central limit theorem for sequential monte carlo methods and its application to bayesian inference the term sequential monte carlo methods or equivalently particle filters refers to a general class of iterative algorithms that performs monte carlo approximations of a given sequence of distributions of interest t we establish in this paper a central limit theorem for the monte carlo estimates produced by these computational methods this result holds under minimal assumptions on the distributions t and applies in a general framework which encompasses most of the sequential monte carlo methods that have been considered in the literature including the resample move algorithm of gilks and berzuini j r stat soc ser b stat methodol 63 2001 127 146 and the residual resampling scheme the corresponding asymptotic variances provide a convenient measurement of the precision of a given particle filter we study in particular in some typical examples of bayesian applications whether and at which rate these asymptotic variances diverge in time in order to assess the long term reliability of the considered algorithm

on line bayesian estimation of signals in symmetric spl alpha stable noise in this paper we describe on line bayesian filtering methods for time series models with heavy tailed spl alpha stable noise the filtering methodology is based on a scale mixtures of normals smin representation of the spl alpha stable distribution which reexpresses the intractable stable distribution in a conditionally gaussian form we describe how the method can be used for estimation of time varying autoregressive signals buried in symmetric spl alpha stable noise efficiently implemented using an adaptation to an existing rao blackwellized particle filter the methodology is shown to work well with both simulated and real corrupted audio data for which the spl alpha stable noise distribution is found to fit the noise data better than other more standard heavy tailed distributions

a monte carlo approach to nonnormal and nonlinear state space modeling abstract a solution to multivariate state space modeling forecasting and smoothing is discussed we allow for the possibilities of nonnormal errors and nonlinear functionals in the state equation the observational equation or both an adaptive monte carlo integration technique known as the gibbs sampler is proposed as a mechanism for implementing a conceptually and computationally simple solution in such a framework the methodology is a general strategy for obtaining marginal posterior densities of coefficients in the model or of any of the unknown elements of the state space missing data problems including the k step ahead prediction problem also are easily incorporated into this framework we illustrate the broad applicability of our approach with two examples a problem involving nonnormal error distributions in a linear model setting and a one step ahead prediction problem in a situation where both the state and observational equations are nonlinear and involve unknown parameters

priors and component structures in autoregressive time series models new approaches to prior specification and structuring in autoregressive time series models are introduced and developed we focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series these new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends quasi periodic subprocesses and high frequency residual noise components of observed series the class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series as it does for predictions of future values posterior analysis involves easily implemented iterative simulation methods developed and described here one motivating field of application is climatology where the evaluation of latent structure especially quasi periodic structure is of critical importance in connection with issues of global climatic variability we explore the analysis of data from the southern oscillation index one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators

on unscented kalman filtering for state estimation of continuous time nonlinear systems this paper considers the application of the unscented kalman filter ukf to continuous time filtering problems where both the state and measurement processes are modeled as stochastic differential equations the mean and covariance differential equations which result in the continuous time limit of the ukf are derived the continuous discrete ukf is derived as a special case of the continuous time filter when the continuous time prediction equations are combined with the update step of the discrete time ukf the filter equations are also transformed into sigma point differential equations which can be interpreted as matrix square root versions of the filter equations

sound speed estimation and source localization with linearization and particle filtering a method is developed for the estimation of source location and sound speed in the water column relying on linearization the jacobian matrix necessary for the proposed linearization approach includes derivatives with respect to empirical orthogonal function coefficients instead of sound speed directly first the inversion technique is tested on synthetic arrival times using gaussian distributions for the errors in the considered arrival times the approach is efficient requiring a few iterations and produces accurate results probability densities of the estimates are calculated for different levels of noise in the arrival times subsequently particle filtering is employed for the estimation of arrival times from signals recorded during the shallow water 06 experiment it has been shown in the past that particle filtering can be employed for the successful estimation of multipath arrival times from short range data and consequently in geometry bathymetry and sound speed inversion here probability density functions of arrival times computed via particle filtering are propagated backward through the proposed inversion process inversion estimates are consistent with values reported in the literature for the same quantities last it is shown that results are consistent with estimates resulting from fast simulated annealing applied to the same data

identification of time varying autoregressive systems using maximum a posteriori estimation time varying systems and nonstationary signals arise naturally in many engineering applications such as speech biomedical and seismic signal processing thus identification of the time varying parameters is of crucial importance in the analysis and synthesis of these systems the present time varying system identification techniques require either demanding computation power to draw a large amount of samples monte carlo based methods or a wise selection of basis functions basis expansion methods in this paper the identification of time varying autoregressive systems is investigated it is formulated as a bayesian inference problem with constraints on the conditional and prior probabilities of the time varying parameters these constraints can be set without further knowledge about the physical system in addition only a few hyper parameters need tuning for better performance based on these probabilistic constraints an iterative algorithm is proposed to evaluate the maximum a posteriori estimates of the parameters the proposed method is computationally efficient since random sampling is no longer required simulation results show that it is able to estimate the time varying parameters reasonably well and a balance between the bias and variance of the estimation is achieved by adjusting the hyperparameters moreover simulation results indicate that the proposed method outperforms the particle filter in terms of estimation errors and computational efficiency

blocking strategies and stability of particle gibbs samplers the authors thank the isaac newton institute for mathematical sciences for support and hospitality during the programme monte carlo inference for complex statistical models when work on this paper was undertaken this work was supported by the engineering and physical sciences research council grant numbers ep k020153 1 ep k032208 1 and the swedish research council contract number 2016 04278

particle markov chain monte carlo methods summary markov chain monte carlo and sequential monte carlo methods have emerged as the two main tools to sample from high dimensional probability distributions although asymptotic convergence of markov chain monte carlo algorithms is ensured under weak assumptions the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and or if highly correlated variables are updated independently we show here how it is possible to build efficient high dimensional proposal distributions by using sequential monte carlo methods this allows us not only to improve over standard markov chain monte carlo schemes but also to make bayesian inference feasible for a large class of statistical models where this was not previously so we demonstrate these algorithms on a non linear state space model and a levy driven stochastic volatility model

a quaternion widely linear model for nonlinear gaussian estimation this paper deals with the nonlinear minimum mean squared error estimation problem by using a quaternion widely linear model on the basis of the information supplied by a gaussian signal and its square a quaternion observation process is defined and then by applying a widely linear processing an optimal estimator for the continuous time setting is provided the special structure of the estimator proves its superiority over the complex valued widely linear solution the continuous discrete version of the problem is also studied where the solution takes the form of a suboptimal estimator useful in practical applications in addition the particular case of signal plus noise is considered in which the suboptimal solution and its associated error can be implemented through an iterative algorithm two numerical simulation examples are presented showing the advantages of the proposed approach

particle based adaptive lag online marginal smoothing in general state space models we present a novel algorithm an adaptive lag smoother approximating efficiently in an online fashion sequences of expectations under the marginal smoothing distributions in general state space models the algorithm evolves recursively a bank of estimators one for each marginal in resemblance with the so called particle based rapid incremental smoother paris each estimator is propagated until a stopping criterion measuring the fluctuations of the estimates is met the presented algorithm is furnished with theoretical results describing its asymptotic limit and memory usage

path integral control and state dependent feedback in this paper we address the problem of computing state dependent feedback controls for path integral control problems to this end we generalize the path integral control formula and utilize this to construct parametrized state dependent feedback controllers in addition we show a relation between control and importance sampling better control in terms of control cost yields more efficient importance sampling in terms of effective sample size the optimal control provides a zero variance estimate

wittrick williams algorithm proof of bracketing and convergence theorems for eigenvalues of constrained structures with positive and negative penalty parameters the well established wittrick williams algorithm is used to derive novel and general proofs that show that the eigenvalues of systems with constraints can be bracketed by replacing the constraints by positive and negative pairs of either ordinary or inertial penalty parameters it is also shown that convergence occurs from both above and below when the numerical values of these parameters are increased towards infinity the proofs are applicable in many contexts but are derived in that of structural systems for which the eigenvalues are either buckling load factors or the squares of natural frequencies of vibration ordinary penalty parameters are stiffnesses of translational and rotational springs and inertial penalty parameters are either masses or rotary inertias the penalty parameters can be used to constrain a system or to impose constraints between systems it is shown that the use of inertial penalty parameters has several advantages compared with using ordinary ones then the pth eigenvalue of a system with n constraints is bounded closely from above by the p n th eigenvalue of the system with very large positive inertial penalty parameters and from below by the pth eigenvalue when large negative values are used instead this work is expected to enhance the versatility of numerical eigenproblem methods e g the rayleigh ritz method copyright 2007 john wiley sons ltd

adaptive importance sampling for control and inference path integral pi control problems are a restricted class of non linear control problems that can be solved formally as a feynman kac pi and can be estimated using monte carlo sampling in this contribution we review pi control theory in the finite horizon case we subsequently focus on the problem how to compute and represent control solutions we review the most commonly used methods in robotics and control within the pi theory the question of how to compute becomes the question of importance sampling efficient importance samplers are state feedback controllers and the use of these requires an efficient representation learning and representing effective state feedback controllers for non linear stochastic control problems is a very challenging and largely unsolved problem we show how to learn and represent such controllers using ideas from the cross entropy method we derive a gradient descent method that allows to learn feed back controllers using an arbitrary parametrisation we refer to this method as the path integral cross entropy method or pice we illustrate this method for some simple examples the pi control methods can be used to estimate the posterior distribution in latent state models in neuroscience these problems arise when estimating connectivity from neural recording data using em we demonstrate the pi control method as an accurate alternative to particle filtering

variance reduction for simulated diffusions this article develops some variance reduction techniques for the monte carlo integration of functionals of the solutions of ito stochastic differential equations sdes the monte carlo method for sdes offers a means of calculating solutions to certain types of parabolic partial differential equation and so has applications in various fields including stochastic control particle physics and econometrics it involves the representation of the required integrals as means of random variables defined on infinite dimensional wiener spaces which cannot be simulated directly sthey must at some stage be approximated by variables defined on high but finite dimensional spaces the approach taken here is to construct variance reduced random variables on the infinite dimensional spaces which can subsequently be approximated by any of a number of known finite difference methods the methods of control variates and importance sampling are developed in both cases a perfect variate i e one which is unbiased and h

a recursive recomputation approach for smoothing in nonlinear state space modeling an attempt for reducing space complexity in this paper we develop a new generic implementation scheme for numerical smoothing in nonlinear and bayesian state space modeling our new generic implementation scheme which we call recursive recomputation scheme reduces the space complexity from o mt to o m log t at the cost of o log t times computation of filtering distributions in time complexity this reduction is accomplished by employing carefully designed recursive recomputation the japanese stock market price time series data with t 956 is taken up as an instance to demonstrate advantage of the proposed scheme the path sampling particle smoother is implemented with the scheme to smooth the whole interval estimating the change of volatility the number of particles is 3 000 000 and the whole interval is smoothed with 5 3 gb storage accomplishing saving of storage by a factor of 1 20 the computed smoothing distribution is compared with the ones computed with the existing two other well known smoothers the forward backward smoother and the smoother based on two filter formula it turns out that among the three ours is the only method which succeeded in computing a reliable and plausible smoothing distribution in the situation

robust filtering and smoothing with gaussian processes we propose a principled algorithm for robust bayesian filtering and smoothing in nonlinear stochastic dynamic systems when both the transition function and the measurement function are described by non parametric gaussian process gp models gps are gaining increasing importance in signal processing machine learning robotics and control for representing unknown system functions by posterior probability distributions this modern way of system identification is more robust than finding point estimates of a parametric function representation our principled filtering smoothing approach for gp dynamic systems is based on analytic moment matching in the context of the forward backward algorithm our numerical evaluations demonstrate the robustness of the proposed approach in situations where other state of the art gaussian filters and smoothers can fail

following a moving target monte carlo inference for dynamic bayesian models markov chain monte carlo mcmc sampling is a numerically intensive simulation technique which has greatly improved the practicality of bayesian inference and prediction however mcmc sampling is too slow to be of practical use in problems involving a large number of posterior target distributions as in dynamic modelling and predictive model selection alternative simulation techniques for tracking moving target distributions known as particle filters which combine importance sampling importance resampling and mcmc sampling tend to suffer from a progressive degeneration as the target sequence evolves we propose a new technique based on these same simulation methodologies which does not suffer from this progressive degeneration

improved particle approximations to the joint smoothing distribution using markov chain monte carlo particle filtering and smoothing algorithms approximate posterior state distributions with a set of samples drawn from those distributions conventionally samples from the joint smoothing distribution are generated by sequentially resampling from the particle filter results if the number of filtering particles is high this process is limited by computational complexity in addition the support of the smoothing distribution is restricted to the values which appear in the filtering approximation in this paper a metropolis hastings sampling procedure is used to improve the efficiency of the particle smoother achieving comparable error performance but with a lower execution time in addition an algorithm for approximating the joint smoothing distribution without limited support is presented which achieves simultaneous improvements in both execution time and error these algorithms also provide a greater degree of flexibility over existing methods allowing a trade off between execution time and error controlled by the length of the markov chains

a generalized path integral control approach to reinforcement learning with the goal to generate more scalable algorithms with higher efficiency and fewer open parameters reinforcement learning rl has recently moved towards combining classical techniques from optimal control and dynamic programming with modern learning techniques from statistical estimation theory in this vein this paper suggests to use the framework of stochastic optimal control with path integrals to derive a novel approach to rl with parameterized policies while solidly grounded in value function estimation and optimal control based on the stochastic hamilton jacobi bellman hjb equations policy improvements can be transformed into an approximation problem of a path integral which has no open algorithmic parameters other than the exploration noise the resulting algorithm can be conceived of as model based semi model based or even model free depending on how the learning problem is structured the update equations have no danger of numerical instabilities as neither matrix inversions nor gradient learning rates are required our new algorithm demonstrates interesting similarities with previous rl research in the framework of probability matching and provides intuition why the slightly heuristically motivated probability matching approach can actually perform well empirical evaluations demonstrate significant performance improvements over gradient based policy learning and scalability to high dimensional control problems finally a learning experiment on a simulated 12 degree of freedom robot dog illustrates the functionality of our algorithm in a complex robot learning scenario we believe that policy improvement with path integrals pi2 offers currently one of the most efficient numerically robust and easy to implement algorithms for rl based on trajectory roll outs

blind deconvolution via sequential imputations abstract the sequential imputation procedure is applied to adaptively and sequentially reconstruct discrete input signals that are blurred by an unknown linear moving average channel and contaminated by additive gaussian noises a problem known as blind deconvolution in digital communication a rejuvenation procedure for improving the efficiency of sequential imputation is introduced and theoretically justified the proposed method does not require the channel to be nonminimum phase and can be used in real time signal restoration two simulated systems are studied to illustrate the proposed method our result shows that the ideas of multiple imputations and flexible simulation techniques are as powerful in engineering as in survey sampling

particle smoothing algorithms for variable rate models standard state space methods assume that the latent state evolves uniformly over time and can be modeled with a discrete time process synchronous with the observations this may be a poor representation of some systems in which the state evolution displays discontinuities in its behavior for such cases a variable rate model may be more appropriate the system dynamics are conditioned on a set of random changepoints which constitute a marked point process in this paper new particle smoothing algorithms are presented for use with conditionally linear gaussian and conditionally deterministic dynamics these are demonstrated on problems in financial modelling and target tracking results indicate that the smoothing approximations provide more accurate and more diverse representations of the state posterior distributions

semi widely simulation and estimation of continuous time bbc eta proper quaternion random signals the repercussions of quaternion bbc eta properness on the continuous time simulation and estimation problems are studied as a first step a semi widely linear quaternion karhunen loeve expansion is derived then this series representation is used in both problems to give the solutions the simulation technique proposed is useful for transformations of nonstationary gaussian quaternion signals the estimation problem is also analyzed in a continuous discrete setting and by considering two different formulations the cambanis formulation and the signal plus noise model initially the estimators are developed on the basis of a semi widely linear processing afterwards the solutions are improved by incorporating the information supplied by the square quaternion observations and by using a semi widely nonlinear processing in this last scenario we assume gaussian observations and unknown probability distributions of the signal of interest

partial non gaussian state space summary in this paper we suggest the use of simulation techniques to extend the applicability of the usual gaussian state space filtering and smoothing techniques to a class of nongaussian time series models this allows a fully bayesian or maximum likelihood analysis of some interesting models including outlier models discrete markov chain components multiplicative models and stochastic variance models finally we discuss at some length the use of a non gaussian model to seasonally adjust the published money supply figures

rao blackwellized particle filters with out of sequence measurement processing this paper addresses the out of sequence measurement oosm problem for mixed linear nonlinear state space models which is a class of nonlinear models with a tractable conditionally linear substructure we develop two novel algorithms that utilize the linear substructure the first algorithm effectively employs the rao blackwellized particle filtering framework for updating with the oosms and is based on storing only a subset of the particles and their weights over an arbitrary predefined interval the second algorithm adapts a backward simulation approach to update with the delayed out of sequence measurements resulting in superior tracking performance extensive simulation studies show the efficacy of our approaches in terms of computation time and tracking performance both algorithms yield estimation improvements when compared with recent particle filter algorithms for oosm processing in the considered examples they achieve up to 10 enhancements in estimation accuracy in some cases the proposed algorithms even deliver accuracy that is similar to the lower performance bounds because the considered setup is common in various estimation scenarios the developed algorithms enable improvements in different types of filtering applications

a study on smoothing for particle filtered 3d human body tracking stochastic models have become the dominant means of approaching the problem of articulated 3d human body tracking where approximate inference is employed to tractably estimate the high dimensional 30d posture space of these approximate inference techniques particle filtering is the most commonly used approach however filtering only takes into account past observations almost no body tracking research employs smoothing to improve the filtered inference estimate despite the fact that smoothing considers both past and future evidence and so should be more accurate in an effort to objectively determine the worth of existing smoothing algorithms when applied to human body tracking this paper investigates three approximate smoothed inference techniques particle filtered backwards smoothing variational approximation and gibbs sampling results are quantitatively evaluated on both the humaneva dataset as well as a scene containing occluding clutter surprisingly it is found that existing smoothing techniques are unable to provide much improvement on the filtered estimate and possible reasons as to why are explored and discussed

likelihood analysis of non gaussian measurement time series summary in this paper we provide methods for estimating non gaussian time series models these techniques rely on markov chain monte carlo to carry out simulation smoothing and bayesian posterior analysis of parameters and on importance sampling to estimate the likelihood function for classical inference the time series structure of the models is used to ensure that our simulation algorithms are efficient

on approximate maximum likelihood methods for blind identification how to cope with the curse of dimensionality we discuss approximate maximum likelihood methods for blind identification and deconvolution these algorithms are based on particle approximation versions of the expectation maximization em algorithm we consider three different methods which differ in the way the posterior distribution of the symbols is computed the first algorithm is a particle approximation method of the fixed interval smoothing the two filter smoothing and the novel joined two filter smoothing involve an additional backward information filter because the state space is finite it is furthermore possible at each step to consider all the offsprings of any given particle it is then required to construct a novel particle swarm by selecting among all these offsprings particle positions and computing appropriate weights we propose here a novel unbiased selection scheme which minimizes the expected loss with respect to general distance functions we compare these smoothing algorithms and selection schemes in a monte carlo experiment we show a significant performance increase compared to the expectation maximization viterbi algorithm emva a fixed lag smoothing algorithm and the block constant modulus algorithm cma

solving quantum stochastic lqr optimal control problem in fock space and its application in finance abstract this paper is an attempt for solving operator valued quantum stochastic optimal control problems in fock space for this purpose the dynamics of the classical system state is described by hudson parthasarathy type quantum stochastic differential equation qsde in fock space and then by associating a quadratic performance criterion with the qsde a quantum stochastic linear quadratic regulator qs lqr optimal control problem is formulated also an algorithm for solving the qs lqr optimal control problem is designed for solving the resulting optimal control problem a new hjb equation is obtained thereby the operator valued control process is obtained two theorems are proved to facilitate the algorithm in this paper for the first time the optimal strategy for trading stock is designed via the presented method for this purpose merton portfolio allocation problem is solved the simulation results show that portfolio optimal performances minimum risk and maximum return are achieved via presented method

fixed lag smoothing for bayes optimal knowledge exploitation in target tracking in this work we are interested in the improvements attainable when multiscan processing of external knowledge is performed over a moving time window we propose a novel algorithm that enforces the state constraints by using a fixed lag smoothing procedure within the prediction step of the bayesian recursion for proving the improvements we utilize differential entropy as a measure of uncertainty and show that the approach guarantees a lower or equal posterior differential entropy than classical single step constrained filtering simulation results using examples for single target tracking are presented to verify that a sequential monte carlo implementation of the proposed algorithm guarantees an improved tracking accuracy

closed form solutions to forward backward smoothing we propose a closed form gaussian sum smoother and more importantly closed form smoothing solutions for increasingly complex problems arising from practice including tracking in clutter joint detection and tracking in clutter and multiple target tracking in clutter via the probability hypothesis density the solutions are based on the corresponding forward backward smoothing recursions that involve forward propagation of the filtering densities followed by backward propagation of the smoothed densities the key to the exact solutions is the use of alternative forms of the backward propagations together with standard gaussian identities simulations are also presented to verify the proposed solutions

monte carlo smoothing for nonlinear time series we develop methods for performing smoothing computations in general state space models the methods rely on a particle representation of the filtering distributions and their evolution through time using sequential importance sampling and resampling ideas in particular novel techniques are presented for generation of sample realizations of historical state sequences this is carried out in a forward filtering backward smoothing procedure that can be viewed as the nonlinear non gaussian counterpart of standard kalman filter based simulation smoothers in the linear gaussian case convergence in the mean squared error sense of the smoothed trajectories is proved showing the validity of our proposed method the methods are tested in a substantial application for the processing of speech signals represented by a time varying autoregression and parameterized in terms of time varying partial correlation coefficients comparing the results of our algorithm with those from a simple smoother based on the filte

approximate smoothing and parameter estimation in high dimensional state space models we present approximate algorithms for performing smoothing in a class of high dimensional state space models via sequential monte carlo methods particle filters in high dimensions a prohibitively large number of monte carlo samples particles growing exponentially in the dimension of the state space are usually required to obtain a useful smoother employing blocking approximations we exploit the spatial ergodicity properties of the model to circumvent this curse of dimensionality we thus obtain approximate smoothers that can be computed recursively in time and parallel in space first we show that the bias of our blocked smoother is bounded uniformly in the time horizon and in the model dimension we then approximate the blocked smoother with particles and derive the asymptotic variance of idealized versions of our blocked particle smoother to show that variance is no longer adversely effected by the dimension of the model finally we employ our method to successfully perform maximum likelihood estimation via stochastic gradient ascent and stochastic expectation maximization algorithms in a 100 dimensional state space model

estimation of delays and other parameters in nonlinear functional differential equations we discuss a spline based approximation scheme for nonlinear nonautonomous delay differential equations convergence results using dissipative type estimates on the underlying nonlinear operators are given in the context of parameter estimation problems which include estimation of multiple delays and initial data as well as the usual coefficient type parameters a brief summary of some of our related numerical findings is also given

exact solutions for sturm liouville problems on trees via novel substitute systems and the wittrick williams algorithm since 1970 the wittrick williams algorithm has been applied with increasing sophistication in structural mechanics to guarantee that eigenvalues cannot be missed and are calculated accurately the underlying theorem enables its application to any discipline requiring eigenvalues of self adjoint systems of differential equations its value in mathematics was recently illustrated by studying sturm liouville equations on large homogeneous trees with dirichlet boundary conditions and n 43 levels recursive subsysteming was applied n 1 times to assemble the tree progressively from sub trees hence numerical results confirmed the recent theoretical bounds of sobolev solomyak for n in addition a structural mechanics analogy yielded a proof that many eigenvalues had high multiplicities determined by n and the branching number b inspired by the structural mechanics analogy we now prove that all eigenvalues of the tree are obtainable from n substitute chains r 1 2 n which involve only r consecutively linked differential equations and which have only singlefold eigenvalues equations are also derived for the multiplicities these eigenvalues have for the tree hence double precision calculations on a pc readily gave eigenvalues for n 10 6 and b 10 i e 10 999 999 linked sturm liouville equations moreover a simple equation is derived which gives all the eigenvalues of uniform trees with dirichlet conditions at both ends and band gap spectra are numerically demonstrated and theoretically justified for trees with the dirichlet conditions at either end replaced by neumann ones additionally even if each multiple eigenvalue would be counted as if it were singlefold the proportion of eigenvalues that are multiple is proved to approach unity as n

efficient markov chain monte carlo methods for decoding neural spike trains stimulus reconstruction or decoding methods provide an important tool for understanding how sensory and motor information is represented in neural activity we discuss bayesian decoding methods based on an encoding generalized linear model glm that accurately describes how stimuli are transformed into the spike trains of a group of neurons the form of the glm likelihood ensures that the posterior distribution over the stimuli that caused an observed set of spike trains is log concave so long as the prior is this allows the maximum a posteriori map stimulus estimate to be obtained using efficient optimization algorithms unfortunately the map estimate can have a relatively large average error when the posterior is highly nongaussian here we compare several markov chain monte carlo mcmc algorithms that allow for the calculation of general bayesian estimators involving posterior expectations conditional on model parameters an efficient version of the hybrid monte carlo hmc algorithm was significantly superior to other mcmc methods for gaussian priors when the prior distribution has sharp edges and corners on the other hand the hit and run algorithm performed better than other mcmc methods using these algorithms we show that for this latter class of priors the posterior mean estimate can have a considerably lower average error than map whereas for gaussian priors the two estimators have roughly equal efficiency we also address the application of mcmc methods for extracting nonmarginal properties of the posterior distribution for example by using mcmc to calculate the mutual information between the stimulus and response we verify the validity of a computationally efficient laplace approximation to this quantity for gaussian priors in a wide range of model parameters this makes direct model based computation of the mutual information tractable even in the case of large observed neural populations where methods based on binning the spike train fail finally we consider the effect of uncertainty in the glm parameters on the posterior estimators

applications of the cross entropy method to importance sampling and optimal control of diffusions we study the cross entropy method for diffusions one of the results is a versatile cross entropy algorithm that can be used to design efficient importance sampling strategies for rare events or to solve optimal control problems the approach is based on the minimization of a suitable cross entropy functional with a parametric family of exponentially tilted probability distributions we illustrate the new algorithm with several numerical examples and discuss algorithmic issues and possible extensions of the method

filtering via simulation auxiliary particle filters this article analyses the recently suggested particle approach to filtering time series we suggest that the algorithm is not robust to outliers for two reasons the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution here we tackle the first of these problems

optimal smoothing of non linear dynamic systems via monte carlo markov chains we consider the smoothing problem of estimating a sequence of state vectors given a nonlinear state space model with additive white gaussian noise and measurements of the system output the system output may also be nonlinearly related to the system state often obtaining the minimum variance state estimates conditioned on output data is not analytically intractable to tackle this difficulty a markov chain monte carlo technique is presented the proposal density for this method efficiently draws samples from the laplace approximation of the posterior distribution of the state sequence given the measurement sequence this proposal density is combined with the metropolis hastings algorithm to generate realizations of the state sequence that converges to the proper posterior distribution the minimum variance estimate and confidence intervals are approximated using these realizations simulations of a fed batch bioreactor model are used to demonstrate that the proposed method can obtain significantly better estimates than the iterated kalman bucy smoother

the simulation smoother for time series models summary recently suggested procedures for simulating from the posterior density of states given a gaussian state space time series are refined and extended we introduce and study the simulation smoother which draws from the multivariate posterior distribution of the disturbances of the model so avoiding the degeneracies inherent in state samplers the technique is important in gibbs sampling with non gaussian time series models and for performing bayesian analysis of gaussian time series

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

particle efficient importance sampling abstract the efficient importance sampling eis method is a general principle for the numerical evaluation of high dimensional integrals that uses the sequential structure of target integrands to build variance minimising importance samplers despite a number of successful applications in high dimensions it is well known that importance sampling strategies are subject to an exponential growth in variance as the dimension of the integration increases we solve this problem by recognising that the eis framework has an offline sequential monte carlo interpretation the particle eis method is based on non standard resampling weights that take into account the construction of the importance sampler as a sequential approximation to the state smoothing density we apply the method for a range of univariate and bivariate stochastic volatility specifications we also develop a new application of the eis approach to state space models with student s t state innovations our results show that the particle eis method strongly outperforms both the standard eis method and particle filters for likelihood evaluation in high dimensions we illustrate the efficiency of the method for bayesian inference using the particle marginal metropolis hastings and importance sampling squared algorithms

rao blackwellization of particle markov chain monte carlo methods using forward filtering backward sampling smoothing in state space models amounts to computing the conditional distribution of the latent state trajectory given observations or expectations of functionals of the state trajectory with respect to this distribution in recent years there has been an increased interest in monte carlo based methods often involving particle filters for approximate smoothing in nonlinear and or non gaussian state space models one such method is to approximate filter distributions using a particle filter and then to simulate using backward kernels a state trajectory backwards on the set of particles we show that by simulating multiple realizations of the particle filter and adding a metropolis hastings step one obtains a markov chain monte carlo scheme whose stationary distribution is the exact smoothing distribution this procedure expands upon a similar one recently proposed by andrieu doucet holenstein and whiteley we also show that simulating multiple trajectories from each realization of the particle filter can be beneficial from a perspective of variance versus computation time and illustrate this idea using two examples

series expansion approximations of brownian motion for non linear kalman filtering of diffusion processes in this paper we describe a novel application of sigma point methods to continuous discrete filtering the nonlinear continuous discrete filtering problem is often computationally intractable to solve assumed density filtering methods attempt to match statistics of the filtering distribution to some set of more tractable probability distributions filters such as these are usually decompose the problem into two sub problems the first of these is a prediction step in which one uses the known dynamics of the signal to predict its state at time tk 1 given observations up to time tk in the second step one updates the prediction upon arrival of the observation at time tk 1 the aim of this paper is to describe a novel method that improves the prediction step we decompose the brownian motion driving the signal in a generalised fourier series which is truncated after a number of terms this approximation to brownian motion can be described using a relatively small number of fourier coefficients and allows us to compute statistics of the filtering distribution with a single application of a sigma point method assumed density filters that exist in the literature usually rely on discretisation of the signal dynamics followed by iterated application of a sigma point transform or a limiting case thereof iterating the transform in this manner can lead to loss of information about the filtering distribution in highly non linear settings we demonstrate that our method is better equipped to cope with such problems

a sequential smoothing algorithm with linear computational cost in this paper we propose a new particle smoother that has a computational complexity of o n where n is the number of particles this compares favourably with the o n2 computational cost of most smoothers the new method also overcomes some degeneracy problems in existing algorithms through simulation studies we show that substantial gains in efficiency are obtained for practical amounts of computational cost it is shown both through these simulation studies and by the analysis of an athletics dataset that our new method also substantially outperforms the simple filter smoother the only other smoother with computational cost that is o n

on gibbs sampling for state space models summary we show how to use the gibbs sampler to carry out bayesian inference on a linear state space model with errors that are a mixture of normals and coefficients that can switch over time our approach simultaneously generates the whole of the state vector given the mixture and coefficient indicator variables and simultaneously generates all the indicator variables conditional on the state vectors the states are generated efficiently using the kalman filter we illustrate our approach by several examples and empirically compare its performance to another gibbs sampler where the states are generated one at a time the empirical results suggest that our approach is both practical to implement and dominates the gibbs sampler that generates the states one at a time

an iterative bayesian filtering framework for fast and automated calibration of dem models abstract the nonlinear history dependent macroscopic behavior of a granular material is rooted in the micromechanics between constituent particles and irreversible plastic deformations reflected by changes in the microstructure the discrete element method dem can predict the evolution of the microstructure resulting from interparticle interactions however micromechanical parameters at contact and particle levels are generally unknown because of the diversity of granular materials with respect to their surfaces shapes disorder and anisotropy the proposed iterative bayesian filter consists in recursively updating the posterior distribution of model parameters and iterating the process with new samples drawn from a proposal density in highly probable parameter spaces over iterations the proposal density is progressively localized near the posterior modes which allows automated zooming towards optimal solutions the dirichlet process gaussian mixture is trained with sparse and high dimensional data from the previous iteration to update the proposal density as an example the probability distribution of the micromechanical parameters is estimated conditioning on the experimentally measured stress strain behavior of a granular assembly four micromechanical parameters i e contact level young s modulus interparticle friction rolling stiffness and rolling friction are chosen as strongly relevant for the macroscopic behavior the a priori particle configuration is obtained from 3d x ray computed tomography images the a posteriori expectation of each micromechanical parameter converges within four iterations leading to an excellent agreement between the experimental data and the numerical predictions as new result the proposed framework provides a deeper understanding of the correlations among micromechanical parameters and between the micro and macro parameters quantities of interest including their uncertainties therefore the iterative bayesian filtering framework has a great potential for quantifying parameter uncertainties and their propagation across various scales in granular materials

an estimator for the relative entropy rate of path measures for stochastic differential equations we address the problem of estimating the relative entropy rate rer for two stochastic processes described by stochastic differential equations for the case where the drift of one process is known analytically but one has only observations from the second process we use a variational bound on the rer to construct an estimator

numerical integration of stochastic differential equations several numerical methods for treating stochastic differential equations are considered both the convergence in the mean square limit and the convergence of the moments is discussed and the generation of appropriate random numbers is treated the necessity of simulations at various time steps with an extrapolation to time step zero is emphasized and demonstrated by a simple example

invariance results for delay and volterra equations in fractional order sobolev spaces on deduit l invariance des trajectoires d equations de type volterra et a retard infini dans des espaces de sobolev d ordre fractionnaire pour des hypotheses minimales sur les donnees du probleme

approximation of infinite delay and volterra type equations we consider a class of infinite delay equations of neutral type which includes volterrra type integral und integrodifferential equations using abstract approximation results cf trotter kato type for strongly continuous semigroups we develop an approximation scheme which is based on approximation of the system state by laguerre and legendre polynomials numerical examples demonstrate the feasibility of the scheme and show infinite order convergence for smooth data

an asymptotic numerical method for a class of singular sturm liouville problems a class of sturm liouville problems has been discovered for which the method of approximating the coefficients by piecewise constant functions has considerable difficulty we characterize this class give examples and derive a modification of the method based on local asymptotic approximations which surmounts the difficulties

numerical methods for stochastic differential equations stochastic differential equations sde s play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability a general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here high order numerical methods are developed for the integration of stochastic differential equations with strong solutions we demonstrate the accuracy of the resulting integration schemes by computing the errors in approximate solutions for sde s which have known exact solutions

estimation of affine term structure models with spanned or unspanned stochastic volatility we develop new procedures for maximum likelihood estimation of affine term structure models with spanned or unspanned stochastic volatility our approach uses linear regression to reduce the dimension of the numerical optimization problem yet it produces the same estimator as maximizing the likelihood it improves the numerical behavior of estimation by eliminating parameters from the objective function that cause problems for conventional methods we find that spanned models capture the cross section of yields well but not volatility while unspanned models fit volatility at the expense of fitting the cross section

system identification of nonlinear state space models this paper is concerned with the parameter estimation of a general class of nonlinear dynamic systems in state space form more specifically a maximum likelihood ml framework is employed and an expectation maximisation em algorithm is derived to compute these ml estimates the expectation e step involves solving a nonlinear state estimation problem where the smoothed estimates of the states are required this problem lends itself perfectly to the particle smoother which provides arbitrarily good estimates the maximisation m step is solved using standard techniques from numerical optimisation theory simulation examples demonstrate the efficacy of our proposed solution

recursive monte carlo filters algorithms and theoretical analysis recursive monte carlo filters also called particle filters are a powerful tool to perform the computations in general state space models we discuss and compare the accept reject version with the more common sampling importance resampling version of the algorithm in particular we show how auxiliary variable methods and stratification can be used in the accept reject version and we compare dierent resampling techniques in a second part we show laws of large numbers and a central limit theorem for these monte carlo filters by simple induction arguments that need only weak conditions we also show that under stronger conditions the required sample size is independent of the length of the observed series ams 2000 subject classifications primary 62m09 secondary 60g35 60j22 65c05

remarks on semilinear partial functional differential equations with infinite delay abstract in this paper we will be concerned with the abstract semilinear equation x t ax t f t xt t x where a represents the infinitesimal generator of a linear c0 semigroup in a banach space and x denotes the segment of the solution x due to the initial function taken from up to t conditions for existence continuous dependence and regularity of solutions are given which are independent of the topology of the phase space as well as conditions which guarantee the compactness of bounded orbits even if exp at is not compact for t 0

unscented rauch tung striebel smoother this note considers the application of the unscented transform to optimal smoothing of nonlinear state space models in this note a new rauch tung striebel type form of the fixed interval unscented kalman smoother is derived the new smoother differs from the previously proposed two filter formulation based unscented kalman smoother in the sense that it is not based on running two independent filters forward and backward in time instead a separate backward smoothing pass is used which recursively computes corrections to the forward filtering result the smoother equations are derived as approximations to the formal bayesian optimal smoothing equations the performance of the new smoother is demonstrated with a simulation

an analysis of temporal difference learning with function approximation we discuss the temporal difference learning algorithm as applied to approximating the cost to go function of an infinite horizon discounted markov chain the algorithm we analyze updates parameters of a linear function approximator online during a single endless trajectory of an irreducible aperiodic markov chain with a finite or infinite state space we present a proof of convergence with probability one a characterization of the limit of convergence and a bound on the resulting approximation error furthermore our analysis is based on a new line of reasoning that provides new intuition about the dynamics of temporal difference learning in addition to proving new and stronger positive results than those previously available we identify the significance of online updating and potential hazards associated with the use of nonlinear function approximators first we prove that divergence may occur when updates are not based on trajectories of the markov chain this fact reconciles positive and negative results that have been discussed in the literature regarding the soundness of temporal difference learning second we present an example illustrating the possibility of divergence when temporal difference learning is used in the presence of a nonlinear function approximator

consistent estimation of the filtering and marginal smoothing distributions in nonparametric hidden markov models in this paper we consider the filtering and smoothing recursions in nonparametric finite state space hidden markov models hmms when the parameters of the model are unknown and replaced by estimators we provide an explicit and time uniform control of the filtering and smoothing errors in total variation norm as a function of the parameter estimation errors we prove that the risk for the filtering and smoothing errors may be uniformly upper bounded by the mathrm l 1 risk of the estimators it has been proved very recently that statistical inference for finite state space nonparametric hmms is possible we study how the recent spectral methods developed in the parametric setting may be extended to the nonparametric framework and we give explicit upper bounds for the mathrm l 2 risk of the nonparametric spectral estimators in the case where the observation space is compact this provides explicit rates for the filtering and smoothing errors in total variation norm the performance of the spectral method is assessed with simulated data for both the estimation of the nonparametric conditional distribution of the observations and the estimation of the marginal smoothing distributions

particle filters for partially observed boolean dynamical systems abstract partially observed boolean dynamical systems pobds are a general class of nonlinear models with application in estimation and control of boolean processes based on noisy and incomplete measurements the optimal minimum mean square error mmse algorithms for pobds state estimation namely the boolean kalman filter bkf and boolean kalman smoother bks are intractable in the case of large systems due to computational and memory requirements to address this we introduce approximate mmse filtering and smoothing algorithms based on the auxiliary particle filter apf method which are called apf bkf and apf bks respectively for joint state and parameter estimation the apf bkf is used jointly with maximum likelihood ml methods for simultaneous state and parameter estimation in pobds models in the case the unknown parameters are discrete the proposed ml adaptive filter consists of multiple apf bkfs running in parallel in a manner reminiscent of the multiple model adaptive estimation mmae method in classical linear filtering theory in the presence of continuous parameters the proposed ml adaptive filter is based on an efficient particle based expectation maximization em algorithm for the pobds model which is based on a modified forward filter backward simulation ffbsi in combination with the apf bks the performance of the proposed particle based adaptive filters is assessed through numerical experiments using a pobds model of the well known cell cycle gene regulatory network observed through noisy rna seq time series data

importance sampling for continuous time bayesian networks a continuous time bayesian network ctbn uses a structured representation to describe a dynamic system with a finite number of states which evolves in continuous time exact inference in a ctbn is often intractable as the state space of the dynamic system grows exponentially with the number of variables in this paper we first present an approximate inference algorithm based on importance sampling we then extend it to continuous time particle filtering and smoothing algorithms these three algorithms can estimate the expectation of any function of a trajectory conditioned on any evidence set constraining the values of subsets of the variables over subsets of the time line we present experimental results on both synthetic networks and a network learned from a real data set on people s life history events we show the accuracy as well as the time efficiency of our algorithms and compare them to other approximate algorithms expectation propagation and gibbs sampling

a self organizing state space model a self organizing filter and smoother for the general nonlinear non gaussian state space model is proposed an expanded state space model is defined by augmenting the state vector with the unknown parameters of the original state space model the state of the augmented state space model and hence the state and the parameters of the original state space model are estimated simultaneously by either a non gaussian filter smoother or a monte carlo filter smoother in contrast to maximum likelihood estimation of model parameters in ordinary state space modeling for which the recursive filter computation has to be done many times model parameter estimation in the proposed self organizing filter smoother is achieved with only two passes of the recursive filter and smoother operations examples such as automatic tuning of dispersion and the shape parameters adaptation to changes of the amplitude of a signal in seismic data state estimation for a nonlinear state space model with unknown parameters and seasonal adjustment with a nonlinear model with changing variance parameters are shown to exemplify the usefulness of the proposed method

two filter formulae for discrete time non linear bayesian smoothing two filler formulae for the bayes solution of the fixed interval discrete time nonlinear smoothing problem are obtained the smoothed a posteriori density is computed under the assumptions of a general markov signal observed through a general memoryless noisy channel the case where there is feedback from the observation to the signal is also considered the derived algorithms complement a two pass algorithm obtained under somewhat more restrictive assumptions by askar and derin 1981 known smoothing results for the linear gaussian case are interpreted in the light of the general bayesian results

on the solution of s ω x 0 by a newtonian procedure the method 1 developed for the solution of e a x 0 where e and a are real symmetric matrices and a is positive definite is extended to deal with s x 0 where s is a real symmetric structural dynamic stiffness matrix whose elements are trancendental functions of the radian natural frequency the method enables all natural frequencies latent roots 2 and mode shapes latent vectors x across a prescribed frequency range to be determined infallibly and in ascending frequency order at a speed approaching twice that associated with bisection and sign counting methods 3 it is ideally suited for use on micro and minicomputers where single precision working is the norm and on the commodore 4032 computer can cope with matrices s of order n and bandwidth p provided that np 4200 much larger problems of course may be dealt with on mainframe machines

feedback particle filter the feedback particle filter introduced in this paper is a new approach to approximate nonlinear filtering motivated by techniques from mean field game theory the filter is defined by an ensemble of controlled stochastic systems the particles each particle evolves under feedback control based on its own state and features of the empirical distribution of the ensemble the feedback control law is obtained as the solution to an optimal control problem in which the optimization criterion is the kullback leibler divergence between the actual posterior and the common posterior of any particle the following conclusions are obtained for diffusions with continuous observations 1 the optimal control solution is exact the two posteriors match exactly provided they are initialized with identical priors 2 the optimal filter admits an innovation error based gain feedback structure 3 the optimal feedback gain is obtained via a solution of an euler lagrange boundary value problem the feedback gain equals the kalman gain in the linear gaussian case numerical algorithms are introduced and implemented in two general examples and a neuroscience application involving coupled oscillators in some cases it is found that the filter exhibits significantly lower variance when compared to the bootstrap particle filter

constrained dirichlet boundary control in l 2 for a class of evolution equations optimal dirichlet boundary control based on the very weak solution of a parabolic state equation is analyzed this approach allows us to consider the boundary controls in l 2 which has advantages over approaches which consider control in sobolev spaces involving fractional derivatives pointwise constraints on the boundary are incorporated by the primal dual active set strategy its global and local superlinear convergences are shown a discretization based on space time finite elements is proposed and numerical examples are included

the scan sampler for time series models summary an algorithm called the scan sampler is developed and discussed the scan sampler has a variety of uses for time series analysis based on the state space model with nongaussian observations the algorithm is based on the kalman filter smoothing algorithm it can be used for bayesian inference using markov chain monte carlo and to find posterior modes

particle filters for partially observed diffusions in this paper we introduce novel particle filters for a class of partially observed continuous time dynamic models where the signal is given by a multivariate diffusion process we consider a variety of observation schemes including diffusion observed with error observation of a subset of the components of the multivariate diffusion and arrival times of a poisson process whose intensity is a known function of the diffusion cox process unlike currently available methods our particle filters do not require approximations of the transition and or the observation density using time discretisations instead they build on recent methodology for the exact simulation of the diffusion process and the unbiased estimation of the transition density as described in beskos et al 2006 in particular we introduce the generalised poisson estimator which generalises the poisson estimator of beskos et al 2006 thus our filters avoid the systematic biases caused by time discretisations and they have significant computational advantages over alternative continuous time filters these advantages are supported theoretically by a central limit theorem

parameter estimation for differential equations a generalized smoothing approach summary we propose a new method for estimating parameters in models that are defined by a system of non linear differential equations such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation the paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non linear differential equations the approach is based on a modification of data smoothing methods along with a generalization of profiled estimation we derive estimates and confidence intervals and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology the performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus

particle smoothing for hidden diffusion processes adaptive path integral smoother smoothing methods are used for inference of stochastic processes given noisy observations the estimation of the marginal posterior distribution given all observations is typically a computationally intensive task we propose a novel algorithm based on path integral control theory to efficiently estimate the smoothing distribution of continuous time diffusion processes from partial observations in particular we use an adaptive importance sampling method to improve the effective sampling size of the posterior and the reliability of the estimation of the marginals this is achieved by estimating a feedback controller together with an adaptive initialization and an annealing scheme to sample efficiently from the joint smoothing distribution we compare the results with estimations obtained from the standard forward filter backward simulator ffbsi for two diffusion processes of different complexity we show that the proposed method gives more accurate estimates than the standard ffbsi

particle smoothing in continuous time a fast approach via density estimation we consider the particle smoothing problem for state space models where the transition density is not available in closed form in particular for continuous time nonlinear models expressed via stochastic differential equations sdes conventional forward backward and two filter smoothers for the particle filter require a closed form transition density with the linear gaussian euler maruyama discretization usually applied to the sdes to achieve this we develop a pair of variants using kernel density approximations to relieve the dependence and in doing so enable use of faster and more accurate discretization schemes such as runge kutta in addition the new methods admit arbitrary proposal distributions providing an avenue to deal with degeneracy issues experimental results on a functional magnetic resonance imaging fmri deconvolution task demonstrate comparable accuracy and significantly improved runtime over conventional techniques

an ell _ 1 laplace robust kalman smoother robustness is a major problem in kalman filtering and smoothing that can be solved using heavy tailed distributions e g l1 laplace this paper describes an algorithm for finding the maximum a posteriori map estimate of the kalman smoother for a nonlinear model with gaussian process noise and l1 laplace observation noise the algorithm uses the convex composite extension of the gauss newton method this yields convex programming subproblems to which an interior point path following method is applied the number of arithmetic operations required by the algorithm grows linearly with the number of time points because the algorithm preserves the underlying block tridiagonal structure of the kalman smoother problem excellent fits are obtained with and without outliers even though the outliers are simulated from distributions that are not l1 laplace it is also tested on actual data with a nonlinear measurement model for an underwater tracking experiment the l1 laplace smoother is able to construct a smoothed fit without data removal from data with very large outliers

nonlinear gaussian smoothers with colored measurement noise this paper is concerned with the gaussian approximation ga smoothing estimation for the nonlinear system with the colored measurement noise modeled as an autoregressive process firstly based on measurement differencing scheme designing the ga smoothers with the colored measurement noise is transformed into deriving the ga ones with delayed state secondly the novel fixed interval fixed point and fixed lag ga smoothers are proposed via the recursive operation of analytical computation and nonlinear integrals as the general and unifying frameworks they are applicable for both linear and nonlinear systems by setting the noise correlation parameter as zero they can automatically reduce to the standard ga smoothers with uncorrelated white noises many implementations of the ga frameworks can be developed through utilizing different numerical technologies for computing such nonlinear integrals e g the cubature rule based cubature kalman smoothers ckss with the colored measurement noise finally the superior performance in estimation accuracy and computation efficiency of the proposed smoothing methods is demonstrated with a multi sensor target tracking example

application of the wittrick williams algorithm to the sturm liouville problem on homogeneous trees a structural mechanics analogy the wittrickwilliams ww algorithm was developed over 30 years ago and has been applied with increasing sophistication to problems in structural mechanics ever since much wider applications to a

the iterated auxiliary particle filter we present an offline iterated particle filter to facilitate statistical inference in general state space hidden markov models given a model and a sequence of observations the associated margina

differentiability and convergence rates of approximating semigroups for retarded functional differential equations in this paper the averaging approximations i e semidiscrete finite difference scheme for linear retarded functional differential equations are considered in the context of the semigroups on the hilbert space r n times l 2 it is shown that the approximating semigroups are all differentiable uniformly with respect to the index n determining the mesh size this fact makes it possible to prove the existence of a uniform exponential decay or growth rate for the approximating semigroups which can be made arbitrarily close to the decay respectively growth rate of the original semigroup more importantly the uniform differentiability makes it possible to establish convergence rates that improve with time and to prove the convergence of the approximating semigroups in the uniform operator topology the dependence of the convergence rate on the initial conditions and system parameters is also established