Régression locale | EPFL Graph Search

La régression locale, ou LOESS, est une méthode de régression non paramétrique fortement connexe qui combine plusieurs modèles de régression multiple au sein d'un méta-modèle qui repose sur la méthode des k plus proches voisins. « LOESS » est, en anglais, l'acronyme de « LOcally Estimated Scatterplot Smoothing ». La régression locale est une alternative possible aux méthodes habituelles de régression, comme la régression par les moindres carrés linéaire ou non linéaire, dans les cas où ces dernières s'avèrent mal adaptées. Elle combine la simplicité de régression linéaire par les moindres carrés avec la flexibilité de la régression non linéaire, en effectuant une régression simple sur des sous-ensembles locaux de données. L'un des principaux avantages de cette méthode est qu'elle rend inutile la définition d'une unique fonction globale qui décrirait le modèle de régression, puisque la méthode consiste à calculer autant de fonctions locales qu'il y a de segments de données. Le princip

Covariance Estimation for Random Surfaces beyond Separability

Tomas Masák

This thesis focuses on non-parametric covariance estimation for random surfaces, i.e.~functional data on a two-dimensional domain. Non-parametric covariance estimation lies at the heart of functional data analysis, andconsiderations of statistical and computational efficiency often compel the use of separability of the covariance, when working with random surfaces. We seek to provide efficient alternatives to this ambivalent assumption.In Chapter 2, we study a setting where the covariance structure may fail to be separable locally -- either due to noise contamination or due to the presence of a non-separable short-range dependent signal component. That is, the covariance is an additive perturbation of a separable component by a non-separable but banded component. We introduce non-parametric estimators hinging on shifted partial tracing -- a novel concept enjoying strong denoising properties. We illustrate the usefulness of the proposed methodology on a data set of mortality surfaces.In Chapter 3, we propose a distinctive decomposition of the covariance, which allows us to understand separability as an unconventional form of low-rankness. From this perspective, a separable covariance has rank one. Allowing for a higher rank suggests a structured class in which any covariance can be approximated up to an arbitrary precision. The key notion of the partial inner product allows us to generalize the power iteration method to general Hilbert spaces and estimate the aforementioned decomposition from data. Truncation and retention of the leading terms automatically induces a non-parametric estimator of the covariance, whose parsimony is dictated by the truncation level. Advantages of this approach, allowing for estimation beyond separability, are demonstrated on the task of classification of EEG signals.While Chapters 2 and 3 propose several generalizations of separability in the densely sampled regime, Chapter 4 deals with the sparse regime, where the latent surfaces are observed only at few irregular locations. Here, a separable covariance estimator based on local linear smoothers is proposed, which is the first non-parametric utilization of separability in the sparse regime. The assumption of separability reduces the intrinsically four-dimensional smoothing problem into several two-dimensional smoothers and allows the proposed estimator to retain the classical minimax-optimal convergence rate for two-dimensional smoothers. The proposed methodology is used for a qualitative analysis of implied volatility surfaces corresponding to call options, and for prediction of the latent surfaces based on information from the entire data set, allowing for uncertainty quantification. Our quantitative results show that the proposed methodology outperforms the common approach of pre-smoothing every implied volatility surface separately.Throughout the thesis, we put emphasis on computational aspects, since those are the main reason behind the immense popularity of separability. We show that the covariance structures of Chapters 2 and 3 come with no (asymptotic) computational overhead relative to assuming separability. In fact, the proposed covariance structures can be estimated and manipulated with the same asymptotic costs as the separable model. In particular, we develop numerical algorithms that can be used for efficient inversion, as required e.g.~for prediction. All the methods are implemented in R and available on~GitHub.

EPFL2022

Inference and Computation for Sparsely Sampled Random Surfaces

Tomas Masák, Victor Panaretos, Tomas Rubin

Nonparametric inference for functional data over two-dimensional domains entails additional computational and statistical challenges, compared to the one-dimensional case. Separability of the covariance is commonly assumed to address these issues in the densely observed regime. Instead, we consider the sparse regime, where the latent surfaces are observed only at few irregular locations with additive measurement error, and propose an estimator of covariance based on local linear smoothers. Consequently, the assumption of separability reduces the intrinsically four-dimensional smoothing problem into several two-dimensional smoothers and allows the proposed estimator to retain the classical minimax-optimal convergence rate for two-dimensional smoothers. Even when separability fails to hold, imposing it can be still advantageous as a form of regularization. A simulation study reveals a favorable bias-variance tradeoff and massive speed-ups achieved by our approach. Finally, the proposed methodology is used for qualitative analysis of implied volatility surfaces corresponding to call options, and for prediction of the latent surfaces based on information from the entire dataset, allowing for uncertainty quantification. Our cross-validated out-of-sample quantitative results show that the proposed methodology outperforms the common approach of pre-smoothing every implied volatility surface separately. Supplementary materials for this article are available online.

TAYLOR & FRANCIS INC2022

Sparsely Observed Functional Time Series: Theory and Applications

Tomas Rubin

Functional time series is a temporally ordered sequence of not necessarily independent random curves. While the statistical analysis of such data has been traditionally carried out under the assumption of completely observed functional data, it may well happen that the statistician only has access to a relatively low number of sparse measurements for each random curve. These discrete measurements may be moreover irregularly scattered in each curve's domain, missing altogether for some curves, and be contaminated by measurement noise. This sparse sampling protocol escapes from the reach of established estimators in functional time series analysis and therefore requires development of a novel methodology. The core objective of this thesis is development of a non-parametric statistical toolbox for analysis of sparsely observed functional time series data. Assuming smoothness of the latent curves, we construct a local-polynomial-smoother based estimator of the spectral density operator producing a consistent estimator of the complete second order structure of the data. Moreover, the spectral domain recovery approach allows for prediction of latent curve data at a given time by borrowing strength from the estimated dynamic correlations in the entire time series across time. Further to predicting the latent curves from their noisy point samples, the method fills in gaps in the sequence (curves nowhere sampled), denoises the data, and serves as a basis for forecasting. A classical non-parametric apparatus for encoding the dependence between a pair of or among a multiple functional time series, whether sparsely or fully observed, is the functional lagged regression model. This consists of a linear filter between the regressors time series and the response. We show how to tailor the smoother based estimators for the estimation of the cross-spectral density operators and the cross-covariance operators and, by means of spectral truncation and Tikhonov regularisation techniques, how to estimate the lagged regression filter and predict the response process. The simulation studies revealed the following findings: (i) if one has freedom to design a sampling scheme with a fixed number of measurements, it is advantageous to sparsely distribute these measurements in a longer time horizon rather than concentrating over a shorter time horizon to achieve dense measurements in order to diminish the spectral density estimation error, (ii) the developed functional recovery predictor surpasses the static predictor not exploiting the temporal dependence, (iii) neither of the two considered regularisation techniques can, in general, dominate the other for the estimation in functional lagged regression models. The new methodologies are illustrated by applications to real data: the meteorological data revolving around the fair-weather atmospheric electricity measured in Tashkent, Uzbekistan, and at Wank mountain, Germany; and a case study analysing the dependence of the US Treasury yield curve on macroeconomic variables. As a secondary contribution, we present a novel simulation method for general stationary functional time series defined through their spectral properties. A simulation study shows universality of such approach and superiority of the spectral domain simulation over the temporal domain in some situations.

EPFL2021