a local linearization approach to nonlinear filtering the local linearization approach to nonlinear filtering of time series generated from continuous time stochastic dynamical systems is discussed and a simple and powerful method for the nonlinear filtering is introduced some numerical results of applications of the filter are shown

relation of signal set choice to the performance of optimal non gaussian detectors the optimal procedure for detecting the presence of discrete time signals in additive noise can be derived from the likelihood ratio test when the noise has statistically independent identically distributed components the dependence of the detector s performance on signal characteristics can be related to the kullback leibler kl distance between the distributions governing the hypotheses performance predictions based on the central limit theorem are shown to be poor approximations to the true performance performance of the optimal detector has long been known to increase exponentially with increasing kl distance symmetric noise amplitude distributions yield a symmetric dependence on the difference between the signals amplitudes at each time index small signal locally optimal detection performance is shown to depend on signal energy whereas large signal performance depends on the signal waveform when a distance measure can be defined performance depends on a different measure than that used in the detector with one exception the gaussian

local linearization filters for non linear continuous discrete state space models with multiplicative noise in this paper the local linearization method for the approximate computation of the prediction and filtering estimates of continuous discrete state space models is extended to the general case of non linear non autonomous models with multiplicative noise the approximate prediction and filter estimates are obtained by applying the optimal linear filter to the piecewise linear state space model that emerges from a local linearization of both the non linear state equation and the non linear measurement equation in addition the solutions of the differential equations that describe the evolution of the first two conditional moments between observations are obtained and an algorithm for their numerical computation is also given the performance of the ll filters is illustrated by mean of numerical experiments

on the relationship between measures of discrimination and the performance of suboptimal detectors the problem of designing and analyzing suboptimal detectors via statistical distance measures is considered as a preliminary result we show that only the minimum and maximum probability of error are valid measures of discrimination between the input statistics this result would seem then to imply that the use of distance measures in this context can be inappropriate however to overcome this apparent obstacle we demonstrate explicit relationships between various f divergences and the loss in performance of an arbitrary detector relative to the optimal detector in particular we establish both upper and lower bounds on the performance loss of a suboptimal detector in terms of the distance between the pertinent statistics of both the optimal and suboptimal detectors while designing detectors by minimizing these upper bounds can be an elusive task in many practical cases the lower bound presented herein holds with equality in this case minimizing the separation of the output statistics of the detector with respect to a particular f divergence equivalently minimizes the resulting probability of error of the detector to facilitate design other researchers have established conditions under which one may design arbitrary detection strategies with respect to a specific f divergence kullback leibler distance being a principal example we extend this approach by deriving necessary and sufficient conditions under which one may design detection strategies with respect to an arbitrarily chosen f divergence thus when these conditions are met one may optimize a detector with respect to the most analytically tractable distance measure to obtain the minimum probability of error detector over a selected class of detection strategies examples demonstrating the utility of this theory for the problem of designing optimal linear detectors and optimal signal sets are presented

coloring non gaussian sequences we propose a novel method for generating random sequences matching both a target non gaussian distribution and a target autocorrelation in the proposed scheme samples matching the target distribution are drawn independently and then suitably rearranged to match the target autocorrelation our method overcomes the difficulties of the standard method in generating sequences with unknown inverse distribution many examples are included

reliability analysis of randomly vibrating structures with parameter uncertainties the problem of reliability analysis of randomly driven stochastically parametered vibrating structures is considered the excitation components are assumed to be jointly stationary gaussian random processes the randomness in structural parameters is modelled through a vector of mutually correlated non gaussian random variables the structure is defined to be safe if a set of response quantities remain within prescribed thresholds over a given duration of time an approximation for the failure probability conditioned on structure randomness is first formulated in terms of recently developed multivariate extreme value distributions an estimate of the unconditional failure probability is subsequently obtained using a method based on taylor series expansion this has involved the evaluation of gradients of the multivariate extreme value distribution conditioned on structure random variables with respect to the basic variables analytical and numerical techniques for computing these gradients have been discussed issues related to the accuracy of the proposed method have been examined the proposed method has applications in reliability analysis of vibrating structural series systems where different failure modes are mutually dependent illustrative examples presented include the seismic fragility analysis of a piping network in a nuclear power plant

langevin equations from time series we discuss the link between the approach to obtain the drift and diffusion of one dimensional langevin equations from time series and pope and ching s relationship for stationary signals the two approaches are based on different interpretations of conditional averages of the time derivatives of the time series at given levels the analysis provides a useful indication for the correct application of pope and ching s relationship to obtain stochastic differential equations from time series and shows its validity in a generalized sense for nondifferentiable processes originating from langevin equations

smoothing data with local instabilities for the identification of chaotic systems chaotic systems are characterized by the local divergence of nearby orbits in state space this provokes a sensitive dependence on initial conditions and in turn drastically limits the accuracy of long term predictions this has important implications in the filtering of data generated by such systems in the present paper the use of global smoothers for chaotic data is investigated the ultimate objective is to be able to identify dynamically valid models from smoothed data when the identification from the original noisy data has completely failed the objective is to produce identified models that faithfully reproduce the dynamical invariants of the original system such as the geometry of the attractors in state space the largest lyapunov exponent fractal dimensions and poincare sections numerical examples are included which illustrate the main points of the paper

generation of a random sequence having a jointly specified marginal distribution and autocovariance we consider the problem of generating a random sequence with a specified marginal distribution and autocovariance the proposed scheme for generating such a sequence consists of a white gaussian noise source input to a linear digital filter followed by a zero memory nonlinearity zmnl the zmnl is chosen so that the desired distribution is exactly realized and the digital filter is designed so that the desired autocovariance is closely approximated both analytic results and examples are included the proposed scheme should prove useful in simulations involving non gaussian processes

convergence rate of weak local linearization schemes for stochastic differential equations with additive noise there exists a diversity of weak local linearization ll schemes for the integration of stochastic differential equations with additive noise which differ in the algorithms employed for the numerical implementation of the weak local linear discretizations despite convergence results for these discretizations have been already developed the convergence of the weak ll schemes has not been considered up to date in this work a general result concerning the convergence rate of the weak ll schemes is presented as well as specificities for a number of particular schemes as an application the convergence of weak ll schemes for equations driven by poisson processes is presented in addition

efficient computation of phi functions in exponential integrators abstract a crucial point for the computational efficiency of the exponential type integrators for differential equations is the effective evaluation of the integrals involving matrix exponential called phi functions focusing on exponential integrators for differential equations of small dimensions for which the krylov based approximations for the phi functions are not viable in this article refined algorithms are introduced for speeding up both the evaluation of several phi functions simultaneously and the computation of a linear combination of phi functions times matrices the algorithms are derived from a meticulous approximation to the exponential of certain partitioned matrix by adapting the conventional pade method to the special structure of this matrix numerical simulations are provided to evaluate the performance of the proposed algorithms and compare them with others in the literature

generalized divergences from generalized entropies abstract several quantifiers of information also known as entropies have been introduced in different contexts and from different motivations for almost each one of these entropies a measure of the loss or gain of information has been introduced in this work we introduce generalized weighted divergences associated with an arbitrary entropy the resulting measures are closely related to bregman divergences we study the main formal properties of the resulting divergences we extend them to weighted probability distributions and we apply some of them to the analysis of simulated and real time series

rate of convergence of local linearization schemes for initial value problems there is a large variety of local linearization ll schemes for the numerical integration of initial value problems which differ with respect to the algorithm that is used in the numerical implementation of the local linear discretization however in contrast with the ll discretization the order of convergence of the ll schemes have not been studied so far in this paper a general result about that matter is given in addition a brief survey of the main implementations of the ll method is also presented

angular billiard and algebraic birkhoff conjecture abstract in this paper we introduce a new dynamical system which we call angular billiard it acts on the exterior points of a convex curve in euclidean plane in a neighborhood of the boundary curve this system turns out to be dual to the birkhoff billiard using this system we get new results on algebraic birkhoff conjecture on integrable billiards

a state space model of the hemodynamic approach nonlinear filtering of bold signals in this paper a new procedure is presented which allows the estimation of the states and parameters of the hemodynamic approach from blood oxygenation level dependent bold responses the proposed method constitutes an alternative to the recently proposed friston neuroimage 16 2002 513 method and has some advantages over it the procedure is based on recent groundbreaking time series analysis techniques that have been in this case adopted to characterize hemodynamic responses in functional magnetic resonance imaging fmri this work represents a fundamental improvement over existing approaches to system identification using nonlinear hemodynamic models and is important for three reasons first our model includes physiological noise previous models have been based upon ordinary differential equations that only allow for noise or error to enter at the level of observation secondly by using the innovation method and the local linearization filter not only the parameters but also the underlying states of the system generating responses can be estimated these states can include things like a flow inducing signal triggered by neuronal activation de oxyhemoglobine cerebral blood flow and volume finally radial basis functions have been introduced as a parametric model to represent arbitrary temporal input sequences in the hemodynamic approach which could be essential to understanding those brain areas indirectly related to the stimulus hence thirdly by inferring about the radial basis parameters we are able to perform a blind deconvolution which permits both the reconstruction of the dynamics of the most likely hemodynamic states and also to implicitly reconstruct the underlying synaptic dynamics induced experimentally which caused these states variations from this study we conclude that in spite of the utility of the standard discrete convolution approach used in statistical parametric maps spm nonlinear bold phenomena and unspecific input temporal sequences must be included in the fmri analysis

representation and generation of non gaussian wide sense stationary random processes with arbitrary psds and a class of pdfs a new method for representing and generating realizations of a wide sense stationary non gaussian random process is described the representation allows one to independently specify the power spectral density and the first order probability density function of the random process the only proviso is that the probability density function must be symmetric and infinitely divisible the method proposed models the sinusoidal component frequencies as random variables a key departure from the usual representation a of wide sense stationary random process by the spectral theorem ergodicity in the mean and autocorrelation is also proven under certain conditions an example is given to illustrate its application to the k distribution which is important in many physical modeling problems in radar and sonar

a parametric approach to suboptimal signal detection in alpha stable noise this paper deals with the detection of a known deterministic signal embedded in alpha stable noise the implementation of the optimal receiver requires the explicit expression of the probability density function pdf of the noise unfortunately since there exists no closed form for the pdf of alpha stable distributed random variables numerical integrations are required to avoid such numerical approximations we suggest a low complexity parametric suboptimal detector well matched to essential properties of alpha stable noises this receiver does not require the explicit expression of the noise pdf in addition parameter optimization is fast for several optimization criteria and the selected receiver allows retrieval of the optimal gaussian detector matched filter as well as the locally optimal detector in the cauchy context the performance of the detector is studied and a comparison with the optimal solution along with a variety of classical detectors is given the robustness of the detector against the signal amplitude and the stability index alpha of the noise is discussed

stochastic differential equations an approach to the generation of continuous non gaussian processes the generation of continuous random processes with jointly specified probability density and covariation functions is considered the proposed approach is based on the interpretation of the simulated process as a stationary output of a nonlinear dynamic system excited by white gaussian noise and described by a system of a first order stochastic differential equations sde the authors explore how the statistical characteristics of the equation s solution depends on the form of its operator and on the intensity of the input noise some aspects of the approximate synthesis of stochastic differential equations and examples of their application to the generation of non gaussian continuous processes are considered the approach should be useful in signal processing when it is necessary to translate the available a priori information on the real random process into the language of its markov model as well as in simulation of continuous correlated processes with the known probability density function

statistical properties and uses of the wavelet variance estimator for the scale analysis of time series abstract many physical processes are an amalgam of components operating on different scales and scientific questions about observed data are often inherently linked to understanding the behavior at different scales we explore time scale properties of time series through the variance at different scales derived using wavelet methods the great advantage of wavelet methods over ad hoc modifications of existing techniques is that wavelets provide exact scale based decomposition results we consider processes that are stationary nonstationary but with stationary dth order differences and nonstationary but with local stationarity we study an estimator of the wavelet variance based on the maximal overlap undecimated discrete wavelet transform the asymptotic distribution of this wavelet variance estimator is derived for a wide class of stochastic processes not necessarily gaussian or linear the variance of this distribution is estimated using spectral methods simulations confirm the theoretical result

robust quantization of e contaminated data the problem of robust quantization of data with uncertain statistical properties is considered uncertainty in the statistics of the data is modeled by assuming that the data have a probability density function of the e contaminated form and a minimax approach to robust design is adopted an approximation is developed for the asymptotic worst case distortion over the e contaminated class produced by an arbitrary companded quantizer and the quantizer design which minimizes this worst case distortion is derived the robustness of the resulting design is verified numerically for the particular problem of quantizing e contaminated gaussian data

multichannel detection for correlated non gaussian random processes based on innovations the paper addresses the problem of multichannel signal detection in additive correlated non gaussian noise using the innovations approach although this problem has been addressed extensively for the case of additive gaussian noise the corresponding problem for the non gaussian case has received limited attention this is due to the fact that there is no unique specification for the joint probability density function pdf of n correlated non gaussian random variables the authors overcome this problem by using the theory of spherically invariant random processes sirps and derive the innovations based detector it is found that the optimal estimators for obtaining the innovations processes are linear and that the resulting detector is canonical for the class of pdfs arising from sirps the authors also present a performance analysis of the innovations based detector for the case of a k distributed sirp

spherically invariant vector random fields in space and time this paper is concerned with spherically invariant or elliptically contoured vector random fields in space and or time which are formulated as scale mixtures of vector gaussian random fields while a spherically invariant vector random field may or may not have second order moments a spherically invariant second order vector random field is determined by its mean and covariance matrix functions just like the gaussian one this paper explores basic properties of spherically invariant second order vector random fields and proposes an efficient approach to develop covariance matrix functions for such vector random fields

on polynomially integrable planar outer billiards and curves with symmetry property we show that every polynomially integrable planar outer convex billiard is elliptic we also prove an extension of this statement to non convex billiards

a class of lower bounds on the bayesian probability of error abstract in this paper a class of lower bounds is considered which unifies and extends some well known bounds on the bayesian probability of error this class is obtained by considering the divergence between two hypotheses both the symmetric and the nonsymmetric case are studied

memoryless discrete time signal detection in long range dependent noise the problem of designing optimum memoryless detectors for known signals in long range dependent lrd noise is considered in particular under the performance criterion of asymptotic relative efficiency are optimum memoryless detection in lrd noise is investigated by exploiting the hermite expansions the detectors considered have the form of a nonlinearity followed by an accumulator and threshold comparator it is shown that when the noise is lrd and gaussian all nonlinearities with hermite rank one are asymptotically equivalent in terms of efficiency and are most powerful moreover the optimum nonlinearities with hermite rank greater than one are given by the corresponding hermite polynomials the case of non gaussian lrd noise that can be derived by nonlinear transformation of gaussian noise is also considered in this case the globally optimum nonlinearity is very difficult to obtain in general instead we proposed a suboptimal nonlinearity which is given by a linear combination of the corresponding locally optimum detector and the inverse of the transform function generating the noise simulations show that the proposed detector outperforms the locally optimal detector for lrd non gaussian noise

martingale integrals over poissonian processes and the ito type equations with white shot noise the construction of the ito type stochastic integrals and differential equations for compound poisson processes is provided the general martingale and nonanticipating properties of the ordinary gaussian ito theory are conserved these properties appear particularly important if the stochastic description has to be proposed according to game theory or the linear relaxation or the exponential growth requirements in contrast to the ordinary ito theory the uncorrelated parametric fluctuation of a definite sign can be still modeled by asymmetric white shot noise so the general scope of applications is not restricted by the positivity requirements the possible use of the developed formalism in econophysics is addressed

ito versus stratonovich a survey is given of the facts and fancies concerning the nonlinear langevin or ito equation actually it is merely a pre equation which becomes an equation when an interpretation rule is added the rules of ito and stratonovich differ but both are mathematically consistent and therefore equally admissible conventions the reason why they seem to lead to physical differences is that the langevin approach used to arrive at the equation involves a tacit assumption for systems with external noise this assumption can be justified and it is then clear that the stratonovich rule applies systems with internal noise however can only be properly described by a master equation and the ito stratonovich controversy never enters afterward one is free to model the resulting fluctuations either with an ito or a stratonovich scheme but that does not lead to any new information

dynamic properties of the local linearization method for initial value problems some dynamic properties of the local linearization ll scheme for the numerical integration of initial value problems in ordinary differential equations odes are investigated specifically the general conditions under which this scheme preserves the stationary points and periodic orbits of the odes and the local stability at these steady states are studied these dynamic properties are also examined by means of numerical experiments and the results are compared with those achieved by other numerical schemes in addition a brief review of the computational implementations of the ll scheme is also presented

a new laplace second order autoregressive time series model nlar 2 a time series model for laplace double exponential variables having second order autoregressive structure nlar 2 is presented the model is markovian and extends the second order process in exponential variables near 2 to the case where the marginal distribution is laplace the properties of the laplace distribution make it useful for modeling in some cases where the normal distribution is not appropriate the time series model has four parameters and is easily simulated the autocorrelation function for the process is derived as well as third order moments to further explore dependency in the process the model can exhibit a broad range of positive and negative correlations and is partially time reversible joint distributions and the distribution of differences are presented for the first order case nlar 1

generation and analysis of non gaussian markov time series correlated non gaussian markov sequences can be considered as filtered white noise independent identically distributed sequences of random variables the filter being a nonlinear system in general the authors discuss the applicability of linear models and nonlinear methods based on the diagonal series expansion of bivariate densities for analyzing this system non gaussian sequences exhibit different properties in the forward and backward directions of time the authors explore the connection to system modeling of this temporal asymmetry and some of its consequences as an example they analyze a first order linear autoregressive model with hyperbolic secant amplifier distribution at its output

on the generation of the baseband and narrowband non gaussian processes one of the widely used classifications of random processes implies their separation from the baseband and narrowband ones exponent and exponentially decaying cosine waves represent the most popular approximations of the observed processes autocovariation functions acf we consider the generation of such processes with any specified marginal probability density function pdf the approach is based on the representation of the process considered to be a stationary solution of certain stochastic differential equations sde with the white gaussian noise in the right hand side it removes the known pearson restrictions on the kind of available process pdf and in the part related to the generation of the processes with strictly exponential acf is very close to that considered by haddad 1970 concerning the generation of the narrowband non gaussian processes with the exact exponential envelope of its acf synthesis of the corresponding sde may be considered to be principally novel several examples of nonlinear dynamical systems generating baseband and narrowband stationary continuous processes with the given pdf are considered some aspects of the sde numerical simulation are presented the generation method proposed may be useful in communication and signal processing applications where a proper interference simulation appears to be really important

local linearization method for numerical integration of delay differential equations in this paper a new approach for the numerical computation of delay differential equations ddes is introduced the essential idea consists of obtaining numerical integrators that use a code expressly developed for linear ddes in contrast with the conventional approach of using a code for ordinary differential equations specifically two numerical schemes of this new class of integrators are proposed and their numerical viability analyzed it includes the estimation of the convergence rate the evaluation of the computational cost of the schemes and a simulation study it is proved that these one step explicit integrators converge uniformly with order two to the solution of nonlinear ddes and are able to integrate stiff equations in a satisfactory way with low computational cost

rate distortion bound for a class of non gaussian sources with memory the rate distortion performance of a class of non gaussian source with memory is studied the source model is generated from a correlated gaussian source through a memoryless transform which actually constitutes a special class of the frequently used gibbs model in speech and image processing applications its entropy and a rate distortion bound are evaluated to obtain further insights into this class of source models several examples are also given to illustrate the usefulness and tightness of the rate distortion bound

fast exact synthesis of gaussian and nongaussian long range dependent processes 1 f noise and statistically self similar random processes such as fractional brownian motion fbm and fractional gaussian noise fgn are fundamental models for a host of real world phenomena from network traffic to dna to the stock market synthesis algorithms play a key role by providing the feedstock of data necessary for running complex simulations and accurately evaluating analysis techniques unfortunately current algorithms to correctly synthesize these long range dependent lrd processes are either abstruse or prohibitively costly which has spurred the wide use of inexact approximations to fill the gap we develop a simple fast o n log n operations for a length n signal framework for exactly synthesizing a range of gaussian and nongaussian lrd processes as a bonus we introduce and study a new bi scaling fbm process featuring a kinked correlation function that exhibits distinct scaling laws at coarse and fine scales

arma signals with specified symmetric marginal probability distribution except in the case of normal i e gaussian distribution it is very difficult to calculate the marginal probability distribution of arma signals by using a particular form of modeling such signals with random coefficients we show that the problem can be solved and we present an algorithm allowing the numerical generation of arma signals with arbitrary specified marginal distribution function their properties are analyzed in computer experiments the experimental results show in general a very good agreement with theoretical calculations

symmetric binary random signals with given spectral properties the usual known methods for generating signals with given marginal probability and spectral properties cannot be applied to binary signals widely used in theoretical problems and communications systems to overcome this difficulty we first present some structural properties of power spectral densities enabling the precise definition of the concept of spectral properties this allows us to introduce a new method valid for symmetric binary random signals this method uses some specific properties of filters with random impulse responses results of computer simulations show clearly the good performance of this method some extensions by using random thinning can further improve its performance

some comments on the use of markov models in communications in their recent article feng field and haykin rediscovered an approach to the channel modelling based on stochastic differential equations sde in the following we would like to bring readers attention to a body of literature which deals with similar subjects using sde approach