Nonlinear system identification using Wiener type Laguerre-Wavelet network model

Compact nonlinear black box models, capable of representing highly nonlinear systems, are of high demand both in industry and academia. In this paper it has been shown that the use of Laguerre basis filters coupled with a wavelet network in Wiener type model structure are capable of modeling highly nonlinear systems with acceptable accuracy. Although individual merits of orthonormal basis functions (especially Laguerre filters) and multiresolution wavelet decompositions and/or wavelet network have been well documented in various literatures on system identification in the past, their combinational advantages in nonlinear system identification has never been explored. Laguerre filter models have the ability to approximate linear systems (even with time delay) with a model order lower than the traditional ARX (e.g., FIR, AR) modeling. Use of Laguerre models for mildly nonlinear system is possible only with a piece-wise linear models. Wavelet basis functions have the property of localization in both time and frequency and they can approximate even severe nonlinearities with appreciable accuracy with fewer model terms. But wavelet approximations fail miserably, in terms of model parsimony, if used for approximating linear or mildly nonlinear systems. In the present work a Laguerre-Wavelet network model has been proposed which combines the Laguerre and Wavelet approximations. In the said model, merits of both these approximations are retained whereas their demerits are suppressed.     

Testing for long-range dependence in the presence of shifting means or a slowly declining trend, using a variance-type estimator

In this paper we examine the effects of certain types of non- stationarity on the detection of long-range dependence and on the estimation of the Hurst parameter H , when using a variance-type estimator. The resulting estimate of H can be misleading when the series has either a jump in the mean or a slow trend. In such a case, plotting the logarithm of the variance versus the logarithm of the level of aggregation gives a curve which is quite different from a straight line. A method for distinguishing between the effects of long-range dependence and these types of non-stationarity is developed.     

Bootstrapping Time Series Regressions with Integrated Processes

This paper studies the bootstrap procedures for time series regressions with integrated processes. Both estimation and hypothesis testing are studied. It is shown that the suggested bootstrap approximations to the distribution of the least squares estimator and the regression test statistic are asymptotically valid. A Monte Carlo experiment is conducted to evaluate the finite sample performance of these bootstrap procedures. The simulation results indicate that the bootstrap method provides reasonably good approximation to the distribution of the least squares estimator, and gives proper size and satisfactory power.     

SPECTRAL MAXIMUM LIKELIHOOD ESTIMATION OF A SIGNAL-TO-NOISE RATIO LYING IN THE VICINITY OF ZERO

Abstract. A time series model representing a decomposition into permanent plus transient components contains a deterministic component when the signal-to-noise ratio is equal to zero; otherwise, the permanent component is said to be stochastic. This distinction has important consequences in the analysis of economic phenomena. On the other hand, the absence of a stochastic permanent component in residuals from a time series regression may indicate cointegration. This paper considers the frequency domain estimation of the signal-to-noise ratio in a representative of the unobserved components model class. The sampling properties of the estimator from the resulting approximate spectral likelihood differ from those observed in the time domain and they vary substantially depending on whether the overall slope must be estimated or not. Further, it is shown that spectral estimates are T -consistent—instead of T ²-consistent in the time domain. These results may explain some of the differences in estimators from frequency domain approximations to the likelihood and exact maximum likelihood estimators, and may be of use when testing for deterministic trends.     

Wavelet transform analysis of pressure fluctuation signals in a pressurized bubble column

Several industrial bubble column reactors are operated at high pressures and it is important to understand its influence on the hydrodynamics and mass transfer. Pressure fluctuation signals in a pressurized bubble column are very complex since a bubble column behaves highly irregularly. The characteristics of the pressure fluctuations have been studied by several analytical methods. The recently developed technique of wavelet transform based on localized wavelet functions is applicable to the analysis of nonlinear or nonstationary pressure fluctuation signals: the time series of pressure fluctuation signals have been analyzed by means of discrete wavelet transform coefficients and multiresolution decomposition. By resorting to this wavelet transform, a pressure signal in a pressurized bubble column can be decomposed into its approximations and details at different resolutions. The energy of the details provides a measure of irregularity of the signal at various resolutions. And this wavelet transform enables us to obtain the frequency content of objects in a pressurized bubble column locally in time. In this work, the wavelet transform technique has been applied to the characterization of the fluctuating pressure signals generated in a pressurized bubble column, which are directly connected with the bubble phenomena in it.     

A Modified Nonparametric Prewhitened Covariance Estimator

Abstract. This paper proposes a fully modified version of the spectral matrix estimator (and the long‐run variance estimator as a special case) proposed originally by Xiao and Linton [Journal of Time Series Analysis (2002) Vol. 23, pp. 215–250], and derives its asymptotic results. A striking feature of the modified spectral matrix estimator is to achieve the convergence rate of O(T ^?8/9) in the mean squared error (MSE), which is usually achieved under the fourth‐order spectral window. However, this estimator does not sacrifice the positive definiteness of the resulting estimate for the rate improvement; it is Hermitian and positive definite in finite samples by construction. The faster convergence rate is established by a multiplicative bias correction of the crude spectral estimator under the second‐order spectral window. The approximations to some sensible definitions of the MSE of the estimator and the bandwidths that minimize the asymptotic MSEs are also derived. Monte Carlo results indicate that for a wide variety of processes the modified spectral matrix estimator reduces the bias without inflating the variance and thus improves the MSE, compared with the crude, bias‐uncorrected estimator.     

High-order approximations for intra-particle mass transfer

It is shown that the orthogonal collocation approach can be compared to the different approximations found in the literature using a straightforward quantitative criterion based on eigenvalue analysis. This work introduces a systematic and theoretically sound approach to the long-running discussion on LDF approximations, which started in 1955 with Glueckauf.     

Recursive versions Of the algorithm by krasker and welsch

Krasker and Welsch proposed an estimator related to GM-esti-mators for robust estimation in the classical linear model. In this paper we use standard methods for creating recursive algorithms to formulate a recursive version of their estimator. We also observe that a slight modification of the basic estimator by Krasker and Welsch has some symmetry properties which can be utilized to formulate an alternative recursive algorithm. The performance of the two algorithms for estimation of AR-parameters is studied in simulations. These indicate that both methods work well, though the second one seems to be slightly better. Hence the modified estimator introduced in this paper may be better suited for recursive versions.     

Banded and tapered estimates for autocovariance matrices and the linear process bootstrap

We address the problem of estimating the autocovariance matrix of a stationary process. Under short range dependence assumptions, convergence rates are established for a gradually tapered version of the sample autocovariance matrix and for its inverse. The proposed estimator is formed by leaving the main diagonals of the sample autocovariance matrix intact while gradually down‐weighting off‐diagonal entries towards zero. In addition, we show the same convergence rates hold for a positive definite version of the estimator, and we introduce a new approach for selecting the banding parameter. The new matrix estimator is shown to perform well theoretically and in simulation studies. As an application, we introduce a new resampling scheme for stationary processes termed the linear process bootstrap (LPB). The LPB is shown to be asymptotically valid for the sample mean and related statistics. The effectiveness of the proposed methods are demonstrated in a simulation study.     

A simple example of an indirect estimator with discontinuous limit theory in the MA(1) model

Indirect estimators usually emerge from two‐step optimization procedures. Each step in such a procedure may induce complexities in the asymptotic theory of the estimator. In this note, we are occupied with a simple example in which the estimator defined by the inversion of the binding function has a ‘discontinuous’ limit theory even in cases where the auxiliary one does not. This example lives in the framework of estimation of the MA (1) parameter. The ‘discontinuities’ involve the dependence of the rate of convergence on the parameter, the non‐continuity of the limit distribution w.r.t. the parameter and the estimator's non‐regularity. We are also occupied with a more complex example where the discontinuities occur because of complexities induced in any step of the defining procedure. We present some Monte Carlo evidence on the quality of the approximations from the limit distributions. Copyright © 2014 Wiley Publishing Ltd     

Validity of sequential bootstrap for subcritical and critical branching processes

Sequential bootstrapping for branching processes with immigration is proposed. The asymptotic validity of a sequential bootstrap estimator of the offspring mean in is estabilished for all values of m ? (0, 1]. This result is to be constrasted with the standard (non-sequential)bootstrapping where the bootstrap estimator estimator has been shown to asymptotically invalid (see, Sriram (1992)).     

Online composition estimation and experiment validation of distillation processes with switching dynamics

This study explores both the theoretical and experimental investigations of applying a continuum and noncontinuum state estimator to composition estimation in a distillation process with switching dynamics. In a hybrid distillation modeling, the column compositions are considered as continuum states while the operating modes are modeled as noncontinuum states. A moving horizon estimator (MHE), which has the capability to handle process constraints is developed for composition estimation in a distillation process under known switching mode criteria using the available temperature measurements. The performance of a MHE is shown to be better than that of EKF in handling process and measurement noise under switching dynamics. For some situations where the system operating mode transition is unknown, a new approach to state estimation under unknown switching functions is investigated. The proposed method combines a MHE for composition estimation with a mode change detector to detect a change in the system operating mode and an operating mode estimator to identify the functioning mode. In the presence of both the measurement noise and plant-model mismatch, the developed estimator is shown to be effective in estimating both the column composition and the system operating mode accurately.     

A simplified method for analyzing mold filling dynamics. Part II: Extensions and comparisons with experiment

A dimensional analysis based on four parameters has been developed previously to predict injection pressure; clamp force, and bulk temperature for the injection molding of amorphous materials in center-gated disk-shaped cavities. In this paper geometric and semicrystalline-materials approximations are introduced and tested for extending the previous analysis to include multigated thin cavities and semicrystalline materials. The combination of these approximations and the previous analysis, known hereafter as the Radial Flow Method (RFM), greatly simplifies the analysis of mold filling. The geometric approximation, which is based on a simple model for the axial stress distribution in the cavity, is shown to give reasonable predictions when compared with experimental data and a numerical two-directional flow simulation for the filling of an off-center-gated rectangular cavity with acrylonitrilebutadiene-styrene copolymer (ABS). The semicrystallinematerials approximation, in which heat capacity and viscosity changes during crystallization are neglected, is shown to give good agreement with experimental data for the filling of a center-gated disk-shaped cavity with polypropylene. As an illustration, the Radial Flow Method is used to analyze the molding of a large, thin-wall automobile interior trim panel. The inlet melt temperature, mold-wall temperature, part thickness, injection rate, viscosity and gate locations are varied in a series of calculations to determine the relative effectiveness of these variables in lowering the injection pressure and Clamp force. The results obtained with the Radial Flow Method are in good agreement with those obtained by a finiteelement simulation of two-directional flow.     

High-order approximations for noncyclic and cyclic adsorption in a biporous adsorbent

An accurate and realistic model for transient diffusion and adsorption in a biporous pellet is typically represented by two coupled second-order partial differential equations. The model, however, has been rarely used in practice because of its mathematical complexity and bulky numerical computation, and approximations of the model have been used instead. But the accuracy of the available approximations has been limited and not enough for detailed analysis and simulation of the mass transfer process. Therefore, in this study, we develop for the first time high-order approximations, of up to third order, for noncyclic and cyclic adsorption in a biporous pellet, respectively. The approximations are in the form of a state equation which consists of first-order differential equations ; the number of the equations is the same as the approximation order. The approximations are easy to use and their accuracy dramatically increases with increasing approximation order, so that the second-or the third-order approximations can effectively substitute the complex biporous diffusion model.     

On Kay's Frequency Estimator

Kay has proposed a technique for estimating the frequency of a complex sinusoid in additive noise, the real and imaginary parts of which are independent and normally distributed with means zero and the same variance. For fixed sample size the estimator achieves the Cramer–Rao lower bound for unbiased estimators of the frequency in the limit as the signal to noise ratio approaches infinity. It has been noted by Lovell and Williamson, however, that the estimator is biased. It is shown in this paper that under Kay's assumptions the estimator is not consistent (for fixed signal to noise ratio the estimator converges almost surely, as the sample size N increases, to a number that is not the true frequency, no matter how large the signal to noise ratio). A class of distributions for the additive noise is proposed under which the technique is strongly consistent and has the correct order of asymptotic variance, namely N ⁻³, for the case where there is some a priori knowledge concerning the range of the frequency. For this class a normal central limit theorem is developed.     

The Influence of Water Coating on the Optical Scattering Properties of Fractal Soot Aggregates

The effect of water coating of constituent monomers on the optical single-scattering properties of fractal soot aggregates is investigated numerically using core-mantle theory and approximations involving two effective medium theories. A cluster–cluster aggregation algorithm is used to numerically generate fractal aggregates, and the core-mantle Generalized Multi-particle Mie (GMM) method is used to compute the exact single-scattering properties of soot aggregates with water-coated monomers. Comparisons are then made with results obtained using approximations that combine either the GMM method or the Rayleigh–Debye–Gans (RDG) method with either the Maxwell-Garnett or the Bruggeman effective medium approximation (a total of four approximation methods). The optical properties calculated are the extinction and absorption cross sections, the single-scattering albedo, and the phase matrix of water-coated fractal aggregates; these calculations are done for two wavelengths, 0.628 μm and 1.1 μm. Water coating of the fractal aggregates is shown to increase the extinction and absorption cross sections, the single-scattering albedo, and forward scattering, but decrease backward scattering. The combination GMM + Maxwell-Garnett gives approximations that are quite good over a range of coating thicknesses and aggregate size. The combination GMM + Bruggeman performs less well, overestimating the extinction and absorption cross sections and underestimating the single-scattering albedo. In the case of RDG, the better combination is with the Bruggeman approximation, but the errors involved are greater than with the GMM + Maxwell-Garnett combination. Results from simple idealized calculations indicate that the differences between results with the Maxwell-Garnett and Bruggeman approximations should be even larger in cases of aerosol cores that are less absorptive than soot.

Copyright 2012 American Association for Aerosol Research     

BIAS CALCULATIONS FOR ADAPTIVE URN DESIGNS

A clinical trial model is considered in which two treatments with immediate binary responses are to be compared. An adaptive urn design is used to assign patients to the treatments. The bias and variance of the maximum likelihood estimators of the probabilities of success are derived by differentiating the fundamental identity of sequential analysis. By embedding the design in a continuous-time process, probability generating functions are then calculated to obtain approximations for the bias and variance. Simulation is used to assess the accuracy of the approximations. It is shown that the bias cannot be ignored, and that the adaptive rules which are subcritical in nature have the most mathematically tractable bias and are the least variable. Methods for correcting for the bias are also addressed.     

Polynomial Trend Regression With Long‐memory Errors

Abstract. For a time series generated by polynomial trend with stationary long‐memory errors, the ordinary least squares estimator (OLSE) of the trend coefficients is asymptotically normal, provided the error process is linear. The asymptotic distribution may no longer be normal, if the error is in the form of a long‐memory linear process passing through certain nonlinear transformations. However, one hardly has sufficient information about the transformation to determine which type of limiting distribution the OLSE converges to and to apply the correct distribution so as to construct valid confidence intervals for the coefficients based on the OLSE. The present paper proposes a modified least squares estimator to bypass this drawback. It is shown that the asymptotic normality can be assured for the modified estimator with mild trade‐off of efficiency even when the error is nonlinear and the original limit for the OLSE is non‐normal. The estimator performs fairly well when applied to various simulated series and two temperature data sets concerning global warming.     

Mechanical and electrical analysis of carbon black networking in elastomers under strain

The influence of quasi‐static straining and relaxation at fixed strain on carbon black networking in elastomers is investigated by simultaneous stress and piezo‐resistivity measurements. The mechanical and the electrical response are analyzed in the framework of the dynamic flocculation model considering the cyclic breakdown and re‐aggregation of filler–filler bonds. In this model, the mechanical hysteresis is related to the stress contribution of stretched filler clusters, which is positive in the up‐cycle, but negative in the down‐cycle because of stretching of clusters in lateral direction. The analysis is supported by dielectric relaxation spectroscopy delivering information about the morphology of the filler network and the gap distance between adjacent filler particles, i.e., the length of filler–filler bond. This is shown to be relevant for a qualitative understanding of piezo‐resistivity and mechanical relaxation behavior of the investigated rubbers. During the relaxation phase at fixed strain, the mechanical stress decreases during the up‐cycle and increases during the down‐cycle, which in both cases results from the relaxation of cluster stress leading to a homogenization of the stress field. The decrease of the resistivity in the relaxation phase independent of cycle direction is shown to be related to a retarded re‐aggregation of filler–filler bonds in compression directions. POLYM. ENG. SCI., 2011. © 2011 Society of Plastics Engineers     

COMPARISON OF TIME AND CROSS-SECTIONAL AGGREGATION UNDER A TIME SERIES RANDOM COMPONENT MODEL

Abstract. Small-area estimation under a stationary time series random component model is considered. Cross-sectional aggregation and varying degrees of time aggregation are treated as competing prediction methods. An estimated mean-squared prediction error criterion is used to compare these methods. Some exact and asymptotic properties of this criterion are developed, a consistent estimator of the associated asymptotic variance is presented and simultaneous approximate confidence intervals for the mean-squared prediction errors are discussed. Time aggregation of a single series is considered as a special case. In addition, an extension to the assessment of mean-squared prediction errors of synthetic small-area predictors is outlined.