a hybrid lower bound for parameter estimation of signals with multiple change points change point estimation has received much attention in the literature as it plays a significant role in several signal processing applications however the study of the optimal estimation performance in such context is a difficult task since the unknown parameter vector of interest may contain both continuous and discrete parameters namely the parameters associated with the noise distribution and the change point locations in this paper we handle this by deriving a lower bound on the mean square error of these continuous and discrete parameters specifically we propose a hybrid cramer rao weiss weinstein bound and derive its associated closed form expressions numerical simulations assess the tightness of the proposed bound in the case of gaussian and poisson observations

estimability and regulability of linear systems a linear state space system is said to be estimable if in estimating its state from its output the posterior error covariance matrix is strictly smaller than the prior covariance matrix it is said to be regulatable if the quadratic cost of the state feedback control is strictly smaller than the cost when no feedback is used estimability and regulability are shown to be dual properties equivalent to the nonreducibility of the kalman filter and of the optimal linear quadratic regulator respectively

oscillator death in populations of all to all coupled nonlinear oscillators abstract the behavior of finitely many and infinitely many coupled weakly nonlinear oscillators is studied and it is shown that the uniform rest state can become stable if the coupling is sufficiently strong and the frequencies are sufficiently disparate numerical results show that the phenomenon is robust and occurs when many of the hypotheses are relaxed

nature of the phase flip transition in the synchronized approach to amplitude death we study the dynamics of time delay coupled limit cycle oscillators in the amplitude death regime through a detailed analysis of the jacobian at the fixed point we show that the phase flip transition namely the abrupt change from in phase synchronized dynamics to antiphase synchronized dynamics is associated with an interchange of the imaginary parts of complex pairs of eigenvalues at an avoided crossing of lyapunov exponents as a parameter is varied an order parameter for the transition is constructed through the eigenvectors of the jacobian

stochastic processes originating in deterministic microscopic dynamics we investigate the probability distribution of the scaled trajectory of a test particle moving in an equilibrium fluid according to the laws of classical mechanics i e ifq t is the displacement of the test particle we letqa t q at a and consider the distribution of the trajectory qa t in the limit a the randomness of the motion is due entirely to the randomness of the initial state of the fluid test particle or both and the process is generally non markovian nevertheless it can be proven in some cases and we expect it to be true in many more that qa t looks like brownian motion in the limit a some results for simple model systems are presented

parameter bounds on estimation accuracy under model misspecification when the assumed data distribution differs from the true distribution the model is said to be misspecified or mismatched model misspecification at some level is an inevitability of engineering practice while huber s celebrated work assesses maximum likelihood ml performance under misspecification no simple theory for bounding parameter estimation exists the class of parameter bounds emerging from the covariance inequality or equivalently the minimum norm theorem is revisited the expectation operator is well known to form an inner product space flexibility in the choice of expectation integrand and measure for integration exists however to establish a class of parameter bounds under a general form of model misspecification i e distribution mismatch the cramer rao bound crb primarily and secondarily the barankin hammersley chapman robbins bhattacharyya and bobrovsky mayer wolf zakai bounds under misspecification are considered huber s sandwich covariance is readily established as a special case of the misspecified crb subject to ml constraints and generalizations of the slepian and bangs formulae under misspecification are obtained

a new class of bayesian cyclic bounds for periodic parameter estimation many practical signal processing applications involve estimation of parameters with periodic nature such as phase frequency and direction of arrival estimation the commonly used mean squared error mse risk does not take periodicity into account and thus is inappropriate for periodic parameter estimation in which one is interested in the modulo t error rather than the plain error value as a result mse lower bounds are not valid for periodic estimation in addition conventional bayesian mse lower bounds such as the bayesian cramer rao bound bcrb and the bobrovsky zakai bound bzb require restrictive regularity conditions and usually do not exist in periodic settings an alternative risk which is commonly used in periodic parameter estimation problems is the mean cyclic error mce in this paper we establish a new class of bayesian lower bounds on the mce of any estimator the new class includes cyclic versions of the bcrb and the bzb that have less restrictive regularity conditions than those of the conventional bcrb and bzb respectively the tightest bound in the proposed class is derived and its tightness is discussed in addition the proposed class is extended to mixed vector parameter estimation with both periodic and nonperiodic parameters the new cyclic lower bounds are compared with the mce performance of the minimum mce and maximum a posteriori probability estimators for von mises parameter estimation and for frequency estimation

a general class of lower bounds in parameter estimation a general class of bayesian lower bounds on moments of the error in parameter estimation is formulated and it is shown that the cramer rao the bhattacharyya the bobrovsky zakai and the weiss weinstein lower bounds are special cases in the class the bounds can be applied to the estimation of vector parameters and any given function of the parameters the extension of these bounds to multiple parameter is discussed

time delay effects on coupled limit cycle oscillators at hopf bifurcation abstract we present a detailed study of the effect of time delay on the collective dynamics of coupled limit cycle oscillators at hopf bifurcation for a simple model consisting of just two oscillators with a time delayed coupling the bifurcation diagram obtained by numerical and analytical solutions shows significant changes in the stability boundaries of the amplitude death phase locked and incoherent regions a novel result is the occurrence of amplitude death even in the absence of a frequency mismatch between the two oscillators similar results are obtained for an array of n oscillators with a delayed mean field coupling and the regions of such amplitude death in the parameter space of the coupling strength and time delay are quantified some general analytic results for the n thermodynamic limit are also obtained and the implications of the time delay effects for physical applications are discussed

the fractal properties of generalised random walks in one dimension the properties of a simple random walk and random walks with persistence of velocity in one dimension are reported the finite fractal properties of the walks are obtained with exact expressions for indefinitely long walks remarkably simple expressions are found for the length l epsilon at scale epsilon computer simulations of these walks are reported for finite trajectories which help to explain the properties of atomic trajectories in realistic liquids determined by molecular dynamics simulation some new molecular dynamics simulations are reported for a typical liquid its coexisting vapour and the crystal all at the same temperature the general form of the fractal properties of the three states of matter are well represented by the present one dimensional model indeed even the quantitative nature is quite well represented which suggests that results for random walks in three dimensions will be substantially the same

lyapunov exponents in disordered chaotic systems avoided crossing and level statistics the behavior of the lyapunov exponents les of a disordered system consisting of mutually coupled chaotic maps with different parameters is studied the les are demonstrated to exhibit avoided crossing and level repulsion qualitatively similar to the behavior of energy levels in quantum chaos recent results for the coupling dependence of the les of two coupled chaotic systems are used to explain the phenomenon and to derive an approximate expression for the distribution functions of le spacings the depletion of the level spacing distribution is shown to be exponentially strong at small values the results are interpreted in terms of the random matrix theory

on lower bounds for nonstandard deterministic estimation we consider deterministic parameter estimation and the situation where the probability density function p d f parameterized by unknown deterministic parameters results from the marginalization of a joint p d f depending on random variables as well unfortunately in the general case this marginalization is mathematically intractable which prevents from using the known standard deterministic lower bounds lbs on the mean squared error mse actually the general case can be tackled by embedding the initial observation space in a hybrid one where any standard lb can be transformed into a modified one fitted to nonstandard deterministic estimation at the expense of tightness however furthermore these modified lbs mlbs appears to include the submatrix of hybrid lbs which is an lb for the deterministic parameters moreover since in the nonstandard estimation maximum likelihood estimators mles can be no longer derived suboptimal nonstandard mles nsmles are proposed as being a substitute we show that any standard lb on the mse of mles has a nonstandard version lower bounding the mse of nsmles we provide an analysis of the relative performance of the nsmles as well as a comparison with the mlbs for a large class of estimation problems last the general approach introduced is exemplified among other things with a new look at the well known gaussian complex observation models

second order parameter estimation this work provides a general framework for the design of second order blind estimators without adopting any approximation about the observation statistics or the a priori distribution of the parameters the proposed solution is obtained minimizing the estimator variance subject to some constraints on the estimator bias the resulting optimal estimator is found to depend on the observation fourth order moments that can be calculated analytically from the known signal model unfortunately in most cases the performance of this estimator is severely limited by the residual bias inherent to nonlinear estimation problems to overcome this limitation the second order minimum variance unbiased estimator is deduced from the general solution by assuming accurate prior information on the vector of parameters this small error approximation is adopted to design iterative estimators or trackers it is shown that the associated variance constitutes the lower bound for the variance of any unbiased estimator based on the sample covariance matrix the paper formulation is then applied to track the angle of arrival aoa of multiple digitally modulated sources by means of a uniform linear array the optimal second order tracker is compared with the classical maximum likelihood ml blind methods that are shown to be quadratic in the observed data as well simulations have confirmed that the discrete nature of the transmitted symbols can be exploited to improve considerably the discrimination of near sources in medium to high snr scenarios

the interacting multiple model algorithm for systems with markovian switching coefficients an important problem in filtering for linear systems with markovian switching coefficients dynamic multiple model systems is the management of hypotheses which is necessary to limit the computational requirements a novel approach to hypotheses merging is presented for this problem the novelty lies in the timing of hypotheses merging when applied to the problem of filtering for a linear system with markovian coefficients the method is an elegant way to derive the interacting multiple model imm algorithm evaluation of the imm algorithm shows that it performs well at a relatively low computational load these results imply a significant change in the state of the art of approximate bayesian filtering for systems with markovian coefficients

weiss weinstein bound on multiple change points estimation in the context of multiple change points estimation performance analysis of estimators such as the maximum likelihood is often difficult to assess since the regularity assumptions are not met focusing on the estimators variance one can however use lower bounds on the mean square error in this paper we derive the so called weiss weinstein bound wwb that is known to be an efficient tool in signal processing to obtain a fair overview of the estimation behavior contrary to several works about performance analysis in the change point literature our study is adapted to multiple changes first useful formulas are given for a general estimation problem whatever the considered distribution of the data second closed form expressions are given in the cases of gaussian observations with changes in the mean and or the variance and changes in the mean rate of a poisson distribution furthermore a semidefinite programming formulation of the minimization procedure is given in order to compute the tightest wwb specifically it consists of finding the unique minimum volume covering the set constituted by hyperellipsoid elements that are generated using the derived candidate wwb matrices w r t the so called loewner partial ordering finally simulation results are provided to show the good behavior of the proposed bound

dynamical effects of integrative time delay coupling we study coupled dynamical systems wherein the influence of one system on the other is cumulative coupling signals are integrated over a time interval t a major consequence of integrative coupling is that amplitude death occurs over a wider range and in a single region in parameter space for coupled limit cycle oscillators the landau stuart model we obtain an analytic estimate for the boundary of this region while for coupled chaotic lorenz oscillators numerical results are presented for given t we find that there is a critical coupling strength at which the frequency of oscillations changes discontinuously

optimal linear mean square filter for continuous time jump linear systems we consider a class of hybrid systems which is modeled by continuous time linear systems with markovian jumps in the parameters lsmjp our aim is to derive the best linear mean square estimator for such systems the approach adopted here produces a filter which bears those desirable properties of the kalman filter a recursive scheme suitable for computer implementation which allows some offline computation that alleviates the computational burden apart from the intrinsic theoretical interest of the problem in its own right and the application oriented motivation of getting more easily implementable filters another compelling reason why the study here is pertinent has to do with the fact that the optimal nonlinear filter for our estimation problem is not computable via a finite computation the filter is infinite dimensional our filter has dimension nn with n denoting the dimension of the state vector and n the number of states of the markov chain

asymptotic design of quantizers for decentralized mmse estimation conceptual and practical encoding decoding aimed at accurately reproducing remotely collected observations has been heavily investigated since the pioneering works by shannon about source coding however when the goal is not to reproduce the observables but making inference about an embedded parameter and the scenario consists of many unconnected remote nodes the landscape is less certain we consider a multiterminal system designed for efficiently estimating a random parameter according to the minimum mean square error mmse criterion the analysis is limited to scalar quantizers followed by a joint entropy encoder and it is performed in the high resolution regime where the problem can be more easily mathematically tackled focus is made on the peculiarities deriving from the estimation task as opposed to that of reconstruction as well as on the multiterminal as opposite to centralized character of the inference the general form of the optimal nonuniform quantizer is derived and examples are given

phase flip bifurcation induced by time delay we present a general bifurcation in the synchronized dynamics of time delay coupled nonlinear oscillators the relative phase between the oscillators jumps from zero to ensuremath pi as a function of the coupling this phase flip bifurcation is accompanied by a discontinuous change in the frequency of the synchronized oscillators this phenomenon is of broad relevance being observed in regimes of oscillator death as well as in periodic quasiperiodic and chaotic dynamics time delay coupling is necessary for the phase flip bifurcation we illustrate the phenomenon and present analytical results for paradigmatic nonlinear systems possible applications are discussed

a fresh look at the bayesian bounds of the weiss weinstein family minimal bounds on the mean square error mse are generally used in order to predict the best achievable performance of an estimator for a given observation model in this paper we are interested in the bayesian bound of the weiss weinstein family among this family we have bayesian cramer rao bound the bobrovsky mayerwolf zakai bound the bayesian bhattacharyya bound the bobrovsky zakai bound the reuven messer bound and the weiss weinstein bound we present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator mmsee and on a constrained optimization problem with this approach we obtain a useful theoretical framework to derive new bayesian bounds for that purpose we propose two bounds first we propose a generalization of the bayesian bhattacharyya bound extending the works of bobrovsky mayer wolf and zakai second we propose a bound based on the bayesian bhattacharyya bound and on the reuven messer bound representing a generalization of these bounds the proposed bound is the bayesian extension of the deterministic abel bound and is found to be tighter than the bayesian bhattacharyya bound the reuven messer bound the bobrovsky zakai bound and the bayesian cramer rao bound we propose some closed form expressions of these bounds for a general gaussian observation model with parameterized mean in order to illustrate our results we present simulation results in the context of a spectral analysis problem

amplitude death in the absence of time delays in identical coupled oscillators we study the dynamics of oscillators that are mutually coupled via dissimilar or conjugate variables and find that this form of coupling leads to a regime of amplitude death analytic estimates are obtained for coupled landau stuart oscillators and this is supplemented by numerics for this system as well as for coupled lorenz oscillators time delay does not appear to be necessary to cause amplitude death when conjugate variables are employed in coupling identical systems coupled chaotic oscillators also show multistability prior to amplitude death and the basins of the coexisting attractors appear to be riddled this behavior is quantified an appropriately defined uncertainty exponent in the coupled lorenz system is shown to be zero

optimal quadratic control of jump linear systems with separately controlled transition probabilities abstract the optimal control of discrete time jump systems that would be linear but for randomly jumping parameters are considered these parameter jumps which are indexed by a finite state form process can be used to model systems subject to component failures or structural changes the control of such systems when the jump probabilities can be controlled by choice of a finite valued input is considered this input form control can be thought of as corresponding to a maintenance or repair policy the optimal control problem with quadratic costs whose weighting matrices depend on the jumping parameters and form control is formulated for such systems the optimal solution for finite and infinite time horizon problems is developed in addition sufficient conditions for the existence of a steady state stabilizing solution to the infinite time problem are given in terms of appropriately defined controllability and observability properties

bayesian bounds for matched field parameter estimation matched field methods concern estimation of source locations and or ocean environmental parameters by exploiting full wave modeling of acoustic waveguide propagation because of the nonlinear parameter dependence of the signal field the estimate is subject to ambiguities and the sidelobe contribution often dominates the estimation error below a threshold signal to noise ratio snr to study the matched field performance three bayesian lower bounds on mean square error are developed the bayesian crame spl acute r rao bound bcrb the weiss weinstein bound wwb and the ziv zakai bound zzb particularly for a multiple frequency multiple snapshot random signal model a closed form minimum probability of error associated with the likelihood ratio test is derived which facilitates error analysis in a wide scope of applications analysis and example simulations demonstrate that 1 unlike the local crb the bcrb is not achieved by the maximum likelihood estimate mle even at high snr if the local performance is not uniform across the prior parameter space 2 the zzb gives the closest mle performance prediction at most snr levels of practical interest 3 the zzb can also be used to determine the necessary number of independent snapshots achieving the asymptotic performance of the mle at a given snr 4 incoherent frequency averaging which is a popular multitone processing approach reduces the peak sidelobe error but may not improve the overall performance due to the increased ambiguity baseline and finally 5 effects of adding additional parameters e g environmental uncertainty can be well predicted from the parameter coupling

universality in chaos lyapunov spectrum and random matrix theory we propose the existence of a new universality in classical chaotic systems when the number of degrees of freedom is large the statistical property of the lyapunov spectrum is described by random matrix theory we demonstrate it by studying the finite time lyapunov exponents of the matrix model of a stringy black hole and the mass deformed models the massless limit which has a dual string theory interpretation is special in that the universal behavior can be seen already at t 0 while in other cases it sets in at late time the same pattern is demonstrated also in the product of random matrices

on the generalized cramer rao bound for the estimation of the location it is shown that the generalized gaussian distribution maximizes the generalized cramer rao cr bound for the pth absolute central moment of any classical location parameter unbiased estimator the underlying maximization is taken over the class of distributions with fixed and finite pth order moment and exhibits particular utility in minimax designs as well as in worst case performance analysis the relationship between the generalized gaussian density and the generalized cr bound is further examined for the model of a mixture of generalized gaussian distributions as well as for scenarios where multiple independent generalized gaussian observations are involved

extended ziv zakai lower bound for vector parameter estimation the bayesian ziv zakai bound on the mean square error mse in estimating a uniformly distributed continuous random variable is extended for arbitrarily distributed continuous random vectors and for distortion functions other than mse the extended bound is evaluated for some representative problems in time delay and bearing estimation the resulting bounds have simple closed form expressions and closely predict the simulated performance of the maximum likelihood estimator in all regions of operation

a crame spl acute r rao type estimation lower bound for systems with measurement faults a crame spl acute r rao type lower bound is presented for systems with measurements prone to discretely distributed faults which are a class of hybrid systems lower bounds for both the state and the markovian interruption variables fault indicators of the system are derived using the recently presented sequential version of the crame spl acute r rao lower bound crlb for general nonlinear systems because of the hybrid nature of the systems addressed the crlb cannot be directly applied due to violation of its associated regularity conditions to facilitate the calculation of the lower bound the hybrid system is first approximated by a system in which the discrete distribution of the fault indicators is replaced by an approximating continuous one the lower bound is then obtained via a limiting process applied to the approximating system the results presented herein facilitate a relatively simple calculation of a nontrivial lower bound for the state vector of systems with fault prone measurements the crlb type lower bound for the interruption process variables turns out to be trivially zero however a nontrivial non crlb type bound for these variables has been recently presented elsewhere by the authors the utility and applicability of the proposed lower bound are demonstrated via a numerical example involving a simple global positioning system gps aided navigation system where the gps measurements are fault prone due to their sensitivity to multipath errors

a class of weiss weinstein bounds and its relationship with the bobrovsky mayer wolf zakai bounds a fairly general class of bayesian large error lower bounds of the weiss weinstein family essentially free from regularity conditions on the probability density functions support and for which a limiting form yields a generalized bayesian cramer rao bound bcrb is introduced in a large number of cases the generalized bcrb appears to be the bobrovsky mayer wolf zakai bound bmzb interestingly enough a regularized form of the bobrovsky zakai bound bzb applicable when the support of the prior is a constrained parameter set is obtained modified weiss weinstein bound and bzb which limiting form is the bmzb are proposed in expectation of an increased tightness in the threshold region some of the proposed results are exemplified with a reference problem in signal processing the gaussian observation model with parameterized mean and uniform prior

analytic sequential weiss weinstein bounds in this paper we explore a sequential bayesian bound for state space models focusing on hybrid continuous and discrete random states we provide an analytic recursion for the sequential weiss weinstein sww bound for linear state space models with solutions for gaussian uniform and exponential distributions as derived as well as for a combination of these we compare the sww bound for discretized states with the corresponding bound for the continuous states the sww bound is contrasted with the sequential cramer rao bound for gaussian distributions practical issues of sww bounds are discussed and numerical simulation results provide insights into their behavior

a lower bound on the mean square error in random parameter estimation corresp a new lower bound on mean square error in parameter estimation is presented the bound is tighter than the cramer rao and bobrovsky zakai lower bounds it requires no bias or regularity assumptions it is computationally simple and it can be applied to estimates of vector parameters or functions of the parameters

a comment on the weiss weinstein bound for constrained parameter sets the weiss weinstein bound wwb provides a lower limit on the mean squared error mse achievable by an estimator of an unknown random parameter in this correspondence it is shown that some previously proposed simplified versions of the bound do not always hold for constrained parameters i e parameters whose distribution has finite support these simplifications can produce results which are no longer lower bounds on the mse sufficient conditions are provided for the reductions to be valid

brief paper detection and estimation for abruptly changing systems the problem of state estimation and system structure detection for discrete stochastic dynamical systems with parameters which may switch among a finite set of values is considered the switchings are modelled by a markov chain with known transition probabilities a brief survey and a unified treatment of the existing suboptimal algorithms are provided the optimal algorithms require exponentially increasing memory and computations with time simulation results comparing the various suboptimal algorithms are presented

general classes of performance lower bounds for parameter estimation part ii bayesian bounds in this paper a new class of bayesian lower bounds is proposed derivation of the proposed class is performed via projection of each entry of the vector function to be estimated on a hilbert subspace of l2 this hilbert subspace contains linear transformations of elements in the domain of an integral transform applied on functions used for computation of bounds in the weiss weinstein class the integral transform generalizes the traditional derivative and sampling operators used for computation of existing performance lower bounds such as the bayesian cramer rao bayesian bhattacharyya and weiss weinstein bounds it is shown that some well known bayesian lower bounds can be derived from the proposed class by specific choice of the integral transform kernel a new lower bound is derived from the proposed class using the fourier transform kernel the proposed bound is compared with other existing bounds in terms of signal to noise ratio snr threshold region prediction in the problem of frequency estimation the bound is shown to be computationally manageable and provides better prediction of the snr threshold region exhibited by the maximum a posteriori probability map and minimum mean square error mmse estimators

a general class of outage error probability lower bounds in bayesian parameter estimation in this paper a new class of lower bounds on the outage error probability in bayesian parameter estimation is proposed the outage error probability is an important criterion in parameter estimation that provides meaningful information even in the presence of large errors and is useful for prediction of the system operation region computation of the minimum outage error probability is usually not tractable and thus lower bounds on this probability can be very useful for performance analysis the proposed class of lower bounds on the outage error probability is derived using holder s inequality several bounds in the proposed class are presented it is shown that the ziv zakai lower bound on the outage error probability can be obtained from a subclass in the proposed class of bounds the proposed class of bounds is utilized to derive a new class of bayesian bounds on the mean square error it is shown that for unimodal posterior probability density functions the tightest lower bound on the probability of outage error in the proposed class attains the minimum probability of outage error the proposed bounds are exemplified in linear gaussian model parameter estimation and time delay estimation

a new estimation error lower bound for interruption indicators in systems with uncertain measurements optimal mean square error estimators of systems with interrupted measurements are infinite dimensional because these systems belong to the class of hybrid systems this renders the calculation of a lower bound for the estimation error of the interruption process in these systems of particular interest recently it has been shown that a crame spl acute r rao type lower bound on the interruption process estimation error is trivially zero in the present work a nonzero lower bound for a class of systems with markovian interruption variables is proposed derivable using the well known weiss weinstein bound this lower bound can be easily evaluated using a simple recursive algorithm the proposed lower bound is shown to depend on a measure of the interruption chain transitional determinism the measurement noise sensitivity to interruption process switchings and a measure of the system s state estimability in some cases identified in this correspondence the proposed bound is tight the use of the lower bound is illustrated via a simple numerical example

amplitude death induced by dynamic coupling the present paper shows that dynamic coupling induces amplitude death in coupled identical oscillators for a simple limit cycle oscillator our theoretical analysis provides the necessary and sufficient condition for amplitude death furthermore we guarantee that amplitude death never occurs if each oscillator satisfies the odd number property that is known in the field of delayed feedback control of chaos

distribution of the ratio of consecutive level spacings in random matrix ensembles we derive expressions for the probability distribution of the ratio of two consecutive level spacings for the classical ensembles of random matrices this ratio distribution was recently introduced to study spectral properties of many body problems as contrary to the standard level spacing distributions it does not depend on the local density of states our wigner like surmises are shown to be very accurate when compared to numerics and exact calculations in the large matrix size limit quantitative improvements are found through a polynomial expansion examples from a quantum many body lattice model and from zeros of the riemann zeta function are presented

the distribution of lyapunov exponents exact results for random matrices simple exact expressions are derived for all the lyapunov exponents of certainn dimensional stochastic linear dynamical systems in the case of the product of independent random matrices each of which has independent gaussian entries with mean zero and variance 1 n the exponents have an exponential distribution asn in the case of the time ordered product integral of exp n 1 2dw where the entries of then n matrixw t are independent standard wiener processes the exponents are equally spaced for fixedn and thus have a uniform distribution as n

linear minimum mean square error estimation for discrete time markovian jump linear systems the linear minimum mean square error estimator lmmse for discrete time linear systems subject to abrupt changes in the parameters modeled by a markov chain spl theta k spl epsiv 1 n is considered the filter equations are derived from geometric arguments in a recursive form resulting in an on line algorithm suitable for computer implementation the author s approach is based on estimating x k 1 sub spl theta k i instead of estimating directly x k the uncertainty introduced by the markovian jumps increases the dimension of the filter to n n 1 where n is the dimension of the state variable an example where the dimension of the filter can be reduced to n is presented as well as a numerical comparison with the imm filter

a lower bound on the bayesian mse based on the optimal bias function a lower bound on the minimum mean squared error mse in a bayesian estimation problem is proposed in this paper this bound utilizes a well known connection to the deterministic estimation setting using the prior distribution the bias function which minimizes the cramer rao bound can be determined resulting in a lower bound on the bayesian mse the bound is developed for the general case of a vector parameter with an arbitrary probability distribution and is shown to be asymptotically tight in both the high and low signal to noise ratio snr regimes a numerical study demonstrates several cases in which the proposed technique is both simpler to compute and tighter than alternative methods

ziv zakai bounds on time delay estimation in unknown convolutive random channels using the ziv zakai bound zzb methodology we develop a bayesian mean square error bound on time delay estimation tde in convolutive random channels and compare it with time delay estimator performance and a crame r rao bound the channel is modeled as a tapped delay line whose taps are gaussian random variables that may be nonzero mean and correlated a model widely adopted in many applications such as wideband fading in a multipath channel the time delay has a uniform prior distribution a zzb is developed that incorporates the prior on the random time delay as well as the convolutive random gaussian channel and does not assume the receiver has knowledge of the channel realization the zzb provides good performance prediction for maximum a posteriori map time delay estimation tracking the low medium and high signal to noise ratio snr regimes the convolutive channel model includes important special cases such as narrowband gaussian channels corresponding to rayleigh rician fading wideband multipath channels with a power decay profile such as exponential decay and known channels we show that the associated crame r rao bound is tight only at high snr whereas the zzb predicts threshold behavior and tde breakdown as the snr decreases when compared to a zzb that assumes knowledge of the channel realization the zzb developed here provides a more realistic and tighter bound revealing the performance loss due to lack of channel knowledge the map estimator incorporates knowledge of the channel statistics and so performs much better than a maximum likelihood estimator that minimizes mean square error but does not use knowledge of the random channel statistics several examples illustrate the estimator and bound behaviors

isotropic brownian motions over complex fields as a solvable model for may wigner stability analysis we consider matrix valued stochastic processes known as isotropic brownian motions and show that these can be solved exactly over complex fields while these processes appear in a variety of questions in mathematical physics our main motivation is their relation to a may wigner like stability analysis for which we obtain a stability phase diagram the exact results establish the full joint probability distribution of the finite time lyapunov exponents and may be used as a starting point for a more detailed analysis of the stability instability phase transition our derivations rest on an explicit formulation of a fokker planck equation for the lyapunov exponents this formulation happens to coincide with an exactly solvable class of models of the calgero sutherland type originally encountered for a model of phase coherent transport the exact solution over complex fields describes a determinantal point process of biorthogonal type similar to recent results for products of random matrices and is also closely related to hermitian matrix models with an external source

explicit ziv zakai lower bound for bearing estimation the extended ziv zakai bound for vector parameters is used to develop a lower bound on the mean square error in estimating the 2 d bearing of a narrowband planewave signal using planar arrays of arbitrary geometry the bound has a simple closed form expression that is a function of the signal wavelength the signal to noise ratio snr the number of data snapshots the number of sensors in the array and the array configuration analysis of the bound suggests that there are several regions of operation and expressions for the thresholds separating the regions are provided in the asymptotic region where the number of snapshots and or snr are large estimation errors are small and the bound approaches the inverse fisher information this is the same as the asymptotic performance predicted by the local cramer rao bound for each value of bearing in the a priori performance region where the number of snapshots or snr is small estimation errors are distributed throughout the a priori parameter space and the bound approaches the a priori covariance in the transition region both small and large errors occur and the bound varies smoothly between the two extremes simulations of the maximum likelihood estimator mle demonstrate that the bound closely predicts the performance of the mle in all regions

mean squared error and threshold snr prediction of maximum likelihood signal parameter estimation with estimated colored noise covariances an interval error based method mie of predicting mean squared error mse performance of maximum likelihood estimators mles is extended to the case of signal parameter estimation requiring intermediate estimation of an unknown colored noise covariance matrix an intermediate step central to adaptive array detection and parameter estimation the successful application of mie requires good approximations of two quantities 1 interval error probabilities and 2 asymptotic snrrarrinfin local mse performance of the mle exact general expressions for the pairwise error probabilities that include the effects of signal model mismatch are derived herein that in conjunction with the union bound provide accurate prediction of the required interval error probabilities the crameacuter rao bound crb often provides adequate prediction of the asymptotic local mse performance of mle the signal parameters however are decoupled from the colored noise parameters in the fisher information matrix for the deterministic signal model rendering the crb incapable of reflecting loss due to colored noise covariance estimation a new modification of the crb involving a complex central beta random variable different from but analogous to the reed mallett and brennan beta loss factor provides a working solution to this problem facilitating mse prediction well into the threshold region with remarkable accuracy

fractal properties of model trajectories with exponential velocity autocorrelation functions abstract the length l at scale has been determined for four different models for atomic motion in a fluid three distinct types of random walk with constant speed and the langevin model this has been done for trajectories in one two three and sometimes four dimensions all these motions have exactly exponential velocity autocorrelation functions however the functional form of l is different for the different models contrary to a recent assertion s toxvaed phys lett a 114 1986 159 a theory for l is lacking except for the limits of small and very large our results can be used as experimental results for the testing of such theories which are quite different from the conventional analysis of molecular motion in fluids a comparison is made of these model results for l with those for simulated trajectories in realistic models of liquids and for the actual trajectories of suspended particles in solution

mmse bounds for additive noise channels under kullback leibler divergence constraints on the input distribution upper and lower bounds on the minimum mean square error for additive noise channels are derived when the input distribution is constrained to be close to a gaussian reference distribution in terms of the kullback leibler divergence the upper bound is tight and is attained by a gaussian distribution whose mean is identical to that of the reference distribution and whose covariance matrix is defined implicitly via a system of non linear equations the estimator that attains the upper bound is identified as a minimax optimal estimator that is robust against deviations from the assumed prior the lower bound provides an alternative to well known inequalities in estimation and information theory such as the cramer rao lower bound stam s inequality or the entropy power inequality that is potentially tighter and defined for a larger class of input distributions several examples of applications in signal processing and information theory illustrate the usefulness of the proposed bounds in practice

frequency domain interpretation of the cramer rao bound for carrier and clock synchronization the cramer rao bound is a fundamental lower bound on the variance of unbiased estimates applied to carrier phase and clock phase synchronization for digital communications it is shown that the corresponding bounds can be derived from the power density spectrum of the transmitted signal irrespective of the particular modulation scheme this allows the localization of the available information about the parameters in the frequency domain if the incoming signal is only partly exploited the resulting performance loss is now easily attainable

weiss weinstein lower bounds for markovian systems part 1 theory being essentially free from regularity conditions the weiss weinstein estimation error lower bound can be applied to a larger class of systems than the well known crameacuter rao lower bound thus this bound is of special interest in applications involving hybrid systems i e systems with both continuously and discretely distributed parameters which can represent in practice fault prone systems however the requirement to know explicitly the joint distribution of the estimated parameters with all the measurements makes the application of the weiss weinstein lower bound to markovian dynamic systems cumbersome a sequential algorithm for the computation of the crameacuter rao lower bound for such systems has been recently reported in the literature along with the marginal state distribution the algorithm makes use of the transitional distribution of the markovian state process and the distribution of the measurements at each time step conditioned on the appropriate states both easily obtainable from the system equations a similar technique is employed herein to develop sequential weiss weinstein lower bounds for a class of markovian dynamic systems in particular it is shown that in systems satisfying the crameacuter rao lower bound regularity conditions the sequential weiss weinstein lower bound derived herein reduces for a judicious choice of its parameters to the sequential crameacuter rao lower bound

a bound on mean square estimation error with background parameter mismatch in typical parameter estimation problems the signal observation is a function of the parameter set to be estimated as well as some background environmental system parameters assumed known the assumed background could differ from the true one leading to biased estimates even at high signal to noise ratio snr to analyze this background mismatch problem a ziv zakai type lower bound on the mean square error mse is developed based on the mismatched likelihood ratio test mlrt at high snr the bound incorporates the increase in mse due to estimation bias at low snr it includes the threshold effect due to estimation ambiguity the kernel of the bound s evaluation is the error probability associated with the mlrt a closed form expression for this error probability is derived under a random signal model typical of the bearing estimation passive source localization problem the mismatch is then analyzed in terms of the related ambiguity functions examples of bearing estimation with system array shape mismatch demonstrate that the developed bound describes the simulations of the maximum likelihood estimate well including the sidelobe introduced threshold behavior and the bias at high snr

posterior cramer rao bounds for discrete time nonlinear filtering a mean square error lower bound for the discrete time nonlinear filtering problem is derived based on the van trees 1968 posterior version of the cramer rao inequality this lower bound is applicable to multidimensional nonlinear possibly non gaussian dynamical systems and is more general than the previous bounds in the literature the case of singular conditional distribution of the one step ahead state vector given the present state is considered the bound is evaluated for three important examples the recursive estimation of slowly varying parameters of an autoregressive process tracking a slowly varying frequency of a single cisoid in noise and tracking parameters of a sinusoidal frequency with sinusoidal phase modulation

bayesian estimation in the presence of deterministic nuisance parameters part i performance bounds how accurately can one estimate a random parameter subject to unknown deterministic nuisance parameters the hybrid cramer rao bound hcrb provides an answer to this question for a restricted class of estimators the hcrb is the most popular performance bound on the mean square error mse for random parameter estimation problems which involve deterministic parameters the hcrb is useful when one is interested in both the random and the deterministic parameters and in the coupling between their estimation errors this bound refers to locally weak sense unbiased estimators with respect to w r t the deterministic parameters however if these parameters are nuisance it is unnecessary to restrict their estimation as unbiased this paper is the first of a two part study of bayesian parameter estimation in the presence of deterministic nuisance parameters it begins with a study on order relations between existing cramer rao cr type bounds of mean unbiased bayesian estimators then a new cr type bound is developed with no assumption of unbiasedness on the nuisance parameters alternatively lehmann s concept of unbiasedness is employed rather than conventional mean unbiasedness it is imposed on a risk that measures the distance between the estimator and the minimum mse mmse estimator which assumes perfect knowledge of the nuisance parameters in the succeeding paper asymptotic performances of some bayesian estimators with maximum likelihood based estimates for the nuisance parameters are investigated the proposed risk unbiased bound rub is proved to be asymptotically achieved by the mmse estimator with maximum likelihood estimates for the nuisance parameters while the existing cr type bounds are not necessarily achievable

weiss weinstein lower bounds for markovian systems part 2 applications to fault tolerant filtering characterized by sudden structural changes fault prone systems are modeled using the framework of systems with switching parameters or hybrid systems since a closed form mean square optimal filtering algorithm for this class of systems does not exist it is of particular interest to derive a lower bound on the state estimation error covariance the well known crameacuter rao bound is not applicable to fault prone systems because of the discrete distribution of the fault indicators which violates the regularity conditions associated with this bound on the other hand the weiss weinstein lower bound is essentially free from regularity conditions moreover a sequential version of the weiss weinstein bound suitable for markovian dynamic systems is presented by the authors in a companion paper in the present paper this sequential version is applied to several classes of fault prone dynamic systems the resulting bounds can be used to examine fault detectability and identifiability in these systems moreover it is shown that several recently reported lower bounds for fault prone systems are special cases of or closely related to the sequential version of the weiss weinstein lower bound

amplitude response of coupled oscillators abstract we investigate the interaction of a pair of weakly nonlinear oscillators eg each near a hopf bifurcation when the coupling strength is comparable to the attraction of the limit cycles changes in amplitude cannot then be ignored and there are new phenomena we show that even for simple forms of these equations there are parameter regimes in which the interaction causes the system to stop oscillating and the rest state at zero stabilized by the interaction is the only stable solution when the uncoupled oscillators have a local frequency that is independent of amplitude zero shear and the coupling is scalar we give a complete description of the behavior we then analyze more complicated equations to show the effects of amplitude dependent frequency in the uncoupled equations nonzero shear and nonscalar coupling for example when there is shear there can be bistability between a phase locked solution and an unlocked drift solution in which the oscillators go at different average frequencies there can also be bistability between a phase locked solution and an equilibrium nonoscillatory solution the equations for the zero shear case have a pair of degenerate bifurcation points and the new phenomena in the nonzero shear case arise from a partial unfolding of these singularities when the coupling is nonscalar there are a variety of new behaviors including bistability of an in phase and an anti phase solution for two identical oscillators parameters for which only the anti phase solution is stable and phase trapping i e nonlocked solutions in which the oscillators still have the same average frequency it is shown that whether the coupling is diffusive or direct has many effects on the behavior of the system

amplitude death in an array of limit cycle oscillators we analyze a large system of limit cycle oscillators with mean field coupling and randomly distributed natural frequencies we prove that when the coupling is sufficiently strong and the distribution of frequencies has sufficiently large variance the system undergoes amplitude death the oscillators pull each other off their limit cycles and into the origin which in this case is astable equilibrium point for the coupled system we determine the region in couplingvariance space for which amplitude death is stable and present the first proof that the infinite system provides an accurate picture of amplitude death in the large but finite system

stability results for discrete time linear systems with markovian jumping parameters stability results for discrete time linear systems subject to random jumps in the parameters are obtained first necessary and sufficient conditions for mean square stability mss including the case in which the system is driven by an independent wide sense stationary random sequence are derived it is shown that mss is equivalent to the spectral radius of an augmented matrix being less than one or to the existence of a solution of a certain lyapunov equation in addition it is shown that the lyapunov equation can be derived in four equivalent forms and as a by product each one gives risc to an easier to check sufficient condition these results give inter alia a unified and rather complete picture for mss of markovian jump linear systems next we derive sufficient conditions for almost sure stability of the noiseless ease finally we conclude by presenting an illustrative application to adaptive filtering where a mild condition for almost sure convergence is provided

the fractal nature of molecular trajectories in fluids the scaled lengths of molecular trajectories obtained by molecular dynamics simulation of a hard sphere fluid are shown to have the same fractal dimensiond 2 as the random walk self similarity first appears on length scales typically a factor of 25 greater than the mean free path length whereas for the simple random walk with constant step size the onset occurs after only six steps the reason for the slow convergence is shown to be the near exponential distribution of intercollision path lengths of the fluid molecules the influence of density on the scaled path lengths is also discussed

information filtering and array algorithms for discrete time markovian jump linear systems this technical note develops information filter and array algorithms for a linear minimum mean square error estimator of discrete time markovian jump linear systems a numerical example for a two mode markovian jump linear system to show the advantage of using array algorithms to filter this class of systems is provided

frequency discontinuity and amplitude death with time delay asymmetry we consider oscillators coupled with asymmetric time delays namely when the speed of information transfer is direction dependent as the coupling parameter is varied there is a regime of amplitude death within which there is a phase flip transition at this transition the frequency changes discontinuously but unlike the equal delay case when the relative phase difference changes by ensuremath pi here the phase difference changes by an arbitrary value that depends on the difference in delays we consider asymmetric delays in coupled landau stuart oscillators and r ossler oscillators analytical estimates of phase synchronization frequencies and phase differences are obtained by separating the evolution equations into phase and amplitude components eigenvalues and eigenvectors of the jacobian matrix in the neighborhood of the transition also show an avoided crossing as has been observed in previous studies with symmetric delays

active state tracking with sensing costs analysis of two states and methods for n states active state tracking is needed in object classification target tracking medical diagnosis and estimation of sparse signals among other various applications herein active state tracking of a discrete time finite state markov chain is considered noisy gaussian observations are dynamically collected by exerting appropriate control over their information content while incurring a related sensing cost the objective is to devise sensing strategies to optimize the tradeoff between tracking performance and sensing cost a recently proposed kalman like estimator by zois et al is employed for state tracking the associated mean squared error and a generic sensing cost metric are then used in a partially observable markov decision process formulation and the optimal sensing strategy is derived via a dynamic programming recursion the resulting recursion proves to be nonlinear challenging control policy design for two state systems with scalar measurements properties of the related cost functions are derived and sufficient conditions are provided regarding the structure of the optimal control policy enabling characterization of when passive state tracking is optimal to overcome the associated computational burden of the optimal sensing strategy two lower complexity strategies are proposed which apply to n state systems with vector measurements the performance of the proposed strategies is illustrated in a wireless body sensing application where cost savings as high as 70 are demonstrated for a 3 detection error with respect to a static equal allocation sensing strategy

discrete time markovian jump linear quadratic optimal control this paper is concerned with the optimal control of discrete time linear systems that possess randomly jumping parameters described by finite state markov processes for problems having quadratic costs and perfect observations the optimal control laws and expected costs to go can be precomputed from a set of coupled riccati like matrix difference equations necessary and sufficient conditions are derived for the existence of optimal constant control laws which stabilize the controlled system as the time horizon becomes infinite with finite optimal expected cost

optimal control of jump linear gaussian systems this paper investigates the problem of controlling a discrete time linear system with jump parameters a review of the literature is presented as well as a development of the application of dynamic programming to this class of control problems dynamic programming has been applied by many researchers and it was observed that no closed form analytical solution could be constructed because of the dual aspects of the controller the main contribution of the present work is an algorithm suitable for computer implementation for the optimal dual control the construction of the algorithm is based on transforming the dynamic programming relations into a space of sufficient statistics and using a finite dimensional optimization procedure to obtain the optimal control as a function of the statistics this is achieved by first developing a suitable recursive realization of a filter which generates the sufficient statistics for the problem and then embedding this filter into the dynamic programming equations

on the non existence of unbiased estimators in constrained estimation problems we address the problem of existence of unbiased constrained parameter estimators we show that if the constrained set of parameters is compact and the hypothesized distributions are absolutely continuous with respect to one another then there exists no unbiased estimator weaker conditions for the absence of unbiased constrained estimators are also specified we provide several examples which demonstrate the utility of these conditions

amplitude death in coupled chaotic oscillators amplitude death can occur in chaotic dynamical systems with time delay coupling similar to the case of coupled limit cycles the coupling leads to stabilization of fixed points of the subsystems this phenomenon is quite general and occurs for identical as well as nonidentical coupled chaotic systems using the lorenz and rossler chaotic oscillators to construct representative systems various possible transitions from chaotic dynamics to fixed points are discussed

optimal design and placement of pilot symbols for channel estimation the problem of designing and placing pilot symbols for the estimation of frequency selective random channels is considered the channel is assumed to be a block fading model with finite impulse response fir for both single input single output siso and multiple input multiple output mimo channels under the assumption of independent and identical distributed channel taps the cramer rao bound crb on the mean square error mse of semi blind channel estimators is derived and minimized with respect to pilot symbols and their placement it is shown that the optimal strategy is to place pilot symbols satisfying certain orthogonality condition in such a way that data and pilot symbols with higher power are in the middle of the packet the results also indicate that the optimal pilot placements are independent of channel probability distribution for constant modulus symbols we show that the quasi periodic placement and its generalization in the multiuser case turn out to be optimal we further consider estimating channels with correlated taps and show that the previous placement strategy is also optimal among orthogonal pilot sequences

outlier probability of generalized chirpogram based estimators for multicomponent polynomial phase signals outliers are often encountered by estimators that have a nonlinear nature in this paper the probability of outliers is derived concerning the generalized chirpogram gc based estimators for polynomial phase signals pps to derive the approximate theoretical expressions we develop an approach that is based on the extrema theory of complex gaussian envelope and the asymptotic poisson model the semi aliasing effect of pps is also considered in order to predict the outliers more accurately then expressions of the probability of outliers are presented for several typical cases of pps including single tone and chirp signals and their two component versions the gc with the hamming windowing is also incorporated numerical examples of computer simulations are also provided to show the accuracy of the theoretical expressions