Statistical approaches for Functional Data Analysis concern the paradigm for which the individuals are functions or curves rather than finite dimensional vectors. In this paper, we particularly focus on the modeling and the classification of functional data which are temporal curves presenting regime changes over time. More specifically, we propose a new mixture model-based discriminant analysis approach for functional data using a specific hidden process regression model. Our approach is particularly adapted to both handle the problem of complex-shaped classes of curves, where each class is composed of several sub-classes, and to deal with the regime changes within each homogeneous sub-class. The model explicitly integrates the heterogeneity of each class of curves via a mixture model formulation, and the regime changes within each sub-class through a hidden logistic process. The approach allows therefore for fitting flexible curve-models to each class of complex-shaped curves presenting regime changes through an unsupervised learning scheme, to automatically summarize it into a finite number of homogeneous clusters, each of them is decomposed into several regimes. The model parameters are learned by maximizing the observed-data log-likelihood for each class by using a dedicated expectation-maximization (EM) algorithm. Comparisons on simulated data and real data with alternative approaches, including functional linear discriminant analysis and functional mixture discriminant analysis with polynomial regression mixtures and spline regression mixtures, show that the proposed approach provides better results regarding the discrimination results and significantly improves the curves approximation.

A time series consists of a series of values or events obtained over repeated measurements in time. Analysis of time series represents and important tool in many application areas, such as stock market analysis, process and quality control, observation of natural phenomena, medical treatments, etc. A vital component in many types of time-series analysis is the choice of an appropriate distance/similarity measure. Numerous measures have been proposed to date, with the most successful ones based on dynamic programming. Being of quadratic time complexity, however, global constraints are often employed to limit the search space in the matrix during the dynamic programming procedure, in order to speed up computation. Furthermore, it has been reported that such constrained measures can also achieve better accuracy. In this paper, we investigate two representative time-series distance/similarity measures based on dynamic programming, Dynamic Time Warping (DTW) and Longest Common Subsequence (LCS), and the effects of global constraints on them. Through extensive experiments on a large number of time-series data sets, we demonstrate how global constrains can significantly reduce the computation time of DTW and LCS. We also show that, if the constraint parameter is tight enough (less than 10-15% of time-series length), the constrained measure becomes significantly different from its unconstrained counterpart, in the sense of producing qualitatively different 1-nearest neighbor graphs. This observation explains the potential for accuracy gains when using constrained measures, highlighting the need for careful tuning of constraint parameters in order to achieve a good trade-off between speed and accuracy.

In this paper, we propose an outlier-robust regularized kernel-based method for linear system identification. The unknown impulse response is modeled as a zero-mean Gaussian process whose covariance (kernel) is given by the recently proposed stable spline kernel, which encodes information on regularity and exponential stability. To build robustness to outliers, we model the measurement noise as realizations of independent Laplacian random variables. The identification problem is cast in a Bayesian framework, and solved by a new Markov Chain Monte Carlo (MCMC) scheme. In particular, exploiting the representation of the Laplacian random variables as scale mixtures of Gaussians, we design a Gibbs sampler which quickly converges to the target distribution. Numerical simulations show a substantial improvement in the accuracy of the estimates over state-of-the-art kernel-based methods.

We prove a central limit theorem for the components of the largest eigenvectors of the adjacency matrix of a finite-dimensional random dot product graph whose true latent positions are unknown. In particular, we follow the methodology outlined in \citet{sussman2012universally} to construct consistent estimates for the latent positions, and we show that the appropriately scaled differences between the estimated and true latent positions converge to a mixture of Gaussian random variables. As a corollary, we obtain a central limit theorem for the first eigenvector of the adjacency matrix of an Erd\"os-Renyi random graph.

This paper presents a new method for estimating high dimensional covariance matrices. The method, permuted rank-penalized least-squares (PRLS), is based on a Kronecker product series expansion of the true covariance matrix. Assuming an i.i.d. Gaussian random sample, we establish high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity. For covariance matrices of low separation rank, our results establish that PRLS has significantly faster convergence than the standard sample covariance matrix (SCM) estimator. The convergence rate captures a fundamental tradeoff between estimation error and approximation error, thus providing a scalable covariance estimation framework in terms of separation rank, similar to low rank approximation of covariance matrices. The MSE convergence rates generalize the high dimensional rates recently obtained for the ML Flip-flop algorithm for Kronecker product covariance estimation. We show that a class of block Toeplitz covariance matrices is approximatable by low separation rank and give bounds on the minimal separation rank $r$ that ensures a given level of bias. Simulations are presented to validate the theoretical bounds. As a real world application, we illustrate the utility of the proposed Kronecker covariance estimator for spatio-temporal linear least squares prediction of multivariate wind speed measurements.

We propose a probabilistic graphical model realizing a minimal encoding of real variables dependencies based on possibly incomplete observation and an empirical cumulative distribution function per variable. The target application is a large scale partially observed system, like e.g. a traffic network, where a small proportion of real valued variables are observed, and the other variables have to be predicted. Our design objective is therefore to have good scalability in a real-time setting. Instead of attempting to encode the dependencies of the system directly in the description space, we propose a way to encode them in a latent space of binary variables, reflecting a rough perception of the observable (congested/non-congested for a traffic road). The method relies in part on message passing algorithms, i.e. belief propagation, but the core of the work concerns the definition of meaningful latent variables associated to the variables of interest and their pairwise dependencies. Numerical experiments demonstrate the applicability of the method in practice.

In a context of document co-clustering, we define a new similarity measure which iteratively computes similarity while combining fuzzy sets in a three-partite graph. The fuzzy triadic similarity (FT-Sim) model can deal with uncertainty offers by the fuzzy sets. Moreover, with the development of the Web and the high availability of storage spaces, more and more documents become accessible. Documents can be provided from multiple sites and make similarity computation an expensive processing. This problem motivated us to use parallel computing. In this paper, we introduce parallel architectures which are able to treat large and multi-source data sets by a sequential, a merging or a splitting-based process. Then, we proceed to a local and a central (or global) computing using the basic FT-Sim measure. The idea behind these architectures is to reduce both time and space complexities thanks to parallel computation.

Modeling structure in complex networks using Bayesian non-parametrics makes it possible to specify flexible model structures and infer the adequate model complexity from the observed data. This paper provides a gentle introduction to non-parametric Bayesian modeling of complex networks: Using an infinite mixture model as running example we go through the steps of deriving the model as an infinite limit of a finite parametric model, inferring the model parameters by Markov chain Monte Carlo, and checking the model's fit and predictive performance. We explain how advanced non-parametric models for complex networks can be derived and point out relevant literature.

Principal component analysis (PCA) is a statistical technique commonly used in multivariate data analysis. However, PCA can be difficult to interpret and explain since the principal components (PCs) are linear combinations of the original variables. Sparse PCA (SPCA) aims to balance statistical fidelity and interpretability by approximating sparse PCs whose projections capture the maximal variance of original data. In this paper we present an efficient and paralleled method of SPCA using graphics processing units (GPUs), which can process large blocks of data in parallel. Specifically, we construct parallel implementations of the four optimization formulations of the generalized power method of SPCA (GP-SPCA), one of the most efficient and effective SPCA approaches, on a GPU. The parallel GPU implementation of GP-SPCA (using CUBLAS) is up to eleven times faster than the corresponding CPU implementation (using CBLAS), and up to 107 times faster than a MatLab implementation. Extensive comparative experiments in several real-world datasets confirm that SPCA offers a practical advantage.

In this work we address the problem of approximating high-dimensional data with a low-dimensional representation. We make the following contributions. We propose an inverse regression method which exchanges the roles of input and response, such that the low-dimensional variable becomes the regressor, and which is tractable. We introduce a mixture of locally-linear probabilistic mapping model that starts with estimating the parameters of inverse regression, and follows with inferring closed-form solutions for the forward parameters of the high-dimensional regression problem of interest. Moreover, we introduce a partially-latent paradigm, such that the vector-valued response variable is composed of both observed and latent entries, thus being able to deal with data contaminated by experimental artifacts that cannot be explained with noise models. The proposed probabilistic formulation could be viewed as a latent-variable augmentation of regression. We devise expectation-maximization (EM) procedures based on a data augmentation strategy which facilitates the maximum-likelihood search over the model parameters. We propose two augmentation schemes and we describe in detail the associated EM inference procedures that may well be viewed as generalizations of a number of EM regression, dimension reduction, and factor analysis algorithms. The proposed framework is validated with both synthetic and real data. We provide experimental evidence that our method outperforms several existing regression techniques.