Square-root algorithms for least-squares estimation

We present several new algorithms, and more generally a new approach, to recursive estimation algorithms for linear dynamical systems. Earlier results in this area have been obtained by several others, especially Potter, Golub, Dyer and McReynolds, Kaminski, Schmidt, Bryson, and Bierman on what are known as square-root algorithms. Our results are more comprehensive. They also show bow constancy of parameters can be exploited to reduce the number of computations and to obtain new forms of the Chandrasekhar-type equations for computing the filter gain. Our approach is essentially based on certain simple geometric interpretations of the overall estimation problem. One of our goals is to attract attention to non-Riccati-based studies of estimation problems.     

Square-root algorithms for parallel processing in optimal estimation

We give explicit algorithms in square-root form that allow measurements for the standard state estimation problem to be processed in a highly parallel fashion with little communication between processors. After this preliminary processing, blocks of measurements may be incorporated into state estimates with essentially the same computation as usually accompanies the incorporation of a single measurement. This formulation also leads to square-root doubling formulae for calculating the steady-state error-covariance matrix of constant models, and an extension of the class of problems for which Chandrasekhar-type algorithms offer computational reductions to include piecewise constant systems with arbitrary initial conditions.     

Discrete-time fixed-lag smoothing algorithms

Kalman filtering results are applied to yield alternative computationally stable fixed-lag smoothing algorithms including reduced order and minimal order fixed-lag smoothers. The reduced order smoothing algorithms are new and clearly have advantages over the more familiar algorithms. The properties of such fixed-lag smoothers are also studied.     

New square-root smoothing algorithms

This paper presents new square-root smoothing algorithms for the three best-known smoothing formulas: (1) Rauch-Tung-Striebel (RTS) formulas, (2) Desai-Weinert-Yusypchuk (DWY) formulas, called backward RTS formulas, and (3) Mayne-Fraser (MF) formulas, called two-filter formulas. The main feature of the new algorithms is that they use unitary rotations to replace all matrix inversion and backsubstitution steps common in earlier algorithms with unitary operations; this feature enables more efficient systolic array and parallel implementations and leads to algorithms with better numerical stability and conditioning properties     

Parallel filtering and smoothing algorithms

This paper develops algorithms for filtering and smoothing for parallel computers. Numerical results are presented and implementation details are discussed. In the example it is illustrated that parallel methods have better convergence properties than nonparallel methods for nonlinear problems.     

Fuzzy smoothing algorithms for variable structure systems

A variable structure system (VSS) is a control system implementing different control laws in different regions of the state space divided by a set of boundary manifolds. The control input switches from one control law to another when the state crosses the boundary manifolds. In general, the control input may not be smooth when switching at these boundary manifolds and may excite high frequency dynamics. This paper proposes two fuzzy rule based algorithms for smoothing the control input. The merits of these fuzzy smoothing control algorithms are illustrated by two examples: a semiactive suspension system based on optimal control and a direct drive robot arm under discrete time sliding mode control. The controller design for these two examples is a blend of traditional control theoretic approaches and fuzzy rule based approaches     

Recursive algorithms for Bayes smoothing with uncertain observations

Recursive algorithms for the Bayes solution of fixed-interval, fixed-point, and fixed-lag smoothing under uncertain observations are presented. The Bayes smoothing algorithms are obtained for a Markovian system model with Markov uncertainty, a model more general than the one used in linear smoothing algorithms. The Bayes fixed-interval smoothing algorithm is applied to a Gauss-Markov example. The simulation results for this example indicate that the MSE performance of the Bayes smoother is significantly better than that of the linear smoother.     

Performance evaluation of smoothing algorithms for transmittingprerecorded variable-bit-rate video

The transfer of prerecorded, compressed variable-bit-rate video requires multimedia services to support large fluctuations in bandwidth requirements on multiple time scales. Bandwidth smoothing techniques can reduce the burstiness of a variable-bit-rate stream by transmitting data at a series of fixed rates, simplifying the allocation of resources in video servers and the communication network. This paper compares the transmission schedules generated by the various smoothing algorithms, based on a collection of metrics that relate directly to the server, network, and client resources necessary for the transmission, transport, and playback of prerecorded video. Using MPEG-1 and MJPEG video data and a range of client buffer sizes, we investigate the interplay between the performance metrics and the smoothing algorithms. The results highlight the unique strengths and weaknesses of each bandwidth smoothing algorithm, as well as the characteristics of a diverse set of video clips     

Decentralized smoothing algorithms of continuous-time systems in distributed-sensor networks

Some decentralized smoothing algorithms are derived by applying a Rauch-Tung-Striebel fixed-interval smoother formula in continuous-time systems supposing that an estimation structure is comprised of a global processor and of two local processors. Two cases are investigated for the problems of decentralized smoothing: when the local-filtered estimated are available and when the local-smoothed estimates are known. Some features of present algorithms are discussed from the point of view of data transmissions and communication bandwidth, etc.     

On the computational aspects of two recent smoothing algorithms

This note is concerned with a consideration of the computational requirements of fixed-lag and fixed-point smoothing algorithms reported recently by the authors.     

Covariance factorization algorithms for fixed-interval smoothing oflinear discrete dynamic systems

Efficient factorized covariance smoothers designed to work with factorized covariance filters are derived for linear discrete dynamic systems. The approach to factorized covariance smoothers (either U -D or square root) uses outputs from factorized covariance filters and is closely derived from the G.J. Bierman's earlier algorithm (1974), the Dyer-McReynolds covariance smoother. These algorithms are more efficient than the Bierman's newer smoother (1983) based upon rank 1 process noise updates. The efficiency of the new algorithms increases significantly as the order of process noise increases. For full process noise, they can be implemented in a way that avoids the inverse of the transition matrix     

Discrete-time complementary models and smoothing algorithms: The correlated noise case

The concept of complementary models for discrete-time linear finite-dimensional systems with correlated observation and process noise is developed. Using this concept, a new algorithm for the fixed interval smoothing problem is obtained. The new algorithm offers great flexibility with respect to changes in the initial state variancePi_{0}. Next, the relationship among the new smoothing algorithm, the two-filter smoother, and the reversed-time Kalman filter is explored. It is shown that a similarity transformation on the Hamiltonian system simultaneously produces the new smoothing algorithm, as well as the reversed-time Kalman filter.     

均方根嵌入式容积卡尔曼滤波

传统容积卡尔曼滤波(CKF)的基础是三阶球面-径向容积准则,该准则不仅要求计算n维超球体上的面积分,还需将容积准则与扩展高斯-拉盖尔准则配合使用,不易推导出高阶CKF滤波算法.此外,CKF推导所采用的三阶球面容积准则也存在缺陷,这极大地限制了CKF的滤波精度.为避免以上问题,本文基于嵌入式容积准则和均方根滤波技术,提出一种加性噪声环境下,用于非线性动态系统状态估计的全新容积卡尔曼滤波算法-三阶均方根嵌入式容积卡尔曼滤波(SICKF).SICKF具有滤波精度高、数值稳定性强等诸多优点,适用于动态目标跟踪、非线性系统控制等.仿真结果表明,SICKF的滤波精度显著优于传统的非线性滤波算法.     

平方根体积积分卡尔曼滤波

体积积分是一种新的具有较高代数精度的积分方法。为了提高非线性滤波算法的精度和数值稳定性,将体积积分规则和平方根分解引入卡尔曼滤波框架中,提出了平方根体积积分卡尔曼滤波算法（SRCQKF）。新算法采用球半径体积规则和高斯-拉盖尔积分规则计算积分点,利用矩阵的QR分解得到协方差矩阵的平方根并传播平方根。两个典型的非线性系统的实验结果表明,与体积卡尔曼滤波相比,新算法提高了非线性状态的估计精度,具有较高的数值稳定性。     

Analysis of filtering and smoothing algorithms for Lévy-driven stochastic volatility models

Filtering and smoothing algorithms that estimate the integrated variance in Lévy-driven stochastic volatility models are analyzed. Particle filters are algorithms designed for nonlinear, non-Gaussian models while the Kalman filter remains the best linear predictor if the model is linear but non-Gaussian. Monte Carlo experiments are performed to compare these algorithms across different specifications of the model including different marginal distributions and degrees of persistence for the instantaneous variance. The use of realized variance as an observed variable in the state space model is also evaluated. Finally, the particle filter's ability to identify the timing and size of jumps is assessed relative to popular nonparametric estimators.     

Evaluating the effectiveness of smoothing algorithms in the absence of ground reference measurements

Time series of vegetation indices like NDVI are used in numerous applications ranging from ecology to climatology and agriculture. Often, these time series have to be filtered before application. The smoothing removes noise introduced by undetected clouds and poor atmospheric conditions. Ground reference measurements are usually difficult to obtain due to the medium/coarse resolution of the imagery. Hence, new filter algorithms are typically only (visually) assessed against the existing smoother. The present work aims to propose a range of quality indicators that could be useful to qualify filter performance in the absence of ground-based reference measurements. The indicators comprise (i) plausibility checks, (ii) distance metrics and (iii) geostatistical measures derived from variogram analysis. The quality measures can be readily derived from any imagery. For illustration, a large SPOT VGT dataset (1999–2008) covering South America at 1?km spatial resolution was filtered using the Whittaker smoother.     

Bayes smoothing algorithms for segmentation of binary images modeled by markov random fields

A new image segmentation algorithm is presented, based on recursive Bayes smoothing of images modeled by Markov random fields and corrupted by independent additive noise. The Bayes smoothing algorithm yields the a posteriori distribution of the scene value at each pixel, given the total noisy image, in a recursive way. The a posteriori distribution together with a criterion of optimality then determine a Bayes estimate of the scene. The algorithm presented is an extension of a 1-D Bayes smoothing algorithm to 2-D and it gives the optimum Bayes estimate for the scene value at each pixel. Computational concerns in 2-D, however, necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm. In particular, the scene (noiseless image) is modeled as a Markov mesh random field, a special class of Markov random fields, and the Bayes smoothing algorithm is applied on overlapping strips (horizontal/vertical) of the image consisting of several rows (columns). It is assumed that the signal (scene values) vector sequence along the strip is a vector Markov chain. Since signal correlation in one of the dimensions is not fully used along the edges of the strip, estimates are generated only along the middle sections of the strips. The overlapping strips are chosen such that the union of the middle sections of the strips gives the whole image. The Bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel (discrete) or continuous random variables.     

Recursive algorithms for the Bayes solution of the fixed-point and fixed-lag smoothing problems

Recursive algorithms for the Bayes solutions of the fixed-point and fixed-lag smoothing problems are obtained. Recursive algorithms for the respective smoothed a posteriori densities are derived under assumptions that the signal to be estimated is a Markov process and the observation is a signal embedded in independent noise (not necessarily additive) which is also independent of the signal. The recursive algorithm for the fixed-point smoothing is applied to a binary Markov signal corrupted by an independent noise in a nonlinear manner.     

Optimal and suboptimal smoothing algorithms for identification of time-varying systems with randomly drifting parameters

Noncausal estimation algorithms, which involve smoothing, can be used for off-line identification of nonstationary systems. Since smoothing is based on both past and future data, it offers increased accuracy compared to causal (tracking) estimation schemes, incorporating past data only. It is shown that efficient smoothing variants of the popular exponentially weighted least squares and Kalman filter-based parameter trackers can be obtained by means of backward-time filtering of the estimates yielded by both algorithms. When system parameters drift according to the random walk model and the adaptation gain is sufficiently small, the properly tuned two-stage Kalman filtering/smoothing algorithm, derived in the paper, achieves the Cramér-Rao type lower smoothing bound, i.e. it is the optimal noncausal estimation scheme. Under the same circumstances performance of the modified exponentially weighted least-squares algorithm is often only slightly inferior to that of the Kalman filter-based smoother.     

Recursive smoothing algorithms for the estimation of signals from uncertain observations via mixture approximations

Considering discrete-time systems with uncertain observations when the signal model is unknown, but only covariance information is available, and the signal and the observation additive noise are correlated and jointly Gaussian, we present recursive algorithms for suboptimal fixed-point and fixed-interval smoothing estimators. To derive the algorithms, we employ a technique consisting in approximating the conditional distributions of the signal given the observations by Gaussian distributions, taking successive approximations of the mixtures of normal distributions. The expectation of these approximations provides us with the suboptimal estimators. In a numerical simulation example, the performance of the proposed estimators is compared with that of linear ones, via the sample mean square values of the corresponding estimation errors.