This paper gives a precise meaning to the robustness of risk-sensitive filters for problems in which one is uncertain as to the exact value of the probability model. It is shown that risk-sensitive estimators (including filters) enjoy an error bound which is the sum of two terms, the first of which coincides with an upper bound on the error that one would obtain if one knew exactly the underlying probability model, while the second term is a measure of the distance between the true and design probability models. The paper includes a discussion of several approaches to estimation, including H/sub /spl infin// filtering.

Robustness and risk-sensitive filtering

We address the risk-sensitive filtering problem which is minimizing the expectation of the exponential of the squared estimation error multiplied by a risk-sensitive parameter. Such filtering can be more robust to plant and noise uncertainty than minimum error variance filtering. Although optimizing a differently formulated performance index to that of the so-called H/sub /spl infin// filtering, risk-sensitive filtering leads to a worst case deterministic noise estimation problem given from the differential game associated with H/sub /spl infin// filtering. We consider a class of discrete-time stochastic nonlinear state-space models. We present linear recursions in the information state and the result for the filtered estimate that minimizes the risk-sensitive cost index. We also present fixed-interval smoothing results for each of these signal models. In addition, a brief discussion is included on relations of the risk-sensitive estimation problem to minimum variance estimation and a worst case estimation problem in a deterministic noise scenario related to minimax dynamic games. The technique used in this paper is the so-called reference probability method which defines a new probability measure where the observations are independent and translates the problem to the new measure. The optimization problem is solved using simple estimation theory in the new measure, and the results are interpreted as solutions in the original measure.

/pdf/risk-sensitive-filtering-and-smoothing-via-reference-395esmv2sq.pdf

Risk-sensitive filtering and smoothing via reference probability methods

This paper presents new schemes for recursive estimation of the state transition probabilities for hidden Markov models (HMMs) via extended least squares (ELS) and recursive state prediction error (RSPE) methods. Local convergence analysis for the proposed RSPE algorithm is shown using the ordinary differential equation (ODE) approach developed for the more familiar recursive output prediction error (RPE) methods. The presented scheme converges and is relatively well conditioned compared with the previously proposed RPE scheme for estimating the transition probabilities that perform poorly in low noise. The ELS algorithm presented is computationally of order N/sup 2/, which is less than the computational effort of order N/sup 4/ required to implement the RSPE (and previous RPE) scheme, where N is the number of Markov states. Building on earlier work, an algorithm for simultaneous estimation of the state output mappings and the state transition probabilities that requires less computational effort than earlier schemes is also presented and discussed. Implementation aspects of the proposed algorithms are discussed, and simulation studies are presented to illustrate the convergence and convergence rates.

/pdf/adaptive-estimation-of-hmm-transition-probabilities-3ez0vr25fo.pdf

Adaptive estimation of HMM transition probabilities

We address the problem of filtering and fixed-lag smoothing for discrete-time and discrete-state hidden Markov models (HMMs), with the intention of extending some important results in Kalman filtering, notably the property of exponential stability. By appealing to a generalized Perron-Frobenius result for non-negative matrices, we are able to demonstrate exponential forgetting for both the recursive filters and smoothers; furthermore, methods for deriving overbounds on the convergence rate are indicated. Simulation studies for a two-state and two-output HMM verify qualitatively some of the theoretical predictions, and the observed convergence rate is shown to be bounded in accordance with the theoretical predictions.

/pdf/exponential-stability-of-filters-and-smoothers-for-hidden-ez0wbks3ck.pdf

Exponential stability of filters and smoothers for hidden Markov models

In this paper, we address the risk-sensitive filtering and smoothing problem for discrete-time nonlinear and linear Gauss-Markov state-space models. Also, connection between L/sub 2/ filtering (termed here risk-neutral filtering) and risk-sensitive filtering is described via the limiting results when the risk-sensitive parameter tends to zero. The technique used in this paper is the so-called reference probability method which defines a new probability measure where the observations are independent. The optimisation problem is in the new measure and the results are interpreted as solutions in the original measure.

/pdf/risk-sensitive-filtering-and-smoothing-via-reference-to2cedobam.pdf

Hidden Markov models have proved suitable for many in­ teresting applications which can be modelled using some unobservable finite state Markov process, influencing mea­ sured signals. This can be used to describe bursty telecom­ munications traffic, or the faults in a complicated systems, for modelling the activity in neurons, for modelling speech patterns, etc. In all these applications, one has to estimate the unobservable underlying state of the Markov process, using the observed signals. Optimal recursive filters are well known for this estimation problem. Recently risk sensitive filters for the same problem have also been ob­ tained. An important question in studying the quality of such filters is the rate at which arbitrarily assigned initial conditions are forgotten. In this paper we show that the effect of initial conditions on these filters dies out geomet­ rically fast under very reasonable observability assump­ tions. The proof is given in the simplest case of finite state space and of a finite, quantised, observations spacc. How­ ever the method can be extended to more general models by continuity arguments.

R.K. Boel

Papers

Geometric convergence of filters for hidden Markov models

Convergence in mean for ELS-based adaptive control

Recursive Prediction Error Methods in Open-Loop Estimation and Adaptive Control

Convergence in Mean for ELS - Based Adaptive Control