nonparametric neural network estimation of lyapunov exponents and a direct test for chaos this paper derives the asymptotic distribution of the nonparametric neural network estimator of the lyapunov exponent in a noisy system positivity of the lyapunov exponent is an operational definition of chaos we introduce a statistical framework for testing the chaotic hypothesis based on the estimated lyapunov exponents and a consistent variance estimator a simulation study to evaluate small sample performance is reported we also apply our procedures to daily stock return data in most cases the hypothesis of chaos in the stock return series is rejected at the 1 level with an exception in some higher power transformed absolute returns

a recursive procedure for arma modeling this paper presents a two part fast recursive algorithm for arma modeling the algorithm first obtains estimates of the p autoregressive coefficients from a set of p extended yule walker equations an exact recursive lattice algorithm for this estimator is then derived the q 1 numerator spectrum coefficients are then obtained by using one of the output data sequences of this lattice algorithm the complete recursive algorithm is fast in the sense that o p q computations are required for each update moreover an exponential forgetting factor is incorporated to facilitate tracking of time variations in the time series

asymptotic properties of high order yule walker estimates of the ar parameters of an arma time series the high order yule walker equations are used to estimate the autoregressive parameters of an autoregressive moving average time series the asymptotic statistical properties of these estimates are derived it is shown that they are asymptotically unbiased and normal the covariance matrix of the limit distribution is derived the special case of estimating the autoregressive parameters of a noise corrupted autoregressive series is also examined

statistical analysis of tls based prony techniques abstract we present an analysis of parameter variance statistics for the tls prony method applied to damped exponential signals we derive the covariance matrix of the estimated parameters for this method the parameters include the magnitudes and angles of the poles and the magnitudes and angles of the amplitude coefficients we verify the theoretical results using monte carlo simulations studies we also compare the variance results to the corresponding cramer rao bounds for several cases

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

computer vision for robust 3d aircraft recognition with fast library search abstract a fast but accurate classification method is presented which takes the fundamental problems of aircraft recognition in 3d space into consideration elliptic fourier descriptors are used for aircraft classification and pose determination from a two dimensional image recorded at an arbitrary viewing angle the computation model of the nearest neighbor classification rule or simply nnr is analysed to come up with proper necessary conditions used by our method for reducing the set of near neighbors of a test projection for a fast partial library search on the other hand our method is robust to noise by incorporating a perturbation analysis of feature variation into the classifier design both graphics generated data and real data of five aircraft have been used to demonstrate the applicability of the new method the results indicate that the method is almost as accurate as the nnr method but is much faster the method is also shown to outperform other existing classification methods including the decision tree classification method

statistical analysis of music and subspace rotation estimates of sinusoidal frequencies consideration is given to the analysis of the large sample second order properties of multiple signal classification music and subspace rotation sur methods such as esprit for sinusoidal frequency estimation explicit expressions for the covariance elements of the estimation errors associated with either method are derived these expressions of covariances are then used to analyze and compare the statistical performances of the music and sur estimation sure methods both music and sure are based on the eigendecomposition of a sample data covariance matrix the expressions for the estimation error variances derived are used to study the dependence of music and sure performances on the dimension of the data covariance matrix used

minimum variance linear estimation of amplitudes for exponential signal models this article presents the minimum variance consistent linear estimator for amplitude parameters in exponential signal models a simple heuristic algorithm is presented to compute the weighting matrix that minimizes error variance the resulting weighted least squares estimator accounts for the statistics of pole estimation errors additionally analysis of binary diagonal weighting matrices demonstrates that for unweighted least squares the amplitude variance is reduced by truncating the vandermonde system of equations

an algorithm for the n lyapunov exponents of an n dimensional unknown dynamical system abstract an algorithm for estimating lyapunov exponents of an unknown dynamical system is designed the algorithm estimates not only the largest but all lyapunov exponents of the unknown system the estimation is carried out by a multivariate feedforward network estimation technique we focus our attention on deterministic as well as noisy system estimation the performance of the algorithm is very satisfactory in the presence of noise as well as with limited number of observations

rational invariant subspace approximations with applications subspace methods such as music minimum norm and esprit have gained considerable attention due to their superior performance in sinusoidal and direction of arrival doa estimation but they are also known to be of high computational cost in this paper new fast algorithms for approximating signal and noise subspaces and that do not require exact eigendecomposition are presented these algorithms approximate the required subspace using rational and power like methods applied to the direct data or the sample covariance matrix several esprit as well as music type methods are developed based on these approximations a substantial computational saving can be gained comparing with those associated with the eigendecomposition based methods these methods are demonstrated to have performance comparable to that of music yet will require fewer computations to obtain the signal subspace matrix

separation of multiple time delays using new spectral estimation schemes the problem of estimating multiple time delays in presence of colored noise is considered this problem is first converted to a high resolution frequency estimation problem then the sample lagged covariance matrices of the resulting signal are computed and studied in terms of their eigenstructure these matrices are shown to be as effective in extracting bases for the signal and noise subspaces as the standard autocorrelation matrix which is normally used in music and the pencil based methods frequency estimators are then derived using these subspaces the effectiveness of the method is demonstrated on two examples a standard frequency estimation problem in presence of colored noise and a real world problem that involves separation of multiple specular components from the acoustic backscattered from an underwater target

consistency properties of least squares estimates of autoregressive parameters in arma models a unified treatment of the consistency properties of the ordinary least squares estimates in an autoregressive fitting of time series from nonstationary or stationary autoregressive moving average models is given for a given model the orders of autoregressions which produce consistent estimates are obtained and the limiting values hence the biases of the estimates of other autoregressions are investigated

stochastic models for closed boundary analysis representation and reconstruction the analysis of closed boundaries of arbitrary shapes on a plane is discussed specifically the problems of representation and reconstruction are considered a one to one correspondence between the given closed boundary and a univariate or multivariate sequence of real numbers is set up univariate or multivariate circular autoregressive models are suggested for the representation of the sequence of numbers derived from the closed boundary the stochastic model representing the closed boundary is invariant to transformations like sealing translation choice of starting point and rotation over angles that are multiples of 2 pi n where n is the number of observations methods for estimating the unknown parameters of the model are given and a decision rule for choosing the appropriate number of coefficients is included constraints on the estimates are derived so that the estimates are invariant to the transformations of the boundaries the stochastic model enables the reconstruction of a dosed boundary using fft algorithms results of simulations are included and the application to contour coding is discussed

signal plus noise models in shape classification abstract in this study we show how a signal plus noise formulation can be used to improve the performance of auto regressive shape classifiers in environments with time varying noise characteristics variations of this type may be due to changes in atmospheric conditions or the relative effect of discretization errors this approach is compared to existing methodologies using a number of noisy templates in a cluttered environment typical applications are to aircraft identification and automatic signature validation

extinction and periodic solutions for an impulsive sir model with incidence rate stochastically perturbed abstract in this paper we consider an impulsive sir model with incidence rate stochastically perturbed first we prove the existence and uniqueness of the global positive solution by constructing the equivalent system without impulse second we obtain a sufficient condition which determines epidemic to extinct then we demonstrate the existence and global attraction of the boundary periodic solution under the certain condition finally we present a sufficient condition which allows the existence of a positive periodic solution

the overdetermined recursive instrumental variable method a recursive instrumental variable algorithm is derived for the overdetermined case in which the number of instruments is greater than the number of estimated parameters in some applications the algorithm provides improved parameter estimation accuracy compared to the standard recursive instrumental variable method the performance of the algorithm is illustrated by simulation results

new results on employing cumulants for retrieving sinusoids in colored non gaussian noise a novel analysis is presented which shows that signals consisting of sinusoids plus non gaussian noise have a new signal to noise ratio nsnr associated with their fourth order cumulants this nsnr is a multiple of the original snr when the noise is colored and the original snr is equal to or greater than one therefore employing cumulants for harmonic retrieval problems is now an approach capable of handling not only gaussian noise but also colored non gaussian noise the presented analysis is verified via computer simulations

pulse vaccination on seir epidemic model with nonlinear incidence rate in this paper we consider an seir epidemic model with two time delays and nonlinear incidence rate and study the dynamical behavior of the model with pulse vaccination by using the floquet theorem and comparison theorem we prove that the infection free periodic solution is globally attractive when r 1

the asymptotic distribution of nonparametric estimates of the lyapunov exponent for stochastic time series this paper derives the asymptotic distribution of a smoothing based estimator of the lyapunov exponent for a stochastic time series under two general scenarios in the first case we are able to establish root t consistency and asymptotic normality while in the second case which is more relevant for chaotic processes we are only able to establish asymptotic normality at a slower rate of convergence we provide consistent confidence intervals for both cases we apply our procedures to simulated data

sampling moments and reconstructing signals of finite rate of innovation shannon meets strang fix consider the problem of sampling signals which are not bandlimited but still have a finite number of degrees of freedom per unit of time such as for example nonuniform splines or piecewise polynomials and call the number of degrees of freedom per unit of time the rate of innovation classical sampling theory does not enable a perfect reconstruction of such signals since they are not bandlimited recently it was shown that by using an adequate sampling kernel and a sampling rate greater or equal to the rate of innovation it is possible to reconstruct such signals uniquely these sampling schemes however use kernels with infinite support and this leads to complex and potentially unstable reconstruction algorithms in this paper we show that many signals with a finite rate of innovation can be sampled and perfectly reconstructed using physically realizable kernels of compact support and a local reconstruction algorithm the class of kernels that we can use is very rich and includes functions satisfying strang fix conditions exponential splines and functions with rational fourier transform this last class of kernels is quite general and includes for instance any linear electric circuit we thus show with an example how to estimate a signal of finite rate of innovation at the output of an rc circuit the case of noisy measurements is also analyzed and we present a novel algorithm that reduces the effect of noise by oversampling

instrumental variable methods for arma spectral estimation the modified yule walker technique of arma spectral estimation is shown to be a special case of the instrumental variable method of system identification several recursive instrumental variable algorithms are proposed for adaptive spectral estimation including a new algorithm for the overdetermined case an efficient lattice algorithm is presented for solving the modified yule walker equations given the sample correlation coefficients the performance of some of the algorithms is illustrated by simulation results

variance of least squares estimators for a damped sinusoidal process the variance of least squares estimators for the parameter estimation of a damped sinusoidal process is analyzed based on first order perturbation analytical expressions for the variances of the frequency damping factor amplitude and phase estimators are derived explicit expressions are given for both damped and undamped single mode cases the effect of mode separation on the accuracy is investigated through the two mode case the dependence of the variances on number of data points model order signal to noise ratio and mode separation is investigated both analytically and numerically for practical applications extensive monte carlo simulation results are given to verify enhance and illustrate the analytical expressions

large sample identification and spectral estimation of noisy multivariate autoregressive processes when large samples are available identification and spectral estimation problems of a noisy multivariate autoregressive process are solved independently of the probability law governing the observed data the procedure of solving the problems is divided into 1 estimate parameters by solving pertinent block toeplitz and hankel matrices as well as compute the modified akaike information criterion for an assumed order 2 select the order and the parameter estimators corresponding to the ones which minimize the criterion function 3 compute the asymptotic distribution functions of the parameter estimators 4 estimate power spectral density functions and coherency spectra 5 compute the estimate confidence bounds these five steps are numerically very efficient because the matrix equations related to block hankel and toeplitz matrices are well studied in the literature the procedure is also robust because it is independent of the probability law governing the observed data

the effects of pulse vaccination on seir model with two time delays abstract in this paper impulsive vaccination has been applied to control the spread and transmission of an infectious disease an seir epidemic model with two time delays and pulse vaccination is formulated the exact infection free periodic solution of the impulsive epidemic system is obtained moreover we show that if the vaccination rate is larger than the infectious population disappear so the disease dies out while if the vaccination rate is less than the infectious population persist our results indicate that a large vaccination rate or a short period of pulsing or a long latent period of the disease is a sufficient condition for the eradication of the disease

ld sup 2 arma identification algorithm the authors present the ld sup 2 arma identifier a novel algorithm that solves the essentially nonlinear autoregressive moving averaged arma identification problem with a linear procedure in two steps an order selection algorithm followed by an arma parameter estimator the determination of the ar and ma coefficients involves the solution of two dual systems of linear equations these systems decouple the estimation of the autoregressive component from the estimation of the moving average component the selection of the number of poles and of the number of zeros is accomplished by a scheme that minimizes the mismatch of the data to each proposed model simulated experiments on the proposed order selection procedure are presented

instrumental variable approach to array processing in spatially correlated noise fields high performance signal parameter estimation from sensor array data is a problem which has received much attention a number of so called eigenvector ev techniques such as music esprit wsf and mode have been proposed in the literature the ev techniques for array processing require knowledge of the spatial noise correlation matrix that constitutes a significant drawback a novel instrumental variable iv approach to the sensor array problem is proposed the iv technique relies on the same basic geometric properties as the ev methods to obtain parameter estimates however by exploiting the temporal correlation of the source signals no knowledge of the spatial noise covariance is required the asymptotic properties of the iv estimator are examined and an optimal iv method is derived computer simulations are presented to study the properties of the iv estimators in samples of practical length the proposed algorithm is also shown to perform better than music on a full scale passive sonar experiment

a comparison of two linear methods of estimating the parameters of arma models a finite order stationary and minimum phase arma autoregressive moving average p q model is equivalent to an infinite order ar autoregressive model two methods of estimating the parameters of the arma p q model by solving only linear equations are based on or closely related to this equivalence relation one method was derived directly from the equivalence relation by d graupe et al ibid vol ac 20 p 104 107 feb 1975 the other was derived by s li and b w dickinson ibid vol ac 31 p 275 278 mar 1986 and ieee trans acoust speech signal process vol assp 36 p 502 512 apr 1988 based on an iterated least squares regression approach the end results bear close resemblance to those of graupe et al the two methods are compared and ways to improve the parameter estimates are suggested

optimal instrumental variable estimates of the ar parameters of an arma process the modified yule walker myw equations for estimating the ar parameters of an arma process are presented as a special case of an instrumental variable iv method the consistency and accuracy of the ar parameter estimates are studied it is shown that estimation accuracy increases monotonically with the number of myw equations for an optimal choice of the weighting matrix used in the least squares solution of these equations the asymptotic error covariance of the optimal iv method equals that of the prediction error method the results of this paper verify experimental results reported in the literature regarding the performance of the myw method and provide the necessary accuracy analysis furthermore they suggest several simple asymptotically efficient multistep algorithms for estimating the ar parameters which are presented in a companion paper

measurement of the lyapunov spectrum from a chaotic time series the exponential divergence or convergence of nearby trajectories lyapunov exponents is conceptually the most basic indicator of deterministic chaos we propose a new method to determine the spectrum of several lyapunov exponents including positive zero and even negative ones from the observed time series of a single variable we have applied the method to various known model systems and also to the rayleigh b enard experiment and have elucidated the dependence of the lyapunov exponents on the rayleigh number

a new technique for arma system identification and rational approximation the fundamental problem of identifying a linear time invariant system from measured samples of its output response to a known input is addressed utilizing a new and simple deterministic theory founded on well established passive network concepts the analysis together with documented numerical results demonstrates that the proposed method achieves two goals stable rational minimum phase transfer functions can be identified with a priori knowledge of either numerator or denominator degrees and stable rational minimum phase pade like approximations appear to be generated automatically in the nonrational case to substantiate these claims a detailed theoretical exposition of the basic ideas an extensive discussion of numerical results and a summary of related results are given

adaptive algorithms for constrained arma signals in the presence of noise a family of algorithms is developed for adaptive parameter estimation of constrained autoregressive moving average arma signals in the presence of noise these algorithms utilize a priori information about the signal s properties such as its spectral type for example low pass bandpass etc or a spatial domain characteristic special applications include modeling of autoregressions ar and signals of known spectral type in the presence of noise signal deconvolution image deblurring and multipath parameter estimation selected results of simulations are included to demonstrate the performance of the algorithms

a statistical framework for testing chaotic dynamics via lyapunov exponents abstract one common criticism of the existing lyapunov exponents algorithms is the absence of a distributional theory which provides a framework for the statistical hypothesis testing for the calculated lyapunov exponents this paper presents a methodology to calculate the empirical distributions of lyapunov exponents by using a bootstrapping technique the methodology of the paper provides a formal test of chaos under the null hypothesis the numerical examples show that the method works well with small data sets

comparison of some instrumental variable methods consistency and accuracy aspects a general analysis of various instrumental variable methods is given the investigation includes methods for open loop operation as well as methods designed for closed loop operation the instrumental variables are formed as different combinations of inputs delayed inputs delayed outputs filtered inputs and external setpoint variations necessary conditions as well as sufficient conditions for consistency are explicitly derived and discussed the paper also includes some simple counter examples to general consistency for some common instrumental variable methods also asymptotic accuracy properties are reviewed it is e g studied how the use of filters will influence the covariance matrix of the parameter estimates

estimation of the parameters of a random amplitude sinusoid by correlation fitting we consider the best asymptotic accuracy that can be achieved when estimating the parameters of a random amplitude sinusoid from its sample covariances an estimator based on matching in a weighted least squares sense the sample correlation sequence to the theoretical sequence is presented the asymptotic properties of the estimator are analyzed a lower bound on the estimation of the parameters from sample covariances is derived this bound is shown to be attainable by appropriately choosing the weighting matrix however the unweighted nonlinear least squares estimate performance is shown to come close to the lower bound the influence of the number of samples the number of correlation samples and the lowpass envelope characteristics are studied finally a comparison with yule walker yw methods is given

new resampling method to assess the accuracy of the maximal lyapunov exponent estimation abstract in this paper we propose a novel method to assess standard errors and confidence intervals for the maximal lyapunov exponent estimated on time continuous chaotic systems the method is based on resampling the original series by means of spline interpolation in the time domain in such a way new time series of increased size are obtained and the sample distribution of the estimators is constructed the method is explained and tested on the basis of computer simulations both for clean and noisy series we give evidence that the distribution of the maximal lyapunov exponent calculated by this method fairly agrees with the one obtained by true series with different initial conditions an empirical criterion for the choice of the parameters of the resampling is also suggested

modified pisarenko harmonic decomposition for single tone frequency estimation in this correspondence based on an alternative derivation of the pisarenko harmonic decomposition phd method a new asymptotically unbiased estimator for the frequency of a single real tone in white noise is devised with the use of novel sample covariance expressions furthermore extension to sample covariances with higher lags for performance enhancement is investigated while a simple and effective scheme is suggested to resolve the corresponding frequency ambiguity problem the variance of the modified pisarenko s method is also derived which is then utilized to find the best estimate among all admissible solutions from various sets of sample covariances computer simulations are included to corroborate the theoretical development and to demonstrate that the proposed approach outperforms several existing low complexity frequency estimators in terms of nearly uniform performance and estimation accuracy

oversampled a d conversion and error rate dependence of nonbandlimited signals with finite rate of innovation we study the problem of a d conversion and error rate dependence of a class of nonbandlimited signals with finite rate of innovation in particular we analyze a continuous periodic stream of diracs characterized by a finite set of time positions and weights previous research has only considered sampling of this type of signals ignoring the presence of quantization necessary for any practical implementation to this end we first define the concept of consistent reconstruction and introduce corresponding oversampling in both time and frequency high accuracy in a consistent reconstruction is achieved by enforcing the reconstructed signal to satisfy three sets of constraints related to low pass filtering quantization and the space of continuous periodic streams of diracs we provide two schemes to reconstruct the signal for the first one we prove that the estimation mean squared error of the time positions is o 1 r sub t sup 2 r sub f sup 3 where r sub t and r sub f are the oversampling ratios in time and frequency respectively for the second scheme it is experimentally observed that at the cost of higher complexity the estimation accuracy lowers to o 1 r sub t sup 2 r sub f sup 5 our experimental results show a clear advantage of consistent over nonconsistent reconstruction regarding the rate we consider a threshold crossing based scheme where as opposed to previous research both oversampling in time and in frequency influence the coding rate we compare the error rate behavior resulting on the one hand from increasing the oversampling in time and or frequency and on the other hand from decreasing the quantization stepsize

the bootstrap and lyapunov exponents in deterministic chaos inasmuch as lyapunov exponents provide a necessary condition for chaos in a dynamical system confidence bounds on estimated lyapunov exponents are of great interest estimates derived either from observations or from numerical integrations are limited to trajectories of finite length and it is the uncertainties in the distribution of these finite time lyapunov exponents which are of interest within this context a bootstrap algorithm for quantifying sampling uncertainties is shown to be inappropriate for multiplicative ergodic statistics of deterministic chaos this result remains unchanged in the presence of observational noise as originally proposed the algorithm is also inappropriate for general nonlinear stochastic processes a modified version is presented which may prove of value in the case of stochastic dynamics a new approach towards quantifying the minimum duration of observations required to estimate global lyapunov exponents is suggested and is explored in a companion paper

testing chaotic dynamics via lyapunov exponents abstract this paper presents a bootstrap based test statistic for testing the presence of chaotic dynamics from data by using the lyapunov exponents in particular a one sided test statistic in gencay s gencay physica d 89 1996 261 266 framework is designed and its small sample properties are tested on the henon map the numerical examples show that the test statistic has desirable small sample properties

an attractor in a solar time series a time series analysis of observed solar radio pulsations suggests that there must be a low dimensional attractor the power spectrum cannot be interpreted as a superposition of periodic components estimates of the maximum lyapunov exponent and of the kolmogorov entropy give some indications for a deterministic chaos in order to study the limitations inherent in small data samples we include data from the lorenz model and an artificial noise record consequences for the physical modelling of the pulsation event are discussed

two dimensional prony modeling and parameter estimation a new method for estimating the two dimensional 2d exponential modes and amplitude coefficients in a prony model is presented this method involves two parts each utilizing a 1d singular value decomposition based technique and is capable of locating frequencies anywhere in the 2d frequency plane simulations are shown which demonstrate the performance of the algorithm

perturbation analysis of tk method for harmonic retrieval problems the authors present a first order perturbation analysis of the tufts kumaresan tk 1980 method used to estimate frequencies of complex sinusoids in small additive noise several fundamental properties are presented and proved further illustrations are provided through numerical examples

maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements the problem of estimating the frequencies phases and amplitudes of sinusoidal signals is considered a simplified maximum likelihood gauss newton algorithm which provides asymptotically efficient estimates of these parameters is proposed initial estimates for this algorithm are obtained by a variation of the overdetermined yule walker method and periodogram based procedure use of the maximum likelihood gauss newton algorithm is not however limited to this particular initialization method some other possibilities to get suitable initial estimates are briefly discussed an analytical and numerical study of the shape of the likelihood function associated with the sinusoids in noise process reveals its multimodal structure and clearly sets the importance of the initialization procedure some numerical examples are presented to illustrate the performance of the proposed estimation procedure comparison to the performance corresponding to the cramer rao lower bound is also presented using a simple expression for the asymptotic cramer rao bound covariance matrix derived in the paper

a high order yule walker method for estimation of the ar parameters of an arma model abstract the high order yule walker hoyw method is often used to estimate the frequencies of sinusoidal signals from noisy measurements a modification of the hoyw method is proposed for estimating the parameters of the autoregressive part of general arma processes the prime application of the hoyw method introduced in this paper is the estimation of the parameters of autoregressions from noisy measurements the performance of the proposed technique is illustrated by some numerical examples

recursive least squares ladder estimation algorithms recursive least squares ladder estimation algorithms have attracted much attention recently because of their excellent convergence behavior and fast parameter tracking capability compared to gradient based algorithms we present some recently developed square root normalized exact least squares ladder form algorithms that have fewer storage requirements and lower computational requirements than the unnormalized ones a hilbert space approach to the derivations of magnitude normalized signal and gain recursions is presented the normalized forms are expected to have even better numerical properties than the unnormalized versions other normalized forms such as joint process estimators e g adaptive line enhancer and arma pole zero models will also be presented applications of these algorithms to fast or zero startup equalizers adaptive noise and echo cancellers non gaussian event detectors and inverse models for control problems are also mentioned

on stability and root location of linear prediction models simple algebraic proofs are given for some results pertaining to the location of the roots of linear prediction models also called all pole or autoregressive models

a bivariate autoregressive technique for analysis and classification of planar shapes a bivariate autoregressive model is introduced for the analysis and classification of closed planar shapes the boundary coordinate sequence of a digitized binary image is sampled to produce a polygonal approximation to an object s shape this circular sample sequence is then represented by a vector autoregressive difference equation which models the individual cartesian coordinate sequences as well as coordinate interdependencies several classification features which are functions or transformations of the estimated coefficient matrices and the associated residual error covariance matrices are developed these features are shown to be invariant to object transformations such as translation rotation and scaling laboratory experiments involving object sets representative of industrial shapes are presented superior classification results are demonstrated

statistical analysis of pisarenko s method for sinusoidal frequency estimation statistical analysis is performed for pisarenko s method of sinusoidal frequency estimation by using the periodogram technique under the assumption that the data consist of several sinusoids plus white noise we give an asymptotic expression of the error variance of the sinusoidal frequency estimator also a modified method is proposed this is computationally efficient but the statistical property is not improved we also present simulation results and compare them to the theoretical ones lastly we point out that the pisarenko s method can be implemented as an on line bias compensated least squares algorithm thus avoiding the eigenvalue and eigenvector calculations

an efficient method to compute consistent estimates of the ar parameters of an arma model a computationally efficient method for consistent estimation of the ar parameters of an arma model is obtained based on an iterated least squares approach proposed by tsay and tiao the method is derived from the lattice filter structure which generates an orthogonal basis for the hilbert space of interest it is shown that for the arma p q model we can get consistent estimates of the ar parameters by using a least squares lattice estimation algorithm and solving a system of q linear equations

auto regressive shape classifiers in time varying noise abstract good results have been reported in a number of recent studies where auto regressive shape classifiers have been used to allocate objects to a pre determined number of classes it is shown that this approach may give unsatisfactory results when applied to imaging systems whose noise characteristics vary substantially over time such variation may be due to changes in atmospheric conditions or the relative effect of discretization errors observations are based on the reconstruction of a number of noisy synthetic templates

information tradeoffs in using the sample autocorrelation function in arma parameter estimation this paper considers bounds on the statistical efficiency of estimators of the poles and zeros of an arma process based on estimates of the process autocorrelation function acf special attention is paid to autoregressive ar and ar plus white noise processes it is seen that reducing the arma process data to a given set of consecutive lags of the popular lagged product acf estimates prior to parameter estimation increases cramer rao bounds on the generalized error covariance a parametric study of the bound deterioration for some illustrative signal and noise situations reveals some empirical strategies for choosing acf estimate lags to preserve statistical information analysis is based on the relative information index rii 2 and derivations of the large sample fisher s information matrix for the raw data and for the lagged product acf estimate of an arma process are included

optimal high order yule walker estimation of sinusoidal frequencies the asymptotic properties of sinusoidal frequency estimators based on the high order yule walker hoyw equations were analyzed recently the results of that analysis are used to propose two classes of frequency estimators one class uses singular value decomposition and the other uses a sparse matrix solution both classes entail two estimation steps the first step generates initial estimates which are used to obtain an optimal weighting matrix and the second step generates an optimally weighted estimate each two step method produces asymptotically minimum variance estimates over all estimators of their class the implementation of the proposed estimators is described in detail and numerical examples are presented to evaluate their performance

an efficient linear method for arma spectral estimation a three step method for obtaining asymptotically maximum likelihood arma spectral estimates using only linear transformations of the data is presented the computational efficiency of the algorithm is comparable to that of yule walker algorithms but the three step method gives asymptotically statistically efficient estimates the implementation of the algorithm is discussed in detail and numerical examples are presented to illustrate its performance

global stability of multigroup epidemic model with group mixing and nonlinear incidence rates in this paper we introduce a basic reproduction number for a multigroup seir model with nonlinear incidence of infection and nonlinear removal functions between compartments then we establish that global dynamics are completely determined by the basic reproduction number r0 it shows that the basic reproduction number r0 is a global threshold parameter in the sense that if it is less than or equal to one the disease free equilibrium is globally stable and the disease dies out whereas if it is larger than one there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population finally two numerical examples are also included to illustrate the effectiveness of the proposed result

random walk or chaos a formal test on the lyapunov exponent abstract a formal test on the lyapunov exponent is developed to distinguish a random walk model from a chaotic system which is based on the nadaraya watson kernel estimator of the lyapunov exponent the asymptotic null distribution of our test statistic is free of nuisance parameter and simply given by the range of standard brownian motion on the unit interval the test is consistent against the chaotic alternatives a simulation study shows that the test performs reasonably well in finite samples we apply our test to some of the standard macro and financial time series finding no significant empirical evidence of chaos

estimation of the parameters of sinusoidal signals in non gaussian noise accurate estimation of the amplitude and frequency parameters of sinusoidal signals from noisy observations is an important problem in many signal processing applications in this paper the problem is investigated under the assumption of non gaussian noise in general and laplace noise in particular it is proven mathematically that the maximum likelihood estimator derived under the condition of laplace white noise is able to attain an asymptotic cramer rao lower bound which is one half of that achieved by periodogram maximization and nonlinear least squares it is also proven that when applied to non laplace situations the laplace maximum likelihood estimator which may also be referred to as the nonlinear least absolute deviations estimator can achieve an even higher statistical efficiency especially when the noise distribution has heavy tails a computational procedure is proposed to overcome the difficulty of local extrema in the likelihood function simulation results are provided to validate the analytical findings

estimation of directions of arrival of multiple scattered sources we consider the problem of estimating the directions of arrival doa of multiple sources in the presence of local scattering this problem is encountered in wireless communications due to the presence of scatterers in the vicinity of the mobile or when the signals propagate through a random inhomogeneous medium assuming a uniform linear array ula we develop doa estimation algorithms based on covariance matching applied to a reduced size statistic obtained from the sample covariance matrix after redundancy averaging next a computationally efficient estimator based on ar modelling of the coherence loss function is derived a theoretical expression for the asymptotic covariance matrix of this estimator is derived finally the corresponding cramer rao bounds crbs are derived despite its simplicity the ar based estimator is shown to possess performance that is nearly as good as that of the covariance matching method

estimation of the parameters of an arma signal model based on an orthogonal search an iterative procedure for estimating the parameters of an arma signal model input not accessible is empirically compared to the results of several other algorithms the arma identifier reported includes inherent order estimation the empirical results indicate unbiased estimates with less variance than those obtained by the other methods

application of bootstrap to detecting chaos in financial time series abstract a moving blocks bootstrap procedure is used to investigate the dynamics of nominal exchange rates and the return rates of the us dollar against the polish zloty the problem if these financial time series exhibit chaotic behavior is undertaken a possibility of detecting the presence of a positive lyapunov exponent is studied

recursive least squares lattice algorithms a geometrical approach several time recursive least squares algorithms have been developed in recent years in this paper a geometrical formalism is defined which utilizes a nested family of metric spaces indexed by the data time interval this approach leads to a simplified derivation of the so called recursive least squares lattice algorithms recursive in time and order in particular it is found that the resulting structure provides a single framework which encompasses an entire family of fairly complex algorithms as well as providing geometrical insight into their behavior

a statistical approach to estimate the lyapunov spectrum in disc brake squeal abstract the estimation of squeal propensity of a brake system from the prediction of unstable vibration modes using the linear complex eigenvalue analysis cea in the frequency domain has its fair share of successes and failures while the cea is almost standard practice for the automotive industry time domain methods and the estimation of lyapunov spectra have not received much attention in brake squeal analyses one reason is the challenge in estimating the true lyapunov exponents and their discrimination against spurious ones in experimental data a novel method based on the application of the eckmann ruelle matrices is proposed here to estimate lyapunov exponents by using noise in a statistical procedure it is validated with respect to parameter variations and dimension estimates by counting the number of non overlapping confidence intervals for lyapunov exponent distributions obtained by moving a window of increasing size over bootstrapped same length estimates of an observation function a dispersion measure s width is calculated and fed into a bayesian beta binomial model results obtained using this method for benchmark models of white and pink noise as well as the classical henon map indicate that true lyapunov exponents can be isolated from spurious ones with high confidence the method is then applied to accelerometer and microphone data obtained from brake squeal tests estimated lyapunov exponents indicate that the pad s out of plane vibration behaves quasi periodically on the brink to chaos while the microphone s squeal signal remains periodic

application of the lattice filter to robust estimation of ar and arma models a method for estimating the parameters of an autoregressive moving average arma model is proposed a p q stage lattice whitening filter is used to obtained consistent estimates of the ar parameters of an arma p p model it is shown that a set of ma parameter estimates can also be obtained using this approach with little added computation the method is used for estimation of the ar parameters of an arma process in additive white noise it is shown that the lattice filter is also useful in robust estimation of the ar parameter on an arma process with additive outliers

autoregressive spectral estimation in additive noise the estimation of the spectral density of a discrete time stationary gaussian autoregressive process ar p from a finite set of noise observations is considered a modified spectral estimator based on the high order yule walker equations is considered joint asymptotic normality of this spectral estimator is established a precise asymptotic expression for the covariance matrix of the limiting distribution is obtained the special case of ar 1 plus noise is considered in some detail

music maximum likelihood and cramer rao bound the performance of the music and ml methods is studied and their statistical efficiency is analyzed the cramer rao bound crb for the estimation problems is derived and some useful properties of the crb covariance matrix are established the relationship between the music and ml estimators is investigated as well a numerical study is reported of the statistical efficiency of the music estimator for the problem of finding the directions of two plane waves using a uniform linear array an exact description of the results is included

analysis of linear prediction by matrix approximation the authors present a new method of analysis of the effects of data perturbations on the estimates of signal parameters which are obtained by the method of linear prediction the variances of signal parameter obtained from linear prediction methods are computed although the results appear to have wide applicability for simplicity the authors restrict their examples to signals which are linear combinations of complex valued exponential functions

estimating the lyapunov exponent of a chaotic system with nonparametric regression abstract we discuss procedures based on nonparametric regression for estimating the dominant lyapunov exponent 1 from time series data generated by a nonlinear autoregressive system with additive noise for systems with bounded fluctuations 1 0 is the defining feature of chaos thus our procedures can be used to examine time series data for evidence of chaotic dynamics we show that a consistent estimator of the partial derivatives of the autoregression function can be used to obtain a consistent estimator of 1 the rate of convergence we establish is quite slow a better rate of convergence is derived heuristically and supported by simulations simulation results from several implementations one local thin plate splines and three global neural nets radial basis functions and projection pursuit are presented for two deterministic chaotic systems local splines and neural nets yield accurate estimates of the lyapunov exponent however the spline method is sensitive to the choice of the emb

asymptotic properties of the high order yule walker estimates of sinusoidal frequencies the asymptotic properties of the high order yule walker hoyw estimator of sinusoidal frequencies are analyzed an explicit formula for the covariance matrix of the hoyw frequency estimation error is derived the effects on estimation accuracy of the number of yw equations the model order and a certain weighting matrix are investigated analytically the optimally weighted hoyw estimator is described the analysis presented together with the already existing empirical evidence provides enough motivation for the use of an overdetermined and high order yule walker estimator

sinusoidal signals with random amplitude least squares estimators and their statistical analysis the asymptotic properties of constrained and unconstrained least squares estimates of the parameters of a random amplitude sinusoid are analyzed an explicit formula for the asymptotic covariance matrix of the estimation errors is derived for both the constrained and unconstrained estimators accuracy aspects are investigated with the following main results for a certain weighting matrix which is shown to be the same for the constrained and unconstrained methods the estimation errors achieve their lower bounds it is proven that in the optimal case the constrained method always outperforms the unconstrained method it is also proven that the accuracy of the optimal estimators improves as the number of least squares equations increases a formula for the sample length needed for the asymptotic theory to hold is derived and its dependence on the lowpass modulating sequence is stressed simulations provide illustrations of the difference between the constrained and unconstrained estimators as well as the difference between the optimal and basic estimates the influence of the number of least squares equations and the characteristics of the lowpass envelope on the estimation accuracy is also investigated

modeling direct and indirect disease transmission using multi group model abstract the survival of pathogens outside the host in the environment is an important factor for some diseases transmission environment to individuals is an indirect mode of transmission which also plays a role in the dynamics of some disease in this paper we proposed a general multi group epidemic model with nonlinear direct and indirect transmission incidence rates nonlinear pathogen shedding rates and common environmental contamination for indirect transmission under the certain assumptions the basic reproduction number of the model is identified we proved the global stability of the equilibria by using global lyapunov functions with the specific coefficients and graph theoretical approach theorem which are determined by the basic reproduction number the main result of our model is that we give the specific coefficients of global lyapunov functions from the discussion of our model we conclude that our model contains earlier cholera models brucellosis models and other general disease multi group models as special cases

a parameter estimation approach to estimation of frequencies of sinusoids it is shown that with uniform sampling a sample of the sum of m sinusoids at time nt can be uniquely expressed as a linear combination of the 2m samples at times n 1 t n 2m t the 2m coefficients of linear combination are also the coefficients of a 2m order polynomial whose roots are e pmj omega i omega i being the frequencies of the sinusoids given the samples of sinusoids plus noise a consistent estimate of the 2m coefficients are obtained by the instrumental variable method of parameter estimation it is also possible to track time varying frequencies by a recursive algorithm that exponentially weighs out the past data simulation examples of one and two sinusoids are given

analysis of music and esprit frequency estimates for sinusoidal signals with lowpass envelopes in this correspondence the music and esprit frequency estimates of sinusoidal signals with lowpass envelopes are analyzed to achieve computational simplicity the frequency estimation is conducted as if the signal had a constant amplitude the aim of the correspondence is to analyze the degradation of performance induced by the aforementioned mismodeling unified expressions for the bias and variances of the music and esprit frequency estimators are derived under the hypothesis of small bandwidth of the signal envelope numerical simulations illustrate the agreement between theoretical and empirical results and study the influence of the envelope bandwidth on the frequency estimation performance

recursive parameter estimation for noisy autoregressive signals corresp the problem of recursively estimating the unknown parameters of a scalar autoregressive ar signal observed in additive white noise including signal power and noise variance is considered a state space model in a canonical but noninnovations form is used to represent the noisy ar signal an algorithm based on a system identification parameter estimation technique known as the recursive prediction error method is presented for recursive parameter estimation two simulation examples illustrate the effectiveness of the proposed algorithm

on instrumental variable estimation of sinusoid frequencies and the parsimony principle multiple sinusoids in noise can be modeled as an arma process with the ar parameters satisfying certain symmetry relations according to the parsimony principle the constraints on the ar parameters should he taken into account to get improved estimation accuracy it is shown in this note that when estimating the ar parameters by a general instrumental variable method such a parsimony does not necessarily apply however the parsimony principle does hold when an optimal instrumental variable method is used

on the accuracy of the kumaresan tufts method for estimating complex damped exponentials recently kumaresan and tufts kt presented a method for estimating the parameters of damped exponential waveforms in additive white noise the kt method uses singular value decomposition svd of the data matrix with truncation and backward prediction to improve the accuracy of the estimates the kt method was demonstrated to have a very good performance in comparison to traditional methods used for the same problem e g prony s method kumaresan and tufts also showed by numerical simulations that the variances of the estimates obtained by their method approaches the cramer rao lower bounds for selected test cases in this correspondence we provide a quantitative accuracy analysis of the kt method the analysis is based on first order taylor series approximations of the estimated parameters around their true values no assumptions are made on the number of data points used but it is assumed that the noise level is small enough for the first order approximations to be valid results of the analysis are illustrated by some numerical examples these results confirm the good performance of the kt method and show the effect of the user chosen parameters on the accuracy of the estimates

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow