Tag Archives: GLM

Classification from scratch, logistic regression 1/8

Let us start today our series on classification from scratch

The logistic regression is based on the assumption that given covariates \mathbf{x}, Y has a Bernoulli distribution,Y|\mathbf{X}=\mathbf{x}\sim\mathcal{B}(p_{\mathbf{x}}),~~~~p_\mathbf{x}=\frac{\exp[\mathbf{x}^T\mathbf{\beta}]}{1+\exp[\mathbf{x}^T\mathbf{\beta}]}The goal is to estimate parameter \mathbf{\beta}.

Recall that the heuristics for the use of that function for the probability is that\log[\text{odds}(Y=1)]=\log\frac{\mathbb{P}[Y=1]}{\mathbb{P}[Y=0]}=\mathbf{x}^T\mathbf{\beta}

Maximimum of the (log)-likelihood function

The log-likelihood is here\log\mathcal{L} = \sum_{i=1}^n y_i\log p_i+(1-y_i)\log (1-p_i) where p_{i}=(1+\exp[-\mathbf{x}_i^T\mathbf{\beta}])^{-1}. Numerical techniques are based on (numerical) gradient descent to compute the maximum of the likelihood function. The (negative) log-likelihood is the following function

y = myocarde$PRONO
X = cbind(1,as.matrix(myocarde[,1:7]))
negLogLik = function(beta){
 -sum(-y*log(1 + exp(-(X%*%beta))) - (1-y)*log(1 + exp(X%*%beta)))
 }

We use the minus sign since standard optimization routines compute minima, not maxima. Now, to find the minimum of that function, we need a starting point to initiate the algorithm

beta_init = lm(PRONO~.,data=myocarde)$coefficients

Why not start with the parameter of the OLS. Somehow, we might think that at least, sign should be ok for instance. Anyway, we need a starting point, and let us use that one.

logistic_opt = optim(par = beta_init, negLogLik, hessian=TRUE, method = "BFGS", control=list(abstol=1e-9))

Here, we obtain

 logistic_opt$par
 (Intercept)        FRCAR        INCAR        INSYS    
 1.656926397  0.045234029 -2.119441743  0.204023835 
       PRDIA        PAPUL        PVENT        REPUL 
-0.102420095  0.165823647 -0.081047525 -0.005992238

Let us verify here that this output is valid. For instance, what if we change the value of the starting point (randomly)

simu = function(i){
logistic_opt_i = optim(par = rnorm(8,0,3)*beta_init, 
negLogLik, hessian=TRUE, method = "BFGS", 
control=list(abstol=1e-9))
logistic_opt_i$par[2:3]
}
v_beta = t(Vectorize(simu)(1:1000))
plot(v_beta)
par(mfrow=c(1,2))
hist(v_beta[,1],xlab=names(myocarde)[1])
hist(v_beta[,2],xlab=names(myocarde)[2])

Ooops. There is a problem here. Clearly, we cannot rely on numerical optimization here. We can think about using another optimization routine

library(optimx)
logit = function(mX, vBeta) {
  exp(mX %*% vBeta)/(1+ exp(mX %*% vBeta)) 
}
logLikelihoodLogitStable = function(vBeta, mX, vY) {
  -sum(vY*(mX %*% vBeta - log(1+exp(mX %*% vBeta))) + 
(1-vY)*(-log(1 + exp(mX %*% vBeta)))) 
}
likelihoodScore = function(vBeta, mX, vY) {
  return(t(mX) %*% (logit(mX, vBeta) - vY) )
}
optimLogitLBFGS = optimx(beta_init, logLikelihoodLogitStable, 
method = 'L-BFGS-B', gr = likelihoodScore, 
mX = X, vY = y, hessian=TRUE)

The optimum is here

attr(optimLogitLBFGS, "details")[[2]]
              [,1]
       0.066680272
FRCAR  0.003080542
INCAR  0.079031364
INSYS -0.001586194
PRDIA  0.040500697
PAPUL -0.041870705
PVENT -0.014162756
REPUL  0.195632244

Let’s be honest here, I do not feel confortable with those techniques. So, what happened here ?

Here, the technique we use is based on the following idea,\mathbf{\beta}_{new}=\mathbf{\beta}_{old} -\left(\frac{\partial^2\log\mathcal{L}(\mathbf{\beta}_{old})}{\partial\mathbf{\beta}\partial\mathbf{\beta}^T}\right)^{-1}\cdot \frac{\partial\log\mathcal{L}(\mathbf{\beta}_{old})}{\partial\mathbf{\beta}}The problem is that my computer does not know this first and second derivatives. So it will compute them using approximation techniques.

Actually, it is possible to use functions dedicated to such computation

library(numDeriv)
library(MASS)
logit = function(x){1/(1+exp(-x))}
logLik = function(beta, X, y){
 -sum(y*log(logit(X%*%beta)) + 
(1-y)*log(1-logit(X%*%beta)))
}
optim_second = function(beta, num_iter){
  LL = vector()
  for(i in 1:num_iter){
    grad = (t(X)%*%(logit(X%*%beta) - y)) 
    H = hessian(logLik, beta, method = "complex", X = X, y = y)
    beta = beta - ginv(H)%*%grad
    LL[i] = logLik(beta, X, y)
  }
  result = list(beta, H)
return(result)
}

With our OLS starting point, we obtain

opt0 = optim_second(beta_init,500)
opt0[[1]]
             [,1]
[1,]  0.951074420
[2,]  0.018860280
[3,]  0.275428978
[4,]  0.144803636
[5,] -0.058535606
[6,]  0.001182178
[7,] -0.108651776
[8,] -0.002940315

But if we try with another starting point

opt1 = optim_second(beta_init*runif(8),500)
opt1[[1]]
             [,1]
[1,]  0.052894794
[2,]  0.024718435
[3,]  0.167953661
[4,]  0.171662947
[5,] -0.057458066
[6,] -0.011361034
[7,] -0.107532114
[8,] -0.002679064

Clearly, some coefficients are rather close. But other aren’t. From my point of viezw, that is a major problem (keep in mind that we do not deal here with massive data ! There are only 7 explanatory variables, and only 71 observations).

Why not try to be clever, and use the analytical values of those derivatives ? Even if some people claim the oppositive, sometimes, it can actually be usefull to do the maths, instead of considering only numerical values.

Newton (or Fisher) Algorithm

If you open any Econometrics textbooks (one can also try to derive it), you will get \frac{\partial\log\mathcal{L}(\mathbf{\beta}_{old})}{\partial\mathbf{\beta}}=\mathbf{X}^T(\mathbf{y}-\mathbf{p}_{old})
while\frac{\partial^2\log\mathcal{L}(\mathbf{\beta}_{old})}{\partial\mathbf{\beta}\partial\mathbf{\beta}^T}=-\mathbf{X}^T\mathbf{\Delta}_{old}\mathbf{X}

Y=myocarde$PRONO
X=cbind(1,as.matrix(myocarde[,1:7]))
colnames(X)=c("Inter",names(myocarde[,1:7]))
 beta=as.matrix(lm(Y~0+X)$coefficients,ncol=1)
 for(s in 1:9){
   pi=exp(X%*%beta[,s])/(1+exp(X%*%beta[,s]))
   gradient=t(X)%*%(Y-pi)
   omega=matrix(0,nrow(X),nrow(X));diag(omega)=(pi*(1-pi))
   Hessian=-t(X)%*%omega%*%X
   beta=cbind(beta,beta[,s]-solve(Hessian)%*%gradient)}

Observe that here, I use only ten iterations of the algorithm !

 beta[,8:10]
                [,1]          [,2]          [,3]
XInter -10.187641685 -10.187641696 -10.187641696
XFRCAR   0.138178119   0.138178119   0.138178119
XINCAR  -5.862429035  -5.862429037  -5.862429037
XINSYS   0.717084018   0.717084018   0.717084018
XPRDIA  -0.073668171  -0.073668171  -0.073668171
XPAPUL   0.016756506   0.016756506   0.016756506
XPVENT  -0.106776012  -0.106776012  -0.106776012
XREPUL  -0.003154187  -0.003154187  -0.003154187

The thing is that is seems to converge extremely fast. And it is rather robust ! Look at what we get if we change our starting point

beta=as.matrix(lm(Y~0+X)$coefficients,ncol=1)*runif(8)
 for(s in 1:9){
   pi=exp(X%*%beta[,s])/(1+exp(X%*%beta[,s]))
   gradient=t(X)%*%(Y-pi)
   omega=matrix(0,nrow(X),nrow(X));diag(omega)=(pi*(1-pi))
   Hessian=-t(X)%*%omega%*%X
   beta=cbind(beta,beta[,s]-solve(Hessian)%*%gradient)}
 beta[,8:10]
                [,1]          [,2]          [,3]
XInter -10.187641586 -10.187641696 -10.187641696
XFRCAR   0.138178118   0.138178119   0.138178119
XINCAR  -5.862429017  -5.862429037  -5.862429037
XINSYS   0.717084013   0.717084018   0.717084018
XPRDIA  -0.073668172  -0.073668171  -0.073668171
XPAPUL   0.016756508   0.016756506   0.016756506
XPVENT  -0.106776012  -0.106776012  -0.106776012
XREPUL  -0.003154187  -0.003154187  -0.003154187

Nice, isn’t it? Looks like we got our winner, don’t we? And one can use the inverse of the Hessian matrix to get standard deviations.

Weighted Least-Squares

Let us go one step further. We’ve seen that we want to compute something like\mathbf{\beta}_{new} =(\mathbf{X}^T\mathbf{\Delta}_{old}\mathbf{X})^{-1}\mathbf{X}^T\mathbf{\Delta}_{old}\mathbf{z}(if we do substitute matrices in the analytical expressions) where \mathbf{z}=\mathbf{X}\mathbf{\beta}_{old}+\mathbf{\Delta}_{old}^{-1}[\mathbf{y}-\mathbf{p}_{old}]. But actually, that’s simply a standard least-square problem\mathbf{\beta}_{new} = \text{argmin}\left\lbrace(\mathbf{z}-\mathbf{X}\mathbf{\beta})^T\mathbf{\Delta}_{old}^{-1}(\mathbf{z}-\mathbf{X}\mathbf{\beta})\right\rbraceThe only problem here is that weights \mathbf{\Delta}_{old} are functions of unknown \mathbf{\beta}_{old}. But actually, if we keep iterating, we should be able to solve it : given the \mathbf{\beta} we got the weights, and with the weights, we can use weighted OLS to get an updated \mathbf{\beta}. That’s the idea of iteratively reweighted least squares.

The algorithm will be

df = myocarde
beta_init = lm(PRONO~.,data=df)$coefficients
X = cbind(1,as.matrix(myocarde[,1:7]))
beta = beta_init
for(s in 1:1000){
p = exp(X %*% beta) / (1+exp(X %*% beta))
omega = diag(nrow(df))
diag(omega) = (p*(1-p))
df$Z = X %*% beta + solve(omega) %*% (df$PRONO - p)
beta = lm(Z~.,data=df[,-8], weights=diag(omega))$coefficients
}

and the output is here

 beta
  (Intercept)         FRCAR         INCAR         INSYS         PRDIA 
-10.187641696   0.138178119  -5.862429037   0.717084018  -0.073668171 
        PAPUL         PVENT         REPUL 
  0.016756506  -0.106776012  -0.003154187

which is almost what we’ve obtained before. Nice isn’t it ? Actually, here we also have standard deviations of estimators

summary( lm(Z~.,data=df[,-8], weights=diag(omega)))
 
Coefficients:
              Estimate Std. Error t value Pr(>|t|)
(Intercept) -10.187642  10.668138  -0.955    0.343
FRCAR         0.138178   0.102340   1.350    0.182
INCAR        -5.862429   6.052560  -0.969    0.336
INSYS         0.717084   0.503527   1.424    0.159
PRDIA        -0.073668   0.261549  -0.282    0.779
PAPUL         0.016757   0.306666   0.055    0.957
PVENT        -0.106776   0.099145  -1.077    0.286
REPUL        -0.003154   0.004386  -0.719    0.475

The standard glm function

Of course, it is possible to use an R built-in function to get our estimate

summary(glm(PRONO~.,data=myocarde,family=binomial(link = "logit")))
 
Coefficients:
              Estimate Std. Error z value Pr(>|z|)
(Intercept) -10.187642  11.895227  -0.856    0.392
FRCAR         0.138178   0.114112   1.211    0.226
INCAR        -5.862429   6.748785  -0.869    0.385
INSYS         0.717084   0.561445   1.277    0.202
PRDIA        -0.073668   0.291636  -0.253    0.801
PAPUL         0.016757   0.341942   0.049    0.961
PVENT        -0.106776   0.110550  -0.966    0.334
REPUL        -0.003154   0.004891  -0.645    0.519

Application and visualisation

Let us visualize the prediction obtained from the logistic regression, on our second dataset

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(x1=x,x2=y,y=as.factor(z))
reg = glm(y~x1+x2,data=df,family=binomial(link = "logit"))
u = seq(0,1,length=101)
p = function(x,y) predict.glm(reg,newdata=data.frame(x1=x,x2=y),type="response")
v = outer(u,u,p)
image(u,u,v,xlab="Variable 1",ylab="Variable 2",col=clr10,breaks=(0:10)/10)
points(x,y,pch=19,cex=1.5,col="white")
points(x,y,pch=c(1,19)[1+z],cex=1.5)
contour(u,u,v,levels = .5,add=TRUE)


Here level curves – or iso-probabilities – are linear, so the space is divided in two (0 and 1, survival and death, white and black) by a straight line (or an hyperplane in higher dimension). Furthermore, since we have a linear model, if we change the cutoff (the threshold used to create the two classes), we obtain another straight line (or hyperplane) parallel to the first one.

Next time, we will introduce splines to smooth those continuous covariates… to be continued.

Classification from scratch, overview 0/8

Before my course on « big data and economics » at the university of Barcelona in July, I wanted to upload a series of posts on classification techniques, to get an insight on machine learning tools.

According to some common idea, machine learning algorithms are black boxes. I wanted to get back on that saying. First of all, isn’t it the case also for regression models, like generalized additive models (with splines) ? Do you really know what the algorithm is doing ? Even the logistic regression. In textbooks, we can easily find math formulas. But what is really done when I run it, in R ?

When I started working on academia, someone told me something like « if you really want to understand a theory, teach it ». And that has been my moto for more than 15 years. I wanted to add a second part to that statement: « if you really want to understand an algorithm, recode it ». So let’s try this… My ambition is to recode (more or less) most of the standard algorithms used in predictive modeling, from scratch, in R. What I plan to mention, within the next two weeks, will be

I will use two datasets to illustrate. The first one is inspired by the cover of « Foundations of Machine Learning » by Mehryar Mohri, Afshin Rostamizadeh and Ameet Talwalkar. At least, with this dataset, it will be possible to plot predictions (since there are only two – continuous – features)

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(x1=x,x2=y,y=as.factor(z))
plot(x,y,pch=c(1,19)[1+z])

Here is some code to get a visualization of the prediction (here the probability to be a black point)

rmatrix_model = function(model){
u = seq(0,1,length=101)
p = function(x,y) predict(model,newdata=data.frame(x1=x,x2=y),type="response")
v = outer(u,u,p)
return(v)}
nice_graph=function(v){
u = seq(0,1,length=101)
image(u,u,v,xlab="Variable 1",ylab="Variable 2",col=clr10[c(1,10)],breaks=c(0,5,10)/10)
points(x,y,pch=19,cex=1.5,col="white")
points(x,y,pch=c(1,19)[1+z],cex=1.5)
contour(u,u,v,levels = .5,add=TRUE)
}
reg = glm(y~x1+x2,data=df,family=binomial)
nice_graph(rmatrix_model(reg))

Note that colors are defined here as

clr10= c("#ffffff","#f7fcfd","#e5f5f9","#ccece6","#99d8c9","#66c2a4","#41ae76","#238b45","#006d2c","#00441b")

or with some nonlinear model

The second one is a dataset I got from Gilbert Saporta, about heart attacks and decease (our binary variable).

myocarde = read.table("http://freakonometrics.free.fr/myocarde.csv",head=TRUE, sep=";")
myocarde$PRONO = (myocarde$PRONO=="SURVIE")*1
y = myocarde$PRONO
X = as.matrix(cbind(1,myocarde[,1:7]))

So far, I do not plan to talk (too much) on the choice of tunning parameters (and cross-validation), on comparing models, etc. The goal here is simply to understand what’s going on when we call either glm, glmnet, gam, random forest, svm, xgboost, or any function to get a predict model.

Multinomial Logit as an Iterated Logit Regression

For the second section of the course at ENSAE, yesterday, we’ve seen how to run a multinomial logistic regression model. It is simply an extension of the binomial logistic regression. But actually, it is also possible to consider iterative binomial regressions.

Consider here a response variable Y with a multinomial distribution (3 factors to have something more general than the binomial), taking values \{A,B,C\}, with respective probabilities \mathbf{p}=(p_A,p_B,p_C). Here is a code to generate some multinomial variables

msample=function(A,B,C){
Y=rep(NA,B)
for(i in 1:B){Y[i]=sample(A,size=1,prob=C[i,])}
return(Y)
}

and here is a code to generate a dataset with n rows,

generate3=function(n,x,pb=c(-2,0)){
set.seed(x)
X1=runif(n)
X2=runif(n)
X3=runif(n)
s1=pb[1]+X1+X2
s2=pb[2]-X1+X2
P1=exp(s1)/(1+exp(s1)+exp(s2))
P2=exp(s2)/(1+exp(s1)+exp(s2))
Y=msample(0:2,n,cbind(1-P1-P2,P1,P2))
df=data.frame(Y=Y,X1=X1,X2=X2,X3=X3)
return(df)
}

Let us generate a training dataset and a validation one

pb=c(.31,.42)
DF1=generate3(1000,1,pb=pb)
DF2=generate3(500,2,pb=pb)

With a multivariate logistic regression
\mathbb{P}[Y=A|\mathbf{x}]=\frac{\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]+\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}
\mathbb{P}[Y=B|\mathbf{x}]=\frac{\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]+\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}
\mathbb{P}[Y=B|\mathbf{x}]=\frac{1}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{\alpha}]+\exp[\mathbf{x}^{\text{T}}\mathbf{\beta}]}

For convenience, consider the most popular factor in our training dataset

modalite=names(sort(table(DF1$Y),decreasing = TRUE))

Consider a regression model on the simulated dataset (with several covariates), let us estimate it, and let us get predictions.

library(nnet)
reg=multinom(as.factor(Y) ~ ., data = DF1)
mp1=predict (reg, DF1, "probs")
mp2=predict (reg, DF2, "probs")

An alternative can be the following.
consider a first regression model on the Bernoulli variable Y_A=\mathbf{1}(Y=A). Actually, we will consider the most important factor, but for convenience, assume that it is A.
\mathbb{P}[Y_A=A|\mathbf{x}]=\frac{\exp[\mathbf{x}^{\text{T}}\mathbf{a}]}{1+\exp[\mathbf{x}^{\text{T}}\mathbf{a}]}
On our dataset, estimate that model, and get predictions. In the case where Y\neq A, define another Bernoulli variable Y_B=\mathbf{1}(Y=B|Y\neq A). We can estimate that model and derive two probabilities, \mathbb{P}(Y=B|Y\neq A) and \mathbb{P}(Y=C|Y\neq A) (the sum of the two being equal to 1). Based on those two models, it is possible to compute the three probabilities we are looking for. \mathbb{P}[Y=A] is obtained from the first model, and we can derive the other two from \mathbb{P}[Y=B|Y\neq A]\cdot\mathbb{P}[Y\neq A] and \mathbb{P}[Y=C|Y\neq A]\cdot\mathbb{P}[Y\neq A].

reg1=glm((Y==modalite[1])~.,data=DF1,family=binomial)
reg2=glm((Y==modalite[2])~.,data=DF1[-which(DF1$Y==modalite[1]),],family=binomial)
p11=predict (reg1, newdata=DF1, type="response")
p12=predict (reg2, newdata=DF1, type="response")
p21=predict (reg1, newdata=DF2, type="response")
p22=predict (reg2, newdata=DF2, type="response")
mmp1=cbind(p11,(1-p11)*p12,(1-p11)*(1-p12))
mmp2=cbind(p21,(1-p21)*p22,(1-p21)*(1-p22))
colnames(mmp1)=colnames(mmp2)=modalite

Let us compare the predicted probabilites, on the same dataset (here the training dataset)

> mmp1[1:9,c("0","1","2")]
0 1 2
1 0.19728737 0.4991805 0.3035321
2 0.17244580 0.5648537 0.2627005
3 0.19291753 0.5971058 0.2099767
4 0.09087176 0.7787304 0.1303978
5 0.23400225 0.4083022 0.3576955
6 0.18063647 0.6637352 0.1556283
7 0.13188881 0.7402710 0.1278401
8 0.13776970 0.6524959 0.2097344
9 0.12325864 0.6790336 0.1977078
> mp1[1:9,c("0","1","2")]
0 1 2
1 0.19691036 0.5022692 0.3008205
2 0.17123189 0.5680647 0.2607034
3 0.19293066 0.5984402 0.2086291
4 0.08821851 0.7813318 0.1304497
5 0.23470739 0.4109990 0.3542936
6 0.18249687 0.6602168 0.1572863
7 0.13128711 0.7400898 0.1286231
8 0.13525341 0.6553618 0.2093848
9 0.12090016 0.6815915 0.1975084

The two are very close. So yes, it is possible to see the multinomial regression as some sequential binomial regressions.

Variables Catégorielles et Modèle Logistique

Petit complément, suite aux coquilles qu’il y avait dans les slides, sur une propriété de la régression logistique quand on régresse sur des variables catégorielles. On était sur la base des avocats

> avocat <- read.table("http://freakonometrics.free.fr/AutoBI.csv",header=TRUE,sep=",") > avocat$CLMSEX <- factor(avocat$CLMSEX, labels=c("M","F")) > avocat$MARITAL <- factor(avocat$MARITAL, labels=c("M","C","V","D")) > avocat=avocat[!is.na(avocat$CLMSEX),]
> attach(avocat)
> sum((ATTORNEY==2)&(CLMSEX=="F"),na.rm=TRUE)/sum(CLMSEX=="F",na.rm=TRUE)
[1] 0.5256065
> sum((ATTORNEY==2)&(CLMSEX=="M"),na.rm=TRUE)/sum(CLMSEX=="M",na.rm=TRUE)
[1] 0.4453925

Autrement dit, la proportion d’hommes qui se sont fait représentés par un avocat est de 44.539%, et la proportion de femmes 52.56%. On peut visualiser le tableau croisé ci-dessous

> tab=xtabs(~ATTORNEY+CLMSEX,data=avocat)
> require(vcd)
> mosaic(tab, shade=TRUE, legend=TRUE)

Quand on fait une régression linéaire (Gaussienne), on retrouve ces probabilités dans les valeurs de coefficients: pour les femmes, on a 44.539%, et pour les hommes, on a la différence avec les femmes (cette dernière modalité étant la modalité de référence ici)

> reglm = lm((ATTORNEY==2) ~ CLMSEX, data=avocat)
> summary(reglm)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.44539 0.02060 21.620 < 2e-16 ***
CLMSEXF     0.08021 0.02756 2.911  0.00367 **

C’est ce que l’on a ci-dessous

> sum(coefficients(reglm))
[1] 0.5256065
> coefficients(reglm)[1]
(Intercept)
0.4453925

On a un résultat similaire avec une régression logistique

> reglogit = glm((ATTORNEY==2) ~ CLMSEX, data=avocat,family=binomial)
> summary(reglogit)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.21930 0.08312 -2.639 0.00833 **
CLMSEXF      0.32182 0.11097  2.900 0.00373 **

même si les coefficients n’ont pas la même interprétation. En transformant ces coefficients, on retrouve très exactement les mêmes valeurs

> exp(sum(reglogit$coefficients[1:2])) /(1+exp(sum(reglogit$coefficients[1:2])))
[1] 0.5256065
> exp(reglogit$coefficients[1]) /(1+exp(reglogit$coefficients[1]))
(Intercept)
0.4453925

Désolé pour la typo.

Actuariat de l’Assurance non-Vie #4

Lundi prochain, suite du cours d’actuariat de l’assurance non-vie. Nous avons terminé la partie sur la classification (modèle logistique, arbres, forêts, bagging, etc), et nous allons aborder la section sur la modélisation de la fréquence, et la régression de Poisson. Je rajouter quelques slides sur la présentation des GLM, qui seront utiles pour parler un peu de sur-dispersion,

 

Regression on factors

Most of our intuitions about regression models come from the Gaussian standard linear model. One interesting feature is that, when we have a factor explanatory variable, the sum of predictions per class is the sum of observations of the endogeneous variable, per class. To be more specific, consider some factor variable https://latex.codecogs.com/gif.latex?x_1\in\{0,1\}, and a regression model

https://latex.codecogs.com/gif.latex?y_i=\beta_0+\beta_1%20\boldsymbol{1}(x_1=1)+\beta_2%20x_2+\varepsilon_i

Use ordinary least squares to fit that model

https://latex.codecogs.com/gif.latex?\widehat{y}_i=\widehat{\beta}_0+\widehat{\beta}_1%20\boldsymbol{1}(x_1=1)+\widehat{\beta}_2%20x_2

Then for all https://latex.codecogs.com/gif.latex?x\in\{0,1\}

https://latex.codecogs.com/gif.latex?\sum_{i:x_i=x}%20y_i%20=%20\sum_{i:x_i=x}%20\widehat{y}_i

> n=200
> X1=rep(0:1,each=n/2)
> set.seed(1)
> X2=runif(2*n)
> L=X1-X2
> B=data.frame(Y=rnorm(n,L),X1=as.factor(X1),X2=X2)
> pd=aggregate(x=B$Y,by=list(B$X1),mean)$x
> pd
[1] -0.4881735  0.5341301
> fit=lm(Y~X1+X2,data=B)
> B2=data.frame(x=B$X1,y=predict(fit))
> aggregate(x=B2$y,by=list(B2$x),mean)$x
[1] -0.4881735  0.5341301

Continue reading Regression on factors

Simple Distributions for Mixtures?

The idea of GLMs is that given some covariates has a distribution in the exponential family (Gaussian, Poisson, Gamma, etc). But that does not mean that  has a similar distribution… so there is no reason to test for a Gamma model for  before running a Gamma regression, for instance. But are there cases where it might work? That the non-conditional distribution is the same (same family at least) than the conditional ones?

For instance, if  has a joint Gaussien distribution, then both marginals are Gaussian, but also . So, in that case, if the covariate is normally distributed, it is possible to have a Gaussian distribution also for . The econometric interpretation is that with a standard Gaussian linear model, if is normally distributed, not only the conditional distribution  is Gaussian but also the non-conditional distribution of .

> set.seed(1)
> n=1e3
> X=rnorm(n,10,2)
> Y=1+3*X+rnorm(n)
> plot(X,Y,xlim=c(4,20))

Indeed, here the distribution of  is also Gaussian

> library(nortest)
> ad.test(Y)

	Anderson-Darling normality test

data:  Y
A = 0.23155, p-value = 0.802

> shapiro.test(Y)

	Shapiro-Wilk normality test

data:  Y
W = 0.99892, p-value = 0.8293

(not only from a statistical point of view, the thoery of Gaussian random vectors confirms that the non-conditional distribution is Gaussian actually)

Here  is continuous. What if we consider a finite mixture here, i.e. takes only a finite number of values? Actually, Teicher (1963) proved that it is not possible to have a non-conditional Gaussian distribution for . But in practice, would we really reject the Gaussian assumption, for ? If the number of classes is to small, yes. But with a large number of classes (a sufficiently large number of mixture components), it is possible,

> pv=function(k=2){
+ n=1e4
+ X=rnorm(n,10,2)
+ Q=quantile(X,(0:k)/k)
+ Q[1]=0
+ Xc=cut(X,Q,labels=1:k)
+ XcN=tapply(X,Xc,mean)
+ Xn=XcN[as.numeric(Xc)]
+ Y=1+3*Xn+rnorm(n)
+ ad.test(Y)$p.value}
 
> plot(2:100,Vectorize(pv)(2:100),type="l")
> abline(h=.05,col="red")

So here, it could be possible to have also a Gaussian distribution, for . As least to accept that assumption, statistically.

In the context of a Poisson regression, it is well know that it’s not possible to have at the same time  that is Poisson distributed (that’s a Poisson regression) and also  that is Poisson distributed. That simply comes from the fact that

while

and because of the conditional Poisson distribution, then

Thus,

So  cannot be Poisson distribution. But again, it could be possible, if heterogeneity is not too large, to accept the null assumption of a Poisson distribution for .

More generally, it is very difficult to have a distribution family for   that is also the distribution of the non-conditional variable . In the context of a finite mixture ( takes a finite number of values),Teicher (1963) proved that it was not not possible, neither for the Gaussian distribution nor the Gamma distribution. An to go further, check Monfrini (2002) (thanks Romuald for point out the reference).

Hence, as a keep saying, before running a regression model on with some given family, it is never a good idea to check if the non-conditional distribution  has the same distribution. Because there is no reason, usually, to remain in the same family.