Category Archives: Statistics

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


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

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

and let us plot the ratio of the two variances

for(i in 1:19) points(rep((((-9):9)/10)[i],500),M[,i],col="light blue")

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,


we get our samples as previously

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

and we plot those ratios

for(i in 1:19) points(rep((((-9):9)/10)[i],500),M[,i],col="light blue")

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


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


Let us see what’s going on…


(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…


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


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


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 ith 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_ii.e. we do note have to estimate (at each round) n models

  mean( (residuals(model)/1-h)^2 ))}

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,


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\rbraceIn 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, iff(\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,nThese 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}}]^2and 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\rbraceas 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_iLearning 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}\|^2so 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)

Probabilistic Foundations of Econometrics, part 3

This post is the third one of our series on the history and foundations of econometric and machine learning models. Part 2 is online here.

Exponential family and linear models

The Gaussian linear model is a special case of a large family of linear models, obtained when the conditional distribution of Y (given the covariates) belongs to the exponential family f(y_i|\theta_i,\phi)=\exp\left(\frac{y_i\theta_i-b(\theta_i)}{a(\phi)}+c(y_i,\phi)\right) with \theta_i=\psi(\mathbf{x}_i^T \beta). Functions a, b and c are specified according to the type of exponential law (studied extensively in statistics since Darmoix (1935), as Brown (1986) reminds us), and \psi is a one-to-one mapping that the user must specify. Log-likelihood then has a simple expression \log\mathcal{L}(\mathbf{\theta},\phi|\mathbf{y}) =\frac{\sum_{i=1}^ny_i\theta_i-\sum_{i=1}^nb(\theta_i)}{a(\phi)}+\sum_{i=1}^n c(y_i,\phi) and the first order condition is then written \frac{\partial \log \mathcal{L}(\mathbf{\theta},\phi|\mathbf{y})}{\partial \mathbf{\beta}} = \mathbf{X}^T\mathbf{W}^{-1}[\mathbf{y}-\widehat{\mathbf{y}}]=\mathbf{0} based on Müller’s (2011) notations, where \mathbf{W} is a weight matrix (which depends on \beta). Given the link between \theta and the expectation of Y, instead of specifying the function \psi(\cdot) , we will tend to specify the link function g(\cdot) defined by \widehat{y}=m(\mathbf{x})=\mathbb{E}[Y|\mathbf{X}=\mathbf{x}]=g^{-1} (\mathbf{x}^T \beta) For the Gaussian linear regression we consider an identity link, while for the Poisson regression, the natural link (called canonical) is the logarithmic link. Here, as \mathbf{W} depends on \beta (with \mathbf{W}=diag(\nabla g(\widehat{\mathbf{y}})Var[\mathbf{y}]) there is generally no explicit formula for the maximum likelihood estimator. But an iterative algorithm makes it possible to obtain a numerical approximation. By setting \mathbf{z}=g(\widehat{\mathbf{y}})+(\mathbf{y}-\widehat{\mathbf{y}})\cdot\nabla g(\widehat{\mathbf{y}}) corresponding to the error term of a Taylor development in order 1 of g, we obtain an algorithm of the form\widehat{\beta}_{k+1}=[\mathbf{X}^T \mathbf{W}_k^{-1} \mathbf{X}]^{-1} \mathbf{X}^T \mathbf{W}_k^{-1} \mathbf{z}_kBy iterating, we will define \widehat{\beta}=\widehat{\beta}_{\infty}, and we can show that – with some additional technical assumptions (detailed in Müller (2011)) – this estimator is asymptotically Gaussian, with \sqrt{n}(\widehat{\beta} -\beta)\overset{\mathcal{L}}{\rightarrow} \mathcal{N}(\mathbf{0},I(β)^{-1}) where numerically I(\beta)=\varphi\cdot[\mathbf{X}^T \mathbf{W}_\infty^{-1} \mathbf{X}] .

From a numerical point of view, the computer will solve the first-order condition, and actually, the law of Y does not really intervene. For example, one can estimate a “Poisson regression” even when observations are not integers (but they need to be positive). In other words, the law of Y is only an interpretation here, and the algorithm could be introduced in a different way (as we will see later on), without necessarily having an underlying probabilistic model.

Logistic Regression

Logistic regression is the generalized linear model obtained with a Bernoulli’s law, and a link function which is the quantile function of a logistic law (which corresponds to the canonical link in the sense of the exponential family). Taking into account the form of Bernoulli’s law, econometrics proposes a model for y_i\in\{0,1\}, in which the logarithm of the odds follows a linear model: \log\left(\frac{\mathbb{P}[Y=1\vert \mathbf{X}=\mathbf{x}]}{\mathbb{P}[Y\neq 1\vert \mathbf{X}=\mathbf{x}]}\right)=\beta_0+\mathbf{x}^T\beta or \mathbb{E}[Y|\mathbf{X}=\mathbf{x}]=\mathbb{P}[Y=1|\mathbf{X}=\mathbf{x}]=\frac{e^{\beta_0+\mathbf{x}^T\beta}}{1+ e^{\beta_0+\mathbf{x}^T\beta}}=H(\beta_0+\mathbf{x}^T\beta) where H(\cdot)=\exp(\cdot)/(1+exp(\cdot)) is the cumulative distribution function of the logistic law. The estimation of (\beta_0,\beta) is performed by maximizing the likelihood: \mathcal{L}=\prod_{i=1}^n \left(\frac{e^{\mathbf{x}_i^T\mathbf{\beta}}}{1+e^{\boldsymbol{x}_i^T\mathbf{\beta}}}\right)^{y_i}\left(\frac{1}{1+e^{\mathbf{x}_i^T\mathbf{\beta}}}\right)^{1-y_i} It is said to be a linear models because isoprobability curves here are the parallel hyperplanes b+\mathbf{x}^T\beta . Rather than this model, popularized by Berkson (1944), some will prefer the probit model (see Berkson, 1951), introduced by Bliss (1934). In this model: \mathbb{E}[Y|\mathbf{X}=\mathbf{x}]=\mathbb{P}[Y=1|\mathbf{X}=\mathbf{x}]=\Phi (\beta_0+\mathbf{x}^T\beta)

where \Phi denotes the distribution function of the reduced centred normal distribution. This model has the advantage of having a direct link with the Gaussian linear model, since y_i=\mathbf{1}(y_i^\star>0) with y_i^\star=\beta_0+\mathbf{x}^T \beta+\varepsilon_i where the residuals are Gaussian, \mathcal{N}(0,\sigma^2). An alternative is to have centered residuals of unit variance, and to consider a latent modeling of the form y_i=\mathbf{1}(y_i^\star>\xi) (where \xi will be fixed). As we can see, these techniques are fundamentally linked to an underlying stochastic model. In the body of the article, we present several alternative techniques – from the learning literature – for this classification problem (with two classes, here 0 and 1).

Regression in high dimension

As we mentioned earlier, the first order condition \mathbf{X}^T (\mathbf{X}\widehat{\beta}-\mathbf{y})=\mathbf{0} is solved numerically by performing a QR decomposition, at a cost which consists in O(np^2) operations (where p is the rank of \mathbf{X}^T \mathbf{X}). Numerically, this calculation can be long (either because p is large or because n is large), and a simpler strategy may be to sub-sample. Let n_s\ll n, and consider a sub-sample size n_s of \{1,\cdots,n\}. Then \widehat{\beta}_s=(\mathbf{X}_s^T \mathbf{X}_s )^{-1} \mathbf{X}_s^T\mathbf{y}_s is a good approximation of \beta as shown by Dhillon et al. (2014). However, this algorithm is dangerous if some points have a high leverage (i.e. L_i=\mathbf{x}_i(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{x}_i^T). Tropp (2011) proposes to transform the data (in a linear way), but a more popular approach is to do non-uniform sub-sampling, with a probability related to the influence of observations (defined by I_i=\widehat{\varepsilon}_iL_i/(1-L_i)^2 , and which unfortunately can only be calculated once the model is estimated).

In general, we will talk about massive data when the data table of size does not fit in the RAM memory of the computer. This situation is often encountered in statistical learning nowadays with very often p\ll n. This is why, in practice, many libraries of algorithms assimilated to machine learning use iterative methods to solve the first-order condition. When the parametric model to be calibrated is indeed convex and semi-differentiable, it is possible to use, for example, the stochastic gradient descent method as suggested by Bottou (2010). This last one allows to free oneself at each iteration from the calculation of the gradient on each observation of our learning base. Rather than making an average descent at each iteration, we start by drawing (without replacement) an observation \mathbf{x}_i among the n available. The model parameters are then corrected so that the prediction made from \mathbf{x}_i is as close as possible to the true value y_i. The method is then repeated until all the data have been reviewed. In this algorithm there is therefore as much iteration as there are observations. Unlike the gradient descent algorithm (or Newton’s method) at each iteration, only one gradient vector is calculated (and no longer n). However, it is sometimes necessary to run this algorithm several times to increase the convergence of the model parameters. If the objective is, for example, to minimize a loss function \ell between the estimator m_\beta (\mathbf{x}) and y (like the quadratic loss function, as in the Gaussian linear regression) the algorithm can be summarized as follows:

  • Step 0: Mix the data
  • Iteration step: For t=1,\cdots, n, we pull i\in\{1,\cdots,n\} without replacement, and we set \beta^{t+1} = \beta^{t} - \gamma_t\frac{ \partial{\ell(y_i,m_{\beta^t}(X_i)) } }{ \partial{ \beta}}

This algorithm can be repeated several times as a whole depending on the user’s needs. The advantage of this method is that at each iteration, it is not necessary to calculate the gradient on all observations (more sum). It is therefore suitable for large databases. This algorithm is based on a convergence in probability towards a neighborhood of the optimum (and not the optimum itself).

(references will be given in the very last post of that series) To be continued

Probabilistic Foundations of Econometrics, part 2

This post is the second one of our series on the history and foundations of econometric and machine learning models. Part 1 is online here.

Geometric Properties of this Linear Model

Let’s define the scalar product in \mathbb{R}^n, ⟨\mathbf{a},\mathbf{b}⟩=\mathbf{a}^T\mathbf{b}, and let’s note \|\cdot\| the associated Euclidean standard, \|\mathbf{a}\|=\sqrt{\mathbf{a}^T\mathbf{a}} (denoted \|\cdot\|_{\ell_2} in the next post). Note \mathcal{E}_X the space generated by all linear combinations of the \mathbf{X} components (adding the constant). If the explanatory variables are linearly independent, \mathbf{X} is a full (column) rank matrix and \mathcal{E}_X is a space of dimension p+1. Let’s assume from now on that the variables \mathbf{x}  and y are centered here. Note that no law hypothesis is made in this section, the geometric properties are derived from the properties of expectation and variance in the set of finite variance variables.

With this notation, it should be noted that the linear model is written m(\mathbf{x})=⟨\mathbf{x},\beta⟩. The space H_z=\{\mathbf{x}\in\mathbb{R}^{p+1}:m(\mathbf{x})=z\} is a hyperplane (affine) that separates the space in two. Let’s define the orthogonal projection operator on \mathcal{E}_X, \Pi_X =\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1} \mathbf{X}^T. Thus, the forecast that can be made for it is: \widehat{\mathbf{y}}=\mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1} \mathbf{X}^T\mathbf{y}=\Pi_X\mathbf{y}. As, \widehat{\varepsilon}=\mathbf{y}-\widehat{\mathbf{y}}=(\mathbb{I}-\Pi_X)\mathbf{y}=\Pi_{X^\perp}\mathbf{y}, we note that \widehat{\varepsilon}\perp\mathbf{x}, which will be interpreted as meaning that residuals are a term of innovation, unpredictable in the sense that \Pi_{X }\widehat{\varepsilon}=\mathbf{0}. The Pythagorean theorem is written here: \Vert \mathbf{y} \Vert^2=\Vert \Pi_{ {X}}\mathbf{y} \Vert^2+\Vert \Pi_{ {X}^\perp}\mathbf{y} \Vert^2=\Vert \Pi_{ {X}}\mathbf{y}\Vert^2+\Vert \mathbf{y}-\Pi_{ {X}}\mathbf{y}\Vert^2=\Vert\widehat{\mathbf{y}}\Vert^2+\Vert\widehat{\mathbf{\varepsilon}}\Vert^2which is classically translated in terms of the sum of squares: \underbrace{\sum_{i=1}^n y_i^2}_{n\times\text{total variance}}=\underbrace{\sum_{i=1}^n \widehat{y}_i^2}_{n\times\text{explained variance}}+\underbrace{\sum_{i=1}^n (y_i-\widehat{y}_i)^2}_{n\times\text{residual variance}} The coefficient of determination, R^2, is then interpreted as the square of the cosine of the angle \theta between \mathbf{y} and \Pi_X \mathbf{y} : R^2=\frac{\Vert \Pi_{{X}} \mathbf{y}\Vert^2}{\Vert \mathbf{y}\Vert^2}=1-\frac{\Vert \Pi_{ {X}^\perp} \mathbf{y}\Vert^2}{\Vert \mathbf {y}\Vert^2}=\cos^2(\theta)An important application was obtained by Frish & Waugh (1933), when the explanatory variables are divided into two groups, \mathbf{X}=[\mathbf{X}_1 |\mathbf{X}_2], so that the regression becomes y=\beta_0+\mathbf{X}_1 β_1+\mathbf{X}_2 β_2+\varepsilon. Frish & Waugh (1933) showed that two successive projections could be considered. Indeed, if \mathbf{y}_2^\star=\Pi_{X_1^\perp} \mathbf{y} and X_2^\star=\Pi_{X_1^\perp}\mathbf{X}_2, we can show that \widehat{\beta} _2=[{\mathbf{X}_2^\star}^T \mathbf{X}_2^\star]^{-1}{\mathbf{X}_2^\star}^T \mathbf{y}_2^\star In other words, the overall estimate is equivalent to the combination of independent estimates of the two models if \mathbf{X}_2^\star=\mathbf{X}_2, i.e. \mathbf{X}_2\in \mathcal{E}_{X_1}^\perp, which can be noted \mathbf{x}_1\perp\mathbf{x}_2 We obtain here the Frisch-Waugh theorem which guarantees that if the explanatory variables between the two groups are orthogonal, then the overall estimate is equivalent to two independent regressions, on each of the sets of explanatory variables. This is a theorem of double projection, on orthogonal spaces. Many results and interpretations are obtained through geometric interpretations (fundamentally related to the links between conditional expectation and the orthogonal projection in space of variables of finite variance).

This geometric interpretation might help to get a better understanding of the problem of under-identification, i.e. the case where the real model would be y_i=\beta_0+ \mathbf{x}_1^T \beta_1+\mathbf{x}_2^T \beta_2+\varepsilon_i, but the estimated model is y_i=b_0+\mathbf{x}_1^T \mathbf{b}_1+\eta_i. The maximum likelihood estimator of \mathbf{b}_1 is \widehat{\mathbf{b}}_1=\mathbf {\beta}_1 + \underbrace{ (\mathbf {X}_1^T\mathbf {X}_1)^{-1} \mathbf {X}_1^T \mathbf {X}_{2} \mathbf{\beta}_2}_{\mathbf{\beta}_{12}}+\underbrace{(\mathbf{X}_1^{T}\mathbf{X}_1)^{-1} \mathbf{X}_1^T\varepsilon}_{\nu}so that \mathbb{E}[\widehat{\mathbf{b}}_1]=\beta_1+\beta_{12}, the bias ( \beta_{12}) being null only in the case where \mathbf{X}_1^T \mathbf{X}_2=\mathbf{0} (i. e. \mathbf{X}_1\perp \mathbf{X}_2 ): we find here a consequence of the Frisch-Waugh theorem.

On the other hand, over-identification corresponds to the case where the real model would be y_i=\beta_0+\mathbf{x}_1^T \beta_1+\varepsilon_i, but the estimated model is y_i=b_0+ \mathbf{x}_1^T \mathbf{b} _1+\mathbf{x}_2^T \mathbf{b}_2+\eta_i. In this case, the estimate is unbiased, in the sense that \mathbb{E}[\widehat{\mathbf{b}}_1]=\beta_1 but the estimator is not efficient. Later on, we will discuss an effective method for selecting variables (and avoid over-identification).

From parametric to non-parametric

We can rewrite equation (4) in the form \widehat{\mathbf{y}}=\Pi_X\mathbf{y} which helps us to see the forecast directly as a linear transformation of the observations. More generally, a linear predictor can be obtained by considering m(\mathbf{x})=\mathbf{s}_{\mathbf{x}}^T \mathbf{y}, where \mathbf{s}_{\mathbf{x}} is a weight vector, which depends on \mathbf{x}, interpreted as a smoothing vector. Using the vectors \mathbf{s}_{\mathbf{x}_i}, calculated from the observations \mathbf{x}_i, we obtain a matrix \mathbf{S} of size n\times n, and \widehat{\mathbf{y}}=\mathbf{S}\mathbf{y}. In the case of the linear regression described above, \mathbf{s}_{\mathbf{x}}=\mathbf{X}[\mathbf{X}^T\mathbf{X}]^{-1}\mathbf{x}, and in that case \text{trace}(\mathbf{S}) is the number of columns in the \mathbf{X} matrix (the number of explanatory variables). In this context of more general linear predictors, \text{trace}(\mathbf{S}) is often seen as equivalent to the number of parameters (or complexity, or dimension, of the model), and \nu=n-\text{trace}(\mathbf{S}) is then the number of degrees of freedom (see Ruppert et al., 2003; Simonoff, 1996). The principle of parsimony says that we should minimize this dimension (the trace of the matrix \mathbf{S}) as much as possible. But in the general case, this dimension is more to obtain, explicitely.

The estimator introduced by Nadaraya (1964) and Watson (1964), in the case of a simple non-parametric regression, is also written in this form since\widehat{m}_h(x)=\mathbf{s}_{x}^T\mathbf{y}=\sum_{i=1}^n \mathbf{s}_{x,i}y_iwhere\mathbf{s}_{x,i}=\frac{K_h(x-x_i)}{K_h(x-x_1)+\cdots+K_h(x-x_n)} where K(\cdot) is a kernel function, which assigns a value that is lower the closer x_i is to x, and h>0 is the bandwidth. The introduction of this metaparameter h is an important issue, as it should be chosen wisely. Using asymptotic developments, we can show that if X has density f, \text{biais}[\widehat{m}_h(x)]=\mathbb{E}[\widehat{m}_h(x)]-m(x)\sim {h^2}\left(\frac{C_1 }{2}m''(x)+C_2 m'(x)\frac{f'(x)}{f(x)}\right)and \displaystyle{{\text{Var}[\widehat{m}_h(x)]\sim\frac{C_3}{{nh}}\frac{\sigma(x)}{f(x)}}}for some constants that can be estimated (see Simonoff (1996) for a discussion). These two functions evolve inversely with h, as shown in Figure 1 (where the metaparameter on the x-axis is here, actually, h^{-1}). Keep in ming that we will see a similar graph in the context of machine learning models.

Figure 1. Choice of meta-parameter and the Goldilocks problem: it must not be too large (otherwise there is too much variance), nor too small (otherwise there is too much bias).

The natural idea is then to try to minimize the mean square error, the MSE, defined as bias[\widehat{m}_h (x)]^2+Var[\widehat{m}_h (x)], and them integrate over x, which gives an optimal value for h of the form h^\star=O(n^{-1/5}) , and reminds us of Silverman’s rule – see Silverman (1986). In larger dimensions, for continuous \mathbf{x} variables, a multivariate kernel with matrix bandwidth \mathbf{H} can be used, and \mathbb{E}[\widehat{m}_{\mathbf{H}}(\mathbf{x})]\sim m(\mathbf{x})+\frac{C_1}{2}\text{trace}\big(\mathbf{H}^Tm''(\mathbf{x})\mathbf{H}\big)+C_2\frac{m'(\boldsymbol{x})^T\mathbf{H}\mathbf{H}^T \nabla f(\mathbf{x})}{f(\mathbf{x})}while\text{Var}[\widehat{m}_{\mathbf{H}}(\mathbf{x})]\sim\frac{C_3}{n~\text{det}(\mathbf{H})}\frac{\sigma(\mathbf{x})}{f(\mathbf{x})}
If \mathbf{H} is a diagonal matrix, with the same term h  on the diagonal, then h^\star=O(n^{-1/(4+dim(\mathbf{x}))}. However, in practice, there will be more interest in the integrated version of the quadratic error, MISE(\widehat{m}_{h})=\mathbb{E}[MSE(\widehat{m}_{h}(X))]=\int MSE(\widehat{m}_{h}(x))dF(x)and we can prove that MISE[\widehat{m}_h]\sim \overbrace{\frac{h^4}{4}\left(\int x^2k(x)dx\right)^2\int\big[m''(x)+2m'(x)\frac{f'(x)}{f(x)}\big]^2dx}^{\text{bias}^2} +\overbrace{\frac{\sigma^2}{nh}\int k^2(x)dx \cdot\int\frac{dx}{f(x)}}^{\text{variance}}as n→∞ and nh→∞. Here we find an asymptotic relationship that again recalls Silverman’s (1986) order of magnitude, h^\star =n^{-\frac{1}{5}}\left(\frac{C_1\int \frac{dx}{f(x)}}{C_2\int \big[m''(x)+2m'(x)\frac{f'(x)}{f(x)}\big]dx}\right)^{\frac{1}{5}}The main problem here, in practice, is that many of the terms in the expression above are unknown. Automatic learning offers computational techniques, when the econometrician used to searching for asymptotic (mathematical) properties.

To be continued (references mentioned above are online here)…

Probabilistic Foundations of Econometrics, part 1

In a series of posts, I wanted to get into details of the history and foundations of econometric and machine learning models. It will be some sort of online version of 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. This is the first one…

The importance of probabilistic models in economics is rooted in Working’s (1927) questions and the attempts to answer them in Tinbergen’s two volumes (1939). The latter have subsequently generated a great deal of work, as recalled by Duo (1993) in his book on the foundations of econometrics, and more particularly in the first chapter “The Probability Foundations of Econometrics”. It should be recalled that Trygve Haavelmo was awarded the Nobel Prize in Economics in 1989 for his “clarification of the foundations of the probabilistic theory of econometrics”. Because as Haavelmo (1944) (initiating a profound change in econometric theory in the 1930s, as recalled in Morgan’s Chapter 8 (1990)) showed, econometrics is fundamentally based on a probabilistic model, for two main reasons. First, the use of statistical quantities (or “measures”) such as means, standard errors and correlation coefficients for inferential purposes can only be justified if the process generating the data can be expressed in terms of a probabilistic model. Second, the probability approach is relatively general, and is particularly well suited to the analysis of “dependent” and “non-homogeneous” observations, as they are often found on economic data.We will then assume that there is a probabilistic space (\Omega,\mathcal{F},\mathbb{P}) such that observations (y_i,\mathbf{x}_i) are seen as realizations of random variables (Y_i, \mathbf{X}_i) . In practice, however, we are not very interested in the joint law of the couple (Y, \mathbf{X}) : the law of \mathbf{X} is unknown, and it is the law of Y conditional on \mathbf{X} that will be interested in. In the following, we will note x a single observation, \mathbf{x} a vector of observations, X a random variable, and \mathbf{X} a random vector. Abusively, \mathbf{X} may also designate the matrix of individual observations (denoted \mathbf{x}_i), depending on the context.

Foundations of mathematical statistics

As recalled in Vapnik’s (1998) introduction, inference in parametric statistics is based on the following belief: the statistician knows the problem to be analyzed well, in particular, he knows the physical law that generates the stochastic properties of the data, and the function to be found is written via a finite number of parameters[1]. To find these parameters, the maximum likelihood method is used. The purpose of the theory is to justify this approach (by discovering and describing its favorable properties). We will see that in learning, philosophy is very different, since we do not have a priori reliable information on the statistical law underlying the problem, nor even on the function we would like to approach (we will then propose methods to construct an approximation from the data at our disposal, as in (1998)). A “golden age” of parametric inference, from 1930 to 1960, laid the foundations for mathematical statistics, which can be found in all statistical textbooks, including today. As Vapnik (1998) states, the classical parametric paradigm is based on the following three beliefs:

  1. To find a functional relationship from the data, the statistician is able to define a set of functions, linear in their parameters, that contain a good approximation of the desired function. The number of parameters describing this set is small.
  2. The statistical law underlying the stochastic component of most real-life problems is the normal law. This belief has been supported by reference to the central limit theorem, which stipulates that under large conditions the sum of a large number of random variables is approximated by the normal law.
  3. The maximum likelihood method is a good tool for estimating parameters.

In this section we will come back to the construction of the econometric paradigm, directly inspired by that of classical inferential statistics.

Conditional laws and likelihood

Linear econometrics has been constructed under the assumption of individual data, which amounts to assuming independent variables (Y_i, \mathbf{X}_i) (if it is possible to imagine temporal observations – then we would have a process (Y_t, \mathbf{X}_t) – but we will not discuss time series here). More precisely, we will assume that, conditionally to the explanatory variables \mathbf{X}_i, the variables Y_i are independent. We will also assume that these conditional laws remain in the same parametric family, but that the parameter is a function of \mathbf{x}. In the Gaussian linear model it is assumed that: (Y\vert \mathbf{X}=\mathbf{x})\overset{\mathcal{L}}{\sim}\mathcal{N}(\mu(\mathbf{x}),\sigma^2)~~~~ (1)where \mu(\mathbf{x})=\beta_0+\mathbf{x}^T\mathbf{\beta} and \mathbf{\beta}\in\mathbb{R}^{p}.

It is usually called a ‘linear’ model since \mathbb{E}[Y\vert \mathbf{X}=\mathbf{x}]=\beta_0+\mathbf{x}^T\mathbf{\beta} is a linear combination of covariates[2]. It is said to be a homoscedastic model if Var[Y|\mathbf{X}=\mathbf{x}]=\sigma^2, where \sigma^2 is a positive constant. To estimate the parameters, the traditional approach is to use the Maximum Likelihood estimator, as initially suggested by Ronald Fisher. In the case of the Gaussian linear model, log-likelihood is written:  \log\mathcal{L}(\beta_0, \mathbf{\beta},\sigma^2\vert \mathbf{y},\mathbf{x}) = -\frac{n}{2}\log[2\pi\sigma^2] - \frac{1}{2\sigma^2}\sum_{i=1}^n (y_i-\beta_0-\mathbf{x}_i^T\mathbf{\beta})^2Note that the term on the right, measuring a distance between the data and the model, will be interpreted as deviance in generalized linear models. Then we will set: (\widehat{\beta}_0,\widehat{\mathbf{\beta}},\widehat{\sigma}^2)=\text{argmax}\left\lbrace\log\mathcal{L}(\beta_0, \mathbf{\beta},\sigma^2\vert \mathbf{y},\mathbf{x})\right\rbraceThe maximum likelihood estimator is obtained by minimizing the sum of the error squares (the so-called “least squares” estimator) that we will find in the “machine learning” approach.

The first order conditions allow to find the normal equations, whose matrix writing is \mathbf{X}^T[\mathbf{y}-\mathbf{X}\mathbf{\beta}]=\mathbf{0}, which can also be written (\mathbf{X}^T \mathbf{X})\mathbf{\beta}=\mathbf{X}^T \mathbf{y}. If \mathbf{X} is a full (column) rank matrix, then we find the classical estimator:\widehat{\mathbf{\beta}}=(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T\mathbf{y}=\mathbf{\beta}+(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^{-1}\mathbf{\varepsilon}~~~(2)using residual-based writing (as often in econometrics), y=\mathbf{x}^T\mathbf{\beta}+\varepsilon. Gauss Markov’s theorem ensures that this estimator is the unbiased linear estimator with minimum variance. It can then be shown that \widehat{\mathbf{\beta}}\sim\mathcal{N}(\mathbf{\beta},\sigma^2(\mathbf{X}^T\mathbf{X})^{-1}), and in particular, if we simply need the first two moments : \mathbb{E}[\widehat{\mathbf{\beta}}]=\mathbf{\beta}~~~Var[\widehat{\mathbf{\beta}}]=\sigma^2 [\mathbf{X}^T\mathbf{X}]^{-1}In fact, the normality hypothesis makes it possible to make a link with mathematical statistics, but it is possible to construct this estimator given by equation (2) without that Gaussian assumption. Hence, if we assume that Y|\mathbf{X} has the same distribution as \mathbf{x}^T\mathbf{\beta}+\varepsilon, where \mathbb{E}[\varepsilon]=0, Var[\varepsilon]=\sigma^2 and Cov[X_j,\varepsilon]=0 for all j, then \widehat{\mathbf{\beta}} is an unbiased estimator of \mathbf{\beta} with smallest variance[3] among unbiased linear estimators. Furthermore, if we cannot get normality at finite distance, asymptotically this estimator is Gaussian, with \sqrt{n}(\widehat{\mathbf{\beta}}-\mathbf{\beta})\overset{\mathcal{L}}{\rightarrow}\mathcal{N}(\mathbf{0},\mathbf{\Sigma})as n\rightarrow\infty, for some matrix \mathbf{\Sigma}.
The condition of having a full rank \mathbf{X} matrix can be (numerically) strong in large dimensions. If it is not satisfied, (\mathbf{X}^T \mathbf{X})^{-1}\mathbf{X}^T does not exist. If \mathbb{I} denotes the identity matrix, however, it should be noted that (\mathbf{X}^T \mathbf{X}+\lambda\mathbb{I})^{-1}\mathbf{X}^T still exists, whatever \lambda>0. This estimator is called the ridge estimator of level \lambda (introduced in the 1960s by Hoerl (1962), and associated with a regularization studied by Tikhonov (1963)). This estimator naturally appears in a Bayesian econometric context.


It is not uncommon to introduce the linear model from the distribution of the residuals, as we mentioned earlier. Also, equation (1) is written as often: y_i=\beta_0+\mathbf{x}_i^T\mathbf{\beta}+\varepsilon_i~~~~(3)where \varepsilon_i’s are realizations of independent and identically distributed random variables (i.i.d.) from some \mathcal{N}(0,\sigma^2) distribution. With a vector notation, we will write \mathbf{\varepsilon}\overset{\mathcal{L}}{\sim}\mathcal{N}(\mathbf{0},\sigma^2\mathbb{I}) . The estimated residuals are defined as: \widehat{\varepsilon}_i =y_i-[\widehat{\beta}_0+\mathbf{x}_i^T\widehat{\mathbf{\beta}}] Those (estimated) residuals are basic tools for diagnosing the relevance of the model.

An extension of the model described by equation (1) has been proposed to take into account a possible heteroscedastic character: (Y\vert \mathbf{X}=\mathbf{x})\overset{\mathcal{L}}{\sim}\mathcal{N}(\mu(\mathbf{x}),\sigma^2(\mathbf{x}))where \sigma^2(\mathbf{x}) is a positive function of the explanatory variables. This model can be rewritten as: y_i=\beta_0+\mathbf{x}_i^T\mathbf{\beta}+\sigma^2(\mathbf{x}_i)\cdot\varepsilon_iwhere residuals are always i.i.d., with unit variance, \varepsilon_i=\frac{y_i-[\beta_0+\mathbf{x}_i^T\mathbf{\beta}]}{\sigma(\mathbf{x}_i)} While residuals based equations are popular in linear econometrics (when the dependent variable is continuous), it is no longer popular in counting models, or logistic regression.

However, writing using an error term (as in equation (3)) raises many questions about the representation of an economic relationship between two quantities. For example, it can be assumed that there is a relationship (linear to begin with) between the quantities of a traded good, q and its price p. This allows us to imagine a supply equationq_i=\beta_0+\beta_1 p_i+u_i(u_i being an error term) where the quantity sold depends on the price, but in an equally legitimate way, one can imagine that the price depends on the quantity produced (what one could call a demand equation), p_i=\alpha_0+\alpha_1 q_i+v_i(v_i denoting another error term). Historically, the error term in equation (3) could be interpreted as an idiosyncratic error on the variable y, the so-called explanatory variables being assumed to be fixed, but this interpretation often makes the link between an economic relationship and a complicated economic model difficult, the economic theory speaking abstractly about a relationship between a magnitude, the econometric model imposing a specific shape (what magnitude is y and what magnitude is x) as shown in more detail in Morgan (1990) Chapter 7.

(references mentioned above are online here). To be continued…

[1] This approach can be compared to structural econometrics, as presented for example in Kean (2010).

[2] Here, we will try to distinguish \beta_0, the intercept, and the other parameters \mathbf{\beta}, since they are considered differently in many extensions (e.g. regularization). Nevertheless, in many expressions \mathbf{\beta} will denote the joint vector (\beta_0, \mathbf{\beta}), for general formulas, to avoid too heavy notations.

[3] In the sense that the difference between variance matrices is a positive matrix.

Histogramme et densité en échelle logarithmique

Il y a presque 20 ans, Paul-André Rosental, Gilles Postel-Vinay, Akiko Suwa-Eisenmann et Jérôme Bourdieu publiaient Migrations et transmissions inter-générationnelles dans la France
du XIXe et du début du XXe siècle qui présentait le graphique ci-dessous,

On est tombé sur ce graphique avec Ewen Gallic en écrivant notre article Using Collaborative Genealogy Data to Study Migration (qui paraîtra très bientôt dans le journal The History of the Family). Classiquement dans un article académique, on a besoin de se positionner par rapport à la littérature existante. Avec nos propres données, et nous avions construit l’histogramme suivant

histoire de montrer que nous retrouvions cette forme bimodale. Depuis le début, je suis pas très confortable avec ce graphique, mais on l’a laissé non pas pour ses vertus sémiologiques, mais parce qu’il valide notre approche, en obtenant des résultats comparables à d’autres, déjà établis dans la littérature.

Je repensais à tout ça avant hier, en mentionnant sur twitter le graphique suivant

qui contient quelque chose de similaire, à savoir une espèce d’histogramme en bleu, sauf que la largeur des bandes décroit ici de manière logarithmique…

Je suis très mal à l’aise face à ces graphiques, parce que je ne sais pas comment les interpréter.  Mais peut-être faut-il revenir à la base, pour comprendre ce qu’on fait.

On a un échantillon \{x_1,\cdots,x_n\} disons de montant d’impôt payé. On a un échantillon comme on dit en statistique descriptive. En statistique mathématique, on suppose que les x_i sont des réalisations de variables aléatoires. On pourrait ainsi dire que x_i=X(\omega_i)X est une variable aléatoire définie sur un espace probabilisé (\Omega,\mathcal{A},\mathbb{P}). Pour rappel, \Omega, c’est l’univers, une espèce d’espace fondamental abstrait. Sur le site les-mathématiques, on nous dit que \Omega est l’espace des observables, ce qui me déplait assez car pour moi, justement \Omega est un espace abstrait. En théorie de la décision, si on reprend par exemple Probability and Uncertainty in Economic Modeling, on parle d’états de la nature (c’est aussi la terminologie que l’on retrouve dans States of Nature and the Nature of States). Et assez souvent, on oublie cet espace, grâce au théorème de transfert : en effet, si on a une variable aléatoire réelle, X:\Omega \rightarrow \mathbb {R} , alors {\displaystyle \mathbb {E} \left[\varphi (X)\right]=\int _{\Omega }\varphi {\big (}X(\omega ){\big )}\mathbb {P} (\mathrm {d} \omega )=\int _{\mathbb {R} }\varphi (x)\mathbb {P} _{X}(\mathrm {d} x)} ce qui veut dire, de manière polie, qu’on ne se place jamais sur ces espace fondamental, mais on regarde juste leur “transfert” sur la droite réelle (ici, je vais me contenter du cas unidimensionnel), c’est à dire la “valeur” prise par la variable aléatoire (je reviendrai tout à l’heure sur ces écritures sous forme intégrales). C’est un abus de langage que l’on fait quand on dit que pour un lancer de dé, l’univers est \Omega = \{1, 2, 3, 4, 5, 6\}, correspondant aux valeurs des faces du dé. En fait, l’espace fondamental, des “états de la nature” peut être plus compliqué que ça, mais pour mieux comprendre, on va le “transférer” sur l’espace comptable \{1, 2, 3, 4, 5, 6\}. Mais je commence à m’écarter du sujet, surtout que je m’étais promis que je ne ferais pas (trop) de théorie de la mesure. L’autre raison est que je pense que ce n’est pas exactement la formalisation qu’on utilise en statistique. Je pense que x_i=X_i(\omega) où les variables X_1,\cdots,X_n sont des variables aléatoires indépendantes, et de même loi. Pourquoi ? Avec les notations précédentes, on dispose de l’échantillon suivant

que l’on peut voir comme une réalisation des variables aléatoires suivantes

Une statistique sera alors une fonction construit sur notre échantillon, \hat{\theta}=h(x_1,\cdots,x_n) , par exemple la moyenne (empirique)

qui est alors un nombre. Mais ça peut être aussi une variable aléatoire, \hat{\theta}=h(X_1,\cdots,X_n) , soit ici, pour la moyenne

La principale difficulté est qu’en statistique mathématique, on utilise la même notation, \widehat{\theta} pour deux objets de nature très différentes. Et donc avoir des abus de langage, en pouvait dire en cours “ici la moyenne vaut 37.5%” et ensuite demander de “calculer la variance de la moyenne”. Mais là encore, je m’écarte. Revenons à notre variable aléatoire X. Souvent, on va essayer de décrire sa loi de probabilité.

Le plus simple est de passer par la fonction de répartition, définie sans ambiguïté par F(x) = \mathbb{P}[\{\omega\in\Omega:X(\omega)\leq x\}] (ce qui a du sens cas la mesure de probabilité \mathbb{P} est effectivement définie sur l’espace des états de la nature \Omega. Mais par abus de langage, on se contentera de noter F(x) = \mathbb{P}[X\leq x] . Supposons maintenant que notre variable X est absolument continue. Alors dans ce cas – ça correspond je pense au premier théorème de l’analyseF(x)=\int _{-\infty}^{x}f(t) dt, où f est alors la densité de la variable aléatoire. Le point un peu subtile ici est que l’intégrale est ici définie par rapport à la mesure de Lebesgue (ce qui permet de définir proprement ce qu’est ce dt à la fin de l’intégrale). Mais oublions ce point un instant.

On arrive enfin au premier point auquel je voulais arriver : pour les variables (absolument) continue, on peut “représenter” leur loi par leur densité. C’est le dessin ci-dessus, si X est une variable qui suit une loi lognormale


L’interprétation est simple : comme la probabilité \mathbb {P} (a<X\leq b) se calcule alors par la relation suivante : \mathbb {P} \left(a<X\leq b\right)=\int _{a}^{b}f(t)dtla probabilité \mathbb {P} (a<X\leq b) se lit comme l’aire sous la courbe sur l’intervalle [a, b].

Parmi les estimateurs usuels de la densité, on peut utiliser un histogramme. Même si c’est un objet que tout le monde manipule depuis ses cours au secondaire, la construction formelle de cet objet est un peu technique. En effet, le premier point est qu’il faut découper l’ensemble des valeurs prises par les observations x_i – disons (a,b] – en une partition, c’est à dire un ensemble d’intervalles disjoints qui vont recouvrir  (a,b], I_1,\cdots,I_k. Le plus simple est d’avoir une partition régulière, I_1=(a,a+(b-a)/k], I_2=(a+(b-a)/k,a+2(b-a)/k], etc. Généralement, I_j=(a+(b-a)(j-1)/k,a+(b-a)j/k]. Et classiquement, on construit un histogramme en comptant le nombre d’observations dans chacun des intervalles,


C’est l’histogramme classique, avec un pas de 100.


Pour avoir une densité (i.e. une “fonction en escalier qui va s’intégrer à 1”) il faut normaliser la hauteur (en divisant par le pas). On peut d’ailleurs superposer la densité de la loi lognormale.

Dans les graphiques que je présentais, on utilise en échelle logarithmique en abscisse. On peut le faire avec R en demandant


ce qui correspond aux graphiques précédant. On peut aussi, à la main, dire qu’on veut non plus voir \{x,f(x)\} mais \{\log(x),f(x)\}.


L’aire précédente devient alors

Visuellement, on raisonne par comparaison. Quand on dit que la densité est une aire sous la courbe, en réalité, c’est la proportion de l’aire sous la courbe (par rapport à l’aire totale) qui nous parle. Mais autant le faire avec deux aires, pour mieux comprendre. A droite, en rouge, on a la probabilité d’avoir une valeur entre 200 et 600, soit ici environ 30%.

[1] 0.3015131

et en bleu, on a la probabilité d’être entre 92 et 200, qui vaut là aussi environ 30%

[1] 0.3010197

Ici les deux aires sont égales. On notera que ce n’est pas forcément évident, au premier coup d’œil, et l’asymétrie (vers la droite) laisse à croire que l’aire rouge est peut-être plus grande. Mais passons.

Considérons maintenant notre échelle logarithmique, en abscisse,

ou alors la version sur le logarithme des abscisses,

Comme on le notait auparavant, on aime ce graphique car on reconnait une forme de densité de “loi normale” (ce qui pourrait faire du sens, car on obtient une loi lognormale justement en prenant l’exponentielle d’une loi normale…). Cela dit, on a clairement un problème d’échelle ici, car l’aire totale sous la courbe vaut moins de 1%… autrement dit, on est loin de 1. Mais on peut quand même retrouver une loi normale,


(on pourra noter d’ailleurs que non seulement, il a fallu diviser par 90, mais la moyenne est ici 4, et pas 5). Comme on a la densité d’une loi normale (à une transformation près), on peut facilement faire des calculs, en particulier, calculer les deux aires, rouges et bleues

[1] 5.300315 6.396263
[1] 0.9032535 0.9917184
[1] 0.08846485
[1] 0.2019666

Aussi, l’aire bleue vaut ici plus du double de l’aire rouge. Autrement dit, si on interprète la courbe comme une densité sur le graphique ci-dessous

on a intuitivement envie de dire qu’il y a 2 fois plus de chance d’avoir une valeur entre 92 et 200 qu’entre 200 et 600. Ce qui est faux.

Si on essaye de comprendre un peu mieux, je vais revenir sur deux points… La première c’est que, aussi étrange que ça puisse paraître, c’est presque un coup de chance qu’en prenant une échelle logarithmique, on obtient une densité lognormale. On avait écrit un article avec Emmanuel Flachaire (Log-Transform Kernel Density Estimation of Income Distribution) qui revenait sur l’importance de cette transformation logarithmique. Mais ici, c’est un peu différent. On a en effet deux graphiques : le premier c’est \{x,f(x)\} alors le second c’est \{\log(x),f(x)\}, soit, en faisant un changement de variable y=\log(x) (ou x=e^y), c’est \{y,f(e^y)\}. Or dans le premier cas, on avait une loi lognormale, autrement dit f(x )=\frac {1}{x\sigma {\sqrt {2\pi }}}\exp \left(-\frac {(\log x-\mu )^{2}}{2\sigma ^{2}}\right)={\frac {1}{x}}\phi(\log(x);\mu ,\sigma^2 )\phi est la densité de la loi normale centrée réduite. Si on regarde le second graphique, \{y,f(e^y)\}, on représente alors \{y,e^{-y}\phi(y;\mu ,\sigma^2 )\}. La magie va opérer parce qu’on peut rentrer le e^{-y} dans la densité de la loi normale, ce qui donnera deux choses (1) une translation de la moyenne, (2) un facteur multiplicatif sur la densité. Ce sont les deux phénomème que nous avions observé ici. Si on détaille un peu e^{-y}\exp \left(-\frac {(y-\mu )^{2}}{2\sigma ^{2}}\right)=\exp \left(-\frac {(y-[\mu-\sigma^2] )^{2}}{2\sigma ^{2}}+\star\right)où on retrouve la translation de 1 de moyenne (centrée sur 4 et non plus sur 5, mais c’est logique puisqu’on avait pris \sigma^2 ayant pour valeur 1. Je laisse les plus courageux calculer le \star qui va donner le facteur multiplicatif. Le second point est un peu plus technique, il est lié à un problème de mesure. Dans le premier cas, quand on faisait un calcul d’intégrale, on avait un dx et dans le second cas, on calcule des aires avec un dy, mais ce n’est pas la bonne transformation. En effet, si y=\log(x) (ou x=e^x) alors dx=e^ydy. Plus formellement, dans le premier cas, quand on calculait \mathbb{P}[X\in[a,b]] on calculait l’intégrale \int _{a}^{b}f(x) dx. Dans le second cas, on calcule \int _{\alpha}^{\beta}f(e^y) dy puisqu’on visualise la courbe \{y,f(e^y)\}. Mais si on fait un changement de variable propre \int _{\alpha}^{\beta}f(e^y) dy=\int _{a}^{b}f(x) \frac{dx}{x}qui n’est plus du tout l’intégrale qu’on cherche à calculer. Je pense que cette histoire de dy correspondant à x^{-1}dx pourrait avoir une interprétation en terme de changement de mesure (la mesure de référence n’est plus la mesure de Lebesgue, qui nous donne une relative uniformité sur l’axe des abscisses, et permet de faire un lien avec l’intégrale “classique” – au sens de Rieman).

Bref, cette histoire de mettre une échelle logarithmique en abscisse pour visualiser une densité est très perturbant. L’intuition que l’on peut en avoir est biaisée, et l’objet mathématique que l’on créé est complexe…

The “probability to win” is hard to estimate…

Real-time computation (or estimation) of the “probability to win” is difficult. We’ve seem that in soccer games, in elections… but actually, as a professor, I see that frequently when I grade my students.

Consider a classical multiple choice exam. After each question, imagine that you try to compute the probability that the student will pass. Consider here the case where we have 50 questions. Students pass when they have 25 correct answers, or more. Just for simulations, I will assume that students just flip a coin at each question… I have n students, and 50 questions


Let X_{i,j} denote the score of student i at question j. Let S_{i,j} denote the cumulated score, i.e. S_{i,j}=X_{i,1}+\cdots+X_{i,j}. At step j, I can get some sort of prediction of the final score, using \hat{T}_{i,j}=50\times S_{i,j}/j. Here is the code


We can actually plot it

for(i in 2:n) lines(NB[,i],type="s",col="light blue")

But that’s simply the prediction of the final score, at each step. That’s not the computation of the probability to pass !

Let’s try to see how we can do it… If after j questions, the students has 25 correct answer, the probability should be 1 – i.e. if S_{i,j}\geq 25 – since he cannot fail. Another simple case is the following : if after j questions, the number of points he can get with all correct answers until the end is not sufficient, he will fail. That means if S_{i,j}+(50-i+1)< 25 the probability should be 0. Otherwise, to compute the probability to sucess, it is quite straightforward. It is the probability to obtain at least 25-S_{i,j} correct answers, out of 50-j questions, when the probability of success is actually S_{i,j}/j. We recognize the survival probability of a binomial distribution. The code is then simply

for(i in 1:50){
  for(j in 1:n){
    if(SM[i,j]&gt;=25) PB[i,j]=1
    if(SM[i,j]+(50-i+1)&lt;25)   PB[i,j]=0
    if((SM[i,j]&lt;25)&amp;(SM[i,j]+(50-i+1)&gt;=25)) PB[i,j]=1-pbinom(25-SM[i,j],size=(50-i),prob=SM[i,j]/i)

So if we plot it, we get

for(i in 2:n) lines(PB[,i],type="s",col="light blue")

which is much more volatile than the previous curves we obtained ! So yes, computing the “probability to win” is a complicated exercice ! Don’t blame those who try to find it hard to do !

Of course, things are slightly different if my students don’t flip a coin… this is what we obtain if half of the students are good (2/3 probability to get a question correct) and half is not good (1/3 chance),

If we look at the probability to pass, we usually do not have to wait until the end (the 50 questions) to know who passed and who failed

PS : I guess a less volatile solution can be obtained with a Bayesian approach… if I find some spare time this week, I will try to code it…

Solving the chinese postman problem

Some pre-Halloween post today. It started actually while I was in Barcelona : kids wanted to go back to some store we’ve seen the first day, in the gothic part, and I could not remember where it was. And I said to myself that would be quite long to do all the street of the neighborhood. And I discovered that it was actually an old problem. In 1962, Meigu Guan was interested in a postman delivering mail to a number of streets such that the total distance walked by the postman was as short as possible. How could the postman ensure that the distance walked was a minimum?

A very close notion is the concept of traversable graph, which is one that can be drawn without taking a pen from the paper and without retracing the same edge. In such a case the graph is said to have an Eulerian trail (yes, from Euler’s bridges problem). An Eulerian trail uses all the edges of a graph. For a graph to be Eulerian all the vertices must be of even order.

An algorithm for finding an optimal Chinese postman route is:

  1. List all odd vertices.
  2. List all possible pairings of odd vertices.
  3. For each pairing find the edges that connect the vertices with the minimum weight.
  4. Find the pairings such that the sum of the weights is minimised.
  5. On the original graph add the edges that have been found in Step 4.
  6. The length of an optimal Chinese postman route is the sum of all the edges added to the total found in Step 4.
  7. A route corresponding to this minimum weight can then be easily found.

For the first steps, we can use the codes from Hurley & Oldford’s Eulerian tour algorithms for data visualization and the PairViz package. First, we have to load some R packages


Then use the following function from stackoverflow,

make_eulerian = function(graph){
  info = c("broken" = FALSE, "Added" = 0, "Successfull" = TRUE)
  is.even = function(x){ x %% 2 == 0 }
  search.for.even.neighbor = !is.even(sum(!is.even(degree(graph))))
  for(i in V(graph)){
    set.j = NULL
    uneven.neighbors = !is.even(degree(graph, neighbors(graph,i))) 
if(sum(uneven.neighbors) == 0){ 
if(sum(!is.even(degree(graph))) &gt; 0){
          info["Broken"] = TRUE
          uneven.candidates &lt;- !is.even(degree(graph, V(graph)))
          if(sum(uneven.candidates) != 0){
            set.j &lt;- V(graph)[uneven.candidates][[1]]
            info["Successfull"] &lt;- FALSE
        set.j &lt;- neighbors(graph, i)[uneven.neighbors][[1]]
    }else if(search.for.even.neighbor == TRUE &amp; is.null(set.j)){
      info["Added"] &lt;- info["Added"] + 1     
      set.j &lt;- neighbors(graph, i)[ !uneven.neighbors ][[1]]
      if(!is.null(set.j)){search.for.even.neighbor &lt;- FALSE}
      if(i != set.j){
        graph &lt;- add_edges(graph, edges=c(i, set.j))
        info["Added"] &lt;- info["Added"] + 1
  (list("graph" = graph, "info" = info))}

Then, consider some network, with 12 nodes

g1 = graph(c(1,2, 1,3, 2,4, 2,5, 1,5, 3,5, 
4,7, 5,7, 5,8, 3,6, 6,8, 6,9, 9,11, 8,11, 
8,10, 8,12, 7,10, 10,12, 11,12), directed = FALSE)

To plot that network, use


Then we convert it to some traversable graph by adding 5 vertices

eulerian = make_eulerian(g1)
     broken       Added Successfull 
          0           5           1 
g = eulerian$graph

as shown below


We cut those 5 vertices in two part, and therefore, we add 5 artificial nodes

newA=lower.tri(A, diag = FALSE)*A1+upper.tri(A, diag = FALSE)*A
for(i in 1:sum(newA==2)) newA = cbind(newA,0)
for(i in 1:sum(newA==2)) newA = rbind(newA,0)
for(i in 1:nrow(A)){
      for(j in Aj){

We get the following graph, where all nodes have an even number of vertices !

for(i in 13:17){

Our network is now the following (new nodes are small because actually, they don’t really matter, it’s just for computational reasons)


Now we can get the optimal path

n &lt;- LETTERS[1:nrow(newA)]
g_2 &lt;- new("graphNEL",nodes=n) for(i in 1:nrow(newA)){ for(j in which(newA[i,]&gt;0)){
    g_2 &lt;- addEdge(n[i],n[j],g_2,1) 
 [1] "A" "B" "D" "G" "E" "A" "C" "E" "H" "F" "I" "K" "H" "J" "G" "P" "J" "L" "K" "Q" "L" "H" "O" "F" "C"
[26] "N" "E" "B" "M" "A"


edg=attr(E(newg), "vnames")
for(i in 1:length(parcours)){
 [1] "A|B" "B|D" "D|G" "E|G" "A|E" "A|C" "C|E" "E|H" "F|H" "F|I" "I|K" "H|K" "H|J" "G|J" "G|P" "J|P"
[17] "J|L" "K|L" "K|Q" "L|Q" "H|L" "H|O" "F|O" "C|F" "C|N" "E|N" "B|E" "B|M" "A|M"
 [1]  1  3  8  9  4  2  6 10 11 12 16 15 14 13 26 27 18 19 28 29 17 25 24  7 22 23  5 21 20

Let us try now on a real network of streets. Like Missoula, Montana.

I will not try to get the shapefile of the city, I will just try to replicate the photography above.

If you look carefully, you will see some problem : 10 and 93 have an odd number of vertices (3 here), so one strategy is to connect them (which explains the grey line).

But actually, to be more realistic, we start in 93, and we end in 10. Here is the optimal (shortest) path which goes through all vertices.

Now, we are ready for Halloween, to go through all streets in the neighborhood !

« Dans toute statistique, l’inexactitude du nombre est compensée par la précision des décimales »

Le statisticien et économiste Alfred Sauvy est resté dans les mémoires pour avoir inventé en 1952 le terme “tiers-monde”. Mais on lui a aussi attribué la paternité de la phrase suivante « dans toute statistique, l’inexactitude du nombre est compensée par la précision des décimales ». J’ai du l’entendre alors que j’étais étudiant, et depuis, elle me suit partout.

J’y repensais l’autre jour, quand Mathieu Gallard mentionnait sur Twitter le graphique suivant (correspondant a la popularité du président de la république, en France, dans les 18 mois qui suivent l’élection).

Le tweet disait (entre autres) “ est à ce stade de son quinquennat légèrement moins populaire que “. Pour rappel, ces courbes “de popularité” sont construites par un sondage, avec a chaque fois environ 1000 personnes interrogées (“sur la taille de l’échantillon on est toujours entre 950 et 970 interviews” me disait Mathieu). Bon, tous ceux qui ont des souvenirs de cours de stats se souviennent qu’avec 1000 personnes interrogées, 2 points de différence, c’est rarement significatif. Mais plus globalement, compte tenu de la marge d’erreur, je me suis demande pourquoi les courbes n’étaient pas lissées ? Ça éviterait les discussions stériles pour une variation de 2 points par exemple..

Si on reprend les données brutes (merci Mathieu), on a ici

lines(rate[,5],type="b",col="dark green")
text(18.15,rate[17,5],"JC",col="dark green")

ce qui donne le même que dans le tweet (même si je n’ose pas interpoler linéairement les valeurs manquantes – il y en a deux dans mon fichier)

Prenons la courbe la plus récente, celle d’Emmanuel Macron (avec en plus les valeurs manquantes pour pimenter un peu) et rajoutons les intervalles de confiance ponctuels.


Personnellement, j’aurais bien voulu (1) lisser tout ça, (2) rajouter quelque chose qui s’apparente a des bandes de confiance. Mais avec des erreurs de mesure (c’est comme ça qu’on peut interpréter le fait que les points viennent d’un sondage), je ne sais pas trop quoi faire. J’ai tenté la méthode suivante : le tire au hasard des points dans l’intervalle de confiance, puis je lisse sur ce nouveau jeu de points. Et je répète mille fois

for(s in 1:1000){
  if(s&lt;100) lines(seq(0,18,by=.25),yp,col="light blue")
lines(seq(0,18,by=.25),apply(Y,1,function(x) quantile(x,.95)),col="red",lty=2)
lines(seq(0,18,by=.25),apply(Y,1,function(x) quantile(x,.05)),col="red",lty=2)

On voit que notre courbe lissée est réaliste, voire même les pseudo-bandes de confiance autour. Pour obtenir ces trois courbes, on peut utiliser la fonction suivante

for(s in 1:1000){
upr=apply(Y,1,function(x) quantile(x,.975)),
lwr=apply(Y,1,function(x) quantile(x,.025)))

Sur les quatre colonnes de notre tableau, ça donne

lines(Y$x,Y$pred,col="dark green",lwd=2)
text(18.65,Y$pred[73],"JC",col="dark green")

Pourquoi les instituts de sondages, qui produisent les courbes de popularité, ne montrent pas ce genre de courbes ? Elles sont – a mon avis – aussi justes que celles qu’ils fournissent, au centième près, jouant sur une précision que l’incertitude ne devrait pas autoriser…

(A brief) history of randomness, and simulation techniques

Hearing “there is a 10% chance of rain today” or “the medical test has a positive predictive value of 75%” shows that the probabilities are now everywhere. A probability is a quantity that is difficult to grasp, but essential when trying to theorize and measure chance, or randomness. And if mathematical theory finally came very late, as Hacking (2006) points out, this did not prevent insurance from developing early enough, and from having the first (actuarial) mortality tables even before the “probability of death” or “life expectancy” had a mathematical basis. And in the same way, many techniques were invented to “generate randomness“, before the explosion of the so-called Monte Carlo methods, in parallel with the development of computing (and the fact that a machine could generate chance). Continue reading (A brief) history of randomness, and simulation techniques

On the robustness of LASSO

Probably the last post on lasso, before the summer break… More specifically, I was wondering about the interpretation of graphs \lambda\mapsto\widehat{\beta}_\lambda. We use them for variable selection, but my major concern was about confidence intervals : how can we trust those lines ?

As usual, a natural way is to use simulations on generated datasets. Consider for instance

Sigma = matrix(c(1,.8,.2,.8,1,.4,.2,.4,1),3,3)
n = 1000
X = rmnorm(n,rep(0,3),Sigma)
df = data.frame(X1=X[,1],X2=X[,2],X3=X[,3],X4=rnorm(n),
df$Y = 1+df$X1-df$X4+5*(df$X7=="A")+rnorm(n)

One can use other simulations of datasets, and store the output

vlambda = exp(seq(-8,1,length=201))
lasso = glmnet(x=X,y=df[,"Y"],family="gaussian",alpha=1,
VLASSO[[s]] = as.matrix(lasso$beta)

To visualize confidence bands, one can compute quantiles

for(i in 1:nrow(Q05)){
  for(j in 1:ncol(Q05)){
    v = unlist(lapply(VLASSO,function(x) x[i,j]))
    Q05[i,j] = quantile(v,.05)
    Q95[i,j] = quantile(v,.95)
    Qm[i,j]  = mean(v)

and get get the graph


An alternative (more realistic on real data) is to use bootstrapped version of the dataset

id = sample(1:nrow(X),size=nrow(X),replace=TRUE)
lasso = glmnet(x=X[id,],y=df[id,"Y"],family="gaussian",alpha=1,

So far, it looks it’s working very well. Now, what if we have a smaller dataset

n = 100

On simulated new samples, we get

while the bootstrap version is

There is more uncertainty, clearly, but the conclusion is not ambiguous here.

Now, what about real data. Consider the following

chicago = read.table("",header=TRUE,sep=";")
   Fire   X_1 X_2    X_3
42  4.8 0.152  19 13.323
43 10.4 0.408  25 12.960
44 15.6 0.578  28 11.260
45  7.0 0.114   3 10.080
46  7.1 0.492  23 11.428
47  4.9 0.466  27 13.731

with one variable of interest (the number of fires, per unhabitants) and 3 features. We can here use bootstrap to generate samples, and then fit a lasso regression. On the original dataset, the regression is

X = model.matrix(lm(Fire~.,data=chicago))
 id = sample(1:nrow(X),size=nrow(X),replace=TRUE)
 vlambda = exp(seq(-4,2,length=201))
 lasso = glmnet(x=X[id,],y=chicago[id,"Fire"],family="gaussian",alpha=1,

And if we just plot lines \lambda\mapsto\widehat{\beta}_\lambda we get

Now, consider bootstrap samples.

for(s in 1:100){

We get here

The interpretation here is much more difficult

What about the order ?

for(s in 1:100000){
        1,function(x) sum(x!=0))))}

The ordering that was obtained on the original dataset was the same in 56% of the scenarios,

mean(apply(N,1,function(x) paste(x,collapse="")=="(Intercept)X_1X_2X_3"))
[1] 0.5693

We can look at all the cases,

g=function(y) mean(apply(N,1,function(x) paste(x,collapse="")==y))