non-Gaussian Mixtures Research Articles

Simultaneous dimension reduction and clustering via the NMF-EM algorithm

Mixture models are among the most popular tools for clustering. However, when the dimension and the number of clusters is large, the estimation of the clusters become challenging, as well as their interpretation. Restriction on the parameters can be used to reduce the dimension. An example is given by mixture of factor analyzers for Gaussian mixtures. The extension of MFA to non-Gaussian mixtures is not straightforward. We propose a new constraint for parameters in non-Gaussian mixture model: the $K$ components parameters are combinations of elements from a small dictionary, say $H$ elements, with $H \ll K$. Including a nonnegative matrix factorization (NMF) in the EM algorithm allows us to simultaneously estimate the dictionary and the parameters of the mixture. We propose the acronym NMF-EM for this algorithm, implemented in the R package {\tt nmfem}. This original approach is motivated by passengers clustering from ticketing data: we apply NMF-EM to data from two Transdev public transport networks. In this case, the words are easily interpreted as typical slots in a timetable.

Flexible High-Dimensional Unsupervised Learning with Missing Data.

The mixture of factor analyzers (MFA) model is a famous mixture model-based approach for unsupervised learning with high-dimensional data. It can be useful, inter alia, in situations where the data dimensionality far exceeds the number of observations. In recent years, the MFA model has been extended to non-Gaussian mixtures to account for clusters with heavier tail weight and/or asymmetry. The generalized hyperbolic factor analyzers (MGHFA) model is one such extension, which leads to a flexible modelling paradigm that accounts for both heavier tail weight and cluster asymmetry. In many practical applications, the occurrence of missing values often complicates data analyses. A generalization of the MGHFA is presented to accommodate missing values. Under a missing-at-random mechanism, we develop a computationally efficient alternating expectation conditional maximization algorithm for parameter estimation of the MGHFA model with different patterns of missing values. The imputation of missing values under an incomplete-data structure of MGHFA is also investigated. The performance of our proposed methodology is illustrated through the analysis of simulated and real data.

Conditional channel simulation

In this work we design a specific simulation tool for quantum channels which is based on the use of a control system. This allows us to simulate an average quantum channel which is expressed in terms of an ensemble of channels, even when these channel-components are not jointly teleportation-covariant. This design is also extended to asymptotic simulations, continuous ensembles, and memory channels. As an application, we derive relative-entropy-of-entanglement upper bounds for private communication over various channels, including non-Gaussian mixtures of bosonic lossy channels. Among other results, we also establish the two-way quantum and private capacity of the so-called “dephrasure” channel.

A stochastic downscaling approach for generating high-frequency solar irradiance scenarios

Solar power is increasingly cost viable with solar photovoltaic (PV) installations becoming commonplace. PV planning and operational studies, however, require high-frequency solar irradiance scenarios to understand potential electric grid impacts due to the variability and uncertainty of the underlying solar resource. Existing remote sensing solar data products are often available over large spatial domains, but are limited in temporal resolution. For example, the global horizontal irradiance (GHI) component contained within the National Solar Radiation Database (NSRDB) is available at time resolution of 30 min on an approximately four-kilometer grid. In contrast, substantial solar variability is present at finer time scales and this article describes an algorithm to stochastically generate one-minute GHI from widely available sub-hourly NSRDB. A generalized linear modeling (GLM) framework is proposed, which includes non-Gaussian mixtures, and extends the literature involving the synthesis of GHI data. The model is trained on a set of sample locations around Oregon, USA, and validated across the USA using both the Surface Radiation Budget Network (SURFRAD) dataset and Solar Radiation Monitoring Laboratory (SRML) network. Simulated ensembles show good coverage properties and temporal correlation structure. The resulting downscaled ensembles allow for understanding the unpredictable variability inherent in GHI at locations without direct measurements. Future work can leverage the algorithm as part of a stochastic optimization of electric grid operations with high-penetration PV systems.

Investigating Einstein-Podolsky-Rosen steering of continuous-variable bipartite states by non-Gaussian pseudospin measurements

EPR steering is an asymmetric form of correlations which is intermediate between quantum entanglement and Bell nonlocality, and can be exploited for quantum communication with one untrusted party. In particular, steering of continuous variable Gaussian states has been extensively studied as a manifestation of the EPR paradox. While most of these studies focused on quadrature measurements for steering detection, two recent works revealed that there exist Gaussian states which are only steerable by non-Gaussian measurements. In this paper we perform a systematic investigation of EPR steering of bipartite Gaussian states by pseudospin measurements, complementing and extending previous findings. We first derive the density matrix elements of two-mode squeezed thermal states in the Fock basis, which may be of independent interest. We then use such a representation to investigate steering of these states as detected by a nonlinear criterion, based on second moments of the pseudospin correlation matrix. This analysis reveals previously unexplored regimes where non-Gaussian measurements are more effective than Gaussian ones to witness steering of Gaussian states in the presence of local noise. We further consider an alternative set of pseudospin observables, whose expectation value can be expressed compactly in terms of Wigner functions for all two-mode Gaussian states. However, according to the adopted criterion, these observables are found to be always less sensitive than Gaussian observables for steering detection. Finally, we investigate continuous variable Werner states, which are non-Gaussian mixtures of Gaussian states, and find that pseudospin measurements are always more effective than Gaussian ones to reveal their steerability. Our results provide useful insights on the role of non-Gaussian measurements in characterizing quantum correlations of Gaussian and non-Gaussian states.

Statistical Analysis of a Local Quadratic Criterion for Blind Speech Extraction

This letter aims at complementing previous empirical work regarding a certain beamforming technique for blind speech extraction that uses a local quadratic approximation of a Kurtosis expression. It is shown here that the proposed method possesses a fixed-point property which means that it remains at an optimal solution once this solution has been reached. The proposed method's fixed-point property is valid for a range of source signals including Gaussian sources. This is an improvement over the FastICA method which diverges at the optimal points that correspond to a Gaussian source. In a real application, it cannot be assured that non-Gaussian mixtures are constantly observed; hence, the proposed method is a viable alternative in that case. The fixed-point property further implies that the approximative Kurtosis expression is identical to the true Kurtosis value at an optimal point which, in turn, means that the approximation error is zero. In addition, the convergence towards an optimal solution is always in the direction of a local minimum point even though the optimal solution that correspond to a super-Gaussian source is always a maximum solution which harmonizes with the concept of Kurtosis maximization.

Efficient identification of non-Gaussian mixtures

Non-Gaussian mixtures are commonly used as parametric models for impulsive interference superimposed on a Gaussian background. In this study, the problem of efficient batch and recursive estimation of the parameters of such mixtures via minimization of the Kullback-Leibler distance is considered. The maximum-likelihood estimator (MLE) provides the starting point for the development of these estimators. First, it is shown that the MLE yields consistent estimates of the parameters, despite the existence of multiple roots in the Kullback-Leibler distance function. Since direct implementation of the MLE is difficult, an alternative estimator, designed with the objective of maximizing the likelihood function, is proposed. A simulation study of the proposed estimator reveals that it performs very well in terms of attaining the Cramer-Rao lower bound. Stochastic approximation is also considered in this study. A modification to the standard recursion is presented, which greatly facilitates its implementation. Three initiating estimators are developed for these recursions: an iterative likelihood-based scheme, a batch estimator that uses a histogram of the data, and an iterative scheme that integrates the concept of complete data into the method-of-moments estimator. Upon initiating the modified stochastic approximation recursion with these estimators, it is seen that excellent estimates of the parameters can be obtained.

Zero-crossing rates of mixtures and products of Gaussian processes

Formulas for the expected zero-crossing rate of non-Gaussian mixtures and products of Gaussian processes are obtained. The approach we take is to first derive the expected zero-crossing rate in discrete time and then obtain the rate in continuous time by an appropriate limiting argument. The processes considered, which are non-Gaussian but derived from Gaussian processes, serve to illustrate the variability of the zero-crossing rate in terms of the normalized autocorrelation function p(t) of the process. For Gaussian processes, Rice's formula gives the expected zero-crossing rate in continuous time as 1//spl pi//spl radic/(-/spl rho/"(0)). We show processes exist with expected zero-crossing rates given by /spl kappa///spl pi//spl radic/(-/spl rho/"(0)) with either /spl kappa//spl Gt/1 or /spl kappa//spl Lt/1. Consequently, such processes can have an arbitrarily large or small zero-crossing rate as compared to a Gaussian process with the same autocorrelation function.