Bispectral Density Research Articles

The bispectral representation of Markov switching bilinear models

This article formulae for the third-order theoretical moments for superdiagonal and subdiagonal of the Markov-switching bilinear $X_{t}=c(s_{t})X_{t-k}e_{t-l}+e_{t}$, $k,l \in\mathrm{N}$, and an expression for the bispectral density function are obtained.

Effect of Fuzzy Time Series on Smoothing Estimation of the INAR(1) Process

In this paper, the effect of fuzzy time series on estimates of the spectral, bispectral and normalized bispectral density functions are studied. This study is conducted for one of the integer autoregressive of order one (INAR(1)) models. The model of interest here is the dependent counting geometric INAR(1) which is symbolized by (DCGINAR(1)). A realization is generated for this model of size n = 500 for estimation. Based on fuzzy time series, the forecasted observations of this model are obtained. The estimators of spectral, bispectral and normalized bispectral density functions are smoothed by different one- and two-dimensional lag windows. Finally, after the smoothing, all estimators are studied in the case of generated and forecasted observations of the DCGINAR(1) model. We investigate the contribution of the fuzzy time series to the smoothing of these estimates through the results.

Higher Order Moments, Cumulants, Spectral and Bispectral Density Functions of the ZTPINAR(1) Process

Higher Order Moments, Cumulants, Spectral and Bispectral Density Functions of the ZTPINAR(1) Process

Spectral analysis for GARCH processes through a bilinear representation

In this paper, we deal with the frequency domain approach to describe some probabilistic properties of the processes using the associated transfer functions which seem more informative. So, in the first part, we propose an appropriate bilinear representation of model governed by a second-order Hermite polynomial with leading coefficient 1, and we give a necessary and sufficient condition for the causal and second-order stationary solution. In a second part, we establish the autocovariance and the associated spectral density function of such a representation, and for its squared version in terms of transfer functions. As a consequence, it is observed that the second-order properties are similar to an ARMA process with uncorrelated innovations and hence the resort to higher-order properties however becomes inevitable. Therefore, the explicit expressions of bispectral densities of the process and its squared version are given and several illustrative examples are emphasized. Our results are related to the work of Terdik (2000, §3) for the GARCH(1, 1) model.

Bispectral Density Estimation of Continuous Time Series with Missed Observations

Bispectral Density Estimation of Continuous Time Series with Missed Observations

Bispectral unfolding of the skewness of correlated additive and multiplicative noise processes.

Correlated additive and multiplicative (CAM) noise processes are well-established as general "null hypothesis" models of non-Gaussian variability in atmospheric and oceanic quantities. In this study, analytic expressions for the bispectral density (which partitions the third statistical moment into triad frequency interactions in a manner analogous to the partitioning of variance by the spectral density) are developed for discrete and continuous-time CAM processes. It is then demonstrated that under lowpass filtering, while the absolute skewness of a discrete-time CAM process may increase or decrease with decreasing cutoff frequency, the absolute skewness of continuous-time CAM processes decreases monotonically. This second result provides a test to assess the degree to which an observed time series is consistent with continuous-time CAM dynamics.

ВІДТВОРЕННЯ СИГНАЛІВ ТА ЗОБРАЖЕНЬ ЗА БІСПЕКТРОМ У ПРИСУТНОСТІ ШУМІВ

In a number of practical applications of digital signal processing, the process under study may include correlated spectral components or phase coupling. Extracting the phase relationships provides very important and useful information for the correct understanding, analysis, and description of the properties of physical phenomena generating these processes. However, such information is irretrievably lost when using classical methods of signal processing using energy statistics, i.e. second-order statistics. Obtaining estimates of signal parameters and analyzing them using third-order correlation functions and bispectrum makes it possible to learn much more about signal properties than when using conventional correlation functions. Estimating the bispectral density (third order spectral density), in contrast to estimating the energy spectrum, makes it possible not only to describe the characteristics of the observed process correctly, but also to preserve and, if necessary, extract the phase characteristics of the component, which includes the observed process. Therefore, in a number of applied tasks of telecommunications, as well as tasks of image processing and other bisection analysis, often serves as an effective tool of signal processing. The aim of the article is to study the feasibility of using a recursive algorithm when restoring a waveform and image by bispectrum in the noise environment. The following types of signals were selected for the study: rectangular, triangular, Gaussian impulses and signal-triplet. They were distorted with additive white Gaussian noise Test image was distorted by additive white Gaussian, pulsed, Poisson and multiplicative noises. Analysis of the signal recovery results indicates that as the noise power increases, the quality of the recovery decreases. The effect of random signal shift does not affect the shape of the recovered signal. Analysis of the image recovery results indicates image recovery, but this algorithm introduces distortions in the form of an offset

Higher Order Moments, Spectral and Bispectral Density Functions for INAR(1)

Higher Order Moments, Spectral and Bispectral Density Functions for INAR(1)

Stochastic characteristics and Second Law violations of atomic fluids in Couette flow

Using Non-equilibrium Molecular Dynamics (NEMD) simulations, we study the statistical properties of an atomic fluid undergoing planar Couette flow, in which particles interact via a Lennard-Jones potential. We draw a connection between local density contrast and temporal fluctuations in the shear stress, which arise naturally through the equivalence between the dissipation function and entropy production according to the fluctuation theorem. We focus on the shear stress and the spatio-temporal density fluctuations and study the autocorrelations and spectral densities of the shear stress. The bispectral density of the shear stress is used to measure the degree of departure from a Gaussian model and the degree of nonlinearity induced in the system owing to the applied strain rate. More evidence is provided by the probability density function of the shear stress. We use the Information Theory to account for the departure from Gaussian statistics and to develop a more general probability distribution function that captures this broad range of effects. By accounting for negative shear stress increments, we show how this distribution preserves the violations of the Second Law of Thermodynamics observed in planar Couette flow of atomic fluids, and also how it captures the non-Gaussian nature of the system by allowing for non-zero higher moments. We also demonstrate how the temperature affects the band-width of the shear-stress and how the density affects its Power Spectral Density, thus determining the conditions under which the shear-stress acts is a narrow-band or wide-band random process. We show that changes in the statistical characteristics of the parameters of interest occur at a critical strain rate at which an ordering transition occurs in the fluid causing shear thinning and affecting its stability. A critical strain rate of this kind is also predicted by the Loose–Hess stability criterion.

Inference for Random Coefficient INAR(1) Process Based on Frequency Domain Analysis

In this article, we present the explicit expressions for the higher-order moments and cumulants of the first-order random coefficient integer-valued autoregressive (RCINAR(1)) process. The spectral and bispectral density functions are also obtained, which can characterize the RCINAR(1) process in the frequency domain. We use a frequency domain approach which is named Whittle criterion to estimate the parameters of the process. We propose a test statistic which is based on the frequency domain approach for the hypothesis test, H0: α = 0⟷H1: 0 < α < 1, where α is the mean of the random coefficient in the process. The asymptotic distribution of the test statistic is obtained. We compare the proposed test statistic with other statistics that can test serial dependence in time series of count via a typically numerical simulation, which indicates that our proposed test statistic has a good power.

Smoothing of the bispectral density estimate based on digital nonlinear filtering and the Kravchenko weight functions

The methods of smoothing of the bispectral density estimate when solving the problems of restoration of signals with an unknown shape in the interference environment and random signal delay are considered for the first time. The performed analysis of statistical characteristics of noise that presents in the bispectrum estimate shows that these statistical characteristics have a rather complex unsteady behavior. An ambiguous selection of a filter optimal by the criterion of the minimum of the root-mean-square error and the minimum of dynamic distortions introduced by the filter seems to be problematic because of the unsteady behavior of counts of the bispectral density estimate and the absence of a priori data on the parameters of the restored signal. Therefore, statistic investigations were performed with the use of linear and nonlinear digital filters with variations of the sliding window sizes. It is shown that the advantages of the proposed approach most pronouncedly manifest themselves with the use of the nonlinear digital filtration and small signal/noise ratios at the input and/or with a small sampling volume of observed implementations. The Kravchenko weight functions are proposed to smooth the bispectrum of the multifrequency signal with a large dynamic range of variations in amplitudes of spectral components. The presented results are of practical interest for use in applications such as radiolocation, hydrolocation, and digital communication.

Bispectral-based methods for clustering time series

Distinguishing among linear and nonlinear time series or between nonlinear time series generated by different underlying processes is challenging, as second-order properties are generally insufficient for the task. Different nonlinear processes have different nonconstant bispectral signatures, whereas the bispectral density function of a Gaussian or linear time series is constant. Based on this, we propose a procedure to distinguish among various nonlinear time series and between nonlinear and linear time series through application of a hierarchical clustering algorithm based on distance measures computed from the square modulus of the estimated normalized bispectra. We find that clustering using a distance measure computed by averaging the ratio of normalized bispectral periodogram ordinates over the intersection of the principle domain of each pair of time series provides good performance, subject to trimming of extreme bispectral values prior to taking the ratios. Additionally, we show through simulation studies that the distance procedure performs better than a significance test that we derive. Moreover, it is robust with respect to the choice of smoothing parameter in estimating the bispectrum. As an example, we apply the method to a set of time series of intensities of gamma-ray bursts, some of which exhibit nonlinear behavior; this enables us to identify gamma-ray bursts that may be emanating from the same type of astral event.

The sample autocorrelation function and the detection of long-memory processes

The detection of long-range dependence in time series analysis is an important task to which this paper contributes by showing that whilst the theoretical definition of a long-memory (or long-range dependent) process is based on the autocorrelation function, it is not possible for long memory to be identified using the sum of the sample autocorrelations, as usually defined. The reason for this is that the sample sum is a predetermined constant for any stationary time series; a result that is independent of the sample size. Diagnostic or estimation procedures, such as those in the frequency domain, that embed this sum are equally open to this criticism. We develop this result in the context of long memory, extending it to the implications for the spectral density function and the variance of partial sums of a stationary stochastic process. The results are further extended to higher order sample autocorrelations and the bispectral density. The corresponding result is that the sum of the third order sample (auto) bicorrelations at lags h,k≥1, is also a predetermined constant, different from that in the second order case, for any stationary time series of arbitrary length.

A new approach for estimation of obstructive sleep apnea syndrome

Obstructive sleep apnea syndrome (OSAS) is a situation where repeatedly upper airway stops off while the respiratory effort continues during sleep at least for 10s. Apart from polysomnography, many researchers have concentrated on exploring alternative methods for OSAS detection. However, not much work has been done on using non-Gaussian and nonlinear behavior of the electroencephalogram (EEG) signals. Bispectral analysis is an advanced signal processing technique particularly used for exhibiting quadratic phase-coupling that may arise between signal components with different frequencies. From this perspective, in this study, a new technique for recognizing patients with OSAS was introduced using bispectral characteristics of EEG signal and an artificial neural network (ANN). The amount of Quadratic phase coupling (QPC) in each subband of EEG (namely; delta, theta, alpha, beta and gamma) was calculated over bispectral density of EEG. Then, these QPCs were fed to the input of the designed ANN. The neural network was configured with two outputs: one for OSAS and one for estimation of normal situation. With this technique a global accuracy of 96.15% was achieved. The proposed technique could be used in designing automatic OSAS identification systems which will improve medical service.

Optimal bispectral detection of weak, quadratic nonlinearities in structural systems

Abstract This work derives the expressions for the bispectrum and bicoherence functions for multi-degree-of-freedom spring–mass systems with quadratic nonlinearities subject to inputs described by a wide class of random processes. The derivation uses a Volterra series model for the system response and yields expressions for both “homogeneous spectra”, where the output of only one degree of freedom is used, and “mixed spectra” where the bispectral density contains multiple response time series. This expression is then used to determine the optimal probability distribution of the input and the optimal bispectrum to compute for the goal of maximizing the probability of detecting the nonlinearity.

An analytical formulation of bispectral densities for multiple degree-of-freedom systems

Expressions for the bispectral density functions for multi-degree-of-freedom spring–mass–damper systems possessing quadratic nonlinearities and subject to Gaussian excitation are derived. The derivation uses a Volterra-series model for the system response and yields expressions for both auto-spectra, where the output of only one degree-of-freedom is used, and cross-spectra, where the bispectral density contains multiple output-response time series. The proposed formulation is used to identify the presence and location of quadratic nonlinearities in multiple degree-of-freedom systems. Results show that the ability to detect and localize nonlinearity is heavily dependent on which particular bispectral density is utilized.

A method for signal reconstruction with the use of normalized spectrum

A new method is suggested for reconstruction of signals of unknown shape by their bispectral density. The new approach is based on application of smooth functions such as cosine and sine — instead of the discontinuous function of the “arctangent of bispectrum argument” type, traditionally accepted in phase measurements. The use of simulation permits to assess effectiveness of the suggested method of reconstruction of signals with the first genus discontinuities at different discretization intervals.

Difference Equations for the Higher Order Moments and Cumulants of the INAR(p) Model

Abstract. Here we obtain difference equations for the higher order moments and cumulants of a time series {Xt} satisfying an INAR(p) model. These equations are similar to the difference equations for the higher order moments and cumulants of the bilinear time series model. We obtain the spectral and bispectral density functions for the INAR(p) process in state–space form, thus characterizing it in the frequency domain. We consider a frequency domain method – the Whittle criterion – to estimate the parameters of the INAR(p) model and illustrate it with the series of the number of epilepsy seizures of a patient.

Difference Equations for the Higher‐Order Moments and Cumulants of the INAR(1) Model

Recently, as a result of the growing interest in modelling stationary processes with discrete marginal distributions, several models for integer value time series have been proposed in the literature. One of these models is the INteger‐AutoRegressive (INAR) model. Here we consider the higher‐order moments and cumulants of the INAR(1) process and show that they satisfy a set of Yule–Walker type difference equations. We also obtain the spectral and bispectral density functions, thus characterizing the INAR(1) process in the frequency domain. We use a frequency domain approach, namely the Whittle criterion, to estimate the parameters of the model. The estimation theory and associated asymptotic theory of this estimation method are illustrated numerically.

Consistent estimation of the bispectral density function of a harmonizable process

In this note, the consistency of a bispectral density estimator of strongly harmonizable processes is obtained. A resampling procedure is used consisting of an α-mixing sequence of sample paths of the process.