Robust Prediction, Filtering and Smoothing

Abstract

Unit 405, Office Block, Hotel Equatorial Shanghai This book describes the classical smoothing, filtering and prediction techniques together with some more recently developed embellishments for improving performance within applications. It aims to present the subject in an accessible way, so that it can serve as a practical guide for undergraduates and newcomers to the field. The material is organised as a ten-lecture course. The foundations are laid in Chapters 1 and 2, which explain minimum-mean-square-error solution construction and asymptotic behaviour. Chapters 3 and 4 introduce continuous-time and discrete-time minimum-variance filtering. Generalisations for missing data, deterministic inputs, correlated noises, direct feedthrough terms, output estimation and equalisation are described. Chapter 5 simplifies the minimum-variance filtering results for steady-state problems. Observability, Riccati equation solution convergence, asymptotic stability and Wiener filter equivalence are discussed. Chapters 6 and 7 cover the subject of continuous-time and discrete-time smoothing. The main fixed-lag, fixedpoint and fixed-interval smoother results are derived. It is shown that the minimum-variance fixed-interval smoother attains the best performance. Chapter 8 attends to parameter estimation. As the above-mentioned approaches all rely on knowledge of the underlying model parameters, maximum-likelihood techniques within expectation-maximisation algorithms for joint state and parameter estimation are described. Chapter 9 is concerned with robust techniques that accommodate uncertainties within problem specifications. An extra term within Riccati equations enables designers to trade-off average error and peak error performance. Chapter 10 rounds off the course by applying the afore-mentioned linear techniques to nonlinear estimation problems. It is demonstrated that step-wise linearisations can be used within predictors, filters and smoothers, albeit by forsaking optimal performance guarantees. How to reference In order to correctly reference this scholarly work, feel free to copy and paste the following:

Evidence for an Optimal Algorithm Underlying Signal Combination in Human Visual Cortex

“…This bears a striking similarity to the optimal combination rule for two inputs (Ernst and Bülthoff 2004), which is a Kalman filter (Kalman and Bucy 1961). The static filter gain ( L ) is given by $L=true\lfloor \frac{{\sigma }_A^2{P}_A}{{\sigma }_A^2{\sigma }_B^2+{\sigma }_A^2{P}_A+{\sigma }_B^2{P}_B}\frac{{\sigma }_B^2{P}_B}{{\sigma }_B^2{\sigma }_A^2+{\sigma }_B^2{P}_B+{\sigma }_A^2{P}_A}true\rfloor ,$ where P A is the response of the channel (or sensor) tuned to input A , and σ A 2 is the variance, with terms bearing the subscript B corresponding to a second channel (Einicke 2012). Hence, the filter weights each input by the inverse of its contribution to the total variance.…”

Comparison of two stochastic techniques for reliable urban runoff prediction by modeling systematic errors

“…This so‐called “state” (or “transition,” or “system”) equation describes the continuous evolution of some “hidden” (or “latent”) states S which, being now stochastic, directly account for modeling errors. This vector of usually unmeasurable variables can be estimated from the measured outputs [ Einicke , ]. is called “diffusion term,” “state noise,” or “level disturbance” and accounts for modeling errors by making the states uncertain or random.…”

“…Forecasting denotes the generation of model outputs (and states) starting from the last observation up to an arbitrary number of time steps in the future. This process is also loosely described as making predictions (in the validation period) [ Dietzel and Reichert , ; Renard et al ., ; Law and Stuart , ; Einicke , ], simulating [ Platen and Bruti‐Liberati , ] or, more precisely, ex‐post hindcasting (when the input is assumed to be known) [ Beven and Young , ].…”

“…It can be useful to predict system output and/or states for points in time where flow data have been employed for parameter inference, in the so‐called “calibration period” (or calibration layout). This retrospective analysis, called smoothing, consists in identifying system states (or output) from all available (noisy) output data [ Einicke , ; Bulygina and Gupta , ].…”

Bounded-observation Kalman filtering of correlation in multivariate neural recordings

“…Fig. 1 shows the estimate of the correlation, i.e., z k , using the proposed filter, as well as the results of naive application of the Extended Kalman Filter (EKF) based on linearization of (2) [10], [11]. Clearly, state estimation using proposed optimal filter is closest to the true state of the system.…”