbinom(n,size = 1,prob = p) auc.tmp=performance(prediction(p,Y),"auc") V_auc[s]=as.numeric(auc.tmp@y.values)} L=list(moy_beta=m, var_beat=v, q05=qbeta(.05,a,b), q95=qbeta(.95,a,b), moy_AUC=mean(V_auc), sd_AUC=sd(V_auc), q05_AUC=quantile(V_auc,.05), q95_AUC=quantile(V_auc,.95)) return(L)} if((a<0)(b<0)){return(list(moy_AUC=NA))}} Vm=seq(.025,.975,by=.025) Vi=seq(.01,.5,by=.01) V=outer(X = Vm,Y = Vi, Vectorize(function(x,y) Sim_AUC_mean_inter(x,y)$moy_AUC)) library("RColorBrewer") image(Vm,Vi,V, xlab="Probability (Average)", ylab="Dispersion (Q95Q5)", col= colorRampPalette(brewer.pal(n = 9, name = "YlGn"))(101)) contour(Vm,Vi,V,add=TRUE,lwd=2)
On the x axis, we have the average probability to claim a loss. Of course, there is a symmetry here. And on the y axis, we have the dispersion : the lower, the less heterogeneity in the portfolio. For instance, with a 30% chance to claim a loss on average, and 20% dispersion (meaning that in the portfolio, 90% of the insured have between 20% and 40% chance to claim a loss, or 15% and 35% chance), we have on average a 60% AUC. With a perfect model ! So with only a few covariates, having 55% should be great !
My point here is that with a low dispersion, we cannot expect to have a great AUC (again, even with a perfect model). In motor insurance, from my experience, 90% of the insured are between 3% chance and 20% chance to claim a loss ! That’s less than 20% dispersion ! and in that case, even if the (average) probability is rather small, it is very difficult to expect an AUC above 60% or 65% !
猜你喜欢

29
After lax supervision led to dicey products and wild expansion, China’s insurance regulator is telling the industry to return to its basics

7
README.md Adversarial Robustness Toolbox (ART v0.1)

0
Learn how to create a training project, add classification tags, upload images, train the project, obtain the endpoint URL, and use that to test an image.

2
Coastal classifiers: using AutoML Vision to assess and track envir...

97
models  Models built with TensorFlow

44
Having a look on the Google Store, it appears many unlocked variants of the Pixel 2 and Pixel 2 XL are available for purchase right now. Over the past couple of weeks, many options have been out of stock, but if you act now, you can score a 64GB...

55
除非特别声明，此文章内容采用知识共享署名 3.0许可，代码示例采用Apache 2.0许可。更多细节请查看我们的服务条款。

62
ATM  Auto Tune Models  A multiuser, multidata system for model selection and tuning.

51
tpu  Reference models and tools for Cloud TPUs.

41
How to use interfaces to make machine learning code more flexible
 上一页
 下一页
关于头条
聚合每日国内外有价值，有趣的链接。