Bayesian quadrature, energy minimization, and space-filling design

A standard objective in computer experiments is to approximate the behavior of an unknown function on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, covering radius) or a discrepancy criterion measuring distance to uniformity. The paper investigates connections between design for integration (quadrature design), construction of the (continuous) best linear unbiased estimator (BLUE) for the location model, space-filling design, and minimization of energy (kernel discrepancy) for signed measures. Integrally strictly positive definite kernels define strictly convex energy functionals, with an equivalence between the notions of potential and directional derivative, showing the strong relation between discrepancy minimization and more traditional design of optimal experiments. In particular, kernel herding algorithms, which are special instances of vertex-direction methods used in optimal design, can be applied to the construction of point sequences with suitable space-filling properties

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstracts

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstract

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstracts

Seventh International Workshop on Simulation, 21-25 May, 2013, Department of Statistical Sciences, Unit of Rimini, University of Bologna, Italy. Book of Abstract

Optimal quadrature-sparsification for integral operator approximation

The design of sparse quadratures for the approximation of integral operators related to symmetric positive-semidefinite kernels is addressed. Particular emphasis is placed on the approximation of the main eigenpairs of an initial operator and on the assessment of the approximation accuracy. Special attention is drawn to the design of sparse quadratures with support included in fixed finite sets of points (that is, quadrature-sparsification), this framework encompassing the approximation of kernel matrices. For a given kernel, the accuracy of a quadrature approximation is assessed through the squared Hilbert--Schmidt norm (for operators acting on the underlying reproducing kernel Hilbert space) of the difference between the integral operators related to the initial and approximate measures; by analogy with the notion of kernel discrepancy, the underlying criterion is referred to as the squared-kernel discrepancy between the two measures. In the quadrature-sparsification framework, sparsity of the approximate quadrature is promoted through the introduction of an $\ell^{1}$-type penalization, and the computation of a penalized squared-kernel-discrepancy-optimal approximation then consists in a convex quadratic minimization problem; such quadratic programs can in particular be interpreted as the Lagrange dual formulations of distorted one-class support-vector machines related to the squared kernel. Error bounds on the induced spectral approximations are derived, and the connection between penalization, sparsity, and accuracy of the spectral approximation is investigated. Numerical strategies for solving large-scale penalized squared-kernel-discrepancy minimization problems are discussed, and the efficiency of the approach is illustrated by a series of examples. In particular, the ability of the proposed methodology to lead to accurate approximations of the main eigenpairs of kernel matrices related to large-scale datasets is demonstrated

Generalized Partial Least Squares Approach for Nominal Multinomial Logit Regression Models with a Functional Covariate

Functional Data Analysis (FDA) has attracted substantial attention for the last two decades. Within FDA, classifying curves into two or more categories is consistently of interest to scientists, but multi-class prediction within FDA is challenged in that most classification tools have been limited to binary response applications. The functional logistic regression (FLR) model was developed to forecast a binary response variable in the functional case. In this study, a functional nominal multinomial logit regression (F-NM-LR) model was developed that shifts the FLR model into a multiple logit model. However, the model generates inaccurate parameter function estimates due to multicollinearity in the design matrix. A generalized partial least squares (GPLS) approach with cubic B-spline basis expansions was developed to address the multicollinearity and high dimensionality problems that preclude accurate estimates and curve discrimination with the F-NM-LR model. The GPLS method extends partial least squares (PLS) and improves upon current methodology by introducing a component selection criterion that reconstructs the parameter function with fewer predictors. The GPLS regression estimates are derived via Iteratively ReWeighted Partial Least Squares (IRWPLS), defining a set of uncorrelated latent variables to use as predictors for the F-GPLS-NM-LR model. This methodology was compared to the classic alternative estimation method of principal component regression (PCR) in a simulation study. The performance of the proposed methodology was tested via simulations and applications on a spectrometric dataset. The results indicate that the GPLS method performs well in multi-class prediction with respect to the F-NM-LR model. The main difference between the two approaches was that PCR usually requires more components than GPLS to achieve similar accuracy of parameter function estimates of the F-GPLS-NM-LR model. The results of this research imply that the GPLS method is preferable to the F-NM-LR model, and it is a useful contribution to FDA techniques. This method may be particularly appropriate for practical situations where accurate prediction of a response variable with fewer components is a priority

Essays in econometrics and energy markets

Cette thèse est organisée en trois chapitres où sont développées des méthodes d’analyse économique et économétrique des marchés de l’énergie. Le Chapitre 1 propose une étude des incitations à la manipulation de marché générées par des opportunités de revenir sur ses engagements. Un modèle théorique est développé pour analyser le comportement d’un monopole face à une frange compétitive en présence d’une demande incertaine, de contraintes de capacité, et de possibilités de trahir ses engagements. Les entreprises se concurrencent avec des fonctions d’offre étant donnés leurs engagements. Le monopole revient sur ses engagements lorsqu’il retire sa production engagée en observant la réalisation de l’incertitude. Il peut ainsi exacerber son pouvoir de marché, réduire l’incertitude autour de la demande, et accroitre sa probabilité de devenir un offreur pivot. À l’équilibre, les stratégies d’offre dépendent du volume de production engagée et du coût d’opportunité de la retirer stratégiquement. En particulier, le monopole peut trouver profitable d’offrir sa production à des prix plus élevés lorsqu’il sait qu’il pourra revenir sur ses engagements si la demande est élevée. Finalement, cette stratégie est présentée comme un comportement de manipulation par perte, et des applications aux marchés de l’électricité sont discutées. Dans le Chapitre 2, nous développons de nouveaux résultats pour les régressions fonctionnelles où le prédicteur Z(t) et la réponse Y (t) sont des fonctions d’espaces de Hilbert, indexés par le temps ou l’espace. Le modèle peut être compris comme une généralisation de la régression multivariée où le coefficient de régression est maintenant un opérateur inconnu Π. Nous proposons d’estimer l’opérateur Π par régularisation de Tikhonov, ce qui revient à appliquer une pénalité sur sa norme L2. Nous dérivons le taux de convergence de l’erreur quadratique moyenne, la distribution asymptotique de l’estimateur, et développons des tests sur Π. Comme les trajectoires ne sont généralement pas complètement observables, nous considérons une situation où les données deviennent de plus en plus fréquentes (asymptotique de remplissage). Nous traitons aussi le cas où Z est endogène et des variables instrumentales sont utilisées afin d’estimer Π. Une application à la consommation d’électricité complète l’article. Le Chapitre 3 propose une nouvelle approche pour l’analyse empirique des enchères à unités multiples, dans lesquelles les participants choisissent des fonctions d’offre ou de demande. Cette approche permet d’évaluer le pouvoir de marché des entreprises dans une cadre d’information privée, en évitant d’avoir à modéliser le mécanisme du marché. Elle repose sur des méthodes économétriques qui traitent les fonctions de mise comme des éléments aléatoires à valeurs fonctionnelles. Notamment, un estimateur fonctionnel à variable instrumentale est développé. La méthode est appliquée au marché de l’électricité de l’état de New York sur des données micro-économiques de mises et de coûts à l’échelle des entreprises pour 2013-2015. J’estime le pouvoir de marché unilatéral des entreprises et compare les comportements observés aux comportements maximisant les profits sous information privée. Je trouve un faisceau d’indices sérieux de comportement optimal, qui suggère que les entreprises sont au courant de leur pouvoir de marché et se comportent en conséquence.This thesis is organized in three chapters which develop economic and econometric methods for the analysis of energy markets. In Chapter 1, we study the incentives for market manipulations created by opportunities to renege on prior commitments. We develop a theoretical framework to analyze the behavior of a monopolist facing a competitive fringe in the presence of demand uncertainty, capacity constraints and reneging opportunities. The firms are assumed to compete in supply functions taking their commitments as sunk decisions. Reneging occurs when the monopolist withdraws its committed output upon observing the realization of demand. By doing so, it can exacerbate its market power, alleviate demand uncertainty, and be more likely to be pivotal. At equilibrium, supply strategies depend on the volume of committed output and the opportunity cost of reneging. In particular, the monopolist may find profitable to offer some of its market output at higher prices in the presence of reneging opportunities. Finally, we present strategic reneging as a loss-based manipulative conduct in a general framework and discuss applications to electricity markets. In Chapter 2, we develop new estimation results for functional regressions where both the regressor Z(t) and the response Y (t) are functions of Hilbert spaces, indexed by the time or a spatial location. The model can be thought as a generalization of the multivariate regression where the regression coefficient is now an unknown operator Π. We propose to estimate the operator Π by Tikhonov regularization, which amounts to apply a penalty on the L2 norm of Π. We derive the rate of convergence of the mean-square error, the asymptotic distribution of the estimator, and develop tests on Π. As trajectories are often not fully observed, we consider the scenario where the data become more and more frequent (infill asymptotics). We also address the case where Z is endogenous and instrumental variables are used to estimate Π. An application to the electricity consumption completes the paper. Chapter 3 proposes a novel approach for the empirical analysis of multiunit auctions, to which participants submit supply or demand functions observable by the researcher. The approach allows for the evaluation of firmlevel market power in a private information setting, and avoids having to model the market mechanism. It relies on econometric methods that treat the observed bid functions as function-valued random elements. Notably, a functional instrumental variable estimator is developed. The method is applied to the New York electricity market using rich data on firm-level bids and marginal costs for 2013-2015. In this market, daily bids are disclosed three months later in order to limit strategic behaviors. I estimate firm-level market power and compare actual bidding behavior to profit-maximizing behavior under private information. I find consistent evidence of optimal bidding, suggesting that firms are well aware of their own market power and behave accordingly. Therefore, the late disclosure of bids is not sufficient to preclude firms from acting strategically, most likely due to the repeated nature of those auctions