Least-squares linear estimation of signals from observations with Markovian delays

The least-squares linear estimation of signals from randomly delayed measurements is addressed when the delay is modeled by a homogeneous Markov chain. To estimate the signal, recursive filtering and fixed-point smoothing algorithms are derived, using an innovation approach, assuming that the covariance functions of the processes involved in the observation equation are known. Recursive formulas for filtering and fixed-point smoothing error covariance matrices are obtained to measure the goodness of the proposed estimators.     

Estimation from a multisensor environment for systems with multiple packet dropouts and correlated measurement noises

This paper addresses the problem of distributed fusion estimation from measurements with packet dropouts and cross-correlated noises acquired from different sensors. Assuming that the packet dropouts are modelled by independent Bernoulli random variables with different characteristics for each sensor and that measurement noises are cross-correlated at the same and at consecutive sampling times, filtering and smoothing algorithms are derived using the distributed fusion method. The distributed fusion filter and smoother are obtained as a matrix-weighted linear combination of corresponding local least-squares linear estimators, verifying that the mean squared error is minimum. The local linear filtering and fixed-point smoothing algorithms are derived using the first and second-order moments of the signal and the noises present in the observation model. Simulation results are provided to illustrate the feasibility of the proposed algorithms, using the error estimation covariance matrices as measure of the quality of the estimators.     

On the Smoothing Estimation Problem for the Intensity of a DSMPP

The smoothing estimation problem for the intensity process of a doubly stochastic multichannel Poisson process is investigated. Recursive procedures for the efficient computation of the fixed-point as well as the fixed-interval smoothing estimators of the intensity process are designed. The only hypothesis imposed is that the covariance function of the intensity process is finite-dimensional and known. Thus, the provided solution is valid for both stationary and non-stationary processes.     

Least-squares Polynomial Estimation from Observations Featuring Correlated Random Delays

The least-squares polynomial filtering and fixed-point smoothing problems of discrete-time signals from randomly delayed observations is addressed, when the Bernoulli random variables modelling the delay are correlated at consecutive sampling times. Recursive estimation algorithms are deduced without requiring full knowledge of the state-space model generating the signal process, but only information about the delay probabilities and the moments of the processes involved. Defining a suitable augmented observation vector, the polynomial estimation problem is reduced to the linear estimation problem of the signal based on the augmented observations, which is solved by using an innovation approach.     

Nonlinear smoothing: approximate algorithms

In this paper the method of stochastic linearization is employed to develop new approximate algorithms for nonlinear smoothing. Both fixed-point and fixed-interval smoothing cases are considered. An example is included to illustrate the use of the algorithms.     

Fixed-point smoothing algorithm for discrete multiplicative systems

A fixed-point smoothing algorithm is proven for discretetime systems with additive and multiplicative noise in the plant and measurement equations. Such systems, although linear, differ in a number of aspects from systems with only additive noise. The algorithm depends on the multiplicative terms, as expected. Steady-state results are derived.     

纵向数据边际模型非参数光滑方法的比较

对于纵向数据边际模型的均值函数, 有很多非参数估计方法, 其中回归样条, 光滑样条, 似乎不相关(SUR)核估计等方法在工作协方差阵正确指定时具有最小的渐近方差. 回归样条的渐近偏差与工作协方差阵无关, 而SUR核估计和光滑样条估计的渐近偏差却依赖于工作协方差阵. 本文主要研究了回归样条, 光滑样条和SUR核估计的效率问题. 通过模拟比较发现回归样条估计的表现比较稳定, 在大多数情况下比光滑样条估计和SUR核估计的效率高.     

The optimal robust finite-horizon Kalman filtering for multiple sensors with different stochastic failure rates

This work deals with the filtering problem for norm-bounded uncertain discrete dynamic systems with multiple sensors having different stochastic failure rates. For tackling the uncertainties of the covariance matrices of state and state estimation error simultaneously, their upper bounds containing a scaling parameter are derived, and then a robust finite-horizon filtering minimizing the upper bound of the estimation error covariance is proposed. Furthermore, an optimal scaling parameter is exploited to reduce the conservativeness of the upper bounds of the state and estimation error covariances, which leads to an optimal robust filtering for all possible missing measurements and all admissible parameter uncertainties. A numerical example illustrates the performance improvement over the traditional Kalman filtering method.     

An Examination of the Use of Adaptive Filtering in Forecasting

Adaptive filtering is a technique for preparing short- to medium-term forecasts based on the weighting of historical observations, in a similar way to moving average and exponential smoothing. However, adaptive filtering, as it has been developed in electrical engineering, attempts to distinguish a signal pattern from random noise, rather than simply smoothing the noise of past data. This paper reviews the technique of adaptive filtering and investigates its applications and limitations for the forecasting practitioner. This is done by looking at the performance of adaptive filtering in forecasting a number of time series and by comparing it with other forecasting techniques.     

Derivation of linear estimation algorithms from measurements affected by multiplicative and additive noises

This paper addresses the problem of estimating signals from observation models with multiplicative and additive noises. Assuming that the state-space model is unknown, the multiplicative noise is non-white and the signal and additive noise are correlated, recursive algorithms are derived for the least-squares linear filter and fixed-point smoother. The proposed algorithms are obtained using an innovation approach and taking into account the information provided by the covariance functions of the process involved.     

Modeling dependence dynamics through copulas with regime switching

Measuring dynamic dependence between international financial markets has recently attracted great interest in financial econometrics because the observed correlations rose dramatically during the 2008–09 global financial crisis. Here, we propose a novel approach for measuring dependence dynamics. We include a hidden Markov chain (MC) in the equation describing dependence dynamics, allowing the unobserved time-varying dependence parameter to vary according to both a restricted ARMA process and an unobserved two-state MC. Estimation is carried out via the inference for the margins in conjunction with filtering/smoothing algorithms. We use block bootstrapping to estimate the covariance matrix of our estimators. Monte Carlo simulations compare the performance of regime switching and no switching models, supporting the regime-switching specification. Finally the proposed approach is applied to empirical data, through the study of the S&P500 (USA), FTSE100 (UK) and BOVESPA (Brazil) stock market indexes.     

Some results on Tchebycheffian spline functions

This report derives explicit solutions to problems involving Tchebycheffian spline functions. We use a reproducing kernel Hilbert space which depends on the smoothness criterion, but not on the form of the data, to solve explicitly Hermite-Birkhoff interpolation and smoothing problems. Sard's best approximation to linear functionals and smoothing with respect to linear inequality constraints are also discussed. Some of the results are used to show that spline interpolation and smoothing is equivalent to prediction and filtering on realizations of certain stochastic processes.     

Nonlinear and Non-Gaussian State Space Modeling Using Sampling Techniques

In this paper, the nonlinear non-Gaussian filters and smoothers are proposed using the joint density of the state variables, where the sampling techniques such as rejection sampling (RS), importance resampling (IR) and the Metropolis-Hastings independence sampling (MH) are utilized. Utilizing the random draws generated from the joint density, the density-based recursive algorithms on filtering and smoothing can be obtained. Furthermore, taking into account possibility of structural changes and outliers during the estimation period, the appropriately chosen sampling density is possibly introduced into the suggested nonlinear non-Gaussian filtering and smoothing procedures. Finally, through Monte Carlo simulation studies, the suggested filters and smoothers are examined.     

Discrete–time nonlinear filtering with marked point process observations: ii. risk–sensitive filters

We discuss in this article the risk–sensitive filtering problem of estimating a nonlinear signal process, with nonadditive non–Gaussian noise, via a marked point process observation. This extends to the risk sensitive case all the risk–neutral results studied in Dufour and Kannan [2].By going into a change of measure, we derive the unnormalized conditional density of the signal conditioned on the observation history. We also derive the unnormalized prediction density. Using these, we present two separate expressions for the optimal estimate of the signal. A similar analysis of the smoothing density of the signal is also studied under both the risk–sensitive and risk–neutral cases. We specialize the above optimal estimation to the linear signal dynamics and marked point process observation under some Gaussian assumptions. We obtain a Kalman type risk-sensitive filter. Due to the special nature of the observation process, the conditional mean and covariance estimates directly depend now on the point process     

Varying-coefficient model for the occurrence rate function of recurrent events

This article mainly considers the recurrent event process with independent censoring mechanism through a more flexible varying-coefficient model. The smoothing estimators for the varying-coefficient functions are also proposed via maximizing the kernel weight version of the log-partial likelihood function with respect to the coefficients at each time point. For the selection of appropriate bandwidths and the construction of confidence intervals, the consistent empirical smoothing estimators for the covariance functions of the estimators and a bias correction method are considered. As for the baseline effect function of recurrent events in the population, two different smoothing estimation methods are suggested and investigated. In this study, the asymptotic properties of the proposed smoothing estimators are derived. The finite sample properties of our methods are examined through a Monte Carlo simulation. Moreover, the procedures are applied to a recurrent sample of AIDS link to intravenous experiences (ALIVE) cohort study.     

Asymptotic Analysis of a Multiple Frequency Estimation Method

A recently proposed method of multiple frequency estimation for mixed-spectrum time series is analyzed. The so-called PF method is a procedure that combines the autoregressive (AR) representation of superimposed sinusoids with the idea of parametric filtering. The gist of the method is to parametrize a linear filter in accord with a certain parametrization property, as suggested by the particular form of the bias encountered by Prony′s least-squares estimator for the AR model. It is shown that for any parametric filter with this property, the least-squares estimator obtained from the filtered data is almost surely contractive as a function of the filter parameter and has a unique multivariate fixed-point in the vicinity of the true AR parameter. The fixed-point, known as the PF estimator, is shown to be stronly consistent for estimating the AR model, and the chronic bias of Prony′s estimator is thus eliminated. The almost sure convergence of an iterative algorithm that calculates the fixed-point and the asymptotic normality of the PF estimator are also established. The all-pole filter is considered as an example and application of the developed theory.     

Adaptive Order Selection for Spline Smoothing

Computational methods are presented for spline smoothing that make it practical to compute smoothing splines of degrees other than just the standard cubic case. Specifically, an order n algorithm is developed that has conceptual and practical advantages relative to classical methods. From a conceptual standpoint, the algorithm uses only standard programming techniques that do not require specialized knowledge about spline functions, methods for solving sparse equation systems or Kalman filtering. This allows for the practical development of methods for adaptive selection of both the level of smoothing and degree of the estimator. Simulation experiments are presented that show adaptive degree selection can improve estimator efficiency over the use of cubic smoothing splines. Run-time comparisons are also conducted between the proposed algorithm and a classical, band-limited, computing method for the cubic case.     

Limiting optimal adaptive filtering with unknown disturbance covariance

The accuracy of estimating the variance of the Kalman-Bucy filter depends essentially on disturbance covariance matrices and measurement noise. The main difficulty in filter design is the lack of necessary statistical information about the useful signal and the disturbance. Filters whose parameters are tuned during active estimation are classified with adaptive filters. The problem of adaptive filtering under parametric uncertainty conditions is studied. A method for designing limiting optimal Kalman-Bucy filters in the case of unknown disturbance covariance is presented. An adaptive algorithm for estimating disturbance covariance matrices based on stochastic approximation is described. Convergence conditions for this algorithm are investigated. The operation of a limiting adaptive filter is exemplified.     

Perturbations and projections of Kalman–Bucy semigroups

We analyse various perturbations and projections of Kalman–Bucy semigroups and Riccati equations. For example, covariance inflation-type perturbations and localisation methods (projections) are common in the ensemble Kalman filtering literature. In the limit of these ensemble methods, the regularised sample covariance tends toward a solution of a perturbed/projected Riccati equation. With this motivation, results are given characterising the error between the nominal and regularised Riccati flows and Kalman–Bucy filtering distributions. New projection-type models are also discussed; e.g. Bose–Mesner projections. These regularisation models are also of interest on their own, and in, e.g., differential games, control of stochastic/jump processes, and robust control.     

Fast and Flexible ADMM Algorithms for Trend Filtering

This article presents a fast and robust algorithm for trend filtering, a recently developed nonparametric regression tool. It has been shown that, for estimating functions whose derivatives are of bounded variation, trend filtering achieves the minimax optimal error rate, while other popular methods like smoothing splines and kernels do not. Standing in the way of a more widespread practical adoption, however, is a lack of scalable and numerically stable algorithms for fitting trend filtering estimates. This article presents a highly efficient, specialized alternating direction method of multipliers (ADMM) routine for trend filtering. Our algorithm is competitive with the specialized interior point methods that are currently in use, and yet is far more numerically robust. Furthermore, the proposed ADMM implementation is very simple, and, importantly, it is flexible enough to extend to many interesting related problems, such as sparse trend filtering and isotonic trend filtering. Software for our method is freely available, in both the C and R languages.