Order theory

Résumé

Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual differenc

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Covariance operators play a fundamental role in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on the space of such operators, as well as with their intrinsic infinite-dimensionality. We will show that we can identify the space of infinite-dimensional covariance operators equipped with the Procrustes size-and-shape metric from shape theory, with that of centred Gaussian processes, equipped with the Wasserstein metric of optimal transportation. We then describe key geometrical and topological aspects of the space of covariance operators endowed with the Procrustes metric. Through the notion of multicoupling of Gaussian measures, we establish existence, uniqueness and stability for the Fréchet mean of covariance operators with respect to the Procrustes metric. Furthermore, we will provide generative models that are canonical for such metric. We then turn to the problem of comparing several samples of stochastic processes with respect to their second-order structure, and we subsequently describe the main modes of variation in this second order structure. These two tasks are carried out via an Analysis of Variance (ANOVA) and a Principal Component Analysis (PCA) of covariance operators respectively. In order to perform ANOVA, we introduce a novel approach based on optimal (multi)transport and identify each covariance with an optimal transport map. These maps are then contrasted with the identity with respect to a norm-induced distance. The resulting test statistic, calibrated by permutation, outperforms the state-of-the-art in the functional case. If the null hypothesis postulating equality of the operators is rejected, thanks to a geometric interpretation of the transport maps we can construct a PCA on the tangent space with the aim of understanding the sample variability. Finally, we provide a further example of use of the optimal transport framework, by applying it to the problem of clustering of operators. Two different clustering algorithms are presented, one of which is innovative. The transportation ANOVA, PCA and clustering are validated both on simulated scenarios and real dataset.

EPFL2019

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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

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Expressing Motivations By Facilitating Other’s Inverse Reinforcement Learning

Wafa Monia Benkaouar Johal, Pierre Dillenbourg, Alexis David Jacq, Ana Paiva

It is often necessary to understand each other’s motivations in order to cooperate. Reaching such a mutual understanding requires two abilities: to build models of other’s motivations in order to understand them, and to build a model of “my” motivations perceived by others in order to be understood. Having a self-image seen by others requires two recursive orders of modeling, known in psychology as the first and second orders of theory of mind. In this paper, we present a second-order theory of mind cognitive architecture that aims to facilitate mutual understanding in multi-agent scenarios. We study different conditions of empathy and gratitude leading to irrational cooperation in iterated prisoner’s dilemma.

Instituto Superiro Technico2017

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EPFL2019

Sparsely Observed Functional Time Series: Theory and Applications

Tomas Rubin

EPFL2021