Régression polynomiale

Polyreg scheffe.svg thumb|Régression sur un nuage de points par un polynôme de degré croissant. La régression polynomiale est une analyse statistique qui décrit la variation d'une variable aléatoire expliquée à partir d'une fonction polynomiale d'une variable aléatoire explicative. C'est un cas particulier de régression linéaire multiple, où les observations sont construites à partir des puissances d'une seule variable. Présentation Si l'on appelle (Xi, Yi) la i-ème réalisation du couple de variables aléatoires, on recherche le polynôme : P_n(x) = a_n x^n + a_{n-1}x^{n-1} + \ldots + a_1 x + a_0, permettant d'écrire : Y_i = P_n(X_i) + \varepsilon_i le résidu εi, ou perturbation, étant « le plus petit » dans le sens des moindres carrés. La régression polynomiale est une régression linéaire multiple : on peut écrire la relation, pour X = ''X : : Y_i = a_n \cdot X_{i,n} + a_{n-1} \cdot X_{i,n-1} + \ldots + a_1\cdot

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Robust inference for generalized linear models

Sahar Hosseinian Ehrensberger

Generalized Linear Models have become a commonly used tool of data analysis. Such models are used to fit regressions for univariate responses with normal, gamma, binomial or Poisson distribution. Maximum likelihood is generally applied as fitting method. In the usual regression setting the least absolute-deviations estimator (L1-norm) is a popular alternative to least squares (L2-norm) because of its simplicity and its robustness properties. In the first part of this thesis we examine the question of how much of these robustness features carry over to the setting of generalized linear models. We study a robust procedure based on the minimum absolute deviation estimator of Morgenthaler (1992), the Lq quasi-likelihood when q = 1. In particular, we investigate the influence function of these estimates and we compare their sensitivity to that of the maximum likelihood estimate. Furthermore we particularly explore the Lq quasi-likelihood estimates in binary regression. These estimates are difficult to compute. We derive a simpler estimator, which has a similar form as the Lq quasi-likelihood estimate. The resulting estimating equation consists in a simple modification of the familiar maximum likelihood equation with the weights wq(μ). This presents an improvement compared to other robust estimates discussed in the literature that typically have weights, which depend on the couple (xi, yi) rather than on μi = h(xiT β) alone. Finally, we generalize this estimator to Poisson regression. The resulting estimating equation is a weighted maximum likelihood with weights that depend on μ only.

EPFL2009

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Using Polynomials to Forecast Univariate Time Series

Polynomial autoregressions have been most of the time discarded as being unrealistic. Indeed, for such processes to be stationary, strong assumptions on the parameters and on the noise are necessary. For example, the distribution of the latter has to have finite support. Nevertheless, the use of polynomials has been advocated by a few authors: Cox (1991), Cobb and Zacks (1988) and Chan and Tong (1994). From a model-free perspective, that is using parametric families of functions (e.g. Fourier series, wavelets, neural networks, MARS, ...) as approximators of the optimal predictor, there is no more concern about stationarity. Still, polynomials have not been used within this framework; the reason is the now well known "curse of dimensionality". Indeed, when high lag orders are used to forecast, a common situation in time series, polynomial autoregressions imply too many parameters to estimate. We will introduce a new family of predictors based on polynomials and on a projection scheme. A censoring procedure allows to solve the instability inherent to polynomials. The forecasting method is parsimonious (that is non-linearity is introduced with a small amount of parameters) and thus allows to forecast noisy time series of short to moderate length.

1996

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Tractability of interpretability via selection of group-sparse models

Luca Baldassarre, Nirav Bhan, Volkan Cevher

Group-based sparsity models are proven instrumental in linear regression problems for recovering signals from much fewer measurements than standard compressive sensing. A promise of these models is to lead to “interpretable” signals for which we identify its constituent groups, however we show that, in general, claims of correctly identifying the groups with convex relaxations would lead to polynomial time solution algorithms for an NP-hard problem. Instead, leveraging a graph-based understanding of group models, we describe group structures which enable correct model identification in polynomial time via dynamic programming. We also show that group structures that lead to totally unimodular constraints have tractable relaxations. Finally, we highlight the non-convexity of the Pareto frontier of group-sparse approximations and what it means for tractability.

2013

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