Fourth post of our series on classification from scratch, following the previous post which was some sort of detour on kernels. But today, we’ll get back on the logistic model.
Formal approach of the problem
We’ve seen before that the classical estimation technique used to estimate the parameters of a parametric model was to use the maximum likelihood approach. More specifically, \widehat{\mathbf{\beta}}=\text{argmax}\lbrace \log\mathcal{L}(\mathbf{\beta}|\mathbf{x},\mathbf{y})\rbraceThe objective function here focuses (only) on the goodness of fit. But usually, in econometrics, we believe something like non sunt multiplicanda entia sine necessitate (“entities are not to be multiplied without necessity”), the parsimony principle, simpler theories are preferable to more complex ones. So we want to penalize for too complex models.
This is not a bad idea. It is mentioned here and there in econometrics textbooks, but usually, for model choice, not about the inference. Usually, we estimate parameters using maximum likelihood techniques, and them we use AIC or BIC to compare two models. Recall that Akaike (AIC) criteria is based on-2\log\mathcal{L}(\widehat{\mathbf{\beta}}|\mathbf{x},\mathbf{y})+2\text{dim}(\widehat{\mathbf{\beta}})We have on the left a measure for the goodness of fit, and on the right, a penalty increasing with the “complexity” of the model.
Very quickly, here, the complexity is the number of variates used. I will not enter into details about the concept of sparsity (and the true dimension of the problem), I will recommend to read the book by Martin Wainwright, Robert Tibshirani and Trevor Hastie on that issue. But assume that we do not make and variable selection, we consider the regression on all covariates. Define\Vert\mathbf{a} \Vert_{\ell_0}=\sum_{i=1}^d \mathbf{1}(a_i\neq 0), ~~\Vert\mathbf{a} \Vert_{\ell_1}=\sum_{i=1}^d |a_i|,~~\Vert\mathbf{a} \Vert_{\ell_2}=\left(\sum_{i=1}^d a_i^2\right)^{1/2}for any \mathbf{a}\in\mathbb{R}^d. One might say that the AIC could be written-2\log\mathcal{L}(\widehat{\mathbf{\beta}}|\mathbf{x},\mathbf{y})+2\|\widehat{\mathbf{\beta}}\|_{\ell_0}And actually, this will be our objective function. More specifically, we will consider
\widehat{\mathbf{\beta}}_{\lambda}=\text{argmin}\lbrace -\log\mathcal{L}(\mathbf{\beta}|\mathbf{x},\mathbf{y})+\lambda\|\mathbf{\beta}\|\rbracefor some norm \|\cdot\|. I will not get back here on the motivation and the (theoretical) properties of those estimates (that will actually be discussed in the Summer School in Barcelona, in July), but in this post, I want to discuss the numerical algorithm to solve such optimization problem, for \|\cdot\|_{\ell_2} (the Ridge regression) and for \|\cdot\|_{\ell_1} (the LASSO regression).
Normalization of the covariates
The problem of \|\mathbf{\beta}\| is that the norm should make sense, somehow. A small \mathbf{\beta}_j is with respect to the “dimension” of x_j‘s. So, the first step will be to consider linear transformations of all covariates x_j to get centered and scaled variables (with unit variance)
y = myocarde$PRONO X = myocarde[,1:7] for(j in 1:7) X[,j] = (X[,j]-mean(X[,j]))/sd(X[,j]) X = as.matrix(X) |
Ridge Regression (from scratch)
Before running some codes, recall that we want to solve something like\widehat{\mathbf{\beta}}_{\lambda}=\text{argmin}\lbrace -\log\mathcal{L}(\mathbf{\beta}|\mathbf{x},\mathbf{y})+\lambda\|\mathbf{\beta}\|_{\ell_2}^2\rbrace In the case where we consider the log-likelihood of some Gaussian variable, we get the sum of the square of the residuals, and we can obtain an explicit solution. But not in the context of a logistic regression.
The heuristics about Ridge regression is the following graph. In the background, we can visualize the (two-dimensional) log-likelihood of the logistic regression, and the blue circle is the constraint we have, if we rewite the optimization problem as a contrained optimization problem : \min_{\mathbf{\beta}:\|\mathbf{\beta}\|^2_{\ell_2}\leq s} \lbrace \sum_{i=1}^n -\log\mathcal{L}(y_i,\beta_0+\mathbf{x}^T\mathbf{\beta}) \rbracecan be written equivalently (it is a strictly convex problem)\min_{\mathbf{\beta},\lambda} \lbrace -\sum_{i=1}^n \log\mathcal{L}(y_i,\beta_0+\mathbf{x}^T\mathbf{\beta}) +\lambda \|\mathbf{\beta}\|_{\ell_2}^2 \rbraceThus, the constrained maximum should lie in the blue disk
LogLik = function(bbeta){ b0=bbeta[1] beta=bbeta[-1] sum(-y*log(1 + exp(-(b0+X%*%beta))) - (1-y)*log(1 + exp(b0+X%*%beta)))} u = seq(-4,4,length=251) v = outer(u,u,function(x,y) LogLik(c(1,x,y))) image(u,u,v,col=rev(heat.colors(25))) contour(u,u,v,add=TRUE) u = seq(-1,1,length=251) lines(u,sqrt(1-u^2),type="l",lwd=2,col="blue") lines(u,-sqrt(1-u^2),type="l",lwd=2,col="blue") |
Let us consider the objective function, with the following code
PennegLogLik = function(bbeta,lambda=0){ b0 = bbeta[1] beta = bbeta[-1] -sum(-y*log(1 + exp(-(b0+X%*%beta))) - (1-y)* log(1 + exp(b0+X%*%beta)))+lambda*sum(beta^2) } |
Why not try a standard optimisation routine ? In the very first post on that series, we did mention that using optimization routines were not clever, since they were strongly relying on the starting point. But here, it is not the case
lambda = 1 beta_init = lm(PRONO~.,data=myocarde)$coefficients vpar = matrix(NA,1000,8) for(i in 1:1000){ vpar[i,] = optim(par = beta_init*rnorm(8,1,2), function(x) PennegLogLik(x,lambda), method = "BFGS", control = list(abstol=1e-9))$par} par(mfrow=c(1,2)) plot(density(vpar[,2]),ylab="",xlab=names(myocarde)[1]) plot(density(vpar[,3]),ylab="",xlab=names(myocarde)[2]) |
Clearly, even if we change the starting point, it looks like we converge towards the same value. That could be considered as the optimum.
The code to compute \widehat{\mathbf{\beta}}_{\lambda} would then be
opt_ridge = function(lambda){ beta_init = lm(PRONO~.,data=myocarde)$coefficients logistic_opt = optim(par = beta_init*0, function(x) PennegLogLik(x,lambda), method = "BFGS", control=list(abstol=1e-9)) logistic_opt$par[-1]} |
and we can visualize the evolution of \widehat{\mathbf{\beta}}_{\lambda} as a function of {\lambda}
v_lambda = c(exp(seq(-2,5,length=61))) est_ridge = Vectorize(opt_ridge)(v_lambda) library("RColorBrewer") colrs = brewer.pal(7,"Set1") plot(v_lambda,est_ridge[1,],col=colrs[1]) for(i in 2:7) lines(v_lambda,est_ridge[i,],col=colrs[i]) |
At least it seems to make sense: we can observe the shrinkage as \lambda increases (we’ll get back to that later on).
Ridge, using Netwon Raphson algorithm
We’ve seen that we can also use Newton Raphson to solve this problem. Without the penalty term, the algorithm was\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}}where
\frac{\partial\log\mathcal{L}(\mathbf{\beta}_{old})}{\partial\mathbf{\beta}}=\mathbf{X}^T(\mathbf{y}-\mathbf{p}_{old})and\frac{\partial^2\log\mathcal{L}(\mathbf{\beta}_{old})}{\partial\mathbf{\beta}\partial\mathbf{\beta}^T}=-\mathbf{X}^T\mathbf{\Delta}_{old}\mathbf{X}where \mathbf{\Delta}_{old} is the diagonal matrix with terms \mathbf{p}_{old}(1-\mathbf{p}_{old}) on the diagonal.
Thus\mathbf{\beta}_{new} = \mathbf{\beta}_{old} + (\mathbf{X}^T\mathbf{\Delta}_{old}\mathbf{X})^{-1}\mathbf{X}^T[\mathbf{y}-\mathbf{p}_{old}]that we can also write\mathbf{\beta}_{new} =(\mathbf{X}^T\mathbf{\Delta}_{old}\mathbf{X})^{-1}\mathbf{X}^T\mathbf{\Delta}_{old}\mathbf{z}where \mathbf{z}=\mathbf{X}\mathbf{\beta}_{old}+\mathbf{\Delta}_{old}^{-1}[\mathbf{y}-\mathbf{p}_{old}]. Here, on the penalized problem, we can easily prove that\frac{\partial\log\mathcal{L}_p(\mathbf{\beta}_{\lambda,old})}{\partial\mathbf{\beta}}=\frac{\partial\log\mathcal{L}(\mathbf{\beta}_{\lambda,old})}{\partial\mathbf{\beta}}-2\lambda\mathbf{\beta}_{old}while\frac{\partial^2\log\mathcal{L}_p(\mathbf{\beta}_{\lambda,old})}{\partial\mathbf{\beta}\partial\mathbf{\beta}^T}=\frac{\partial^2\log\mathcal{L}(\mathbf{\beta}_{\lambda,old})}{\partial\mathbf{\beta}\partial\mathbf{\beta}^T}-2\lambda\mathbb{I}Hence\mathbf{\beta}_{\lambda,new} =(\mathbf{X}^T\mathbf{\Delta}_{old}\mathbf{X}+2\lambda\mathbb{I})^{-1}\mathbf{X}^T\mathbf{\Delta}_{old}\mathbf{z}
The code is then
Y = myocarde$PRONO X = myocarde[,1:7] for(j in 1:7) X[,j] = (X[,j]-mean(X[,j]))/sd(X[,j]) X = as.matrix(X) X = cbind(1,X) colnames(X) = c("Inter",names(myocarde[,1:7])) beta = as.matrix(lm(Y~0+X)$coefficients,ncol=1) for(s in 1:9){ pi = exp(X%*%beta[,s])/(1+exp(X%*%beta[,s])) Delta = matrix(0,nrow(X),nrow(X));diag(Delta)=(pi*(1-pi)) z = X%*%beta[,s] + solve(Delta)%*%(Y-pi) B = solve(t(X)%*%Delta%*%X+2*lambda*diag(ncol(X))) %*% (t(X)%*%Delta%*%z) beta = cbind(beta,B)} beta[,8:10] [,1] [,2] [,3] XInter 0.59619654 0.59619654 0.59619654 XFRCAR 0.09217848 0.09217848 0.09217848 XINCAR 0.77165707 0.77165707 0.77165707 XINSYS 0.69678521 0.69678521 0.69678521 XPRDIA -0.29575642 -0.29575642 -0.29575642 XPAPUL -0.23921101 -0.23921101 -0.23921101 XPVENT -0.33120792 -0.33120792 -0.33120792 XREPUL -0.84308972 -0.84308972 -0.84308972 |
Again, it seems that convergence is very fast.
And interestingly, with that algorithm, we can also derive the variance of the estimator\text{Var}[\widehat{\mathbf{\beta}}_{\lambda}]=[\mathbf{X}^T\mathbf{\Delta}\mathbf{X}+2\lambda\mathbb{I}]^{-1}\mathbf{X}^T\mathbf{\Delta}\text{Var}[\mathbf{z}]\mathbf{\Delta}\mathbf{X}[\mathbf{X}^T\mathbf{\Delta}\mathbf{X}+2\lambda\mathbb{I}]^{-1}where\text{Var}[\mathbf{z}]=\mathbf{\Delta}^{-1}
The code to compute \widehat{\mathbf{\beta}}_{\lambda} as a function of \lambda is then
newton_ridge = function(lambda=1){ beta = as.matrix(lm(Y~0+X)$coefficients,ncol=1)*runif(8) for(s in 1:20){ pi = exp(X%*%beta[,s])/(1+exp(X%*%beta[,s])) Delta = matrix(0,nrow(X),nrow(X));diag(Delta)=(pi*(1-pi)) z = X%*%beta[,s] + solve(Delta)%*%(Y-pi) B = solve(t(X)%*%Delta%*%X+2*lambda*diag(ncol(X))) %*% (t(X)%*%Delta%*%z) beta = cbind(beta,B)} Varz = solve(Delta) Varb = solve(t(X)%*%Delta%*%X+2*lambda*diag(ncol(X))) %*% t(X)%*% Delta %*% Varz %*% Delta %*% X %*% solve(t(X)%*%Delta%*%X+2*lambda*diag(ncol(X))) return(list(beta=beta[,ncol(beta)],sd=sqrt(diag(Varb))))} |
We can visualize the evolution of \widehat{\mathbf{\beta}}_{\lambda} (as a function of \lambda)
v_lambda=c(exp(seq(-2,5,length=61))) est_ridge=Vectorize(function(x) newton_ridge(x)$beta)(v_lambda) library("RColorBrewer") colrs=brewer.pal(7,"Set1") plot(v_lambda,est_ridge[1,],col=colrs[1],type="l") for(i in 2:7) lines(v_lambda,est_ridge[i,],col=colrs[i]) |
and to get the evolution of the variance
v_lambda=c(exp(seq(-2,5,length=61))) est_ridge=Vectorize(function(x) newton_ridge(x)$sd)(v_lambda) library("RColorBrewer") colrs=brewer.pal(7,"Set1") plot(v_lambda,est_ridge[1,],col=colrs[1],type="l") for(i in 2:7) lines(v_lambda,est_ridge[i,],col=colrs[i],lwd=2) |
Recall that when \lambda=0 (on the left of the graphs), \widehat{\mathbf{\beta}}_{0}=\widehat{\mathbf{\beta}}^{mco} (no penalty). Thus as \lambda increase (i) the bias increase (estimates tend to 0) (ii) the variances deacrease.
Ridge, using glmnet
As always, there are R functions availble to run a ridge regression. Let us use the glmnet function, with \alpha=0
y = myocarde$PRONO X = myocarde[,1:7] for(j in 1:7) X[,j] = (X[,j]-mean(X[,j]))/sd(X[,j]) X = as.matrix(X) library(glmnet) glm_ridge = glmnet(X, y, alpha=0) plot(glm_ridge,xvar="lambda",col=colrs,lwd=2) |
as a function of the norm
the \ell_1 norm here, I don’t know why. I don’t know either why all graphs obtained with different optimisation routines are so different… Maybe that will be for another post…
Ridge with orthogonal covariates
An interesting case is obtained when covariates are orthogonal. This can be obtained using a PCA of the covariates.
library(factoextra) pca = princomp(X) pca_X = get_pca_ind(pca)$coord |
Let us run a ridge regression on those (orthogonal) covariates
library(glmnet) glm_ridge = glmnet(pca_X, y, alpha=0) plot(glm_ridge,xvar="lambda",col=colrs,lwd=2) |
plot(glm_ridge,col=colrs,lwd=2) |
We clearly observe the shrinkage of the parameters, in the sense that \widehat{\mathbf{\beta}}_{\lambda}^{\perp}=\frac{\widehat{\mathbf{\beta}}^{mco}}{1+\lambda}
Application
Let us try with our second set of data
df0 = df df0$y=as.numeric(df$y)-1 plot_lambda = function(lambda){ m = apply(df0,2,mean) s = apply(df0,2,sd) for(j in 1:2) df0[,j] = (df0[,j]-m[j])/s[j] reg = glmnet(cbind(df0$x1,df0$x2), df0$y==1, alpha=0,lambda=lambda) u = seq(0,1,length=101) p = function(x,y){ xt = (x-m[1])/s[1] yt = (y-m[2])/s[2] predict(reg,newx=cbind(x1=xt,x2=yt),type='response')} v = outer(u,u,p) image(u,u,v,col=clr10,breaks=(0:10)/10) points(df$x1,df$x2,pch=c(1,19)[1+z],cex=1.5) contour(u,u,v,levels = .5,add=TRUE) } |
We can try various values of \lambda
reg = glmnet(cbind(df0$x1,df0$x2), df0$y==1, alpha=0) par(mfrow=c(1,2)) plot(reg,xvar="lambda",col=c("blue","red"),lwd=2) abline(v=log(.2)) plot_lambda(.2) |
or
reg = glmnet(cbind(df0$x1,df0$x2), df0$y==1, alpha=0) par(mfrow=c(1,2)) plot(reg,xvar="lambda",col=c("blue","red"),lwd=2) abline(v=log(1.2)) plot_lambda(1.2) |
Next step is to change the norm of the penality, with the \ell_1 norm (to be continued…)
