The wavelet-NARMAX representation : a hybrid model structure combining polynomial models with multiresolution wavelet decompositions

A new hybrid model structure combing polynomial models with multiresolution wavelet decompositions is introduced for nonlinear system identification. Polynomial models play an important role in approximation theory, and have been extensively used in linear and nonlinear system identification. Wavelet decompositions, in which the basis functions have the property of localization in both time and frequency, outperform many other approximation schemes and offer a flexible solution for approximating arbitrary functions. Although wavelet representations can approximate even severe nonlinearities in a given signal very well, the advantage of these representations can be lost when wavelets are used to capture linear or low-order nonlinear behaviour in a signal. In order to sufficiently utilise the global property of polynomials and the local property of wavelet representations simultaneously, in this study polynomial models and wavelet decompositions are combined together in a parallel structure to represent nonlinear input-output systems. As a special form of the NARMAX model, this hybrid model structure will be referred to as the WAvelet-NARMAX model, or simply WANARMAX. Generally, such a WANARMAX representation for an input-output system might involve a large number of basis functions and therefore a great number of model terms. Experience reveals that only a small number of these model terms are significant to the system output. A new fast orthogonal least squares algorithm, called the matching pursuit orthogonal least squares (MPOLS) algorithm, is also introduced in this study to determine which terms should be included in the final model

Analysis, Visualization, and Transformation of Audio Signals Using Dictionary-based Methods

date-added: 2014-01-07 09:15:58 +0000 date-modified: 2014-01-07 09:15:58 +0000date-added: 2014-01-07 09:15:58 +0000 date-modified: 2014-01-07 09:15:58 +000

Detecting and quantifying stellar magnetic fields -- Sparse Stokes profile approximation using orthogonal matching pursuit

In the recent years, we have seen a rapidly growing number of stellar magnetic field detections for various types of stars. Many of these magnetic fields are estimated from spectropolarimetric observations (Stokes V) by using the so-called center-of-gravity (COG) method. Unfortunately, the accuracy of this method rapidly deteriorates with increasing noise and thus calls for a more robust procedure that combines signal detection and field estimation. We introduce an estimation method that provides not only the effective or mean longitudinal magnetic field from an observed Stokes V profile but also uses the net absolute polarization of the profile to obtain an estimate of the apparent (i.e., velocity resolved) absolute longitudinal magnetic field. By combining the COG method with an orthogonal-matching-pursuit (OMP) approach, we were able to decompose observed Stokes profiles with an overcomplete dictionary of wavelet-basis functions to reliably reconstruct the observed Stokes profiles in the presence of noise. The elementary wave functions of the sparse reconstruction process were utilized to estimate the effective longitudinal magnetic field and the apparent absolute longitudinal magnetic field. A multiresolution analysis complements the OMP algorithm to provide a robust detection and estimation method. An extensive Monte-Carlo simulation confirms the reliability and accuracy of the magnetic OMP approach.Comment: A&A, in press, 15 pages, 14 figure

Lattice dynamical wavelet neural networks implemented using particle swarm optimisation for spatio-temporal system identification

Starting from the basic concept of coupled map lattices, a new family of adaptive wavelet neural networks, called lattice dynamical wavelet neural networks (LDWNN), is introduced for spatiotemporal system identification, by combining an efficient wavelet representation with a coupled map lattice model. A new orthogonal projection pursuit (OPP) method, coupled with a particle swarm optimisation (PSO) algorithm, is proposed for augmenting the proposed network. A novel two-stage hybrid training scheme is developed for constructing a parsimonious network model. In the first stage, by applying the orthogonal projection pursuit algorithm, significant wavelet-neurons are adaptively and successively recruited into the network, where adjustable parameters of the associated waveletneurons are optimised using a particle swarm optimiser. The resultant network model, obtained in the first stage, may however be redundant. In the second stage, an orthogonal least squares (OLS) algorithm is then applied to refine and improve the initially trained network by removing redundant wavelet-neurons from the network. The proposed two-stage hybrid training procedure can generally produce a parsimonious network model, where a ranked list of wavelet-neurons, according to the capability of each neuron to represent the total variance in the system output signal is produced. Two spatio-temporal system identification examples are presented to demonstrate the performance of the proposed new modelling framework

Feature detection using spikes: the greedy approach

A goal of low-level neural processes is to build an efficient code extracting the relevant information from the sensory input. It is believed that this is implemented in cortical areas by elementary inferential computations dynamically extracting the most likely parameters corresponding to the sensory signal. We explore here a neuro-mimetic feed-forward model of the primary visual area (VI) solving this problem in the case where the signal may be described by a robust linear generative model. This model uses an over-complete dictionary of primitives which provides a distributed probabilistic representation of input features. Relying on an efficiency criterion, we derive an algorithm as an approximate solution which uses incremental greedy inference processes. This algorithm is similar to 'Matching Pursuit' and mimics the parallel architecture of neural computations. We propose here a simple implementation using a network of spiking integrate-and-fire neurons which communicate using lateral interactions. Numerical simulations show that this Sparse Spike Coding strategy provides an efficient model for representing visual data from a set of natural images. Even though it is simplistic, this transformation of spatial data into a spatio-temporal pattern of binary events provides an accurate description of some complex neural patterns observed in the spiking activity of biological neural networks.Comment: This work links Matching Pursuit with bayesian inference by providing the underlying hypotheses (linear model, uniform prior, gaussian noise model). A parallel with the parallel and event-based nature of neural computations is explored and we show application to modelling Primary Visual Cortex / image processsing. http://incm.cnrs-mrs.fr/perrinet/dynn/LaurentPerrinet/Publications/Perrinet04tau

A Multiscale Pyramid Transform for Graph Signals

Multiscale transforms designed to process analog and discrete-time signals and images cannot be directly applied to analyze high-dimensional data residing on the vertices of a weighted graph, as they do not capture the intrinsic geometric structure of the underlying graph data domain. In this paper, we adapt the Laplacian pyramid transform for signals on Euclidean domains so that it can be used to analyze high-dimensional data residing on the vertices of a weighted graph. Our approach is to study existing methods and develop new methods for the four fundamental operations of graph downsampling, graph reduction, and filtering and interpolation of signals on graphs. Equipped with appropriate notions of these operations, we leverage the basic multiscale constructs and intuitions from classical signal processing to generate a transform that yields both a multiresolution of graphs and an associated multiresolution of a graph signal on the underlying sequence of graphs.Comment: 16 pages, 13 figure

Sparse and Nonnegative Factorizations For Music Understanding

In this dissertation, we propose methods for sparse and nonnegative factorization that are specifically suited for analyzing musical signals. First, we discuss two constraints that aid factorization of musical signals: harmonic and co-occurrence constraints. We propose a novel dictionary learning method that imposes harmonic constraints upon the atoms of the learned dictionary while allowing the dictionary size to grow appropriately during the learning procedure. When there is significant spectral-temporal overlap among the musical sources, our method outperforms popular existing matrix factorization methods as measured by the recall and precision of learned dictionary atoms. We also propose co-occurrence constraints -- three simple and convenient multiplicative update rules for nonnegative matrix factorization (NMF) that enforce dependence among atoms. Using examples in music transcription, we demonstrate the ability of these updates to represent each musical note with multiple atoms and cluster the atoms for source separation purposes. Second, we study how spectral and temporal information extracted by nonnegative factorizations can improve upon musical instrument recognition. Musical instrument recognition in melodic signals is difficult, especially for classification systems that rely entirely upon spectral information instead of temporal information. Here, we propose a simple and effective method of combining spectral and temporal information for instrument recognition. While existing classification methods use traditional features such as statistical moments, we extract novel features from spectral and temporal atoms generated by NMF using a biologically motivated multiresolution gamma filterbank. Unlike other methods that require thresholds, safeguards, and hierarchies, the proposed spectral-temporal method requires only simple filtering and a flat classifier. Finally, we study how to perform sparse factorization when a large dictionary of musical atoms is already known. Sparse coding methods such as matching pursuit (MP) have been applied to problems in music information retrieval such as transcription and source separation with moderate success. However, when the set of dictionary atoms is large, identification of the best match in the dictionary with the residual is slow -- linear in the size of the dictionary. Here, we propose a variant called approximate matching pursuit (AMP) that is faster than MP while maintaining scalability and accuracy. Unlike MP, AMP uses an approximate nearest-neighbor (ANN) algorithm to find the closest match in a dictionary in sublinear time. One such ANN algorithm, locality-sensitive hashing (LSH), is a probabilistic hash algorithm that places similar, yet not identical, observations into the same bin. While the accuracy of AMP is comparable to similar MP methods, the computational complexity is reduced. Also, by using LSH, this method scales easily; the dictionary can be expanded without reorganizing any data structures

Generalised additive multiscale wavelet models constructed using particle swarm optimisation and mutual information for spatio-temporal evolutionary system representation

A new class of generalised additive multiscale wavelet models (GAMWMs) is introduced for high dimensional spatio-temporal evolutionary (STE) system identification. A novel two-stage hybrid learning scheme is developed for constructing such an additive wavelet model. In the first stage, a new orthogonal projection pursuit (OPP) method, implemented using a particle swarm optimisation(PSO) algorithm, is proposed for successively augmenting an initial coarse wavelet model, where relevant parameters of the associated wavelets are optimised using a particle swarm optimiser. The resultant network model, obtained in the first stage, may however be a redundant model. In the second stage, a forward orthogonal regression (FOR) algorithm, implemented using a mutual information method, is then applied to refine and improve the initially constructed wavelet model. The proposed two-stage hybrid method can generally produce a parsimonious wavelet model, where a ranked list of wavelet functions, according to the capability of each wavelet to represent the total variance in the desired system output signal is produced. The proposed new modelling framework is applied to real observed images, relative to a chemical reaction exhibiting a spatio-temporal evolutionary behaviour, and the associated identification results show that the new modelling framework is applicable and effective for handling high dimensional identification problems of spatio-temporal evolution sytems

A Greedy Algorithm for a Sparse Scalet Decomposition

International audienceSparse decompositions were mainly developed to optimize the signal or the image compression. The sparsity was first obtained by a coefficient thresholding. The matching pursuit (MP) algorithms were implemented to extract the optimal patterns from a given dictionary. They carried out a new insight on the sparse representations. In this communication, this way is followed. It takes into account the goal to obtain a sparse multiscale decomposition with the different constraints: i/ to get a sparse representation with patterns looking like to Gaussian functions, ii/ to be able to decompose into patterns with only positive amplitudes, iii/ to get a representation from a translated and dilated pattern, iv/ to constrain the representation by a threshold, v/ to separate the sparse signal from a smooth baseline. Different greedy algorithms were built from the use of redundant wavelet transforms (pyramidal and `a trous ones), for 1D signals and 2D images. Experimentations on astronomical images allow one a gain of about two in sparsity compared to a classical DWT thresholding. A fine denoising is obtained. The results do not display any wavy artifacts. This decomposition is an efficient tool for astronomical image analysis