minimality and irreducibility of time invariant linear boundary value systems abstract minimality and irreducibility in the class of time invariant linear systems with well posed boundary conditions are introduced and characterized necessary and sufficient conditions for similarity of irreducible or minimal time invariant systems with well posed boundary conditions are given the problem of stable minimality is analysed some of the results are specified for stationary and displacement systems

adaptive ode solvers in extended kalman filtering algorithms this paper studies numerical methods that are used to treat ordinary differential equations arising in extended kalman filtering algorithms such problems are called the moment equations and constitute an important subclass of differential equations they all possess a number of specific properties that make the standard available software ineffective for some mathematical models in this area for instance we show that built in matlab ode solvers are not able to cope with a number of tests considered here in contrast the hybrid method developed by mazzoni 2007 exhibits much better performance on the same examples the latter means that specially designed numerical schemes are needed for efficient solution of the moment equations instead of the standard software designed for general systems of ordinary differential equations

norm constrained capon beamforming using multirank signal models with kalman filter implementation the performance of the capon beamforming is known to degrade dramatically in the presence of model mismatches especially when the desired signal is present in the training data to improve the robustness the diagonal loading technique was introduced however the major drawback is the selection of the diagonal loading level which is related to the unknown signal powers another way to alleviate performance degradation is to use a more accurate signal model of the array response in this paper a norm constrained capon beamforming using multirank signal models is proposed based on the pseudo observation method the quadratic constraints can be easily constructed the problem is solved by a nonlinear kalman filter which can be implemented on line the simulation results show that the design of the norm constraint value is less sensitive to the signal powers small angle mismatches and number of sensors with a standard linear array further it is shown that the use of a multirank signal model and kalman filter technique result in less self cancellation and performance degradation than that of the rank 1 signal model and the estimation of sample matrix

truncated unscented kalman filtering we devise a filtering algorithm to approximate the first two moments of the posterior probability density function pdf the novelties of the algorithm are in the update step if the likelihood has a bounded support we can use a modified prior distribution that meets bayes rule exactly applying a kalman filter kf to the modified prior distribution referred to as truncated kalman filter tkf can vastly improve the performance of the conventional kalman filter particularly when the measurements are informative relative to the prior the application of the tkf to practical problems in which the measurement noise pdf has unbounded support is achieved by imposing several approximating assumptions which are valid only when the measurements are informative this implies that we adaptively choose between an approximation to the kf or the tkf according to the information provided by the measurement the resulting algorithm based on the unscented transformation is referred to as truncated unscented kf

sparse grid quadrature nonlinear filtering in this paper a novel nonlinear filter named sparse grid quadrature filter sgqf is proposed the filter utilizes weighted sparse grid quadrature points to approximate the multi dimensional integrals in the nonlinear bayesian estimation algorithm the locations and weights of the univariate quadrature points with a range of accuracy levels are determined by the moment matching method then the univariate quadrature point sets are extended to form a multi dimensional grid using the sparse grid theory compared with the conventional point based methods the estimation accuracy level of the sgqf can be flexibly controlled and the number of sparse grid quadrature points for the sgqf is a polynomial of the dimension of the system which alleviates the curse of dimensionality for high dimensional problems the unscented kalman filter ukf is proven to be a subset of the sgqf at the level 2 accuracy the performance of this filter is demonstrated by an orbit estimation problem the simulation results show that the sgqf achieves higher accuracy than the extended kalman filter ekf the ukf and the cubature kalman filter ckf in addition the sgqf is computationally much more efficient than the multi dimensional gauss hermite quadrature filter ghqf with the same performance

a quadratic kalman filter abstract we propose a new filtering and smoothing technique for non linear state space models observed variables are quadratic functions of latent factors following a gaussian var stacking the vector of factors with its vectorized outer product we form an augmented state vector whose first two conditional moments are known in closed form we also provide analytical formulae for the unconditional moments of this augmented vector our new quadratic kalman filter qkf exploits these properties to formulate fast and simple filtering and smoothing algorithms a simulation study first emphasizes that the qkf outperforms the extended and unscented approaches in the filtering exercise showing up to 70 rmses improvement of filtered values second it provides evidence that qkf based maximum likelihood estimates of model parameters always possess lower bias or lower rmses than the alternative estimators

gaussian particle filtering sequential bayesian estimation for nonlinear dynamic state space models involves recursive estimation of filtering and predictive distributions of unobserved time varying signals based on noisy observations this paper introduces a new filter called the gaussian particle filter it is based on the particle filtering concept and it approximates the posterior distributions by single gaussians similar to gaussian filters like the extended kalman filter and its variants it is shown that under the gaussianity assumption the gaussian particle filter is asymptotically optimal in the number of particles and hence has much improved performance and versatility over other gaussian filters especially when nontrivial nonlinearities are present simulation results are presented to demonstrate the versatility and improved performance of the gaussian particle filter over conventional gaussian filters and the lower complexity than known particle filters

fluctuation theorems for quantum master equations a quantum fluctuation theorem for a driven quantum subsystem interacting with its environment is derived based solely on the assumption that its reduced density matrix obeys a closed evolution equation i e a quantum master equation qme quantum trajectories and their associated entropy heat and work appear naturally by transforming the qme to a time dependent liouville space basis that diagonalizes the instantaneous reduced density matrix of the subsystem a quantum integral fluctuation theorem a steady state fluctuation theorem and the jarzynski relation are derived in a similar way as for classical stochastic dynamics

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

steady states invariant measures and response theory using the method of computer simulation we test the predictions of nonlinear response theory for classical systems subject to dissipative external fields we provide convincing numerical evidence that kawasaki methods agree with both the transient time correlation function predictions and with a direct measurement of the nonlinear response furthermore this numerical agreement is observed over a time scale which is sufficiently long for the response to have relaxed to within ensuremath sim 1 of its nonequilibrium steady state value this is in spite of the fact that in the steady state the n particle distribution function ultimately becomes fractal we discuss the normalization of the kawasaki distribution and derive a lagrangian form of the kawasaki response function and show that it is consistent with predictions that are obtained using a natural invariant measure for nonequilibrium steady states c 1995 the american physical society

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

the fluctuation theorem and lyapunov weights the fluctuation theorem ft is a generalisation of the second law of thermodynamics that applies to small systems observed for short times for thermostatted systems it gives the probability ratio that entropy will be consumed rather than produced in this paper we derive the transient and steady state fts using lyapunov weights rather than the usual gibbs weights at long times the fts so derived are identical to those derived using the more standard gibbs weights

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

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

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

gaussian filters for nonlinear filtering problems we develop and analyze real time and accurate filters for nonlinear filtering problems based on the gaussian distributions we present the systematic formulation of gaussian filters and develop efficient and accurate numerical integration of the optimal filter we also discuss the mixed gaussian filters in which the conditional probability density is approximated by the sum of gaussian distributions a new update rule of weights for gaussian sum filters is proposed our numerical tests demonstrate that new filters significantly improve the extended kalman filter with no additional cost and the new gaussian sum filter has a nearly optimal performance

boundary value descriptor systems well posedness reachability and observability in this paper we introduce the class of two point boundary value descriptor systems tpbvds discrete time systems described by possibly linear dynamics and a set of boundary conditions constraining the values of the system state at the two endpoints of the system s interval of definition by introducing a standard form for regular pencils we obtain a new and simple generalized cayley hamilton theorem that simplifies our investigation of well posedness green s function solutions and reachability and observability for tpbvds there are two distinct notions of reachability and observability that one can define for tpbvds associated with processes that propagate inward from and outward toward the boundaries we investigate each of these in detail obtaining among other things far simpler forms for the reachability and observability results than found previously in the literature in addition we describe several methods for the efficient solution of tpbvds one involving recursions from each end of t

work fluctuations in a time dependent harmonic potential rigorous results beyond the overdamped limit we investigate the stochastic motion of a brownian particle in the harmonic potential with a time dependent force constant it may describe the motion of a colloidal particle in an optical trap where the potential well is formed by a time dependent field we use the path integral formalism to solve the langevin equation and the associated fokker planck kramers equation rigorous relations are derived to generate the probability density function for the time dependent nonequilibrium work production beyond the overdamped limit we find that the work distribution exhibits an exponential tail with a power law prefactor accompanied by an interesting oscillatory feature multiple pseudo locking unlocking transitions due to the inertial effect some exactly solvable cases are discussed in the overdamped limit

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

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

on the fluctuation relation for nose hoover boundary thermostated systems we discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures the system is a modified lorentz gas in which the fixed scatterers exchange energy with the gas of particles and the thermostats are modelled by two nose hoover thermostats applied at the boundaries of the system the transient fluctuation relation which holds only for a precise choice of the initial ensemble is verified at all times as expected times longer than the mesoscopic scale needed for local equilibrium to be settled are required if a different initial ensemble is considered this shows how the transient fluctuation relation asymptotically leads to the steady state relation when as explicitly checked in our systems the condition found in d j searles et al j stat phys 128 1337 2007 for the validity of the steady state fluctuation relation is verified for the steady state fluctuations of the phase space contraction rate and of the dissipation function a similar relaxation regime at shorter averaging times is found the quantity satisfies with good accuracy the fluctuation relation for times larger than the mesoscopic time scale the quantity appears to begin a monotonic convergence after such times this is consistent with the fact that and differ by a total time derivative and that the tails of the probability distribution function of are gaussian

accurate numerical implementation of the continuous discrete extended kalman filter this paper addresses an accurate and effective implementation of the continuous discrete extended kalman filtering method the technique under discussion is grounded in numerical solution of the moment differential equations to predict the state mean of the stochastic dynamical system and the corresponding error covariance matrix here we apply an efficient embedded runge kutta pair possessing superior stability and many other attractive features including automatic global error control in order to improve performance of the complex computational procedure consisting of the extended kalman filter and the underlying adaptive ode solver thus we introduce a new continuous discrete adaptive extended kalman filter and show its advantage over the standard variant on two test examples in practice this technique allows for much longer sampling intervals without any loss of accuracy and that improves the applied potential of the extended kalman filtering method significantly

some relations between extended and unscented kalman filters the unscented kalman filter ukf has become a popular alternative to the extended kalman filter ekf during the last decade ukf propagates the so called sigma points by function evaluations using the unscented transformation ut and this is at first glance very different from the standard ekf algorithm which is based on a linearized model the claimed advantages with ukf are that it propagates the first two moments of the posterior distribution and that it does not require gradients of the system model we point out several less known links between ekf and ukf in terms of two conceptually different implementations of the kalman filter the standard one based on the discrete riccati equation and one based on a formula on conditional expectations that does not involve an explicit riccati equation first it is shown that the sigma point function evaluations can be used in the classical ekf rather than an explicitly linearized model second a less cited version of the ekf based on a second order taylor expansion is shown to be quite closely related to ukf the different algorithms and results are illustrated with examples inspired by core observation models in target tracking and sensor network applications

stability stochastic stationarity and generalized lyapunov equations for two point boundary value descriptor systems the authors introduce the concept of internal stability for two point boundary value descriptor systems tpbvdss since tpbvdss are defined only over a finite interval the concept of stability is not easy to formulate the definition which is used consists of requiring that as the length of the interval of definition increases the effect of boundary conditions on states located close to the center of the interval should go to zero stochastic tpbvdss are studied and the property of stochastic stationarity is characterized in terms of a generalized lyapunov equation satisfied by the variance of the boundary vector a second generalized lyapunov equation satisfied by the state variance of a stochastically stationary tpbvds is also introduced and the existence and uniqueness of positive definite solutions to this equation are then used to characterize the property of internal stability

a finite difference method for linearization in nonlinear estimation algorithms abstract linearizations of nonlinear functions that are based on jacobian matrices often cannot be applied in practical applications of nonlinear estimation techniques an alternative linearization method is presented in this paper the method assumes that covariance matrices are determined on a square root factored form a factorization of the output covariance from a nonlinear vector function is directly determined by perturbing the nonlinear function with the columns of the factored input covariance without explicitly calculating the linearization and with no differentiations involved the output covariance is more accurate than that obtained with the ordinary jacobian linearization method it also has an advantage that jacobian matrices do not have to be derived symbolically

fully symmetric interpolatory rules for multiple integrals over infinite regions with gaussian weight fully symmetric interpolatory integration rules are constructed for multidimensional integrals over infinite integration regions with a gaussian weight function the points for these rules are determined by successive extensions of the one dimensional three point gauss hermite rule the new rules are shown to be efficient and only moderately unstable

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

on the asymptotic convergence of the transient and steady state fluctuation theorems nonequilibrium molecular dynamics simulations are used to demonstrate the asymptotic convergence of the transient and steady state forms of the fluctuation theorem in the case of planar poiseuille flow we find that the transient form valid for all times converges to the steady state predictions on microscopic time scales further we find that the time of convergence for the two theorems coincides with the time required for satisfaction of the asymptotic steady state fluctuation theorem

nirk based accurate continuous discrete extended kalman filters for estimating continuous time stochastic target tracking models abstract this paper presents three state estimators grounded in the variable stepsize gauss and lobatto type nested implicit runge kutta nirk formulas of orders 4 and 6 and designed for treating continuous time stochastic systems arisen in radar tracking our filters are built within the extended kalman filtering ekf framework and based on accurate numerical integrations of the corresponding moment differential equations mdes automatic local and global error regulation mechanisms implemented in these methods allow the committed discretization error to be under control and made negligible in automatic mode the latter raises the state estimation accuracy of the constructed filters significantly this also leads to the advanced notion of accurate continuous discrete extended kalman filtering acd ekf developed by kulikov and kulikova in 2013 2016 our novel methods are constructed within the same approach but possess the improved accuracy and efficiency in comparison to their predecessors due to both more effective error control mechanisms implemented for integrating mdes and more accurate iterations used for treating arisen nonlinear equations in the revised filters numerical experiments with the updated state estimators and their comparison to the cited earlier designed acd ekfs are fulfilled in severe conditions of tackling a seven dimensional radar tracking problem where an aircraft executes a coordinated turn in matlab this examination suggests that the novel state estimation algorithms outperform their predecessors and possess a promising potential for solving target tracking tasks in real world applications

cubature kalman filtering for continuous discrete systems theory and simulations in this paper we extend the cubature kalman filter ckf to deal with nonlinear state space models of the continuous discrete kind to be consistent with the literature the resulting nonlinear filter is referred to as the continuous discrete cubature kalman filter cd ckf we use the ito taylor expansion of order 1 5 to transform the process equation modeled in the form of stochastic ordinary differential equations into a set of stochastic difference equations building on this transformation and assuming that all conditional densities are gaussian distributed the solution to the bayesian filter reduces to the problem of how to compute gaussian weighted integrals to numerically compute the integrals we use the third degree cubature rule for a reliable implementation of the cd ckf in a finite word length machine it is structurally modified to propagate the square roots of the covariance matrices the reliability and accuracy of the square root version of the cd ckf are tested in a case study that involves the use of a radar problem of practical significance the problem considered herein is challenging in the context of radar in two respects high dimensionality of the state and increasing degree of nonlinearity the results presented herein indicate that the cd ckf markedly outperforms existing continuous discrete filters

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

analysis of kalman filter approximations for nonlinear measurements a theoretical analysis is presented of the correction step of the kalman filter kf and its various approximations for the case of a nonlinear measurement equation with additive gaussian noise the kf is based on a gaussian approximation to the joint density of the state and the measurement the analysis metric is the kullback leibler divergence of this approximation from the true joint density the purpose of the analysis is to provide a quantitative tool for understanding and assessing the performance of the kf and its variants in nonlinear scenarios this is illustrated using a numerical example

jarzynski equality in van der pol and rayleigh oscillators we have studied the jarzynski equality je in van der pol and rayleigh oscillators which are typical deterministic non hamiltonian models but not expected to rigorously satisfy the je because they are not reversible our simulations that calculate the contribution to the work w of an applied ramp force with a duration show that the je approximately holds for a fairly wide range of including 0 and except for t where t denotes the period of relaxation oscillations in the limit cycle the work distribution function wdf is shown to be non gaussian with the u shaped structure for a strong damping parameter the dependence of r k b tln e w obtained by our simulations is semiquantitatively elucidated with the use of a simple expression for limit cycle oscillations where the bracket expresses an average over the wdf the result obtained in self excited oscillators is in contrast with the fact that the je holds in the nose hoover oscillator which also belongs to deterministic non hamiltonian models

fluctuation of mean lyapunov exponent for a coupled map lattice model abstract fluctuation of mean lyapunov exponent for the coupled map lattice of logistic map is studied theorem by gallavotti and cohen for many particle systems is applied to the coupled map lattice model it is shown that the fluctuation of mean lyapunov exponent for the model satisfies the extended form of the fluctuation theorem by them the concept of the mean lyapunov exponent introduced in this paper turns out to be realistic as well as the one for many particle systems as a result the relation between mean lyapunov exponent and entropy production rate is derived

continuous time random walk for open systems fluctuation theorems and counting statistics we consider continuous time random walks ctrw for open systems that exchange energy and matter with multiple reservoirs each waiting time distribution wtd for times between steps is characterized by a positive parameter alpha which is set to alpha 1 if it decays at least as fast as t 2 at long times and therefore has a finite first moment a wtd with alpha 1 decays as t alpha 1 a fluctuation theorem for the trajectory quantity r defined as the logarithm of the ratio of the probability of a trajectory and the probability of the time reversed trajectory holds for any ctrw however r can be identified as a trajectory entropy change only if the wtds have alpha 1 and satisfy separability also called direction time independence for nonseparable wtds with alpha 1 r can only be identified as a trajectory entropy change at long times and a fluctuation theorem for the entropy change then only holds at long times for wtds with 0 alpha 1 no meaningful fluctuation theorem can be derived we also show that the experimentally accessible nth moments of the energy and matter transfers between the system and a given reservoir grow as t nalpha at long times

adaptive sparse grid gauss hermite filter abstract in this paper a new nonlinear filter based on sparse grid quadrature method has been proposed the proposed filter is named as adaptive sparse grid gauss hermite filter asghf ordinary sparse grid technique treats all the dimensions equally whereas the asghf assigns a fewer number of points along the dimensions with lower nonlinearity it uses adaptive tensor product to construct multidimensional points until a predefined error tolerance level is reached the performance of the proposed filter is illustrated with two nonlinear filtering problems simulation results demonstrate that the new algorithm achieves a similar accuracy as compared to sparse grid gauss hermite filter sghf and gauss hermite filter ghf with a considerable reduction in computational load further in the conventional ghf and sghf any increase in the accuracy level may result in an unacceptably high increase in the computational burden however in asghf a little increase in estimation accuracy is possible with a limited increase in computational burden by varying the error tolerance level and the error weighting parameter this enables the online estimator to operate near full efficiency with a predefined computational budget

diversity effects in kalman filtering over rayleigh fading channels a scalar first order gauss markov signal is sent over d parallel i i d rayleigh fading channels using uncoded transmission assuming perfect channel estimation at the receiver the optimal mmse estimator is the kalman filter with random instantaneous estimation error variance statistical characteristics of the instantaneous estimation error variance are studied to characterize the estimation quality one such characteristic is the estimation error outage which probabilistically depicts how often the value of the instantaneous estimation error variance exceeds a certain threshold in this paper we study the effect of channel diversity on the estimation error outage probability we show that for a certain range of outage thresholds the estimation error outage probability decreases with inverse of d th power of the average signal to noise ratio at the receiver side this means that analogous to the context of digital communication over fading channels our result implies a diversity order of d in estimating the input process

square root quadrature kalman filtering the quadrature kalman filter qkf is a recursive nonlinear filtering algorithm developed in the kalman filtering framework it computes the mean and covariance of all conditional densities using the gauss hermite quadrature rule in this correspondence we develop a square root extension of the quadrature kalman filter using matrix triangularizations the square root quadrature kalman filter sqkf propagates the mean and the square root of the covariance although equivalent to the qkf algebraically the sqkf exhibits excellent numerical characteristics but at the expense of increased computational complexity we also present possible refinements of the generic sqkf

on minimality and stable minimality of time varying linear systems with well posed boundary conditions for time varying linear systems with well posed boundary conditions the concept of stable minimality is introduced a complete characterization is given of stably minimal systems with analytic coefficients

gaussian smoothers for nonlinear systems with one step randomly delayed measurements this technical note is concerned with the nonlinear state smoothing problem in the case that the measurements are randomly delayed by one sampling time under gaussian domain two general gaussian approximation ga and gaussian mixture approximation gma smoothers are proposed in minimum mean square error mmse sense the smoothing implementation is transformed into computing some special posterior covariances which triggers the development of the new unscented kalman smoother uks by applying unscented transformation ut simulation results demonstrate the superior performance of the proposed ga uks and gma uks algorithms

on gaussian optimal smoothing of non linear state space models in this note we shall present a new gaussian approximation based framework for approximate optimal smoothing of non linear stochastic state space models the approximation framework can be used for efficiently solving non linear fixed interval fixed point and fixed lag optimal smoothing problems we shall also numerically compare accuracies of approximations which are based on taylor series expansion unscented transformation central differences and gauss hermite quadrature

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

advanced point mass method for nonlinear state estimation state estimation of discrete time nonlinear non gaussian stochastic systems by point mass approach which is based on discretization of state space by a regular grid and numerical solution of bayesian recursive relations is treated the stress is laid to grid design which is crucial for estimator quality and significantly affects the computational demands of the estimator boundary based grid design thrifty convolution and multigrid design with grid splitting and merging are proposed the main advantages of these techniques are nonnegligible support delimitation time saving computation of convolution and effective processing of multimodal probability density functions respectively the techniques are involved into the basic point mass approach and a new general purpose more sophisticated point mass algorithm is designed computational demands and estimation quality of the designed algorithm are presented and compared with the particle filter in a numerical example

comment on a new method for the nonlinear transformation of means and covariances in filters and estimators with authors reply the above paper julier et al ieee trans automat contr vol 45 pp 477 82 2000 generalizes the kalman filter to nonlinear systems by transforming approximations of the probability distributions through the nonlinear process and measurement functions this comment derives exactly the same estimator by linearizing the process and measurement functions by a statistical linear regression through some regression points in contrast with the extended kalman filter which uses an analytic linearization in one point this insight allows one 1 to understand predict the performance of the estimator for specific applications and 2 to make adaptations to the estimator i e the choice of the regression points and their weights in those cases where the original formulation does not assure good results in reply the authors state that the commenters conclusion is unnecessarily narrow interpretation of results

advances in multidimensional integration this paper gives a personal bird s eye view of some aspects of multivariate numerical integration it will sketch what happened during the past 50 years and point the reader s attention to what did not happen

a class of stable square root nonlinear information filters information filters can process nonlinear systems with uncertain prior knowledge and the particular square root form of adaptive filters can improve numerical stability based on a square root decomposition of information matrix and an extra positive definite matrix the unscented transform and the cubature rule are applied to the information filtering architecture for nonlinear estimation a class of stable square root nonlinear information filters is then proposed in this technical note in addition the boundedness of their estimation errors is also proven results from simulations of filtering a chaotic map demonstrate that the proposed square root nonlinear filters can improve numerical stability and has better filtering performance than other information filters

entropy production fluctuations of finite markov chains for almost every trajectory segment over a finite time span of a finite markov chain with any given initial distribution the logarithm of the ratio of its probability to that of its time reversal converges exponentially to the entropy production rate of the markov chain the large deviation rate function has a symmetry of gallavotti cohen type which is called the fluctuation theorem moreover similar symmetries also hold for the rate functions of the joint distributions of general observables and the logarithmic probability ratio

a new method for the nonlinear transformation of means and covariances in filters and estimators this paper describes a new approach for generalizing the kalman filter to nonlinear systems a set of samples are used to parametrize the mean and covariance of a not necessarily gaussian probability distribution the method yields a filter that is more accurate than an extended kalman filter ekf and easier to implement than an ekf or a gauss second order filter its effectiveness is demonstrated using an example

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

the steady state fluctuation relation for the dissipation function we give a proof of transient fluctuation relations for the entropy production dissipation function in nonequilibrium systems which is valid for most time reversible dynamics we then consider the conditions under which a transient fluctuation relation yields a steady state fluctuation relation for driven nonequilibrium systems whose transients relax producing a unique nonequilibrium steady state although the necessary and sufficient conditions for the production of a unique nonequilibrium steady state are unknown if such a steady state exists the generation of the steady state fluctuation relation from the transient relation is shown to be very general it is essentially a consequence of time reversibility and of a form of decay of correlations in the dissipation which is needed also for e g the existence of transport coefficients because of this generality the resulting steady state fluctuation relation has the same degree of robustness as do equilibrium thermodynamic equalities the steady state fluctuation relation for the dissipation stands in contrast with the one for the phase space compression factor whose convergence is problematic for systems close to equilibrium we examine some model dynamics that have been considered previously and show how they are described in the context of this work

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

reachability observability minimality and extendibility for two point boundary value descriptor systems a deterministic system theory is developed for two point boundary value descriptor systems tpbvdss in particular detailed characterizations of the properties of reachability observability and minimality are obtained in addition extendibility i e the concept of considering tpbvds as being defined on a sequence of intervals of increasing length is defined and studied this system theory generalizes our earlier results for the class of stationary systems for which the input output map weighting pattern is shift invariant

improved cubature kalman filter for gnss ins based on transformation of posterior sigma points error tightly coupled gnss ins has been widely approved as a promising substitute for standalone gnss in urban areas navigation however due to the frequent gnss signal outages the filter used in gnss ins should be insensitive to the less informative observations in this paper a novel sigma points update method is proposed to enhance the robustness of cubature kalman filter ckf under the circumstance of unavailable observations first the problems of existing sampling based filters are analyzed then by transforming the posterior sigma points error matrix from prediction phase of filtering to the posterior domain of update the updated sigma points are expected to capture the covariance more precisely than traditional sigma points finally an improved ckf ickf is developed by embedding these points into the bayesian estimation framework and the upper bounds of error covariance matrices are analyzed theoretically signal outages with different durations are simulated and results demonstrate that ickf outperforms state of the art methods

new developments in state estimation for nonlinear systems based on an interpolation formula accurate state estimators for nonlinear systems can be derived the estimators do not require derivative information which makes them simple to implement

a derivative free implementation of the extended kalman filter a nonlinear estimation paradigm is developed to estimate the mean and covariance of a time evolving state distribution the approach represents uncertainty by an ensemble set of state vectors rather than by the traditional mean and covariance measures avoiding the need for the calculation of matrix partial derivatives jacobian matrices the paradigm is shown to be equivalent to the extended kalman filter in a limiting case implementation of the new filtering approach is illustrated with a simple example and a step by step description the paradigm is not significantly more computationally intensive than traditional filters and proves ideal for the rapid implementation of complex nonlinear system and observation models

transformed unscented kalman filter this technical note concerns the deterministic sampling points construction strategy for unscented kalman filter ukf and cubature kalman filter ckf from the numerical integration viewpoint a new deterministic sampling points set is derived by orthogonal transformation on the cubature points by embedding these points into the ukf framework a modified nonlinear filter named transformed unscented kalman filter tukf is derived the tukf can address the nonlocal sampling problem inherent in ckf while maintaining the virtue of numerical stability for high dimensional problems moreover the methodology proposed in this technical note can be used to construct nonlinear filters with improved accuracy for certain problems the performance of the proposed algorithm is demonstrated through a nonlinear high dimensional problem

cubature kalman filters in this paper we present a new nonlinear filter for high dimensional state estimation which we have named the cubature kalman filter ckf the heart of the ckf is a spherical radial cubature rule which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear bayesian filter specifically we derive a third degree spherical radial cubature rule that provides a set of cubature points scaling linearly with the state vector dimension the ckf may therefore provide a systematic solution for high dimensional nonlinear filtering problems the paper also includes the derivation of a square root version of the ckf for improved numerical stability the ckf is tested experimentally in two nonlinear state estimation problems in the first problem the proposed cubature rule is used to compute the second order statistics of a nonlinearly transformed gaussian random variable the second problem addresses the use of the ckf for tracking a maneuvering aircraft the results of both experiments demonstrate the improved performance of the ckf over conventional nonlinear filters

descriptor systems decomposition into forward and backward subsystems it is shown that a decomposition of the descriptor space with certain properties induces a decomposition of the descriptor system into forward and backward subsystems different decompositions correspond to different apportionments of the required boundary conditions between initial and final times applications include several different but equivalent methods for solving the descriptor system and a clarification of the relations between some previously known methods of solution

high degree cubature kalman filter the cubature kalman filter ckf which is based on the third degree spherical radial cubature rule is numerically more stable than the unscented kalman filter ukf but less accurate than the gauss hermite quadrature filter ghqf to improve the performance of the ckf a new class of ckfs with arbitrary degrees of accuracy in computing the spherical and radial integrals is proposed the third degree ckf is a special case of the class the high degree ckfs of the class can achieve the accuracy and stability performances close to those of the ghqf but at lower computational cost a numerical integration problem and a target tracking problem are utilized to demonstrate the necessity of using the high degree cubature rules to improve the performance the target tracking simulation shows that the fifth degree ckf can achieve higher accuracy than the extended kalman filter the ukf the third degree ckf and the particle filter and is computationally much more efficient than the ghqf

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

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

a systematization of the unscented kalman filter theory in this paper we propose a systematization of the discrete time unscented kalman filter ukf theory we gather all available ukf variants in the literature present corrections to theoretical inconsistencies and provide a tool for the construction of new ukf s in a consistent way this systematization is done mainly by revisiting the concepts of sigma representation unscented transformation ut scaled unscented transformation sut ukf and square root unscented kalman filter srukf inconsistencies are related to 1 matching the order of the transformed covariance and cross covariance matrices of both the ut and the sut 2 multiple ukf definitions 3 issue with some reduced sets of sigma points described in the literature 4 the conservativeness of the sut 5 the scaling effect of the sut on both its transformed covariance and cross covariance matrices and 6 possibly ill conditioned results in srukf s with the proposed systematization the symmetric sets of sigma points in the literature are formally justified and we are able to provide new consistent variations for ukf s such as the scaled srukf s and the ukf s composed by the minimum number of sigma points furthermore our proposed srukf has improved computational properties when compared to state of the art methods

nonequilibrium fluctuations for linear diffusion dynamics we present the theoretical study on nonequilibrium neq fluctuations for diffusion dynamics in high dimensions driven by a linear drift force we consider a general situation in which neq is caused by two conditions i drift force not derivable from a potential function and ii diffusion matrix not proportional to the unit matrix implying nonidentical and correlated multidimensional noise the former is a well known neq source and the latter can be realized in the presence of multiple heat reservoirs or multiple noise sources we develop a statistical mechanical theory based on generalized thermodynamic quantities such as energy work and heat the neq fluctuation theorems are reproduced successfully we also find the time dependent probability distribution function exactly as well as the neq work production distribution p mathcal w in terms of solutions of nonlinear differential equations in addition we compute low order cumulants of the neq work production explicitly in two dimensions we carry out numerical simulations to check out our analytic results and also to get p mathcal w we find an interesting dynamic phase transition in the exponential tail shape of p mathcal w associated with a singularity found in solutions of the nonlinear differential equation finally we discuss possible realizations in experiments

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