State estimation of probabilistic hybrid systems with particle filters

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2004.Includes bibliographical references (p. 117-123).Robotic and embedded systems have become increasingly pervasive in every-day applications, ranging from space probes and life support systems to autonomous rovers. In order to act robustly in the physical world, robotic systems must handle the uncertainty and partial observability inherent in most real-world situations. A convenient modeling tool for many applications, including fault diagnosis and visual tracking, are probabilistic hybrid models. In probabilistic hybrid models, the hidden state is represented with discrete and continuous state variables that evolve probabilistically. The hidden state is observed indirectly, through noisy observations. A challenge is that real-world systems are non-linear, consist of a large collection of concurrently operating components, and exhibit autonomous mode transitions, that is, discrete state transitions that depend on the continuous dynamics. In this thesis, we introduce an efficient algorithm for hybrid state estimation that combines Rao-Blackwellised particle filtering with a Gaussian representation. Conceptually, our algorithm samples trajectories traced by the discrete variables over time and, for each trajectory, estimates the continuous state with a Kalman Filter. A key insight to handling the autonomous transitions is to reuse the continuous estimates in the importance sampling step. We extended the class of autonomous transitions that can be efficiently handled by Gaussian techniques and provide a detailed empirical evaluation of the algorithm on a dynamical system with four continuous state variables. Our results indicate that our algorithm is substantially more efficient than non-RaoBlackwellised approaches. Though not as good as a k-best filter in nominal scenarios, our algorithm outperforms(cont.) a k-best filter when the correct diagnosis has too low a probability to be included in the leading set of trajectories. Through these accomplishments, the thesis lays ground work for a unifying stochastic search algorithm that shares the benefits of both methods.by Stanislav Funiak.M.Eng

Recursive Bayesian inference on stochastic differential equations

This thesis is concerned with recursive Bayesian estimation of non-linear dynamical systems, which can be modeled as discretely observed stochastic differential equations. The recursive real-time estimation algorithms for these continuous-discrete filtering problems are traditionally called optimal filters and the algorithms for recursively computing the estimates based on batches of observations are called optimal smoothers. In this thesis, new practical algorithms for approximate and asymptotically optimal continuous-discrete filtering and smoothing are presented. The mathematical approach of this thesis is probabilistic and the estimation algorithms are formulated in terms of Bayesian inference. This means that the unknown parameters, the unknown functions and the physical noise processes are treated as random processes in the same joint probability space. The Bayesian approach provides a consistent way of computing the optimal filtering and smoothing estimates, which are optimal given the model assumptions and a consistent way of analyzing their uncertainties. The formal equations of the optimal Bayesian continuous-discrete filtering and smoothing solutions are well known, but the exact analytical solutions are available only for linear Gaussian models and for a few other restricted special cases. The main contributions of this thesis are to show how the recently developed discrete-time unscented Kalman filter, particle filter, and the corresponding smoothers can be applied in the continuous-discrete setting. The equations for the continuous-time unscented Kalman-Bucy filter are also derived. The estimation performance of the new filters and smoothers is tested using simulated data. Continuous-discrete filtering based solutions are also presented to the problems of tracking an unknown number of targets, estimating the spread of an infectious disease and to prediction of an unknown time series.reviewe

Modelling Stochastic Volatility with Leverage and Jumps : A Simulated Maximum Likelihood Approach via Particle Filtering

In this paper we provide a unified methodology in order to conduct likelihood-based inference on the unknown parameters of a general class of discrete-time stochastic volatility models, characterized by both a leverage e®ect and jumps in returns. Given the non-linear/non-Gaussian state-space form, approximating the likelihood for the parameters is conducted with output generated by the particle filter. Methods are employed to ensure that the approximating likelihood is continuous as a function of the unknown parameters thus enabling the use of Newton-Raphson type maximization algorithms. Our approach is robust and efficient relative to alternative Markov Chain Monte Carlo schemes employed in such contexts. In addition it provides a feasible basis for undertaking the non-trivial task of model comparison. The technique is applied to daily returns data for various stock price indices. We find strong evidence in favour of a leverage effect in all cases. Jumps are an important component in two out of the four series we consider.Particle filter ; Simulation ; SIR ; State space ; Leverage effect ; Jumps

Modelling stochastic volatility with leverage and jumps: a simulated maximum likelihood approach via particle filtering

In this paper we provide a unified methodology in order to conduct likelihood-based inference on the unknown parameters of a general class of discrete-time stochastic volatility models, characterized by both a leverage effect and jumps in returns. Given the non-linear/non-Gaussian state-space form, approximating the likelihood for the parameters is conducted with output generated by the particle filter. Methods are employed to ensure that the approximating likelihood is continuous as a function of the unknown parameters thus enabling the use of Newton-Raphson type maximization algorithms. Our approach is robust and efficient relative to alternative Markov Chain Monte Carlo schemes employed in such contexts. In addition it provides a feasible basis for undertaking the non-trivial task of model comparison. The technique is applied to daily returns data for various stock price indices. We find strong evidence in favour of a leverage effect in all cases. Jumps are an important component in two out of the four series we consider

Multivariate stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck processes : a quasi-likelihood approach

This paper extends the ordinary quasi-likelihood estimator for stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck (OU) processes to vector processes. Despite the fact that multivariate modeling of asset returns is essential for portfolio optimization and risk management -- major areas of financial analysis -- the literature on multivariate modeling of asset prices in continuous time is sparse, both with regard to theoretical and applied results. This paper uses non-Gaussian OU-processes as building blocks for multivariate models for high frequency financial data. The OU framework allows exact discrete time transition equations that can be represented on a linear state space form. We show that a computationally feasible quasi-likelihood function can be constructed by means of the Kalman filter also in the case of high-dimensional vector processes. The framework is applied to Euro/NOK and US Dollar/NOK exchange rate data for the period 2.1.1989-4.2.2010.Financial support from the Norwegian Research Council ("Finansmarkedsfondet") is gratefully acknowledged

Signal tracking beyond the time resolution of an atomic sensor by Kalman filtering

We study causal waveform estimation (tracking) of time-varying signals in a paradigmatic atomic sensor, an alkali vapor monitored by Faraday rotation probing. We use Kalman filtering, which optimally tracks known linear Gaussian stochastic processes, to estimate stochastic input signals that we generate by optical pumping. Comparing the known input to the estimates, we confirm the accuracy of the atomic statistical model and the reliability of the Kalman filter, allowing recovery of waveform details far briefer than the sensor's intrinsic time resolution. With proper filter choice, we obtain similar benefits when tracking partially-known and non-Gaussian signal processes, as are found in most practical sensing applications. The method evades the trade-off between sensitivity and time resolution in coherent sensing.Comment: 15 pages, 4 figure

Idealized computational models for auditory receptive fields

This paper presents a theory by which idealized models of auditory receptive fields can be derived in a principled axiomatic manner, from a set of structural properties to enable invariance of receptive field responses under natural sound transformations and ensure internal consistency between spectro-temporal receptive fields at different temporal and spectral scales. For defining a time-frequency transformation of a purely temporal sound signal, it is shown that the framework allows for a new way of deriving the Gabor and Gammatone filters as well as a novel family of generalized Gammatone filters, with additional degrees of freedom to obtain different trade-offs between the spectral selectivity and the temporal delay of time-causal temporal window functions. When applied to the definition of a second-layer of receptive fields from a spectrogram, it is shown that the framework leads to two canonical families of spectro-temporal receptive fields, in terms of spectro-temporal derivatives of either spectro-temporal Gaussian kernels for non-causal time or the combination of a time-causal generalized Gammatone filter over the temporal domain and a Gaussian filter over the logspectral domain. For each filter family, the spectro-temporal receptive fields can be either separable over the time-frequency domain or be adapted to local glissando transformations that represent variations in logarithmic frequencies over time. Within each domain of either non-causal or time-causal time, these receptive field families are derived by uniqueness from the assumptions. It is demonstrated how the presented framework allows for computation of basic auditory features for audio processing and that it leads to predictions about auditory receptive fields with good qualitative similarity to biological receptive fields measured in the inferior colliculus (ICC) and primary auditory cortex (A1) of mammals.Comment: 55 pages, 22 figures, 3 table

LMMSE Estimation and Interpolation of Continuous-Time Signals from Discrete-Time Samples Using Factor Graphs

The factor graph approach to discrete-time linear Gaussian state space models is well developed. The paper extends this approach to continuous-time linear systems/filters that are driven by white Gaussian noise. By Gaussian message passing, we then obtain MAP/MMSE/LMMSE estimates of the input signal, or of the state, or of the output signal from noisy observations of the output signal. These estimates may be obtained with arbitrary temporal resolution. The proposed input signal estimation does not seem to have appeared in the prior Kalman filtering literature

Signal tracking beyond the time resolution of an atomic sensor by Kalman filtering

We study causal waveform estimation (tracking) of time-varying signals in a paradigmatic atomic sensor, an alkali vapor monitored by Faraday rotation probing. We use Kalman filtering, which optimally tracks known linear Gaussian stochastic processes, to estimate stochastic input signals that we generate by optical pumping. Comparing the known input to the estimates, we confirm the accuracy of the atomic statistical model and the reliability of the Kalman filter, allowing recovery of waveform details far briefer than the sensor's intrinsic time resolution. With proper filter choice, we obtain similar benefits when tracking partially-known and non-Gaussian signal processes, as are found in most practical sensing applications. The method evades the trade-off between sensitivity and time resolution in coherent sensing.Comment: 15 pages, 4 figure