Backward Prediction Errors Research Articles

Algorithms for Sparse Multichannel Blind Deconvolution

In this paper we present two algorithms for sparse multichannel blind deconvolution. The first algorithm is based on a cascade of a forward and a backward prediction error filter (C-PEF). The second consists in an alternating minimization algorithm for estimating both the reflectivity series and the seismic wavelet (AM-SMBD). We also compare the algorithms with other state-of-the-art sparse blind deconvolution algorithms. Simulation results with synthetic data for different SNR levels showed that the AM-SMBD was outperformed (in terms of the Pearson Correlation Coefficient and the Gini Correlation Coefficient) other estimation methods, like the reduced SMBD, the TS-sparseTLS (Toeplitz-structured sparse total least square) and the SMBD-SPG (Sparse multichannel blind deconvolution via spectral projected-gradient). For the same data, the C-PEF was able to provide better results (in terms of the Gini Correlation Coefficient, visual inspection and frequency gain) when compared with the F-SMBD (Fast sparse multichannel blind deconvolution). In a simulation considering reflectivities with different levels of sparsity, the C-PEF seems to be more robust for less sparse data when compared with AM-SMBD and SMBD-SPG (up to a certain degree of sparsity). Finally, simulations considering a real land acquisition show that both algorithms were able to greatly improve the resolution of the seismic data.

Chaos Theory and ARTFA: Emerging Tools for Interpreting ECG Signals to Diagnose Cardiac Arrhythmias

Timely detection of cardiac abnormalities from an Electrocardiogram (ECG) signal is very essential. This requires its appropriate and efficient processing. In the literature, most of the researchers focussed on linear techniques that were applied on filtered ECG datasets leaving an ample scope for exploring the use of non-linear techniques in the presence and absence of natural noise-processes. Therefore, there is a need of supplementing the existing research on ECG signal interpretation by using non-linear techniques on noisy ECG data. Non-linear techniques are expected to yield supplementary clues about the non-linearities in the underlying cardiovascular system. One such promising non-linear technique, known as chaos theory (analysis), has been considered here to estimate reliable and robust thresholds for R-peak detection using fractal dimension, Approximate Entropy, Sample Entropy, correlation dimension, and Lyapunov exponent based on time-delay dimension (embedding). Also, time–frequency analysis techniques have shown their effectiveness for analyzing such types of non-linear and non-stationary signals due to simultaneous interpretation of the signal in both time and frequency domain. Among existing TFA techniques such as wavelet transform, short time Fourier transform, Hilbert transform, Auto-regressive Time Frequency Analysis (ARTFA) offers good time–frequency resolution. Therefore, Chaos theory and ARTFA have been considered in this paper. First, raw ECG signal was filtered using Savitzky Golay Digital Filtering (SGDF) because it retains all important signal features after filtering. Second, a novel optimal trajectory detection step was proposed on the basis of phase space reconstruction (attractors) in chaos theory. Third, ARTFA has been used to find the spectral components of the extracted features using chaos theory. Here, ARTFA has been used for finding the autoregressive coefficients in the first step and time–frequency description in the second step. Burg method has been considered for Auto-regressive modeling to fit an ARTFA model for analyzing ECG signal by minimizing (least squares) the forward and backward prediction errors. MIT-BIH arrhythmia database has been considered for validating the present research effort. Some real time signals were also tested to explore direct usage of the proposed technique in practical applications. The obtained results show that the proposed technique enhances sensitivity, positive predictive value and accuracy thereby improving the detection of arrhythmias. Computational cost of the proposed technique is reduced to a great extent by using the chaos theory (analysis) yielding efficient detection performance. All results have been obtained in MATLAB environment R2011a. Improved values viz. 99.96% SE, 99.97% PPV, and 99.93% ACC are obtained.

Appraising the Financial Sustainability of a Pension System with Signal Processing

One key issue of the Spanish pension system is its financial sustainability in regard to slumping fertility rate and rising longevity of the Spanish population. The paper presents a versatile and robust model that may help pension managers gain insight into the future Spanish pyramid of ages, from which they will appraise cash inflows and outflows of the pension system. The model forecasts ninety years of the Spanish population for each cohort of the pyramid of ages. Borrowed from the signal processing discipline, the model relies on the Burg method which fits a pth order autoregressive (AR) model to the input signal by minimizing (least squares) the forward and backward prediction errors while constrai-ning the AR parameters to satisfy the Levinson-Durbin recursión, then uses an infinite impulse response prediction error filter. Results add better perspective and insight to the Spanish population projection forecasted by the United Nations Population Division.

When will European Muslim population be majority and in which country?

Purpose The purpose of this paper is to estimate the years the European Muslim population will be majority among 30 European countries. Design/methodology/approach The methodology/approach is to forecast the population of 30 European countries with wavelet analysis combined with the Burg model which fits a pth order autoregressive model to the input signal by minimizing (least squares) the forward and backward prediction errors while constraining the autoregressive parameters to satisfy the Levinson–Durbin recursion, then relies on an infinite impulse response prediction error filter. Three scenarios are considered: the zero-migration scenario where the authors assume that the Muslim population has a higher fertility (one child more per woman, on average) than other Europeans, mirroring a global pattern; a 2017 migration scenario: to the Muslim population obtained in the zero-migration scenario, the authors add a continuous flow of migrants every year based on year 2017; the mid-point migration scenario is obtained by averaging the data of the two previous scenarios. Findings Among three scenarios, the most likely mid-point migration scenario identifies 13 countries where the Muslim population will be majority between years 2085 and 2215: Cyprus (in year 2085), Sweden (2125), France (2135), Greece (2135), Belgium (2140), Bulgaria (2140), Italy (2175), Luxembourg (2175), the UK (2180), Slovenia (2190), Switzerland (2195), Ireland (2200) and Lithuania (2215). The 17 remaining countries will never reach majority in the next 200 years. Originality/value The growing Muslim population will change the face of Europe socially, politically and economically. This paper will provide a better insight and understanding of Muslim population dynamics to European governments, policymakers, as well as social and economic planners.

Will Saudi Arabia get older? Will its pension system be sustainable? Spectral answers

Purpose The purpose of this paper is to answer the following two questions: Will Saudi Arabia get older? Will its pension system be sustainable? Design/methodology/approach The methodology/approach is to forecast KSA’s population with wavelet analysis combined with the Burg model which fits a pth order autoregressive model to the input signal by minimizing (least squares) the forward and backward prediction errors while constraining the autoregressive parameters to satisfy the Levinson-Durbin recursion, then relies on an infinite impulse response prediction error filter. Findings Spectral analysis projections of Saudi age groups are more optimistic than the Bayesian probabilistic model sponsored by the United Nations Population Division: Saudi Arabia will not get older as fast as projected by the United Nations model. The KSA’s pension system will stay sustainable based on spectral analysis, whereas it will not based on the U.N. model. Originality/value Spectral analysis will provide better insight and understanding of population dynamics for Saudi government policymakers, as well as economic, health and pension planners.

Yield Curve Forecasting with the Burg Model

AbstractWe introduce a versatile and robust model that may help policymakers, bond portfolio managers and financial institutions to gain insight into the future shape of the yield curve. The Burg model forecasts a 20‐day yield curve, which fits a pth‐order autoregressive (AR) model to the input signal by minimizing (least squares) the forward and backward prediction errors while constraining the autoregressive parameters to satisfy the Levinson–Durbin recursion. Then, it uses an infinite impulse response prediction error filter. Results are striking when the Burg model is compared to the Diebold and Li model: the model not only significantly improves accuracy, but also its forecast yield curves stick to the shape of observed yield curves, whether normal, humped, flat or inverted. Copyright © 2016 John Wiley & Sons, Ltd.

Stable 1-Norm Error Minimization Based Linear Predictors for Speech Modeling

In linear prediction of speech, the 1-norm error minimization criterion has been shown to provide a valid alternative to the 2-norm minimization criterion. However, unlike 2-norm minimization, 1-norm minimization does not guarantee the stability of the corresponding all-pole filter and can generate saturations when this is used to synthesize speech. In this paper, we introduce two new methods to obtain intrinsically stable predictors with the 1-norm minimization. The first method is based on constraining the roots of the predictor to lie within the unit circle by reducing the numerical range of the shift operator associated with the particular prediction problem considered. The second method uses the alternative Cauchy bound to impose a convex constraint on the predictor in the 1-norm error minimization. These methods are compared with two existing methods: the Burg method, based on the 1-norm minimization of the forward and backward prediction error, and the iteratively reweighted 2-norm minimization known to converge to the 1-norm minimization with an appropriate selection of weights. The evaluation gives proof of the effectiveness of the new methods, performing as well as unconstrained 1-norm based linear prediction for modeling and coding of speech.

New Efficient 2-D Lattice Structures for General Autoregressive Modeling of Random Fields

In this paper, we present a new efficient general 2-D autoregressive (AR) lattice modeling technique of random fields. Our proposed approach is capable of simultaneously providing all possible types of 2-D causal quarter-plane (QP) and asymmetric half-plane (ASHP) AR models for an arbitrary rectangular shape of the prediction support region (PSR). It is shown that it is also possible to obtain various noncausal half-plane AR models without any additional computational cost. This new lattice structure introduces only one row or one column of new observation points into the existing rectangular PSR when the order of the model increases in the horizontal or vertical direction at each of the recursive order incrementation stage, respectively. Starting with a given random data field, a set of auxiliary forward and backward prediction error fields (PEFs) for the horizontal and vertical directions are generated. After recursive order updates to the desired size of the rectangular PSR, all types of causal QP/ASHP and noncausal half-plane 2-D lattice models are obtained by deriving the appropriate PEF from the remaining ones. In addition to developing the basic theory, the presentation includes the derivation of the synthesis model transfer functions from the calculated lattice parameters. It is shown that the proposed auxiliary horizontal and vertical backward prediction error filters inherently generates the mutually orthogonal realization vectors and thus are optimum for all stages. The theory has been confirmed by computer simulations.

Multicomponent f-x seismic random noise attenuation via vector autoregressive operators

We propose an extension of the traditional frequency-space ([Formula: see text]) random noise attenuation method to three-component seismic records. For this purpose, we develop a three-component vector autoregressive (VAR) model in the [Formula: see text] domain that is applied to the multicomponent spatial samples of each individual temporal frequency. VAR model parameters are estimated using the least-squares minimization of forward and backward prediction errors. VAR modeling effectively identifies the potential coherencies between various components of a multicomponent signal. We use the squared coherence spectrum of VAR models as an indicator to determine these coherencies. Synthetic and real data examples are provided to show the effectiveness of the proposed method.

Transitions in neural oscillations reflect prediction errors generated in audiovisual speech

According to the predictive coding theory, top-down predictions are conveyed by backward connections and prediction errors are propagated forward across the cortical hierarchy. Using MEG in humans, we show that violating multisensory predictions causes a fundamental and qualitative change in both the frequency and spatial distribution of cortical activity. When visual speech input correctly predicted auditory speech signals, a slow delta regime (3-4 Hz) developed in higher-order speech areas. In contrast, when auditory signals invalidated predictions inferred from vision, a low-beta (14-15 Hz) / high-gamma (60-80 Hz) coupling regime appeared locally in a multisensory area (area STS). This frequency shift in oscillatory responses scaled with the degree of audio-visual congruence and was accompanied by increased gamma activity in lower sensory regions. These findings are consistent with the notion that bottom-up prediction errors are communicated in predominantly high (gamma) frequency ranges, whereas top-down predictions are mediated by slower (beta) frequencies.

AR LATTICE ℓ-STEP AHEAD OUTPUT UNCERTAINTY PREDICTION SCHEME WITH UNCERTAINTY INTERVALS

In this paper, a lattice ℓ-step ahead output uncertainty prediction algorithm is presented. The proposed prediction scheme is applicable to linear, stable, auto-regressive (AR) systems. The uncertainty predictors utilize the structure of the lattice filters. Subject to the saturation limits of the excitation signal, the measurement of the system output, and the a priori bounds of the reflection coefficients, the set of the feasible predicted output variables is computed. The suggested scheme is recursively computed over an ℓ-step ahead future time window, while the prediction uncertainty intervals are similarly computed over the lattice stages, regarding the forward and the backward prediction errors. Simulation results are presented that prove the efficacy of the employed lattice uncertainty predictors.

Direct multichannel predictive deconvolution

The Levinson principle generally can be used to compute recursively the solution of linear equations. It can also be used to update the error terms directly. This is used to do single-channel deconvolution directly on seismic data without computing or applying a digital filter. Multichannel predictive deconvolution is used for seismic multiple attenuation. In a standard procedure, the prediction-error filter matrices are computed with a Levinson recursive algorithm, using a covariance matrix of the input data. The filtered output is the prediction errors or the nonpredictable part of the data. Starting with the classical Levinson recursion,wehave derived new algorithms for direct recursive calculationof the prediction errors without computing the data covariance-matrix or computing the prediction-error filters. One algorithm generates recursively the one-step forward and backward predic-tion errors and the L-step forward prediction error, computing only the filter matrices with the highest index. A numerically more stable algorithm uses reduced QR decomposition or singular-value decomposition (SVD) in a direct recursive computation of the prediction errors without computing any filter matrix. The new, stable, predictive algorithms require more arithmetic opera-tions in the computer, but the computer programs and data flow are much simpler than for standard predictive deconvolution.

Multichannel fast QRD-RLS adaptive filtering: Block-channel and sequential-channel algorithms based on updating backward prediction errors

Fast QR decomposition recursive least-squares (FQRD-RLS) algorithms are well known for their fast convergence and reduced computational complexity. A considerable research effort has been devoted to the investigation of single-channel versions of the FQRD-RLS algorithms, while the multichannel counterparts have not received the same attention. The goal of this paper is to broaden the study of the efficient and low complexity family of multichannel RLS adaptive filters, and to offer new algorithm options. We present a generalized approach for block-type multichannel FQRD-RLS (MC-FQRD-RLS) algorithms that includes both cases of equal and multiple order. We also introduce new versions for block-channel and sequential-channel processing, details of their derivations, and a comparison in terms of computational complexity. The proposed algorithms are based on the updating of backward a priori and a posteriori error vectors, which are known to be numerically robust.

Method estimating reflection coefficients of adaptive lattice filter and its application to system identification

In this paper, we propose a method of estimating the reflection coefficients of an adaptive lattice filter. In this method, conventional adaptive algorithms, for example, the normalized least mean square (NLMS) algorithm, are used for the estimation. In general, the reflection coefficients are estimated as cross-correlation coefficients between forward and backward prediction errors in each stage of the adaptive lattice filter. Accordingly, two divisions in each stage, and effectively doubling the number of stages, are required. A problem is that the processing cost of division is higher than that of multiplication, especially in cheap digital signal processors (DSPs). Hence, the reduction of the number of divisions is strongly desired in practical use. The proposed technique can decrease the number of divisions to one, provided that the NLMS algorithm is used. Moreover, in the application of the adaptive lattice filter, system identification is also important. In this paper, we present a technique for the application. The technique is derived from the proposed method.

On parameter estimation of ARMAX model via BCLS method

This paper studies the problem of parameter estimation of ARMAX model from a novel point of view. An efficient bias compensation least squares algorithm is proposed to provide consistent parameter estimate for ARMAX model. The main feature of our proposed algorithm is to introduce the auxiliary least squares linear backward predictors to construct the cross-correlations of least-squares (LS) error and backward prediction (BWP) errors. And with the help of the cross-correlations of LS error and BWP errors, estimate of the bias resulted from LS solution can be obtained. Consequently the consistent estimate for ARMAX model can be obtained via compensating the estimated bias of LS estimate. The batch-processing approach and the recursive processing approach for the proposed method are given. Theoretical analysis that compares the proposed method with the other existing methods such as bias-eliminated least-squares (BELS) method proposed by Zheng and instrumental variables (IV) method is carried out. Simulation results are presented to illustrate the effectiveness of the proposed algorithm,

Fast QR Algorithms Based on Backward Prediction Errors: A New Implementation and Its Finite Precision Performance

QR decomposition techniques are well known for their good numerical behavior and low complexity. Fast QRD recursive least squares adaptive algorithms benefit from these characteristics to offer robust and fast adaptive filters. This paper examines two different versions of the fast QR algorithm based on a priori backward prediction errors as well as two other corresponding versions of the fast QR algorithm based on a poste- riori backward prediction errors. The main matrix equations are presented with different versions derived from two distinct deployments of a particular matrix equation. From this study, a new algorithm is derived. The discussed algorithms are compared, and differences in computational complexity and in finite-precision behavior are shown. Ke yw ords: Adaptive system, fast RLS algorithm, QR decomposition.

Blind equalization using least-squares lattice prediction

Second-order statistics of the received signal can be used to equalize a communication channel without knowledge of the transmitted sequence. Blind zero-forcing (ZF) and minimum mean-square error (MMSE) equalization can be achieved with linear prediction error filtering. The equivalence with the equalizers derived by Giannakis and Halford (see ibid., vol.45, p.2277-92, 1997) is shown, and adaptive predictors that result in a lattice filtering structure are applied. The required channel coefficient vector is obtained with adaptive eigen-pair tracking. Either forward or backward prediction errors can be used. The performance of the blind equalizer is examined by simulations. The MMSE of the optimum FSE is approached, and the algorithm exhibits robustness to channels with common subchannel zeros.

Fast convergence algorithms for active noise control in vehicles

When the reference signal for the FXLMS algorithm is taken from an acoustic sensor, convergence can be very slow due to great eigenvalue spread. Using a nonacoustic sensor, such as a tachometer, cancellation of narrow-band noise in the sensed fundamental frequency and harmonically related ones can be achieved very fast, although other periodic noises and the underlying broadband noise will remain. Backward prediction errors resulting at the various stages of an adaptive lattice predictor (ALP) represent a time-domain orthogonalization of the input signal. An ALP structure, with the acoustic reference as input signal, before a FXLMS, makes up the FXGAL algorithm. Due to orthogonalization, FXGAL is significantly faster compared to FXLMS with reference from a microphone. When compared to FXLMS with a tachometer signal, it is not faster but it can cancel every periodic noise, independently of the harmonical relation between them, as well as the underlying broadband noise. Depending on the relative weights of the different components (periodic harmonically related, periodic nonharmonically related, and broadband noise) in the reference signal, the FXGAL algorithm can be an excellent alternative to FXLMS. Comparative results between FXLMS (with acoustic and nonacoustic reference) and FXGAL will be presented. [Work supported by CONSI+D-DGA.]

Improved 2D joint process lattice for adaptive restorationof images

An improved method to obtain the 2D joint process lattice structure using recently developed 2D orthogonal quarter-plane filters for AR modelling is reported. It is shown that a set of orthogonal backward prediction error fields can be used to calculate the corresponding joint process coefficients.

Optimum quarter-plane autoregressive modeling of 2-D fields using four-field lattice approach

A new orthogonal four-field two-dimensional (2-D) quarter-plane lattice structure with a complete set of reflection coefficients is developed by employing appropriately defined auxiliary prediction errors. This work is the generalization of the three-parameter lattice filter proposed by Parker and Kayran (1984). After the first stage, four auxiliary forward and four auxiliary backward prediction errors are generated in order to obtain a growing number of 2-D reflection coefficients at successive stages. The theory has been proven by using a geometrical formulation based on the mathematical concepts of vector space, orthogonal projection, and subspace decomposition. It is shown that all four quarter-plane filters are orthogonal and thus optimum for all stages. In addition to developing the basic theory, a set of orthogonal backward prediction error fields for successive lattice parameter model stages is presented.