Second-order Volterra Research Articles

Volterra kernel based face recognition using artificial bee colonyoptimization

The present paper describes a novel method of implementation of a stochastic optimization technique for the face recognition problem. The method proposed divides the original images into patches in space, and seeks a non-linear functional mapping using second-order Volterra kernels. The artificial bee colony optimization technique, a modern stochastic optimization algorithm, is used to derive optimal Volterra kernels during training to simultaneously maximize inter-class distances and minimize intra-class distances in the feature space. During testing, a voting procedure is used in conjunction with a nearest neighbor classifier to decide to which class each individual patch belongs. Finally, the aggregate classification results of all patches in an image are used to determine the overall recognition outcome for the given image. The utility of the proposed scheme is aptly demonstrated by implementing it on two popular benchmark face recognition datasets, and comparing the effectiveness of the proposed approach vis-à-vis other statistical learning procedures in facial recognition and also several other methods developed so far. The effectiveness of the artificial bee colony optimization technique and its Levy-mutated variation in optimizing Volterra kernels is conclusively proven in this paper by significantly outperforming many popular contemporary algorithms.

Adaptive RSOV filter using the FELMS algorithm for nonlinear active noise control systems

This paper presents a recursive second-order Volterra (RSOV) filter to solve the problems of signal saturation and other nonlinear distortions that occur in nonlinear active noise control systems (NANC) used for actual applications. Since this nonlinear filter based on an infinite impulse response (IIR) filter structure can model higher than second-order and third-order nonlinearities for systems where the nonlinearities are harmonically related, the RSOV filter is more effective in NANC systems with either a linear secondary path (LSP) or a nonlinear secondary path (NSP). Simulation results clearly show that the RSOV adaptive filter using the multichannel structure filtered-error least mean square (FELMS) algorithm can further greatly reduce the computational burdens and is more suitable to eliminate nonlinear distortions in NANC systems than a SOV filter, a bilinear filter and a third-order Volterra (TOV) filter.

Numeric Variable Forgetting Factor RLS Algorithm for Second-Order Volterra Filtering

Nonlinear adaptive filtering techniques for system identification (based on the Volterra model) are widely used for the identification of nonlinearities in many applications. In this correspondence, the improved tracking capability of a numeric variable forgetting factor recursive least squares (NVFF-RLS) algorithm is presented for first-order and second-order time-varying Volterra systems under a nonstationary environment. The nonlinear system tracking problem is converted into a state estimation problem of the time-variant system. The time-varying Volterra kernels are governed by the first-order Gauss–Markov stochastic difference equation, upon which the state-space representation of this system is built. In comparison to the conventional fixed forgetting factor recursive least squares algorithm, the NVFF-RLS algorithm provides better channel estimation as well as channel tracking performance in terms of the minimum mean square error (MMSE) for first-order and second-order Volterra systems. The NVFF-RLS algorithm is adapted to the time-varying signals by using the updating prediction error criterion, which accounts for the nonstationarity of the signal. The demonstrated simulation results manifest that the proposed method has good adaptability in the time-varying environment, and it also reduces the computational complexity.

MMSE design of nonlinear Volterra equalizers using artificial bee colony algorithm

In this paper a novel approach for channel equalization is presented, where a framework for Volterra system is used to model both the channel and the equalizer. We propose development of first-order and second-order Volterra equalizers using minimum mean square error (MMSE) approach and design these equalizers using swarm intelligence based stochastic optimization algorithm which is applied to adapt the equalizer coefficients to the time varying channel. This work proposes to use the artificial bee colony (ABC) algorithm, recently introduced for global optimization, simulating the intelligent foraging behavior of honey bee swarm in a simple, robust, and flexible manner. For comparative analysis, adaptive equalizers like least mean squares (LMSs) equalizer, recursive least squares (RLSs) equalizer and least mean p-Norm (LMP) equalizer and population based optimum equalizers employing PSO are also applied for identical problems and the superiority of the newly proposed algorithm is aptly demonstrated.

Adaptive Extended Pipelined Second-Order Volterra Filter for Nonlinear Active Noise Controller

This correspondence presents an extended pipelined second-order Volterra (EPSOV) filter for active control of nonlinear noise processes. The corresponding nonlinear filtered-x algorithms using the filter bank implementation are also suggested. Compared to the standard SOV filter using the filtered-x least mean square (SOVFXLMS), those modules of the EPSOV filter can be performed simultaneously in a pipelined parallelism fashion, and this would lead to a significant improvement in its total computational efficiency. Results obtained from computer simulations for nonlinear noise processes demonstrate that the proposed method outperforms the SOV.

Nonlinear Long-Term Prediction of Speech Based on Truncated Volterra Series

Previous studies of nonlinear prediction of speech have been mostly focused on short-term prediction. This paper presents long-term nonlinear prediction based on second-order Volterra filters. It will be shown that the presented predictor can outperform conventional linear prediction techniques in terms of prediction gain and “whiter” residuals.

A Generalized Proportionate Subband Adaptive Second-Order Volterra Filter for Acoustic Echo Cancellation in Changing Environments

The performance of a typical linear echo canceller is limited by changes that occur in the acoustic echo path along with nonlinear echoes arising from electrodynamic loudspeaker distortion. This paper presents a subband structure based on proportionately adapted second-order Volterra filters for acoustic echo cancellation under these adverse conditions. The proposed structure allows the time and frequency domain nature of echo path changes to be exploited, resulting in faster convergence and tracking compared to an equivalent fullband structure while requiring significantly less computational complexity. Experimental results, based on measured data from a practical hands-free environment under changing conditions, demonstrate the improved echo cancellation performance of the proposed structure.

Estimating second-order Volterra system parameters from noisy measurements based on an LMS variant or an errors-in-variables method

This paper deals with the identification of a nonlinear SISO system modelled by a second-order Volterra series expansion when both the input and the output are disturbed by additive white Gaussian noises. Two methods are proposed. Firstly, we present an unbiased on-line approach based on the LMS. It includes a bias correction scheme which requires the variance of the input additive noise. Secondly, we suggest solving the identification problem as an errors-in-variables issue, by means of the so-called Frisch scheme. Although its computational cost is high, this approach has the advantage of estimating the Volterra kernels and the variances of both the additive noises and the input signal, even if the signal-to-noise ratios at the input and the output are low.

Multiscale nonlinear model for monthly streamflow forecasting: a wavelet-based approach

The dynamics of the streamflow in rivers involve nonlinear and multiscale phenomena. An attempt is made to develop nonlinear models combining wavelet decomposition with Volterra models. This paper describes a methodology to develop one-month-ahead forecasts of streamflow using multiscale nonlinear models. The method uses the concept of multiresolution decomposition using wavelets in order to represent the underlying integrated streamflow dynamics and this information, across scales, is then linked together using the first- and second-order Volterra kernels. The model is applied to 30 river data series from the western USA. The mean monthly data series of 30 rivers are grouped under the categories low, medium and high. The study indicated the presence of multiscale phenomena and discernable nonlinear characteristics in the streamflow data. Detailed analyses and results are presented only for three stations, selected to represent the low-flow, medium-flow and high-flow categories, respectively. The proposed model performance is good for all the flow regimes when compared with both the ARMA-type models as well as nonlinear models based on chaos theory.

The Two-Dimensional Power Spectral Density: A Connection Between 2-D Rational Functions and Linear Systems

A derivation of the power spectral density (PSD) matrix of a continuous-time, linear, constant-coefficient system with white Gaussian noise input is given. The assumption of wide-sense stationarity is not made. By transforming the two-dimensional (2-D) autocorrelation function (ACF), a 2-D spectral density is obtained that takes the system's transient behavior into account. The 2-D PSD is a non-separable rational function. The results are applicable to second-order Volterra series representations of nonlinear systems. Simulations are presented for a fourth order system that compare the 2-D PSD amplitude (both theoretical and empirical) with its 1-D counterpart on 50 realizations of a short data segment containing an initial transient.

Classification images: A review

Classification images have recently become a widely used tool in visual psychophysics. Here, I review the development of classification image methods over the past fifteen years. I provide some historical background, describing how classification images and related methods grew out of established statistical and mathematical frameworks and became common tools for studying biological systems. I describe key developments in classification image methods: use of optimal weighted sums based on the linear observer model, formulation of classification images in terms of the generalized linear model, development of statistical tests, use of priors to reduce dimensionality, methods for experiments with more than two response alternatives, a variant using multiplicative noise, and related methods for examining nonlinearities in visual processing, including second-order Volterra kernels and principal component analysis. I conclude with a selective review of how classification image methods have led to substantive findings in three representative areas of vision research, namely, spatial vision, perceptual organization, and visual search.

Predictor Using an Error-Convergence Neuron Network and its Application to Electrocardiograms

The output error of a neuron network cannot converge at zero, even if the training for a neuron network is iterated many times. “Error-convergence neuron network” in which the output error of a singleoutput system uses neuron networks with multiplestep convergence, has been designed to resolve this problem. The output error is converged at zero by setting infinite steps of the neuron network. Three types of neuron network systems also have been designed. They are “Error-convergence parallel neuron network,” “Error-convergence recurrent neuron network,” and “Error-convergence parallel recurrent neuron network.” A subsequent prediction can be obtained by recurring the prediction of a predictor to its input if the predictor is free of prediction error. An error-convergence neuron network can be applied to realize this predictor. “Error-convergence neuron network predictor” has been proposed as such a predictor. In this study, its feasibility is investigated by performing prediction training for the errorconvergence neuron network predictor constructed of second-order Volterra neuron networks with two steps, using the nonlinear time series signal of a normal sinus rhythm electrocardiogram. Predictions without any error were obtained.

Chaotic property analysis and prediction model study for heating load time series

In order to reveal the internal dynamics characteristics of heating load time series, the existing chaotic behavior is validated by use of nonlinear analysis method. The data sets taken from heat source and substation of district heat supply are studied by which phase spaces are reconstructed, and the correlation dimensions and the largest Lyapunov exponent are computed to identify the presence of chaos in heat load time series. By the analysis of the results, chaotic characteristics obviously exist in the heat load time series, which is a theoretical basis for the correlative investigation of heat load prediction. According to the existing heat load predictive method almostly based subjective models, a novel predictive approach based on Volterra adaptive filter, which avoids the subjective model assumptions, is presented for heat load prediction. Finally the predictive results are presented, and the simulation results illustrate that the second-order Volterra adaptive filter has high predictive accuracy which can meet the demands of heat energy-saving control and heat dispatching in practical applications.

Reduced-State MLSD Based on Volterra Kernels for Square-Law Detected Multipath Channels

We propose a novel reduced-state maximum-likelihood sequence detection (MLSD) structure using the Viterbi algorithm based on the second-order Volterra kernel modeling nonlinear distortion due to square law detection of multipath channels commonly occurring in chromatic dispersion (CD) or polarization mode dispersion (PMD) in optical communication systems. While all existing MLSD methods for square-law detection receivers are based on direct computation of branch metrics, the proposed algorithm provides an efficient and structured way to implement reduced-state MLSD with almost the same complexity of a MLSD for linear channels. As a result, the proposed algorithm reduces the number of parameters to be estimated and the complexity of computation.

Nonlinear MPC based on a Volterra series model for greenhouse temperature control using natural ventilation

Suitable environmental conditions are a fundamental issue in greenhouse crop growth and can be achieved by advanced climate control strategies. In different climatic zones, natural ventilation is used to regulate both the greenhouse temperature and humidity. In mild climates, the greatest problem faced by far in greenhouse climate control is cooling, which, for dynamical reasons, leads to natural ventilation as a standard tool. This work addresses the design of a nonlinear model predictive control (NMPC) strategy for greenhouse temperature control using natural ventilation. The NMPC strategy is based on a second-order Volterra series model identified from experimental input/output data of a greenhouse. These models, representing the simple and logical extension of convolution models, can be used to approximate the nonlinear dynamic effect of the ventilation and other environmental conditions on the greenhouse temperature. The developed NMPC is applied to a greenhouse and the control performance of the proposed strategy will be illustrated by means of experimental results.

System characterization of neuronal excitability in the hippocampus and its relevance to observed dynamics of spontaneous seizure-like transitions

Most forms of epilepsy are marked by seizure episodes that arise spontaneously. The low-magnesium/high-potassium (low-Mg2+/high-K+) experimental model of epilepsy is an acute model that produces spontaneous, recurring seizure-like events (SLEs). To elucidate the nature of spontaneous seizure transitions and their relationship to neuronal excitability, whole-cell recordings from the intact hippocampus were undertaken in vitro, and the response of hippocampal CA3 neurons to Gaussian white noise injection was obtained before and after treatment with various concentrations of low-Mg2+/high-K+ solution. A second-order Volterra kernel model was estimated for each of the input–output response pairs. The spectral energy of the responses was also computed, providing a quantitative measure of neuronal excitability. Changes in duration and amplitude of the first-order kernel correlated positively with the spectral energy increase following treatment with low-Mg2+/high-K+ solution, suggesting that variations in neuronal excitability are coded by the system kernels, in part by differences to the profile of the first-order kernel. In particular, kernel duration was more sensitive than amplitude to changes in spectral energy, and correlated more strongly with kernel area. An oscillator network model of the hippocampal CA3 was constructed to investigate the relationship of kernel duration to network excitability, and the model was able to generate spontaneous, recurrent SLEs by increasing the duration of a mode function analogous to the first-order kernel. Results from the model indicated that disruption to the dynamic balance of feedback was responsible for seizure-like transitions and the observed intermittency of SLEs. A physiological candidate for feedback imbalance consistent with the network model is the destabilizing interaction of extracellular potassium and paroxysmal neuronal activation. Altogether, these results (1) validate a mathematical model for epileptiform activity in the hippocampus by quantifying and subsequently correlating its behavior with an experimental, in vitro model of epilepsy; (2) elucidate a possible mechanism for epileptogenesis; and (3) pave the way for control studies in epilepsy utilizing the herein proposed experimental and mathematical setup.

Adaptive non-linear filter using a modular polynomial perceptron

This study presents a joint adaptive non-linear filter with pipelined second-order polynomial perceptron (PSOVNN) to reduce the computational complexity and improve the non-linear processing capability of adaptive direct-form second-order Volterra (SOV) filter. The PSOVNN is a nesting modular structure comprising a number of modules that are interconnected in a chained form. Each module is implemented by a small-scale direct-form SOV neural network (SOVNN). These cascaded modules can perform a non-linear mapping from the input space to an intermediate space. In addition, the linear filter of the complete PSOVNN performs a linear mapping from the intermediate space to the output space. A modified real-time recurrent learning (RTRL) algorithm is developed, and its performance is evaluated by a series of simulation experiments. Computer simulations indicate that the proposed non-linear filter exhibits better performance over the direct-form SOV filter with less computational complexity.

Separation of Nonlinear Ultrasound Signals Based on Second-Order Volterra System Identification

This paper introduces a system identification based on second Volterra filter (SVF) as a mathematical model to separate harmonic components of a radio frequency (RF) line from a simulated nonlinear pulse–echo system (NPS). Filter coefficients of both the linear (the first-order Volterra) and quadratic (the second-order Volterra) kernels of the SVF can be determined by processing the input and output data in the frequency domain and solving a system of linear equations. Results from the validation of the approach show that while the linear kernel correctly predicts the linear output in the fundamental energy band, the quadratic kernel appropriately captures energy primarily from the second harmonic and low frequency bands. In addition, it is shown that the system identification based on the SVF is capable of separating the second order nonlinearity embedded under the level of noise signal. This is the significant advantage of the SVF approach over other static models such as linear bandpass filters. In addition, the feasibility of regularization in decreasing the number of input–output sequences used in system identification is presented.

Nonlinear identification in structural dynamics based on Wiener series and Kautz filters

The present paper proposes a method to identify localized nonlinear parameters in structural dynamics using vibration data. The approach is based on the identification of the first and second-order Volterra kernels in an orthogonal basis, namely Wiener kernels, while taking the experimental data as exact values. The focus is to identify an already localized nonlinearity with known structure. An optimization procedure is implemented using a metric involving the difference between the experimental kernel and the simulated orthogonal kernel from model updating, which is function of the unknown nonlinear parameters. In order to reduce the problems of convergence, the Kautz filter is used to decrease the number of samples to estimate. The proposed methodology is illustrated on a numerical example using a cantilever beam and its advantages and drawbacks are discussed in detail.

A novel adaptive bilinear filter based on pipelined architecture

To overcome the computational complexity of the Volterra filter, a novel adaptive joint process filter using pipelined bilinear polynomial architecture (JPBPF) is proposed in this paper. The proposed architecture consists of two subsections: nonlinear subsection performing a nonlinear mapping from the input space to an intermediate space by the bilinear polynomial filter (BPF), and a linear filter performing a linear mapping from the intermediate space to the output space. The corresponding adaptive algorithms are deduced for the nonlinear subsection and linear filter subsection, respectively. To evaluate the performance of the JPBPF, a series of simulations are presented including nonlinear system identification, predicting of speech signals and nonlinear channel equalization. Compared with the conventional second-order Volterra (SOV) filter and BPF, the JPBPF exhibits a slightly better convergence performance in terms of convergence speed and steady-state error. Moreover, since those modules of a JPBPF can be performed simultaneously in a pipelined parallelism fashion, this would lead to a significant improvement in its total computational efficiency.