Interaction patterns of brain activity across space, time and frequency. Part I: methods

We consider exploratory methods for the discovery of cortical functional connectivity. Typically, data for the i-th subject (i=1...NS) is represented as an NVxNT matrix Xi, corresponding to brain activity sampled at NT moments in time from NV cortical voxels. A widely used method of analysis first concatenates all subjects along the temporal dimension, and then performs an independent component analysis (ICA) for estimating the common cortical patterns of functional connectivity. There exist many other interesting variations of this technique, as reviewed in [Calhoun et al. 2009 Neuroimage 45: S163-172]. We present methods for the more general problem of discovering functional connectivity occurring at all possible time lags. For this purpose, brain activity is viewed as a function of space and time, which allows the use of the relatively new techniques of functional data analysis [Ramsay & Silverman 2005: Functional data analysis. New York: Springer]. In essence, our method first vectorizes the data from each subject, which constitutes the natural discrete representation of a function of several variables, followed by concatenation of all subjects. The singular value decomposition (SVD), as well as the ICA of this new matrix of dimension [rows=(NT*NV); columns=NS] will reveal spatio-temporal patterns of connectivity. As a further example, in the case of EEG neuroimaging, Xi of size NVxNW may represent spectral density for electric neuronal activity at NW discrete frequencies from NV cortical voxels, from the i-th EEG epoch. In this case our functional data analysis approach would reveal coupling of brain regions at possibly different frequencies.Comment: Technical report 2011-March-15, The KEY Institute for Brain-Mind Research Zurich, KMU Osak

Group Lasso estimation of high-dimensional covariance matrices

In this paper, we consider the Group Lasso estimator of the covariance matrix of a stochastic process corrupted by an additive noise. We propose to estimate the covariance matrix in a high-dimensional setting under the assumption that the process has a sparse representation in a large dictionary of basis functions. Using a matrix regression model, we propose a new methodology for high-dimensional covariance matrix estimation based on empirical contrast regularization by a group Lasso penalty. Using such a penalty, the method selects a sparse set of basis functions in the dictionary used to approximate the process, leading to an approximation of the covariance matrix into a low dimensional space. Consistency of the estimator is studied in Frobenius and operator norms and an application to sparse PCA is proposed

Nonparametric estimation of covariance functions by model selection

We propose a model selection approach for covariance estimation of a multi-dimensional stochastic process. Under very general assumptions, observing i.i.d replications of the process at fixed observation points, we construct an estimator of the covariance function by expanding the process onto a collection of basis functions. We study the non asymptotic property of this estimate and give a tractable way of selecting the best estimator among a possible set of candidates. The optimality of the procedure is proved via an oracle inequality which warrants that the best model is selected

Adaptive estimation of spectral densities via wavelet thresholding and information projection

In this paper, we study the problem of adaptive estimation of the spectral density of a stationary Gaussian process. For this purpose, we consider a wavelet-based method which combines the ideas of wavelet approximation and estimation by information projection in order to warrants that the solution is a nonnegative function. The spectral density of the process is estimated by projecting the wavelet thresholding expansion of the periodogram onto a family of exponential functions. This ensures that the spectral density estimator is a strictly positive function. Then, by Bochner's theorem, the corresponding estimator of the covariance function is semidefinite positive. The theoretical behavior of the estimator is established in terms of rate of convergence of the Kullback-Leibler discrepancy over Besov classes. We also show the excellent practical performance of the estimator in some numerical experiments

A non parametric approach for calibration with functional data

International audienceA new nonparametric approach for statistical calibration with functional data is studied. The practical motivation comes from calibration problems in chemometrics in which a scalar random variable Y needs to be predicted from a functional random variable X. The proposed predictor takes the form of a weighted average of the observed values of Y in the training data set, where the weights are determined by the conditional probability density of X given Y. This functional density, which represents the data generation mechanism in the context of calibration , is so incorporated as a key information into the estimator. The new proposal is computationally simple and easy to implement. Its statistical consistency is proved, and its relevance is shown through simulations and an application to data

A Simulation Study of Functional Density-Based Inverse Regression

National audienceIn this paper a new nonparametric functional method is introduced for predicting a scalar random variable $Y$ on the basis of a functional random variable $X$. The prediction has the form of a weighted average of the training data $y_{i}$, where the weights are determined by the conditional probability density of $X$ given $Y=y_{i}$, which is assumed to be Gaussian. In this way such a conditional probability density is incorporated as a key information into the estimator. Contrary to some previous approaches, no assumption about the dimensionality of $E(X|Y=y)$ or about the distribution of $X$ is required. The new proposal is computationally simple and easy to implement. Its performance is assessed through a simulation study

A functional density-based nonparametric approach for statistical calibration

International audienceIn this paper a new nonparametric functional method is introduced for predicting a scalar random variable $Y$ from a functional random variable $X$. The resulting prediction has the form of a weighted average of the training data set, where the weights are determined by the conditional probability density of $X$ given $Y$, which is assumed to be Gaussian. In this way such a conditional probability density is incorporated as a key information into the estimator. Contrary to some previous approaches, no assumption about the dimensionality of $\mathbb{E}(X|Y=y)$ is required. The new proposal is computationally simple and easy to implement. Its performance is shown through its application to both simulated and real data

Innovations orthogonalization: a solution to the major pitfalls of EEG/MEG "leakage correction"

The problem of interest here is the study of brain functional and effective connectivity based on non-invasive EEG-MEG inverse solution time series. These signals generally have low spatial resolution, such that an estimated signal at any one site is an instantaneous linear mixture of the true, actual, unobserved signals across all cortical sites. False connectivity can result from analysis of these low-resolution signals. Recent efforts toward "unmixing" have been developed, under the name of "leakage correction". One recent noteworthy approach is that by Colclough et al (2015 NeuroImage, 117:439-448), which forces the inverse solution signals to have zero cross-correlation at lag zero. One goal is to show that Colclough's method produces false human connectomes under very broad conditions. The second major goal is to develop a new solution, that appropriately "unmixes" the inverse solution signals, based on innovations orthogonalization. The new method first fits a multivariate autoregression to the inverse solution signals, giving the mixed innovations. Second, the mixed innovations are orthogonalized. Third, the mixed and orthogonalized innovations allow the estimation of the "unmixing" matrix, which is then finally used to "unmix" the inverse solution signals. It is shown that under very broad conditions, the new method produces proper human connectomes, even when the signals are not generated by an autoregressive model.Comment: preprint, technical report, under license "Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)", https://creativecommons.org/licenses/by-nc-nd/4.0

Modélisation de séries financières, estimation, ajustement de modèles et test d'hypothèses

De nombreux modèles ont été proposés pour expliquer la dynamique complexe des marchés financiers. La plupart des problèmes en finance comme la valorisation d'options, la couverture d'options, l'optimisation de porte-feuille, etc. ont été étudiés dans le cadre de ces nouveaux modèles. Mais pour les applications il est nécessaire de disposer de méthodes précises pour ajuster ces modèles aux données réelles. Dans cette thèse nous étudions plusieurs problèmes inférentiels relatifs aux modèles communément utilisés en finance. Le premier chapitre de cette thèse est une révision commentée de la majorité des méthodes existantes pour estimer les paramètres des lois stables. Entre autres, la méthode appelée de L-moments est de notre crû. Les propriétés de ces méthodes ont été testés sur des données simulées. Finalement nous présentons une application à l'estimation de la VaR (Value at Risk) sur des données financières réelles. Dans le deuxième chapitre nous étudions le problème de l'estimation d'un modèle de diffusion avec sauts observé à temps discret. Nous proposons deux méthodes différentes pour résoudre ce problème inférentiel, et nous prouvons que après une détection préalable des sauts, nous pouvons estimer les paramètres d'une diffusion avec sauts en utilisant des méthodes similaires a celles utilisées dans le cas des diffusions ordinaires. Un étude de simulation vient confirmer les résultats théoriques obtenus. Dans le troisième chapitre nous considérons le problème de la détection des sauts de la volatilité dans un modèle à volatilité stochastique. Nous proposons des estimateurs pour le nombre des sauts de la volatilité, des instants de saut et de la volatilité entre les sauts. Nous démontrons enfin un résultat asymptotique sur ces estimateurs lorsque le pas de discrétisation décroît vers 0.TOULOUSE3-BU Sciences (315552104) / SudocSudocFranceF