Modeling Pandemics (3)

In Statistical Inference in a Stochastic Epidemic SEIR Model with Control Intervention, a more complex model than the one we’ve seen yesterday was considered (and is called the SEIR model). Consider a population of size $N$, and assume that $S$ is the number of susceptible, $E$ the number of exposed, $I$ the number of infectious, and $R$ for the number recovered (or immune) individuals, \displaystyle{\begin{aligned}{\frac {dS}{dt}}&=-\beta {\frac {I}{N}}S\\[8pt]{\frac {dE}{dt}}&=\beta {\frac {I}{N}}S-aE\\[8pt]{\frac {dI}{dt}}&=aE-b I\\[8pt]{\frac {dR}{dt}}&=b I\end{aligned}}Between $S$ and $I$, the transition rate is $\beta I$, where $\beta$ is the average number of contacts per person per time, multiplied by the probability of disease transmission in a contact between a susceptible and an infectious subject. Between $I$ and $R$, the transition rate is $b$ (simply the rate of recovered or dead, that is, number of recovered or dead during a period of time divided by the total number of infected on that same period of time). And finally, the incubation period is a random variable with exponential distribution with parameter $a$, so that the average incubation period is $a^{-1}$.

Probably more interesting, Understanding the dynamics of ebola epidemics suggested a more complex model, with susceptible people $S$, exposed $E$, Infectious, but either in community $I$, or in hospitals $H$, some people who died $F$ and finally those who either recover or are buried and therefore are no longer susceptible $R$.

Thus, the following dynamic model is considered\displaystyle{\begin{aligned}{\frac {dS}{dt}}&=-(\beta_II+\beta_HH+\beta_FF)\frac{S}{N}\\[8pt]\frac {dE}{dt}&=(\beta_II+\beta_HH+\beta_FF)\frac{S}{N}-\alpha E\\[8pt]\frac {dI}{dt}&=\alpha E+\theta\gamma_H I-(1-\theta)(1-\delta)\gamma_RI-(1-\theta)\delta\gamma_FI\\[8pt]\frac {dH}{dt}&=\theta\gamma_HI-\delta\lambda_FH-(1-\delta)\lambda_RH\\[8pt]\frac {dF}{dt}&=(1-\theta)(1-\delta)\gamma_RI+\delta\lambda_FH-\nu F\\[8pt]\frac {dR}{dt}&=(1-\theta)(1-\delta)\gamma_RI+(1-\delta)\lambda_FH+\nu F\end{aligned}}In that model, parameters are $\alpha^{-1}$ is the (average) incubation period (7 days), $\gamma_H^{-1}$ the onset to hospitalization (5 days), $\gamma_F^{-1}$ the onset to death (9 days), $\gamma_R^{-1}$ the onset to “recovery” (10 days), $\lambda_F^{-1}$ the hospitalisation to death (4 days) while $\lambda_R^{-1}$ is the hospitalisation to recovery (5 days), $\eta^{-1}$ is the death to burial (2 days). Here, numbers are from Understanding the dynamics of ebola epidemics (in the context of ebola). The other parameters are $\beta_I$ the transmission rate in community (0.588), $\beta_H$ the transmission rate in hospital (0.794) and $\beta_F$ the transmission rate at funeral (7.653). Thus

epsilon = 0.001 Z = c(S = 1-epsilon, E = epsilon, I=0,H=0,F=0,R=0) p=c(alpha=1/7*7, theta=0.81, delta=0.81, betai=0.588, betah=0.794, blambdaf=7.653,N=1, gammah=1/5*7, gammaf=1/9.6*7, gammar=1/10*7, lambdaf=1/4.6*7, lambdar=1/5*7, nu=1/2*7)

If $\boldsymbol{Z}=(S,E,I,H,F,R)$, if we write $$\frac{\partial \boldsymbol{Z}}{\partial t} = SEIHFR(\boldsymbol{Z})$$where $SEIHFR$ is

SEIHFR = function(t,Z,p){ S=Z[1]; E=Z[2]; I=Z[3]; H=Z[4]; F=Z[5]; R=Z[6] alpha=p["alpha"]; theta=p["theta"]; delta=p["delta"] betai=p["betai"]; betah=p["betah"]; gammah=p["gammah"] gammaf=p["gammaf"]; gammar=p["gammar"]; lambdaf=p["lambdaf"] lambdar=p["lambdar"]; nu=p["nu"]; blambdaf=p["blambdaf"] N=S+E+I+H+F+R dS=-(betai*I+betah*H+blambdaf*F)*S/N dE=(betai*I+betah*H+blambdaf*F)*S/N-alpha*E dI=alpha*E-theta*gammah*I-(1-theta)*(1-delta)*gammar*I-(1-theta)*delta*gammaf*I dH=theta*gammah*I-delta*lambdaf*H-(1-delta)*lambdaf*H dF=(1-theta)*(1-delta)*gammar*I+delta*lambdaf*H-nu*F dR=(1-theta)*(1-delta)*gammar*I+(1-delta)*lambdar*H+nu*F dZ=c(dS,dE,dI,dH,dF,dR) list(dZ)}

We can solve it, or at least study the dynamics from some starting values

library(deSolve) times = seq(0, 50, by = .1) resol = ode(y=Z, times=times, func=SEIHFR, parms=p)

For instance, the proportion of people infected is the following

plot(resol[,"time"],resol[,"I"],type="l",xlab="time",ylab="",col="red") lines(resol[,"time"],resol[,"H"],col="blue")

Modeling pandemics (2)

When introducing the SIR model, in our initial post, we got an ordinary differential equation, but we did not really discuss stability, and periodicity. It has to do with the Jacobian matrix of the system. But first of all, we had three equations for three function, but actually$$\displaystyle{{\frac{dS}{dt}}+{\frac {dI}{dt}}+{\frac {dR}{dt}}=0}$$so it means that our problem is here simply in dimension 2. Hence\displaystyle {\begin{aligned}&X={\frac {dS}{dt}}=\mu(N-S)-{\frac {\beta IS}{N}},\\[6pt]&Y={\frac {dI}{dt}}={\frac {\beta IS}{N}}-(\mu+\gamma)I\end{aligned}}and therefore, the Jacobian of the system is$$\begin{pmatrix}\displaystyle{\frac{\partial X}{\partial S}}&\displaystyle{\frac{\partial X}{\partial I}}\\[9pt]\displaystyle{\frac{\partial Y}{\partial S}}&\displaystyle{\frac{\partial Y}{\partial I}}\end{pmatrix}=\begin{pmatrix}\displaystyle{-\mu-\beta\frac{I}{N}}&\displaystyle{-\beta\frac{S}{N}}\\[9pt]\displaystyle{\beta\frac{I}{N}}&\displaystyle{\beta\frac{S}{N}-(\mu+\gamma)}\end{pmatrix}$$We should evaluate the Jacobian at the equilibrium, i.e. $$S^\star=\frac{\gamma+\mu}{\beta}=\frac{1}{R_0}$$and$$I^\star=\frac{\mu(R_0-1)}{\beta}$$We should then look at eigenvalues of the matrix.

Our very last example was

times = seq(0, 100, by=.1) p = c(mu = 1/100, N = 1, beta = 50, gamma = 10) start_SIR = c(S=0.19, I=0.01, R = 0.8) resol = ode(y=start_SIR, t=times, func=SIR, p=p) plot(resol[,"time"],resol[,"I"],type="l",xlab="time",ylab="")

We can compute values at the equilibrium

mu=p["mu"]; beta=p["beta"]; gamma=p["gamma"] N=1 S = (gamma + mu)/beta I = mu * (beta/(gamma + mu) - 1)/beta

and the Jacobian matrix

J=matrix(c(-(mu + beta * I/N),-(beta * S/N), beta * I/N,beta * S/N - (mu + gamma)),2,2,byrow = TRUE)

Now, if we look at the eigenvalues,

eigen(J)$values [1] -0.024975+0.6318831i -0.024975-0.6318831i or more precisely $2\pi/b$ where $a\pm ib$ are the conjuguate eigenvalues 2 * pi/(Im(eigen(J)$values[1])) [1] 9.943588

we have a damping period of 10 time lengths (10 days, or 10 weeks), which is more or less what we’ve seen above,

The graph above was obtained using

p = c(mu = 1/100, N = 1, beta = 50, gamma = 10) start_SIR = c(S=0.19, I=0.01, R = 0.8) resol = ode(y=start_SIR, t=times, func=SIR, p=p) plot(resol[1:1e5,"time"],resol[1:1e5,"I"],type="l",xlab="time",ylab="",lwd=3,col="red") yi=resol[,"I"] dyi=diff(yi) i=which((dyi[2:length(dyi)]*dyi[1:(length(dyi)-1)])&lt;0) t=resol[i,"time"] arrows(t[2],.008,t[4],.008,length=.1,code=3)

If we look carefully. at the begining, the duration is (much) longer than 10 (about 13)… but it does converge towards 9.94

plot(diff(t[seq(2,40,by=2)]),type="b") abline(h=2 * pi/(Im(eigen(J)values[1])) So here, theoretically, every 10 weeks (assuming that our time length is a week), we should observe an outbreak, smaller than the previous one. In practice, initially it is every 13 or 12 weeks, but the time to wait between outbreaks decreases (until it reaches 10 weeks). Modeling pandemics (1) The most popular model to model epidemics is the so-called SIR model – or Kermack-McKendrick. Consider a population of size $N$, and assume that $S$ is the number of susceptible, $I$ the number of infectious, and $R$ for the number recovered (or immune) individuals, \displaystyle {\begin{aligned}&{\frac {dS}{dt}}=-{\frac {\beta IS}{N}},\\[6pt]&{\frac {dI}{dt}}={\frac {\beta IS}{N}}-\gamma I,\\[6pt]&{\frac {dR}{dt}}=\gamma I,\end{aligned}}so that $$\displaystyle{{\frac{dS}{dt}}+{\frac {dI}{dt}}+{\frac {dR}{dt}}=0}$$which implies that $S+I+R=N$. In order to be more realistic, consider some (constant) birth rate $\mu$, so that the model becomes\displaystyle {\begin{aligned}&{\frac {dS}{dt}}=\mu(N-S)-{\frac {\beta IS}{N}},\\[6pt]&{\frac {dI}{dt}}={\frac {\beta IS}{N}}-(\gamma+\mu) I,\\[6pt]&{\frac {dR}{dt}}=\gamma I-\mu R,\end{aligned}}Note, in this model, that people get sick (infected) but they do not die, they recover. So here, we can model chickenpox, for instance, not SARS. The dynamics of the infectious class depends on the following ratio:$$\displaystyle{R_{0}={\frac {\beta }{\gamma +\mu}}}$$ which is the so-called basic reproduction number (or reproductive ratio). The effective reproductive ratio is $R_0S/N$, and the turnover of the epidemic happens exactly when $R_0S/N=1$, or when the fraction of remaining susceptibles is $R_0^{-1}$. As shown in Directly transmitted infectious diseases:Control by vaccination, if $S/N the disease (the number of people infected) will start to decrease. Want to see it ? Start with mu = 0 beta = 2 gamma = 1/2 for the parameters. Here, $R_0=4$. We also need starting values epsilon = .001 N = 1 S = 1-epsilon I = epsilon R = 0 Then use the ordinary differential equation solver, in R. The idea is to say that $\boldsymbol{Z}=(S,I,R)$ and we have the gradient $$\frac{\partial \boldsymbol{Z}}{\partial t} = SIR(\boldsymbol{Z})$$where $SIR$ is function of the various parameters. Hence, set p = c(mu = 0, N = 1, beta = 2, gamma = 1/2) start_SIR = c(S = 1-epsilon, I = epsilon, R = 0) The we must define the time, and the function that returns the gradient, times = seq(0, 10, by = .1) SIR = function(t,Z,p){ S=Z[1]; I=Z[2]; R=Z[3]; N=S+I+R mu=p["mu"]; beta=p["beta"]; gamma=p["gamma"] dS=mu*(N-S)-beta*S*I/N dI=beta*S*I/N-(mu+gamma)*I dR=gamma*I-mu*R dZ=c(dS,dI,dR) return(list(dZ))} To solve this problem use library(deSolve) resol = ode(y=start_SIR, times=times, func=SIR, parms=p) We can visualize the dynamics below par(mfrow=c(1,2)) t=resol[,"time"] plot(t,resol[,"S"],type="l",xlab="time",ylab="") lines(t,resol[,"I"],col="red") lines(t,resol[,"R"],col="blue") plot(t,t*0+1,type="l",xlab="time",ylab="",ylim=0:1) polygon(c(t,rev(t)),c(resol[,"R"],rep(0,nrow(resol))),col="blue") polygon(c(t,rev(t)),c(resol[,"R"]+resol[,"I"],rev(resol[,"R"])),col="red") We can actually also visualize the effective reproductive number is $R_0S/N$, where R0=p["beta"]/(p["gamma"]+p["mu"]) The effective reproductive number is on the left, and as we mentioned above, when we reach 1, we actually reach the maximum of the infected, plot(t,resol[,"S"]*R0,type="l",xlab="time",ylab="") abline(h=1,lty=2,col="red") abline(v=max(t[resol[,"S"]*R0&gt;=1]),col="darkgreen") points(max(t[resol[,"S"]*R0&gt;=1]),1,pch=19) plot(t,resol[,"S"],type="l",xlab="time",ylab="",col="grey") lines(t,resol[,"I"],col="red",lwd=3) lines(t,resol[,"R"],col="light blue") abline(v=max(t[resol[,"S"]*R0&gt;=1]),col="darkgreen") points(max(t[resol[,"S"]*R0&gt;=1]),max(resol[,"I"]),pch=19) And when adding a $\mu$ parameter, we can obtain some interesting dynamics on the number of infected, times = seq(0, 100, by=.1) p = c(mu = 1/100, N = 1, beta = 50, gamma = 10) start_SIR = c(S=0.19, I=0.01, R = 0.8) resol = ode(y=start_SIR, t=times, func=SIR, p=p) plot(resol[,"time"],resol[,"I"],type="l",xlab="time",ylab="") Richesse et espérance de vie Ce matin, je découvrais un graphique de l’INSEE qui présentait les taux de mortalité par sexe, âge et niveau de vie, avec entre autres le graphique suivant Comme souvent avec l’INSEE, on a accès aux données… pas celles au niveau individuel (malheureusement) mais au moins on peut retravailler la visualisation. En fait, les données sont même encore plus fines, puisque les niveaux de richesses sont définis avec des tranches de 5%, et en plus, on a le détail entre hommes et femmes b = read.csv2("MORT-RICHESSE.csv") plot(b[,1],b[,2]/1000,col="red",type="l",ylab="% de survivants",xlab="Age") lines(b[,1],b[,3]/1000,col="red",type="l",lty=2) lines(b[,1],b[,4]/1000,col="blue",type="l",lty=1) lines(b[,1],b[,5]/1000,col="blue",type="l",lty=2) legend("bottomleft",c("Femmes 95-100%","Hommes 95-100%","Femmes 0-5%","Hommes, 0-5%"), bty="n",col=c("red","blue","red","blue"),lty=c(2,2,1,1)) Je me demandais si on ne pouvait pas tenter une lecture inverse de ce graphique : sur ce graphique, assez naturellement, on regarde à pourcentage donné l’écart entre la courbe en trait plein (les pauvres) et celle en trait pointillé (les riches). Si c’est cette information qu’on veut avoir, on peut alors tenter de la visualiser. Pour ça, il faut inverser notre fonction de survie (je l’ai fait rapidement, avec une simple interpolation linéaire… je pense qu’on peut faire mieux) inversef = function(p,k=2){ y=1-b[,k]/100000 idx=sum(y&lt;=p) y1=y[idx-1] y2=y[idx] w1=(y1-p)/(y1-y2) w2=(p-y2)/(y1-y2) w2*b[idx-1,1]+w1*b[idx,1] } Ensuite, on peut construire les inverses, et mieux, les différences entre les courbes des riches et des pauvres diffF = function(p) inversef(p,3)-inversef(p,2) diffH = function(p) inversef(p,5)-inversef(p,4) u = seq(.01,.99,by=.01) vF = Vectorize(diffF)(u) vH = Vectorize(diffH)(u) plot(u*100,vF,col="red",type="l",xlab="Probabilité (%)",ylab="Nombre d'années",ylim=c(0,max(vF,vH))) lines(u*100,vH,col="blue",type="l",lty=1) legend("topright",c("Femmes","Hommes"), bty="n",col=c("red","blue"),lty=c(1,1)) Je ne suis pas très à l’aise avec le graphique. Tout d’abord parce que je ne sais pas ce que la richesse indique (un pauvre de 20 ans peut devenir un riche de 50 ans, non ?), la richesse était souvent liée à l’âge. Après l’axe des abscisses me semble aussi avoir une interprétation compliquée : quand on regarde à 10%, on regarde les pauvres et les riches qui sont morts relativement jeunes (relativement car je regarde le quantile de la fonction de survie à tranche de richesse donnée). Autrement dit, si je me place à 10%, je compare les jeunes riches morts très jeunes (10% des riches seulement sont morts plus jeunes – c’est l’interprétation d’un quantile) et les jeunes pauvres (mort à un âge que seulement 90% des pauvres ont dépassé), on observe une différence de 20 ans environ. J’ai aussi l’impression qu’on pourrait dire que pour la majorité des hommes, les riches vivent 12 ans de plus que les pauvres, soit le double des femmes (de l’ordre de 6 ans). On pourrait bien sûr se contenter de calculer les différences entre les aires, ce qui donne une différence entre les espérances de vie à la naissance des pauvres et des riches (comme le fait l’INSEE) et qu’on peut visualiser sur le graphique suivant plot(b[,1],b[,2]/1000,col="white",type="l",ylab="% de survivants",xlab="Age") polygon(c(b[,1],rev(b[,1])),c(b[,3]/1000,rev(b[,2]/1000)),col="red",border=NA) et un calcul donne une différence de l’ordre de 8 ans sum(b[,3]-b[,2])/100000 [1] 8.239346 mais la visualisation raconte bien plus qu’un simple calcul d’aire. Par exemple, le graphique ci-dessous donne exactement le même écart entre les espérances de vie des pauvres et des riches diff = sum(b[,3]-b[,2])/1000 y1 = b[,2]/1000 for(i in 1:100){ y1[i] = b[i,2]/1000+min(100-b[i,2]/1000,diff) diff = diff-(y1[i]-b[i,2]/1000) } plot(b[,1],b[,2]/1000,col="white",type="l",ylab="% de survivants",xlab="Age") polygon(c(b[,1],rev(b[,1])),c(y1,rev(b[,2]/1000)),col="red",border=NA) sum(b[,3]-b[,2])/100000 lines(b[,1],b[,2]/1000,col="red") Ici, on dit que la moitié des femmes riches meurent vers 81 ans, et les autres meurent au même âge qu’une femmes pauvre (mais une femme pauvre qui vivrait longtemps. Les distributions sont vraiment différentes, et c’est ça que je cherche à visualiser… Parce que la densité de l’âge au décès ne me semble pas forcément très simple à analyser… Comme toujours, les commentaires sont ouverts si certains ont des idées quant à la visualisation de ces données… Testing for a causal effect (with 2 time series) A few days ago, I came back on a sentence I found (in a French newspaper), where someone was claiming that “… an old variable explains 85% of the change in a new variable. So we can talk about causality” and I tried to explain that it was just stupid : if we consider the regression of the temperature on day $t+1$ against the number of cyclist on day $t$, the $R^2$ exceeds 80%… but it is hard to claim that the number of cyclists on specific day will actually cause the temperature on the next day… Nevertheless, that was frustrating, and I was wondering if there was a clever way to test for causality in that case. A popular one is Granger causality (I can mention a paper we published a few years ago where we use such a test, Tents, Tweets, and Events: The Interplay Between Ongoing Protests and Social Media). To explain that test, consider a bivariate time series (just like the one we have here), $\boldsymbol{z}_t=(x_t,y_t)$, and consider some bivariate autoregressive model $${\displaystyle {\begin{bmatrix}x_{t}\\y_{t}\end{bmatrix}}={\begin{bmatrix}c_{1}\\c_{2}\end{bmatrix}}+{\begin{bmatrix}a_{1,1}&\textcolor{red}{a_{1,2}}\\\textcolor{blue}{a_{2,1}}&a_{2,2}\end{bmatrix}}{\begin{bmatrix}x_{t-1}\\y_{t-1}\end{bmatrix}}+{\begin{bmatrix}u_{t}\\v_{t}\end{bmatrix}}}$$where $\boldsymbol{\varepsilon}_t=(u_t,v_t)$ is some bivariate white noise, in the sense that (i) ${\displaystyle \mathbb{E} (\boldsymbol{\varepsilon}_{t})=\boldsymbol{0}}$ (the noise is centered) (ii) ${\displaystyle \mathbb{E} (\boldsymbol{\varepsilon}_{t}\boldsymbol{\varepsilon}_{t}^\top)=\Omega }$, so the variance matrix is constant, but possibly non-diagonal (iii) ${\displaystyle \mathbb{E} (\boldsymbol{\varepsilon}_{t}\boldsymbol{\varepsilon}_{t-h}^\top)=\boldsymbol{0} }$ for all $h\neq 0$. Note that we can use the simplified expression$${\displaystyle {\boldsymbol{z}_t=\boldsymbol{c}+\boldsymbol{A}\boldsymbol{z}_{t-1}+\boldsymbol{\varepsilon}_t}}$$Now, Granger test is based on several quantities. With off-diagonal terms of matrix $\Omega$, we have a so-called instantaneous causality, and since $\Omega$ is symmetry, we will write $x\leftrightarrow y$. With off-diagonal terms of matrix $\boldsymbol{A}$, we have a so-called lagged causality, with either $\textcolor{blue}{x\rightarrow y}$ or $\textcolor{red}{x\leftarrow y}$ (and possibly both, if both terms are significant). So I wanted to try on my two-variable problem. df = read.csv("cyclistsTempHKI.csv") dfts = cbind(C=ts(dfcyclists,start = c(2014, 1,2), frequency = 365), T=ts(df$meanTemp,start = c(2014, 1,2), frequency = 365)) library(vars) I now have “time series” objects, and we can fit a VAR model, var2 = VAR(dfts, p = 1, type = "const") coefficients(var2)$C Estimate Std. Error t value Pr(&gt;|t|) C.l1 0.8684009 0.02889424 30.054460 8.080226e-107 T.l1 70.3042012 20.07247411 3.502518 5.102094e-04 const 807.6394001 187.75472482 4.301566 2.110412e-05   $T Estimate Std. Error t value Pr(&gt;|t|) C.l1 0.0003865391 6.257596e-05 6.177118 1.540467e-09 T.l1 0.6611135594 4.347074e-02 15.208241 6.086394e-42 const -1.6413074565 4.066184e-01 -4.036481 6.446018e-05 For instant, we can run a causality, to test if the number of cyclists can cause the temperature (on the next day) causality(var2, cause = "C")$Granger   Granger causality H0: C do not Granger-cause T   data: VAR object var2 F-Test = 38.157, df1 = 1, df2 = 842, p-value = 1.015e-09

Here, we should clearly reject $H_0$, which is that there is no causal effect. Which is the way statisticians say that there should be some causal effect between the number of cyclist and the temperature…

So clearly, something is wrong here. Either it is some sort of superpower that cyclists are not aware of. Or this test that was used for forty years (Clive Granger even got a Nobel price for it) is not working. Or we missed something. Actually… I think we missed something here. Possibly because the series are not stationary. We can almost see it with

Phi = matrix(c(coefficients(var2)$C[1:2,1],coefficients(var2)$T[1:2,1]),2,2) eigen(Phi) eigen() decomposition $values [1] 0.9594810 0.5700335 where the highest eigenvalue is very close to one. But actually, we look here at the temperature… plot(dfts) so, at least, we should expect some seasonal unit root here. So let us use two techniques. The first one is a classical one-year difference, $\Delta_{365}\boldsymbol{z}_t=\boldsymbol{z}_t-\boldsymbol{z}_{t-365}$ var2 = VAR(diff(dfts,365), p = 1, type = "const") coefficients(var2)$C Estimate Std. Error t value Pr(&gt;|t|) C.l1 0.8376424 0.07259969 11.537823 1.993355e-16 T.l1 42.2638410 28.58783276 1.478386 1.449076e-01 const -507.5514795 219.40240747 -2.313336 2.440042e-02   $T Estimate Std. Error t value Pr(&gt;|t|) C.l1 0.000518209 0.0003277295 1.5812096 1.194623e-01 T.l1 0.598425288 0.1290511945 4.6371154 2.162476e-05 const 0.547828079 0.9904263469 0.5531235 5.823804e-01 The test on the fited VAR model yields causality(var2, cause = "C")$Granger   Granger causality H0: C do not Granger-cause T   data: VAR object var2 F-Test = 2.5002, df1 = 1, df2 = 112, p-value = 0.1167

i.e., with a 11% $p$-value, we should reject the assumption that the number of cyclists cause the temperature (on the next day), and actually, we should also reject the other way

causality(var2, cause = "T") $Granger Granger causality H0: T do not Granger-cause C data: VAR object var2 F-Test = 2.1856, df1 = 1, df2 = 112, p-value = 0.1421 Nevertheless, if we look at the instantaneous causality, this one makes more sense $Instant   H0: No instantaneous causality between: T and C   data: VAR object var2 Chi-squared = 13.081, df = 1, p-value = 0.0002982

The second idea would be to use a one day difference, $\Delta_{1}\boldsymbol{z}_t=\boldsymbol{z}_t-\boldsymbol{z}_{t-1}$ and to fit a VAR model on that one

VARselect(diff(dfts,1), lag.max = 4, type="const") $selection AIC(n) HQ(n) SC(n) FPE(n) 3 3 2 3 but on that one, a VAR(1) model – with only one lag – might not be sufficient. It might be better to consider a VAR(3) var2 = VAR(diff(dfts,1), p = 3, type = "const") and on that one, one more time, we should reject the causal effect of the number of cyclists on the temperature (on the next day) causality(var2, cause = "C")$Granger   Granger causality H0: C do not Granger-cause T   data: VAR object var2 F-Test = 0.67644, df1 = 3, df2 = 828, p-value = 0.5666

and this time, there could be a (lagged) causal effect of the temperature on the number of cyclists

causality(var2, cause = "T") $Granger Granger causality H0: T do not Granger-cause C data: VAR object var2 F-Test = 7.7981, df1 = 3, df2 = 828, p-value = 3.879e-05$Instant   H0: No instantaneous causality between: T and C   data: VAR object var2 Chi-squared = 55.83, df = 1, p-value = 7.905e-14

but nothing instantaneously… So it looks like Granger causality performs well on that one !

Le R² pour justifier la causalite…

Ce soir, Louis (@LouisdeCharson) me signalait un article assez délirant où, dans une entrevue, un monsieur souvent présenté comme un chercheur (je n’ai pas réussi à trouver où) disait une phrase assez étonnante,

“… une variable ancienne explique 85 % de la variation d’une variable nouvelle. On peut donc parler de causalité”

Histoire de plaisanter un peu, je l’ai pris au mot, et j’ai ressorti un vieux jeu de données que j’avais utilisé dans un précédant billet, avec le traffic cycliste à Helsinki, en fonction de la température. Pour rester dans la classe des modèles linéaires, j’ai enlevé quelques jours d’hiver, et j’ai tenté de régresser la température le jour $j+1$ (la nouvelle variable comme dit dans la discussion) sur le nombre de cyclistes la veille, autrement dit le jour $j$

Si on regarde la régression, on obtient

lm(formula = temp ~ cyclists, data = df0) Coefficients: Estimate Std. Error t value Pr(&gt;|t|) (Intercept) -4.170e+00 4.052e-01 -10.29 &lt;2e-16 *** cyclists 1.066e-03 3.558e-05 29.96 &lt;2e-16 ***   Residual standard error: 2.833 on 212 degrees of freedom Multiple R-squared: 0.809, Adjusted R-squared: 0.8081

autrement dit, $81\%$ de la variation de la température (moyenne) le  jour $j+1$ est expliqué par le nombre de cyclistes le jour $j$, c’est la définition du $R^2$ (“the coefficient of determination, denoted R2, is the proportion of the variance in the dependent variable that is predictable from the independent variable“). Et le monsieur en déduit qu’il y a une relation causale, autrement dit, dans notre exemple, le nombre de cyclistes sur la route le jour $j$ cause la température le jour $j+1$. Loin de moi l’idée de me poser comme un expert sur les modèles causaux et les modèles climatiques, je doute que ce soit le cas… sinon la solution pour stopper le réchauffement climatique est toute trouvée : il faut diminuer le nombre de cyclistes !

Des modeles prédictifs en assurance

Cet post est aussi en ligne sur https://hal.archives-ouvertes.fr/hal-02350006, il a été coécrit avec Laurence Barry et Ewen Gallic.

Les compagnies d’assurance émettent des contrats qui prévoient des paiements d’indemnités en cas de survenance d’évènements aléatoires (accident, maladie, décès, etc.). En contrepartie, l’assuré doit s’acquitter d’une prime, dont le montant est déterminé ex-ante, avant le début de la période de couverture. Cette prime se décompose en deux termes : une prime pure (destinée à couvrir les pertes anticipées) et un chargement (incluant des commissions à des agents, divers frais, mais aussi couvrant contre le risque de variabilité des pertes). La prime pure est souvent calculée par classe de risque, et une classification est alors nécessaire.

Assurer une population hétérogène, ou l’importance de la classification

Le regroupement des risques selon diverses informations telles l’âge de l’assuré, son état de santé ou encore sa profession constitue ce que l’on appelle la classification des risques. Cette pratique de segmentation se justifie (à des fins d’admissibilité mais aussi de tarification) par la supposition que les risques sont placés dans des groupes relativement homogènes, au sein desquels les probabilités de survenance sont similaires. Pour Schauer (2006), cette « généralisation », qui vise à voir l’individu sous le prisme de sa classe de risque, de généraliser son comportement à partir de quelques variables explicatives, est probablement la raison d’être de l’actuaire : « to be an actuary is to be a specialist  in  generalization,  and  actuaries  engage  in  a form of decision-making that is sometimes called actuarial ». Statistiquement on cherche une méthode de classification aussi « discriminatoire » que possible[1], en gardant en mémoire que la discrimination est interdite, ce qui rend l’exercice périlleux, et souvent critiqué (nous y reviendrons plus loin).

Les assureurs évoquent souvent deux arguments pour justifier une segmentation. Le premier est qu’elle serait rendue économiquement nécessaire par la concurrence ; ne pas classifier conduit à une anti-sélection, les risques importants restant seuls chez les assureurs qui ne segmentent pas. Dans une telle situation, l’équilibre de marché ne serait pas possible puisque les risques faibles seraient chez un concurrent ayant segmenté. Si le facteur de risque était observable, tant par les assurés que les assureurs, il y aurait un phénomène d’auto-sélection, les assurés à risque faible ayant les polices les moins chères. Cette situation constitue un équilibre séparant de Nash. Mais si le facteur de risque n’est pas observable, un équilibre sous-optimal peut être atteint, résultant d’une externalité négative de cette information non-accessible, à la manière de Wilson (1977), tel que décrit dans Cummins et al. (1982) dans le cas des contrats d’assurance-vie. Cela dit, Kleindorfer & Kunreuther (1980) montrent qu’accéder à davantage d’information ne conduit pas nécessairement à une amélioration du bien-être des consommateurs. De plus si la classification n’est pas autorisée, l’équilibre est maintenu, les risques faibles subventionnant les risques élevés.

Le second argument avancé pour justifier une segmentation est que cette dernière (et le fait, par conséquent, d’ajuster les primes au risque) serait juste et équitable. Mais cette vision de l’équité n’a pas toujours été de mise et semble portée par les développements techniques. Ainsi la classification est devenue de plus en plus fine, multipliant les classes de risque et conduisant à des tarifs « personnalisés ». En plus des avancées statistiques, des facteurs économiques pourraient justifier cette sophistication : la concurrence de plus en plus forte sur certaines branches.

Incertitude en assurance

Il y a plusieurs manières de caractériser l’incertitude en assurance. Comme bien souvent quand on fait des prévisions, il convient de distinguer l’incertitude associée à l’estimation des probabilités et l’incertitude réelle sur le résultat (aléa de l’évènement). Pour la seconde notion, Hacking (1975) parle de probabilité structurelle, et c’est celle qui est souvent utilisée pour introduire les concepts de probabilité, par exemple avec des dés ou des jeux de cartes : les probabilités sont connues, seule l’issue du jeu est incertaine. Par exemple je sais que la probabilité d’avoir 6 en lançant un dé est 1/6 (compte tenu de la géométrie du cube).

D’un point de vue statistique, la probabilité se mesure quand on peut observer une fréquence, c’est-à-dire une répétition de risques semblables. Les statisticiens ont ainsi défini une notion de probabilité empirique, basée sur la répétition[2]. Si, en lançant mille dés j’obtiens 173 fois la face 6, la probabilité empirique d’avoir 6 est de 17,3%. La loi des grands nombres nous assure que cette fréquence va tendre vers la vraie valeur en répétant l’expérience, et le théorème central limite permet d’en contrôler les fluctuations. C’est la première incertitude dont nous parlions au début de cette section, que nous appellerions l’erreur d’estimation.

On peut enfin mentionner deux notions supplémentaires ; tout d’abord, les probabilités conditionnelles. Cette idée est introduite en assurance par de Moivre, ou de Witt, lorsqu’ils notaient que pour estimer une probabilité de décès, il fallait considérer des personnes de même âge. C’est cette idée que l’on retrouve quand on considère une classification : on veut des risques homogènes, similaires, sans être pour autant identiques. La probabilité que l’on obtient est alors conditionnelle à ce facteur commun qui caractérise la classe observée. Dans notre exemple des dés, cela revient à dire qu’il ne faut pas lancer mille dés, mais mille fois le même dé – ou à défaut des dés semblables.

Enfin, les probabilités subjectives ont été formalisées par Bruno de Finetti et Leonard Savage (ainsi que plus philosophiquement par Frank P. Ramsey) pour comprendre et modéliser la prise de décision. Elles sont relativement populaires en économie de l’incertain, mais difficile à mettre en œuvre dans un contexte de valorisation de contrats d’assurance automobile ou habitation. Il s’agit d’un jugement, qui ne peut être confronté à la réalité, mais envisageable pour l’assurance de risques encore mal connus (McGrayne (2012) évoque ainsi les premiers contrats d’assurance aviation). Une approche bayésienne consiste alors à combiner cette probabilité subjective avec la probabilité comme fréquence observée d’un phénomène : partant d’une croyance a priori, on affine l’estimation par une mise à jour progressive en répétant les expériences. Classiquement, la probabilité d’avoir la face 6 sera une moyenne entre notre croyance (1 chance sur 6) et une probabilité dite historique, obtenue en faisant quelques lancés (3 sur 20 lancers, par exemple). Les poids attribués aux deux dépendant du nombre d’expériences effectuées : on donnera plus de crédit à l’expérience si on fait mille lancés que si on en fait soixante.

Incertitude sur le résultat, ou aléa fondamental

Les probabilités prédictives, utilisées pour calculer la prime d’un contrat d’assurance, sont la première étape d’un problème de classification. Un outil classique pour juger de la pertinence d’un classifieur est la courbe ROC, décrite dans Kuhn (2018)) : on compare la probabilité individuelle (a priori, telle que résultant du modèle de classification) à un seuil, compris entre 0 et 1; si la probabilité est inférieure au seuil, l’estimation est que la personne survit, sinon qu’elle décède.

On compare ensuite cette estimation aux réalisations (ex-post) de survie et de décès. Pour chaque seuil, on peut considérer la matrice classique dite matrice de confusion de théorie de la décision : elle consiste à répartir les observations suivant le résultat observé (en colonne) et l’estimation résultant du modèle en ligne (en fonction de la probabilité estimée pour l’individu et le seuil que l’on s’est fixé). On peut ainsi partager la population entre les classements corrects, et les erreurs (dont les « faux positifs » si la personne a survécu malgré une probabilité estimée de décès supérieure au seuil, et les « faux négatifs » si la personne décède malgré une probabilité estimée inférieure au seuil).

Figure 1: Courbe ROC et classification pour un seuil de probabilité valant 1.5%.

La courbe ROC est obtenue en faisant varier le seuil. Chaque seuil correspond à un point de la courbe, rapportant graphiquement les taux de faux positifs (en abscisse) et de vrais positifs (en ordonnée), comme sur la Figure 1.

Considérons un groupe de 1000 assurés, où 20 personnes sont décédées l’an passé. Supposons un modèle dans lequel on admet que la population est parfaitement homogène, la probabilité estimée de décès est de 2% pour tout le monde. Dans ce cas pour tout seuil supérieur à 2%, on estimera que la totalité de la population survit : on aura un taux de faux positifs de 0% et un taux de vrais positifs de 0%, d’où un point (0,0) sur le graphe. A l’inverse pour tout seuil inférieur à 2%, on estimera que la totalité de la population décède : on aura un taux de faux positifs de 100% et un taux de vrais positifs de 100%, d’où un point (1,1) sur le graphe. La courbe de ROC de ce modèle uniforme à 2% est donc la diagonale du carré sur la figure 1.

Mais on peut aussi imaginer qu’il existe un peu d’hétérogénéité avec, par exemple, une probabilité de décès de 1% pour une moitié de la population et de 3% pour l’autre moitié, ou encore que le modèle produit des probabilités comprises entre 1% et 3% de façon non dichotomique. Les données simulées pour construire la courbe noire sur la Figure 1 suppose que la population a des probabilités de décès variables, comprises entre 1% et 3%, obtenues par une régression logistique. Comme le montre le tableau de droite, on commet des erreurs, et comme le montre celle de gauche, la nature de celle-ci varie en fonction du seuil choisi, qui modifie les taux de faux positifs et de faux négatifs.

Le cas extrême serait celui où le modèle aurait correctement attribué une probabilité de 100% aux 20 personnes qui sont effectivement décédées. C’est la courbe rouge sur la Figure 1. Ce partage est possible ex-post, une fois réalisation de l’aléa : a posteriori, il y a une certitude de décès pour ceux qui sont effectivement morts. Mais cela n’a cependant pas grande réalité dans l’assurance, à moins d’imaginer que l’actuaire serait un oracle, qui saurait avec certitude qui va mourir, et qui va survivre. La réalité est plutôt celle de la situation intermédiaire entre la courbe rouge et la diagonale, avant d’arriver dans la région hachurée, où le taux d’erreur est faible, mais pas nul : on ne peut pas prédire, avec certitude, qui va décéder. L’assurance n’est possible que si cette borne supérieure n’est pas trop élevée.

Incertitude statistique, données et modèles

Une question fondamentale pour la survie de l’assurance est de savoir où se situe cette borne supérieure : jusqu’où peut-on aller, entre les deux cas extrêmes (population homogène avec une probabilité de 2% pour tous, et une population très discriminée, avec 2% de la population ayant 100% de chances de mourir, et l’autre 0%) ? Et de quoi cette borne dépend-elle ? En particulier, des modèles plus complexes, tels que les réseaux de neurones très profonds permettent-ils vraiment d’améliorer la prévision ? Et l’enrichissement de données, tel qu’on l’observe grâce aux objets connectés et la fusion avec toutes sortes d’informations externes, va-t-il déplacer la borne supérieure vers le haut ?

Si l’apprentissage profond – voir Goodfellow et al. (2018) – permet d’avoir des classifieurs d’images avec un taux d’erreur proche de 0%, il est difficile d’imaginer qu’il sera possible de prévoir, presque un an à l’avance (à la signature du contrat), qui décèdera dans l’année, qui aura la grippe, qui aura un dégât des eaux, etc. Les modèles plus complexes permettent d’améliorer les prévisions, en tenant compte de non-linéarités, d’effets croisés entre les variables tarifaires, mais pas au point de faire disparaître l’aléa. Et tant que l’assurance est envisagée ex-ante (la prime est fixée au début de la période de couverture), il est difficile d’imaginer que rajouter de l’information fera aussi disparaître l’aléa. C’est d’ailleurs le cas pour les tests génétiques qui n’expliquent qu’une (petite) partie du risque de cancer, par exemple. Et rajouter des données revient souvent à rajouter du bruit, ce qui rend le travail d’analyse plus complexe. Cependant, force est de constater que des modèles plus complexes et des données plus riches tendent effectivement à « améliorer » la prévision, en remontant la courbe ROC vers le haut. Mais se pose-t-on les bonnes questions ? Que signifie vraiment une borne très éloignée du cas homogène, sur la diagonale ?

Homogénéité, équité et causalité

Comme nous l’avons vu, la tarification en assurance repose sur une répartition des risques (des contrats) en catégories, au sein desquelles la distribution des pertes peut être estimée, afin de fixer un niveau de primes. La répartition se fait à partir des caractéristiques de l’assuré, et du bien assuré. En retraçant l’histoire de l’assurance, Ewald (1986) montre que les mécanismes de prévoyance se sont mis en place en déplaçant la charge des accidents du travail sur la société : on abandonne l’idée d’une responsabilité individuelle de l’accident en faveur de la solidarité. L’assurance distingue « entre le dommage que subit tel ou tel individu — c’est affaire de chance ou de malchance — et la perte liée au dommage dont l’attribution est, quant à elle, toujours collective et sociale ». Ce principe de solidarité sociale, de mutualisation des risques, fait que le risque (en assurance) est toujours pensé collectivement.

Aujourd’hui, les tarifs sont considérés comme « justes », ou « actuariellement équitables » si chaque prime correspond à la perte attendue (pour ne pas dire « espérée », au sens mathématique) pour chaque assuré. Dans cette perception de l’équité, une hypothèse essentielle est que les classes soient « homogènes ». En effet, dans l’hypothèse inverse, les personnes les moins risquées subventionnent les personnes les plus risquées, ce qui est perçu comme socialement injuste.

On peut décrire cette version de l’équité actuarielle à l’aide de la formule de décomposition de la variance. La variance globale se décompose en effet en deux termes, la variance inter-classes et la variance intra-classes : l’ « équité actuarielle » vise à ce que  les classes de risque soient relativement distinctes les unes des autres, donc une variance inter-classes forte, accompagnée d’une homogénéité des classes, donc une variance intra-classes faible. D’un point de vue statistique, chercher à augmenter l’une est équivalente à faire diminuer l’autre. Cette mécanique n’est pas toujours claire pour des observateurs non avertis ; ainsi dans l’affaire Manhart, un des cas les plus documentés sur la discrimination par le genre en assurance, le juge Stevens affirme : « we focus on fairness to individuals rather than on fairness to classes […] even a true generalization about a class is an insufficient reason for disqualifying an individual to whom the generalization does not apply» (cité dans Anzalone (2016)). Autrement dit, pour la justice, un critère statistique de type « true generalization » ne peut être opposable à un individu.

Une autre critique importante, que l’on retrouve dans la « gender directive », est le lien entre discrimination et causalité. En effet, statistiquement, les actuaires vont chercher des facteurs de classification fortement corrélés avec la sinistralité. Mais il est possible que ces facteurs ne soient qu’un proxy de la vraie variable causale, restée elle inobservée, conduisant à une mauvaise estimation du risque pour certains. Comme le notent Antonio et Charpentier (2017), le genre a ainsi été longtemps utilisé en assurance automobile car très corrélé avec des variables associées au style de conduite et à d’autres variables historiquement non observables (mais qui le sont aujourd’hui grâce aux objets connectés, comme le kilométrage, les heures de conduites, les types de routes utilisés, etc).

Ce lien avec les mécanismes causaux est d’ailleurs relativement profond, et Hacking (1975) y voit une connexion avec la « révolution probabiliste » : on peut assez facilement mettre en évidence des corrélations, mais les causes, si elles existent, nous restent plus opaques. Laplace au début du 19e siècle déclare ainsi que « la probabilité est relative en partie à nos connaissances, en partie à notre ignorance », liant les probabilités à la fois à une vision newtonienne déterministe du monde et à notre incapacité à le connaitre parfaitement. Cette dernière composante fait que l’on ne peut pas annoncer la date exacte du décès d’un individu, mais statistiquement, dans un groupe homogène, on peut prédire le nombre de décès au cours d’une année. Et pour revenir à la relation causale, le tabagisme par exemple ne cause pas forcément une mort prématurée mais fumer sera vu comme dangereux car il augmente la probabilité de décès pendant une période donnée. Ainsi nous montre Hacking (1975), la causalité est pensée aujourd’hui dans un contexte probabiliste, et non plus déterministe.

Abraham, K.S. (1985) Efficiency and fairness in insurance risk classification, Virginia Law Review 71: 403-451.

Anderson, A.W. (1978). A Critique of the Manhart Brief. The Actuary, 12:5.

Antonio, K. & Charpentier, A. (2017). La tarification par genre en assurance, corrélation ou causalité ? Risques, 109.

Anzalone, C.A. (2016). U.S. Supreme Court Cases on Gender and Sexual Equality. Routledge.

Bailey, H., Hutchison, T. & Narber, G. (1975) The regulatory challenge to life insurance classification, Drake Law Review Insurance Law Annual 4: 779-827

Barry L. (2019). Justice ou justesse? L’équité de l’assurance. Working paper #15, chaire PARI.

Charpentier, A. & Denuit, M. (2004). Mathématiques de l’Assurance Non-Vie : Principes Généraux de Théorie du Risque. Economica.

Cramer, H. (1946). Mathematical Methods of Statistics. Princeton University Press.

Cummins, J.D., Smith, B.D., Vance, R.N. & VanDerhai, J.L. (1982). Risk Classification in Life Insurance. Kluwer-Nijhoff Publishing.

Ewald F. (1986). L’État providence. Grasset.

Fisher, R. A. (1936).  The Use of Multiple Measurements in Taxonomic Problems. Annals of Eugenics. 7 (2): 179–188.

Frézal, S. & Barry, L. (2019). Fairness in Uncertainty: Some Limits and Misinterpretations of Actuarial Fairness, Journal of Business Ethics.

Goodfellow, U., Bengio, Y. & Courville, A. (2018) L’apprentissage profond. Massot Edition.

Hacking, I. (1975) The Emergence of Probability. Cambridge University Press

Hoy, M. (1982) Categorizing risks in the insurance industry, The Quarterly Journal of Economics 97: 321-336.

Kleindorfer, P. & Kunreuther, H. (1980) Misinformation and Equilibrium in Insurance Markets, in Economic Analysis of Regulated Markets, Jörg Finsinger Editor, Springer Verlag, 67-90

Kuhn, M. & Johnson, K. (2018). Applied Predictive Modeling. Springer Verlag.

Martin,G.D. (1976) Gender Discrimination in Pension Plans, Journal of Risk and Insurance 43.

McGrayne, S.B. (2012) The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy. Yale University Press.

Ramsey P.F. (1926). Truth and Probability.

Schauer, F. (2006) Profiles, Probabilities, and Stereotypes. Harvard University Press.

Von Mises, R. (1957). Probability, Statistics and Truth. Dover publications.

Wilson, C. (1977). A model of insurance markets with incomplete information. Journal of Economic Theory, 16:2, 167-207.

[1] Au sens statistique du mot, dans le sens introduit par Fisher (1936).

[2] Dans cette approche fréquentiste, et notamment pour Ronald Fisher et Richard von Mises, la probabilité d’un évènement unique (dit « one shot ») n’a pas de sens.

Principal Component Analysis: A Generalized Gini Approach

With Stéphane Mussard and Téa Ouraga, we recently uploaded on arxiv a paper Principal Component Analysis: A Generalized Gini Approach,

A principal component analysis based on the generalized Gini correlation index is provided. It is proven that the reduction dimensionality based on the generalized Gini correlation index, that relies on city-block distances, is robust to outliers.

Some codes are also available on a dedicated github repo.

Combining automatically factor levels with trees

Last year, in a post, I discussed how to merge levels of factor variables, using combinatorial techniques (it was for my STT5100 cours, and trees are not in the syllabus), with an extension on trees at the end of the post.

consider the following (simulated dataset)

n=200
set.seed(1)
x1=runif(n)
x2=runif(n)
y=1+2*x1-x2+rnorm(n,0,.2)
LB=sample(LETTERS[1:10])
b=data.frame(y=y,x1=x1,
x2=cut(x2,breaks=
c(-1,.05,.1,.2,.35,.4,.55,.65,.8,.9,2),
labels=LB))
str(b)
'data.frame':	200 obs. of  3 variables:
$y : num 1.345 1.863 1.946 2.481 0.765 ...$ x1: num  0.266 0.372 0.573 0.908 0.202 ...
$x2: Factor w/ 10 levels "I","A","H","F",..: 4 4 6 4 3 6 7 3 4 8 ... table(b$x2)[LETTERS[1:10]]

A  B  C  D  E  F  G  H  I  J
11 12 23 34 23 36 12 32  3 14

Just by looking at the data (see the previous post), we could easily get the feeling that 10 levels was too much.

Following my post, Przemyslaw sent a comment suggesting to use

library(factorMerger)

It is indeed a nice package (unless you have really really big datasets with a lot of categories in your factor variables – as I experienced recently), and you can get great graphs

MF = mergeFactors(response = b$y, factor = b$x2,
family = "gaussian")
plot(MF)

Here is suggests to create three categories. Recall that with student t-tests (changing the reference), we got

Another interesting package, by Piro Polo, is

library(tree.bins)

To use it, we simply call the following function, and we transform automatically our dataset : the continuous variables remain unchanged, and (possibly) categories of categorical variables are merged

b.bins = tree.bins(data=b, y=y)
str(b.bins)
Classes ‘data.table’ and 'data.frame':	200 obs. of  3 variables:
$y : num 1.345 1.863 1.946 2.481 0.765 ...$ x1: num  0.266 0.372 0.573 0.908 0.202 ...
$x2: chr "Group.4" "Group.4" "Group.4" "Group.4" ... - attr(*, ".internal.selfref")= table(b.bins$x2)

Group.1 Group.2 Group.3 Group.4
23      35      26     116

here in four groups. To get the correspondance, use

tree.bins(data=b, y=y, return = "lkup.list")
[[1]]
x2 Categories
1   E    Group.1
2   G    Group.2
3   C    Group.2
4   B    Group.3
5   J    Group.3
6   I    Group.4
7   A    Group.4
8   H    Group.4
9   F    Group.4
10  D    Group.4

(we have a list with one element, one dataframe, since there is only one factor variable). Cool, isn’t it ? I miss Przemyslaw’s plot, but this is rather quick, and efficient..

Estimates on training vs. validation samples

Before moving to cross-validation, it was natural to say “I will burn 50% (say) of my data to train a model, and then use the remaining to fit the model”. For instance, we can use training data for variable selection (e.g. using some stepwise procedure in a logistic regression), and then, once variable have been selected, fit the model on the remaining set of observations. A natural question is usually “does it really matter ?”.

In order to visualize this problem, consider my (simple) dataset

MYOCARDE=read.table( "http://freakonometrics.free.fr/saporta.csv", head=TRUE,sep=";")

Let us generate 100 training samples (where we keep about 50% of the observations). On each of them, we use a stepwise procedure, and we keep the estimates of the remaining variables (and their standard deviation actually)

n=nrow(MYOCARDE) M=matrix(NA,100,ncol(MYOCARDE)) colnames(M)=c("(Intercept)",names(MYOCARDE)[1:7]) S1=S2=M1=M2=M for(i in 1:100){ idx = which(sample(0:1,size=n, replace=TRUE)==1) reg=step(glm(PRONO=="DECES"~.,data=MYOCARDE[idx,])) nm=names(reg$coefficients) M1[i,nm]=reg$coefficients S1[i,nm]=summary(reg)$coefficients[,2] f=paste("PRONO=='DECES'~",paste(nm[-1],collapse="+"),sep="") reg=glm(f,data=MYOCARDE[-idx,]) M2[i,nm]=reg$coefficients S2[i,nm]=summary(reg)$coefficients[,2] } Then, for the 7 covariates (and the constant) we can look at the value of the coefficient in the model fitted on the training sample, and the value on the model fitted on the validation sample (of course, only when they were remaining) for(j in 1:8){ idx=which(!is.na(M1[,j])) plot(M1[idx,j],M2[idx,j]) abline(a=0,b=1,lty=2,col="gray") segments(M1[idx,j]-2*S1[idx,j],M2[idx,j],M1[idx,j]+2*S1[idx,j],M2[idx,j]) segments(M1[idx,j],M2[idx,j]-2*S2[idx,j],M1[idx,j],M2[idx,j]+2*S2[idx,j]) } For instance, with the intercept, we have the following where horizontal segments are confidence intervals of the parameter on the model fitted on the training sample, the vertical on the validation sample. The green part means some sort of consistency, while the red one means that actually, the coefficient was negative with one model, positive with the other one. Which is odd (but in that case, observe that coefficients are rarely significant). We can also visualize the joint distribution of the two estimators, for(j in 1:8){ library(ks) idx = which(!is.na(M1[,j])) Z = cbind(M1[idx,j],M2[idx,j]) H = Hpi(x=Z) fhat = kde(x=Z, H=H) image(fhat$eval.points[[1]], fhat$eval.points[[2]],fhat$estimate) abline(a=0,b=1,lty=2,col="gray") abline(v=0,lty=2) abline(h=0,lty=2) }

which are here, almost on the diagonal,

meaning that the intercept on the two samples is (more or less) the same. We can then look at other parameters (which is actually more interesting).

On that variable, it seems that it is significant on the training dataset (somehow, it is consistent with the fact that it is remaining in the model after the stepwise procedure) but not on the validation sample (or hardly significant).

Others are much more consistent (with some possible outliers)

On the next one, we have again significance on the training sample, but not on the validation sample,

and probably more interesting

where the two are very consistent.

Variance of the slope in a regression model

In my “applied linear models” exam, there was a tricky question (it was a multiple choice, so no details were asked). I was simply asking if the following statement was valid, or not

Consider a linear regression with one single covariate, $y=\beta_0+\beta_1x_1+\varepsilon$ and the least-square estimates. The variance of the slope is $\text{Var}[\widehat{\beta}_1]$ Do we decrease this variance if we add one variable, and consider $y=\beta_0+\beta_1x_1+\beta_2x_2+\varepsilon$ ?

For the exam, the expected answer was simply “no”. In a nutshell, there are two cases where we should expect different changes,

• if $x_1$ and $x_2$ are highly correlated, then we should expect the variance to increase
• if $x_1$ and $x_2$ are not correlated, then we should expect the variance to decrease

We did briefly observed (and discussed) those points on examples during the lecture… but I wanted to go a bit further, since I couldn’t find any analytical results. Let us generate a model $y=\beta_0+\beta_1x_1+\beta_2x_2+\varepsilon$, and then compare the variance $\text{Var}[\widehat{\beta}_1]$ on the two fitted modes, depending on the correlation between $x_1$ and $x_2$

library(mnormt) n=200 s=function(r=0){ S=matrix(c(1,r,r,1),2,2) X=rmnorm(n,c(0,0),S) B=data.frame(y=-2+X[,1]+X[,2]+rnorm(n)/2, x1=X[,1], x2=X[,2]) reg12=lm(y~x1+x2,data=B) reg1=lm(y~x1,data=B) k=summary(reg12)$coefficients[2,2]/summary(reg1)$coefficients[2,2] k}

Let us generate 500 samples for each value of the correlation, from -0.9 to _0.9

M=NULL for(r in ((-9):9)/10) M=cbind(M,Vectorize(s)(rep(r,500)))

and let us plot the ratio of the two variances

plot(0:1,0:1,xlim=c(-1,1),ylim=c(0,2),col="white") for(i in 1:19) points(rep((((-9):9)/10)[i],500),M[,i],col="light blue") VM=apply(M,2,mean) lines((((-9):9)/10),VM,col="red",lwd=2) abline(h=1,lty=2)

If the ratio exceeds 1, the variance increases when adding a covariate.

Indeed, here, when the two variables are independent, the variance is divided by two. But when covariates are highly correlated, the variance is multiplied by two…

Now, what if, actually, $x_2$ is not a real explanatory variable : the true model we generate is $y=\beta_0+\beta_1x_1+\varepsilon$. In that case,

s=function(r=0){ S=matrix(c(1,r,r,1),2,2) X=rmnorm(n,c(0,0),S) B=data.frame(y=-2+X[,1]+rnorm(n)/2, x1=X[,1], x2=X[,2]) reg12=lm(y~x1+x2,data=B) reg1=lm(y~x1,data=B) k=summary(reg12)$coefficients[2,2]/summary(reg1)$coefficients[2,2] k}

we get our samples as previously

M=NULL for(r in ((-9):9)/10) M=cbind(M,Vectorize(s)(rep(r,500)))

and we plot those ratios

plot(0:1,0:1,xlim=c(-1,1),ylim=c(0,2),col="white") for(i in 1:19) points(rep((((-9):9)/10)[i],500),M[,i],col="light blue") VM=apply(M,2,mean) lines((((-9):9)/10),VM,col="red",lwd=2) abline(h=1,lty=2)

In the case we add a useless variable $x_2$, whatever the correlation with $x_1$, it will always, on average, increase the variance of $\widehat{\beta}_1$.

Random thoughts on econometric models with (pure) random features

For my lectures on applied linear models, I wanted to illustrate the fact that the $R^2$ is never a good measure of the goodness of the model, since it’s quite easy to improve it. Consider the following dataset

n=100 df=data.frame(matrix(rnorm(n*n),n,n)) names(df)=c("Y",paste("X",1:99,sep=""))

with one variable of interest $y$, and 99 features $x_j$. All of them being (by construction) independent. And we have 100 observations… Consider here the regression on the first $k$ features, and compute $R_k^2$ of that regression

reg=function(k){ frm=paste("Y~",paste("X",1:k,collapse="+",sep="")) model=lm(frm,data=df) summary(model)$adj.r.squared} Let us see what’s going on… plot(1:99,Vectorize(reg)(1:99)) (actually, it’s not exactly what we have on the graph…. we have the average obtained over 1,000 samples randomly generated, with 90% confidence bands). Oberve that $\mathbb{E}[R^2_k]=k/n$, i.e. if we add some pure random noise, we keep increasing the $R^2$ (up to 1, actually). Good news, as we’ve seen in the course, the adjusted $R^2$ – denoted $\bar R^2$-might help. Observe that $\mathbb{E}[\barR^2_k]=0$, so, in some sense, adding features does not help here… reg=function(k){ frm=paste("Y~",paste("X",1:k,collapse="+",sep="")) model=lm(frm,data=df) summary(model)$r.squared} plot(1:99,Vectorize(reg)(1:99))

We can actually do the same with Akaike criteria $AIC_k$ and Schwarz (bayesian) criteria $BIC_k$.

reg=function(k){ frm=paste("Y~",paste("X",1:k,collapse="+",sep="")) model=lm(frm,data=df) AIC(model)} plot(1:99,Vectorize(reg)(1:99))

For the $AIC$, the intitial increase makes sense : we should not prefer the model with 10 covariates, compared with nothing. The strange thing is the far right behavior : we prefer here 80 random noise features to none ! Which I find hard to interprete… For the $BIC$ the code is simply

reg=function(k){ frm=paste("Y~",paste("X",1:k,collapse="+",sep="")) model=lm(frm,data=df) BIC(model)} plot(1:99,Vectorize(reg)(1:99))

and here also, we have the same pattern, where we prefer a big model with juste pure noise to nothing…

A last one to conclude (or not) : what about the leave-one-out cross validation mean squared error ? More precisely, $$CV=\frac{1}{n}\sum_{i=1}\widehat{\varepsilon}^2_{-i}$$where $\widehat{\varepsilon}^2_{-i}=y_i-\widehat{y}_{-i}$ where $\widehat{y}_{-i}$ is the predicted value obtained with the model is estimated when the $i$th observation is deleted. One can prove that $$\widehat{\beta}_{-i}=\widehat{\beta}-(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{x}_i\hat\varepsilon_i(1-H_{i,i})^{-1}$$where $H$ is the classical hat matrix, thus$$\widehat{\varepsilon}_{-i}=(1-H_{i,i})^{-1}\hat\varepsilon_i$$i.e. we do note have to estimate (at each round) $n$ models

reg=function(k){ frm=paste("Y~",paste("X",1:k,collapse="+",sep="")) model=lm(frm,data=df) h=lm.influence(model)\$hat/2 mean( (residuals(model)/1-h)^2 ))} plot(1:99,Vectorize(reg)(1:99))

Here, it make sense : adding noisy features yields overfit ! So the mean squared error is decreasing !

That’s all nice, but it might not be very realistic… Here, for my model with only one variable, I just pick one, at random…. In practice, we try to get the “best one”… So a more natural idea would be to order the variables according to their correlations with $y$,

df=data.frame(matrix(rnorm(n*n),n,n)) df=df[,rev(order(abs(cor(df)[1,])))] names(df)=c("Y",paste("X",1:99,sep=""))}

and as before, we can plot the evolution of $R^2_k$ as a function of $k$ the number of features considered,

which is increasing, with a higher slope at the beginning… For the $\bar R^2_k$ we might actually prefer a correlated noise to nothing (which makes sense actually). So here since we somehow chose our variables, $\bar R^2_k$ seems to be always positive…

For the $AIC_k$ here also, there is an improvement. Before coming back to the original situation (with about 80 features) and here also, we observe the drop on the far right part of the graph

The $BIC_k$ might like the top three features, but soon, we have a deterioration…. even if here also, we have the drop at the far right (with more than 95 features… for 100 observations).

Finally, observe that here again, our (leave-one-out) cross-validation has not been mesled by our noisy variables : it is always decreasing !

So it seems that cross-validation techniques are more robust than the $AIC$ and $BIC$ (even if we mentioned in a previous post connexions between all those concepts) when we have a lot a noisy (non-relevent) features.

Foundations of Machine Learning, part 5

This post is the nineth (and probably last) one of our series on the history and foundations of econometric and machine learning models. The first fours were on econometrics techniques. Part 8 is online here.

Optimization and algorithmic aspects

In econometrics, (numerical) optimization became omnipresent as soon as we left the Gaussian model. We briefly mentioned it in the section on the exponential family, and the use of the Fisher score (gradient descent) to solve the first order condition $\mathbf{X}^T W(\beta)^{-1})[y-\widehat{y}]=\mathbf{0}$. In learning, optimization is the central tool. And it is necessary to have effective optimization algorithms, to solve problems (described previously) of the form: $$\widehat{\beta}\in\underset{\beta\in\mathbb{R}^p}{\text{argmin}}\left\lbrace\sum_{i=1}^n \ell(y_i,\beta_0+\mathbf{x}^T\beta)+\lambda\Vert\boldsymbol{\beta}\Vert\right\rbrace$$In some cases, instead of global optimization, it is sufficient to consider optimization by coordinates (widely studied in Daubechies et al. (2004)). If $f:\mathbb{R}^d\rightarrow\mathbf{R}$ is convex and differentiable, if $\mathbf{x}$ satisfies $f(\mathbf{x}+h\boldsymbol{e}_i)\geq f(\mathbf{x})$ for any $h>0$ and $i\in\{1,\cdots, d\}$then $f(\mathbf{x})=\min\{f\}$, where $\mathbf{e}=(\mathbf{e}_i)$ is the canonical basis of $\mathbb{R}^d$. However, this property is not true in the non-differentiable case. But if we assume that the non-differentiable part is separable (additively), it becomes true again. More specifically, if$$f(\mathbf{x})=g(\mathbf{x})+\sum_{i=1}^d h_i(x_i)$$with$$\left\lbrace\begin{array}{l}g: \mathbb{R}^d\rightarrow\mathbb{R}\text{ convex-differentiable}\\h_i: \mathbb{R}\rightarrow\mathbb{R}\text{ convex}\end{array}\right.$$This was the case for Lasso regression, $\beta)\mapsto\| \mathbf{y}-\beta_0-\mathbf{X}\beta\|_{\ell_2 }+\lambda\|\beta\|_{\ell_1}$, as shown by Tsen (2001). Getting back to our initial notations, we can use a coordinate descent algorithm: from an initial value $\mathbf{x}^{(0)}$, we consider (by iterating)$$x_j^{(k)}\in\text{argmin}\big\lbrace f(x_1^{(k)},\cdots,x_{k-1}^{(k)},x_k,x_{k+1}^{(k-1)},\cdots,x_n^{(k-1)})\big\rbrace$$ for $j=1,2,\cdots,n$These algorithmic problems and numerical issues may seem secondary to econometricians. However, they are essential in automatic learning: a technique is interesting if there is a stable and fast algorithm, which allows to obtain a solution. These optimization techniques can be transposed: for example, this coordinate descent technique can be used in the case of SVM methods (known as “vector support” methods) when the space is not linearly separable, and the classification error must be penalized (we will come back to this technique in the next section).

In-sample, out-of-sample and cross-validation

These techniques seem intellectually interesting, but we have not yet discussed the choice of the penalty parameter $\lambda$. But this problem is actually more general, because comparing two parameters $\widehat{\beta}_{\lambda_1}$ and $\widehat{\beta}_{\lambda_2}$ is actually comparing two models. In particular, if we use a Lasso method, with different thresholds $\lambda$, we compare models that do not have the same dimension. Previously, we have addressed the problem of model comparison from an econometric perspective (by penalizing overly complex models). In the learning literature, judging the quality of a model on the data used to construct it does not make it possible to know how the model will behave on new data. This is the so-called “generalization” problem. The traditional approach then consists in separating the sample (size $n$) into two parts: a part that will be used to train the model (the training database, in-sample, size $m$) and a part that will be used to test the model (the testing database, out-of-sample, size $n-m$). The latter then makes it possible to measure a real predictive risk. Suppose that the data are generated by a linear model $y_i=\mathbf{x}_i^T \beta_0+\varepsilon_i$ where $\varepsilon_i$ are independent and centred law achievements. The empirical quadratic risk in-sample is here$$\frac{1}{m}\sum_{i=1}^m\mathbb{E}\big([\mathbf{x}_i^T \widehat{\beta}-\mathbf{x}_i^T \beta_0]^2\big)=\mathbb{E}\big([\mathbf{x}_i^T \widehat{\beta}-\mathbf{x}_i^T \beta_0]^2\big),$$for any observation $i$. Assuming the residuals $\varepsilon$ Gaussian, then we can show that this risk is worth $\sigma^2 \text{trace} (\Pi_X)/m$ is $\sigma^2 p/m$. On the other hand, the empirical out-of-sample quadratic risk is here $$\mathbb{E}\big([\mathbf{x}^T \widehat{\beta}-\mathbf{x}^T \beta_0]^2\big)$$where $\mathbf{x}$ is a new observation, independent of the others. It can be noted that $$\mathbb{E}\big([\mathbf{x}^T \widehat{\beta}-\mathbf{x}^T \beta_0]^2\big\vert \mathbf{x}\big)=\text{Var}\big(\mathbf{x}^T \widehat{\beta}\big\vert \mathbf{x}\big)=\sigma^2\mathbf{x}^T(\mathbf{x}^T\mathbf{x})^{-1}\mathbf{x},$$and by integrating with respect to $\mathbf{x}$, $$\mathbb{E}\big([\mathbf{x}^T \widehat{\beta}-\mathbf{x}^T\beta_0]^2\big)=\sigma^2\text{trace}\big(\mathbb{E}[\mathbf{x}\mathbf{x}^T]\mathbb{E}\big[(\mathbf{x}^T\mathbf{x})^{-1}\big]\big).$$The expression is then different from that obtained in-sample, and using the Groves & Rothenberg (1969) increase, we can show that $$\mathbb{E}\big([\mathbf{x}^T \widehat{\beta}-\mathbf{x}^T \beta_0]^2\big) \geq \sigma^2\frac{p}{m}$$which is pretty intuitive, when we start thinking about it. Except in some simple cases, there is no simple (explicit) formula. Note, however, that if $\mathbf{X}\sim\mathcal{N}(0,\sigma^2 \mathbb{I})$, then $\mathbf{x}^T \mathbf{x}$ follows a Wishart law, and it can be shown that $$\mathbb{E}\big([\mathbf{x}^T \widehat{\beta}-\mathbf{x}^T \beta_0]^2\big)=\sigma^2\frac{p}{m-p-1}.$$If we now look at the empirical version: if $\widehat{\beta}$ is estimated on the first $m$ observations,$$\widehat{\mathcal{R}}^{~\text{ IS}}=\sum_{i=1}^m [y_i-\boldsymbol{x}_i^T\widehat{\boldsymbol{\beta}}]^2\text{ and }\widehat{\mathcal{R}}^{\text{ OS}}=\sum_{i=m+1}^{n} [y_i-\boldsymbol{x}_i^T\widehat{\boldsymbol{\beta}}]^2$$and as Leeb (2008) noted, $\widehat{\mathcal{R}}^{\text{IS}}-\widehat{\mathcal{R}}^{\text{OS}}\approx 2\cdot\nu$ where $\nu$ represents the number of degrees of freedom, which is not unlike the penalty used in the Akaike test.

Figure 4 shows the respective evolution of $\widehat{\mathcal{R}}^{\text{IS}}$ and $\widehat{\mathcal{R}}^{\text{OS}}$ according to the complexity of the model (number of degrees in a polynomial regression, number of nodes in splines, etc). The more complex the model, the more $\widehat{\mathcal{R}}^{\text{IS}}$ will decrease (this is the red curve, below). But that’s not what we’re interested in here: we want a model that predicts well on new data (i. e. out-of-sample). As Figure 4 shows, if the model is too simple, it does not predict well (as it does with in-sample data). But what we can see is that if the model is too complex, we are in a situation of “overlearning”: the model will start to model the noise. Of course, this figure should remind us of the one we’ve seen in our second post of that series

Figure 4 : Generalization, under- and over-fitting

Instead of splitting the database in two, with some of the data that will be used to calibrate the model and some to study its performance, it is also possible to use cross-validation. To present the general idea, we can go back to the “jackknife”, introduced by Quenouille (1949) (and formalized by Quenouille (1956) and Tukey (1958)) relatively used in statistics to reduce bias. Indeed, if we assume that $\{y_1,\cdots,y_n\}$ is a sample drawn according to a law $F_\theta$, and that we have an estimator $T_n (\mathbf{y})=T_n (y_1,\cdots,y_n)$, but that this estimator is biased, with $\mathbf{E}[T_n (\mathbf{Y})]=\theta+O(n^{-1})$, it is possible to reduce the bias by considering $$\widetilde{T}_n(\mathbf{y})=\frac{1}{n}\sum_{i=1}^n T_{n-1}(\mathbf{y}_{(i)})\text{ where }\mathbf{y}_{(i)}=(y_1,\cdots,y_{i-1},y_{i+1},\cdots,y_n)$$It can then be shown that $\mathbb{E}[\tilde{T}_n(Y)]=\theta+O(n^{-2})$The idea of cross-validation is based on the idea of building an estimator by removing an observation. Since we want to build a predictive model, we will compare the forecast obtained with the estimated model, and the missing observation$$\widehat{\mathcal{R}}^{\text{ CV}}=\frac{1}{n}\sum_{i=1}^n \ell(y_i,\widehat{m}_{(i)}(\mathbf{x}_i))$$We will speak here of the “leave-one-out” (loocv) method.

This technique reminds us of the traditional method used to find the optimal parameter in exponential smoothing methods for time series. In simple smoothing, we will construct a forecast from a time series as ${}_t\widehat{y}_{t+1} =\alpha\cdot{}_{t-1}\widehat{y}_t +(1-\alpha)\cdot y_t$, where $\alpha\in[0,1]$, and we will consider as “optimal” $$\alpha^\star = \underset{\alpha\in[0,1]}{\text{argmin}}\left\lbrace \sum_{t=2}^T \ell({}_{t-1}\widehat{y}_{t},y_{t}) \right\rbrace$$as described by Hyndman et al (2009).

The main problem with the leave-one-out method is that it requires calibration of n models, which can be problematic in large dimensions. An alternative method is cross validation by $k$-blocks (called “$k$-fold cross validation”) which consists in using a partition of $\{1,\cdots,n\}$ in $k$ groups (or blocks) of the same size, $\mathcal{I}_1,\cdots,\mathcal{I}_k$, and let us note $\mathcal{I}_{\bar j}=\{1,\cdots,n\}\setminus \mathcal{I}_j$. By noting $\widehat{m}_{(j)}$ built on the sample $\mathcal{I}_{\bar j}$, we then set:$$\widehat{\mathcal{R}}^{k-\text{ CV}}=\frac{1}{k}\sum_{j=1}^k \mathcal{R}_j\text{ where }\mathcal{R}_j=\frac{k}{n}\sum_{i\in\mathcal{I}_{{j}}} \ell(y_i,\widehat{m}_{(j)}(\mathbf{x}_i))$$Standard cross-validation, where only one observation is removed each time (loocv), is a special case, with $k=n$. Using $k=5$ or $10$ has a double advantage over $k=n$: (1) the number of estimates to be made is much smaller, 5 or 10 rather than $n$; (2) the samples used for estimation are less similar and therefore less correlated to each other, which tends to avoid excess variance, as recalled by James et al. (2013).

Another alternative is to use boosted samples. Let $\mathcal{I}_b$ be a sample of size $n$ obtained by drawing with replacement in $\{1,\cdots,n\}$ to know which observations $(y_i,\mathbf{x}_i)$ will be kept in the learning population (at each draw). Note $\mathcal{I}_{\bar b}=\{1,\cdots,n\}\setminus\mathcal{I}_b$. By noting $\widehat{m}_{(b)}$ built on sample $\mathcal{I}_b$, we then set :$$\widehat{\mathcal{R}}^{\text{ B}}=\frac{1}{B}\sum_{b=1}^B \mathcal{R}_b\text{ where }\mathcal{R}_b=\frac{n_{\overline{b}}}{n}\sum_{i\in\mathcal{I}_{\overline{b}}} \ell(y_i,\widehat{m}_{(b)}(\mathbf{x}_i))$$where $n_{\bar b}$ is the number of observations that have not been kept in $\mathcal{I}_b$. It should be noted that with this technique, on average $e^{-1}\sim36.7\%$ of the observations do not appear in the boosted sample, and we find an order of magnitude of the proportions used when creating a calibration sample, and a test sample. In fact, as Stone (1977) had shown, the minimization of AIC is to be compared to the cross-validation criterion, and Shao (1997) showed that the minimization of BIC corresponds to $k$-fold cross-validation, with $k=n/\log n$.

All those techniques here are mentioned in the “machine learning” section since they rely on automatic, computational techniques, and no probabilistic foundations are necessary. In many cases we did use the notation $m^\star$ (at least in the first posts on “machine learning” techniques) to highlight the fact that we want some sort of “optimal” model – and to make a distinction with estimators $\widehat{m}$ considered earlier, when we had some probabilistic framework. But of course, it is possible (and necessary) to build bridges between those two cultures…

References are online here. As explained in the introduction, it is some sort of online version of an introduction to our joint paper with Emmanuel Flachaire and Antoine Ly, Econometrics and Machine Learning (initially writen in French), that will actually appear soon in the journal Economics and Statistics (in English and in French).

Foundations of Machine Learning, part 1

This post is the fifth one of our series on the history and foundations of econometric and machine learning models. The first fours were on econometrics techniques. Part 4 is online here.

In parallel with these tools developed by, and for economists, a whole literature has been developed on similar issues, centered on the problems of prediction and forecasting. For Breiman (2001a), a first difference comes from the fact that the statistic has developed around the principle of inference (or to explain the relationship linking $y$ to variables $\mathbf{x}$) while another culture is primarily interested in prediction. In a discussion that follows the article, David Cox states very clearly that in statistic (and econometrics) “predictive success (…) is not the primary basis for model choice“. We will get back here on the roots of automatic learning techniques. The important point, as we will see, is that the main concern of machine learning is related to the generalization properties of a model, i.e. its performance – according to a criterion chosen a priori – on new data, and therefore on non-sample tests.

A learning machine

Today, we speak of “machine learning” to describe a whole set of techniques, often computational, as alternatives to the classical econometric approach. Before characterizing them as much as possible, it should be noted that historically other names have been given. For example, Friedman (1997) proposes to make the link between statistics (which closely resemble econometric techniques – hypothesis testing, ANOVA, linear regression, logistics, GLM, etc.) and what was then called “data mining” (which then included decision trees, methods from the closest neighbours, neural networks, etc.). The bridge between those two cultures corresponds to “statistical learning” techniques described in Hastie et al (2009). But one should keep in mind that machine learning is a very large field of research.

The so-called “natural” learning (as opposed to machine learning) is that of children, who learn to speak, read and play. Learning to speak means segmenting and categorizing sounds, and associating them with meanings. A child also learns simultaneously the structure of his or her mother tongue and acquires a set of words describing the world around him or her. Several techniques are possible, ranging from rote learning, generalization, discovery, more or less supervised or autonomous learning, etc. The idea in artificial intelligence is to take inspiration from the functioning of the brain to learn, to allow “artificial” or “automatic” learning, by a machine. A first application was to teach a machine to play a game (tic-tac-toe, chess, go, etc.). An essential step is to explain the objective it must achieve to win. One historical approach has been to teach the machine the rules of the game. If it allows you to play, it will not help the machine to play well. Assuming that the machine knows the rules of the game, and that it has a choice between several dozen possible moves, which one should it choose? The classical approach in artificial intelligence uses the so-called min-max algorithm using an evaluation function: in this algorithm, the machine searches forward in the possible moves tree, as far as the calculation resources allow (about ten moves in chess, for example). Then, it calculates different criteria (which have been previously indicated to her) for all positions (number of pieces taken, or lost, occupancy of the center, etc. in our example of the chess game), and finally, the machine plays the move that allows it to maximize its gain. Another example may be the classification and recognition of images or shapes. For example, the machine must identify a number in a handwritten handwriting (checks, ZIP code on envelopes, etc). It is a question of predicting the value of a variable $y$, knowing that a priori $y\in\{0,1,2,\cdots,8,9\}$. A classical strategy is to provide the machine with learning bases, in other words here millions of labelled (identified) images of handwritten numbers. A simple (and natural) strategy is to use a decision criterion based on the closest neighbors whose labels are known (using a predefined metric).

The method of the closest neighbors (“$k$-nearest neighbors”) can be described as follows: we consider (as in the previous part) a set of n observations, i. e. pairs $(y_i,\mathbf{x}_i)$ with $\mathbf{x}_i\in\mathbb{R}^p$. Let us consider a distance $\Delta$ on $\mathbb{R}^p$ (the Euclidean distance or the Mahalanobis distance, for example). Given a new observation $\mathbf{x}\in\mathbb{R}^p$, let us assume the ordered observations as a function of the distance between the $\mathbf{x}_i$ and $\mathbf{x}$, in the sense that $$\Delta(\mathbf{x}_1, \mathbf{x})\leq\Delta(\mathbf{x}_2, \mathbf{x})\leq\cdots\leq\Delta(\mathbf{x}_n, \mathbf{x})$$ then we can consider as prediction for y the average of the nearest $k$ neighbours,$$\widehat{m}_k(\mathbf{x})=\frac{1}{k}\sum_{i=1}^k y_i$$Learning here works by induction, based on a sample (called the learning – or training – sample).

Automatic learning includes those algorithms that give computers the ability to learn without being explicitly programmed (as Arthur Samuel defined it in 1959). The machine will then explore the data with a specific objective (such as searching for the nearest neighbours in the example just described). Tom Mitchell proposed a more precise definition in 1998: a computer program is said to learn from experience $E$ in relation to a task $T$ and a performance measure $P$, if its performance on $T$, measured by $P$, improves with experience $E$. Task $T$ can be a defect score for example, and performance $P$ can be the percentage of errors made. The system learns if the percentage of predicted defects increases with experience.

As we can see, machine learning is basically a problem of optimizing a criterion based on data (from now on called learning). Many textbooks on machine learning techniques propose algorithms, without ever mentioning any probabilistic model. In Watt et al (2016) for example, the word “probability” is mentioned only once, with this footnote that will surprise and make smile any econometricians, “the logistic regression can also be interpreted from a probabilistic perspective” (page 86). But many recent books offer a review of machine learning approaches using probabilistic theories, following the work of Vaillant and Vapnik. By proposing the paradigm of “probably almost correct” learning (PAC), a probabilistic flavor has been added to the previously very computational approach, by quantifying the error of the learning algorithm (usually in a classification problem).

To be continued (references are online here)…

Probabilistic Foundations of Econometrics, part 4

This post is the fourth one of our series on the history and foundations of econometric and machine learning models. Part 3 is online here.

Goodness of Fit, and Model

In the Gaussian linear model, the determination coefficient – noted $R^2$ – is often used as a measure of fit quality. It is based on the variance decomposition formula $$\underbrace{\frac{1}{n}\sum_{i=1}^n (y_i-\bar{y})^2}_{\text{total variance}}=\underbrace{\frac{1}{n}\sum_{i=1}^n (y_i-\widehat{y}_i)^2}_{\text{residual variance}}+\underbrace{\frac{1}{n}\sum_{i=1}^n (\widehat{y}_i-\bar{y})^2}_{\text{explained variance}}$$ The $R^2$ is defined as the ratio of explained variance and total variance, another interpretation of the coefficient that we had introduced from the geometry of the least squares $$R^2= \frac{\sum_{i=1}^n (y_i-\bar{y})^2-\sum_{i=1}^n (y_i-\widehat{y}_i)^2}{\sum_{i=1}^n (y_i-\bar{y})^2}$$The sums of the error squares in this writing can be rewritten as a log-likelihood. However, it should be remembered that, up to one additive constant (obtained with a saturated model) in generalized linear models, deviance is defined by ${Deviance}(\widehat{\beta}) = -2\log[\mathcal{L}]$ which can also be noted $Deviance(\widehat{\mathbf{y}})$. A null deviance can be defined as the one obtained without using the explanatory variables $\mathbf{x}$, so that $\widehat{y}_i=\overline{y}$. It is then possible to define, in a more general context (with a non-Gaussian distribution for $y$)$$R^2=\frac{{Deviance}(\overline{y})-{Deviance}(\widehat{\mathbf{y}})}{{Deviance}(\overline{y})}=1-\frac{{Deviance}(\widehat{\mathbf{y}})}{{Deviance}(\overline{y})}$$However, this measure cannot be used to choose a model, if one wishes to have a relatively simple model in the end, because it increases artificially with the addition of explanatory variables without significant effect. We will then tend to prefer the adjusted $R^2$,$$\bar R^2 = {1-(1-R^{2})\cdot{n-1 \over n-p}} = R^{2}-\underbrace{(1-R^{2})\cdot{p-1 \over n-p}}_{\text{penalty}}$$where $p$ is the number of parameters of the model. Measuring the quality of fit will penalize overly complex models.

This idea will be found in the Akaike criterion, where $AIC=Deviance+2\cdot p$ or in the Schwarz criterion, $BIC=Deviance+log(n)\cdot p$. In large dimensions (typically $p>\sqrt{n}$), we will tend to use a corrected $AIC$, defined by $AIC_c=Deviance+2⋅p⋅n/(n-p-1)$.

These criterias are used in so-called “stepwise” methods, introducing the set methods. In the “forward” method, we start by regressing to the constant, then we add one variable at a time, retaining the one that lowers the $AIC$ criterion the most, until adding a variable increases the AIC criterion of the model. In the “backward” method, we start by regressing on all variables, then we remove one variable at a time, removing the one that lowers the $AIC$ criterion the most, until removing a variable increases the $AIC$ criterion from the model.

Another justification for this notion of penalty (we will come back to this idea in machine learning) can be the following. Let us consider an estimator in the class of linear predictors,$$\mathcal{M}=\big\lbrace m:~m(\mathbf{x})=s_h(\mathbf{x})^T\mathbf{y} \text{ where }S=(s(\mathbf{x}_1),\cdots,s(\mathbf{x}_n))^T\text{ is some smoothing matrix}\big\rbrace$$ and assume that $y=m_0 (x)+\varepsilon$, with $\mathbb{E}[\varepsilon]=0$ and $Var[\varepsilon]=\sigma^2\mathbb{I}$, so that $m_0 (x)=\mathbb{E}[Y|X=x]$. From a theoretical point of view, the quadratic risk, associated with an estimated model $\widehat{m}$, $\mathbb{E}\big[(Y-\widehat{m}(\mathbf{X}))^2\big]$, is written$$\mathcal{R}(\widehat{m})=\underbrace{\mathbb{E}\big[(Y-m_0(\mathbf{X}))^2\big]}_{\text{error}}+\underbrace{\mathbb{E}\big[(m_0(\mathbf {X})-\mathbb{E}[\widehat{m}(\mathbf{X})])^2\big]}_{\text{bias}^2}+\underbrace{\mathbb{E}\big[(\mathbb{E}[\widehat{m}(\mathbf{X})]-\widehat{m}(\mathbf{X}))^2\big]}_{\text{variance}}$$if $m_0$ is the true model. The first term is sometimes called “Bayes error”, and does not depend on the estimator selected, $\widehat{m}$.

The empirical quadratic risk, associated with a model $m$, is here:$$\widehat{\mathcal{R}}_n(m)=\frac{1}{n}\sum_{i=1}^n (y_i-m(\mathbf{x}_i))^2$$ (by convention). We recognize here the mean square error, “mse”, which will more generally give the “risk” of the model $m$ when using another loss function (as we will discuss later on). It should be noted that:$$\displaystyle{\mathbb{E}[\widehat{\mathcal{R}}_n(m)]=\frac{1}{n}\|m_0(\mathbf{x})-m(\mathbf{x})\|^2+\frac{1}{n}\mathbb{E}\big(\|{Y}-m_0(\mathbf{X})\|^2\big)}$$We can show that:$$n\mathbb{E}\big[\widehat{\mathcal{R}}_n(\widehat{m})\big]=\mathbb{E}\big(\|Y-\widehat{m}(\mathbf{x})\|^2\big)=\|(\mathbb{I}-\mathbf{S})m_0\|^2+\sigma^2\|\mathbb{I}-\mathbf{S}\|^2$$so that the (real) risk of $\widehat{m}$ is:$${\mathcal{R}}_n(\widehat{m})=\mathbb{E}\big[\widehat{\mathcal{R}}_n(\widehat{m})\big]+2\frac{\sigma^2}{n}\text{trace}(\boldsymbol{S})$$So, if $\text{trace}(\boldsymbol{S})\geq0$ (which is not a too strong assumption), the empirical risk underestimates the true risk of the estimator. Actually, we recognize here the number of degrees of freedom of the model, the right-hand term corresponding to Mallow’s $C_p$, introduced in Mallows (1973) using not deviance but $R^2$.

Statistical Tests

The most traditional test in econometrics is probably the significance test, corresponding to the nullity of a coefficient in a linear regression model. Formally, it is the test of $H_0:\beta_k=0$ against $H_1:\beta_k\neq 0$. The so-called Student test, based on the statistics $t_k=\widehat{\beta}_k/se_{\widehat{β}_k}$, allows to decide between the two alternatives, using the test $p$-value, defined by $\mathbb{P}[|T|>|t_k|]$ avec $T\overset{\mathcal{L}}{\sim} Std_\nu$, where $\nu$ is the number of degrees of freedom of the model ($\nu=p+1$ for the standard linear model). In large dimension, however, this statistic is of very limited interest, given a significant FDR (“False Discovery Ratio”). Classically, with a level of significance $\alpha=0.05$, 5% of the variables are falsely significant. Suppose that we have $p$=100 explanatory variables, but that 5 (only) are really significant. We can hope that these 5 variables will pass the Student test, but we can also expect that 5 additional variables (false positive test) will emerge. We will then have 10 variables perceived as significant, while only half are significant, i.e. an FDR ratio of 50%. In order to avoid this recurrent pitfall in multiple tests, it is natural to use the procedure of Benjamini & Hochberg (1995).

From a correlation to some causal effect

Econometric models are used to implement public policy evaluations. It is therefore essential to fully understand the underlying mechanisms in order to know which variables actually make it possible to act on a variable of interest. But then we move on to another important dimension of econometrics. Jerry Neyman was responsible for the first work on the identification of causal mechanisms, and then Rubin (1974) formalized the test, called the “Rubin causal model” in Holland (1986). The first approaches to the notion of causality in econometrics were based on the use of instrumental variables, models with discontinuity of regression, analysis of differences in differences, and natural or unnatural experiments. Causality is usually inferred by comparing the effect of a policy – or more generally of a treatment – with its counterfactual, ideally given by a random control group. The causal effect of the treatment is then defined as $\Delta=y_1-y_0$, i.e. the difference between what the situation would be with treatment (noted t=1) and without treatment (noted t=0). The concern is that only $y=t\cdot y_1+(1-t)\cdot y_0$ and $t$ are observed. In other words, the causal effect of variable $t$  on $t$  is not observed (since only one of the two potential variables – $y_0$ or $y_1$  is observed for each individual), but it is also individual, and therefore a function of x-covariates. Generally, by making assumptions about the distribution of the triplet $(Y_0,Y_1,T)$, some parameters of the causal effect distribution become identifiable, based on the density of the observable variables $(Y,T)$. Classically, we will be interested in the moments of this distribution, in particular the average effect of treatment in the population, $\mathbb{E}[\Delta]$, or even just the average effect of treatment in the case of treatment $\mathbb{E}[\Delta|T=1]$. If the result $(Y_0,Y_1)$ is independent of the processing access variable $T$, it can be shown that $\mathbb{E}[\Delta]=\mathbb{E}[Y|T=1]- \mathbb{E} [Y|T=0]$. But if this independence hypothesis is not verified, there is a selection bias, often associated with $\mathbb{E}[Y_0|T=1]- \mathbb{E} [Y_0|T=0]$. Rosenbaum & Rubin (1983) propose to use a propensity to be treated score, $p(x)=\mathbb{P}[T=1|X=x]$, noting that if variable $Y_0$\ is independent of access to treatment $T$ conditionally to the explanatory variables $X$, then it is independent of $T$  conditionally to the score $p(X)$ : it is sufficient to match them using their propensity score. Heckman et al (2003) thus proposes a kernel estimator on the propensity score, which simply provides an estimator of the effect of the treatment, provided that it is treated.

To be continued next time, we’ll introduce “machine learning techniques” (references mentioned above are online here)