Hidden State Process Research Articles

Likelihood inference for Markov switching GARCH(1,1) models using sequential Monte Carlo

Markov switching (MS-)GARCH(1,1) models allow for structural changes in volatility dynamics between a finite number of regimes. Since the regimes are not observed, computation of the likelihood requires integrating over an exponentially increasing number of regime paths, which is intractable. An existing smooth likelihood estimation procedure for sequential Monte Carlo (SMC), that is currently limited to hidden Markov models with a one-dimensional state variable, is modified to enable likelihood estimation and maximisation for MS-GARCH(1,1) models, a model which requires two dimensions, volatility and regime, to evolve its hidden state process. Furthermore, the modified SMC procedure is shown to be easily adapted to fitting MS-GARCH(1,1) models even when there are missing observations. The proposed methodology is validated with simulated data and is also illustrated with analysis of two financial time series, the daily returns on the S&P 500 index and on the Henry Hub natural gas spot price, with the latter series containing a gap caused by shutdown in response to hurricane Rita in 2005.

A Markov-switching generalized additive model for compound Poisson processes, with applications to operational loss models

This paper is concerned with modelling the behaviour of random sums over time. Such models are particularly useful to describe the dynamics of operational losses, and to correctly estimate tail-related risk indicators. However, time-varying dependence structures make it a difficult task. To tackle these issues, we formulate a new Markov-switching generalized additive compound process combining Poisson and generalized Pareto distributions. This flexible model takes into account two important features: on the one hand, we allow all parameters of the compound loss distribution to depend on economic covariates in a flexible way. On the other hand, we allow this dependence to vary over time, via a hidden state process. A simulation study indicates that, even in the case of a short time series, this model is easily and well estimated with a standard maximum likelihood procedure. Relying on this approach, we analyse a novel data-set of 819 losses resulting from frauds at the Italian bank UniCredit. We show that our model improves the estimation of the total loss distribution over time, compared to standard alternatives. In particular, this model provides estimations of the 99.9% quantile that are never exceeded by the historical total losses, a feature particularly desirable for banking regulators.

Performance of dynamic texture segmentation using GPU

This work is focused on the assessment of the use of GPU computation in dynamic texture segmentation under the mixture of dynamic textures (MDT) model. In this generative video model, the observed texture is a time-varying process commanded by a hidden state process. The use of mixtures in this model allows simultaneously handling of different visual processes. Nowadays, the use of GPU computing is growing in high-performance applications, but the adaptation of existing algorithms in such a way as to obtain a benefit from its use is not an easy task. In this paper, we made two implementations, one in CPU and the other in GPU, of a known segmentation algorithm based on MDT. In the MDT algorithm, there is a matrix inversion process that is highly demanding in terms of computing power. We make a comparison between the gain in performance obtained by porting to GPU this matrix inversion process and the gain obtained by porting to GPU the whole MDT segmentation process. We also study real-time motion segmentation performance by separating the learning part of the algorithm from the segmentation part, leaving the learning stage as an off-line process and keeping the segmentation as an online process. The results of performance analyses allow us to decide the cases in which the full GPU implementation of the motion segmentation process is worthwhile.

A Monte Carlo method for filtering a marked doubly stochastic Poisson process

The author provides an approximated solution for the filtering of a state-space model, where the hidden state process is a continuous-time pure jump Markov process and the observations come from marked point processes. Each state k corresponds to a different marked point process, defined by its conditional intensity function λk(t). When a state is visited by the hidden process, the corresponding marked point process is observed. The filtering equations are obtained by applying the innovation method and the integral representation theorem of a point process martingale. Since the filtering equations belong to the family of Kushner–Stratonovich equations, an iterative solution is calculated. The theoretical solution is approximated and a Monte Carlo integration technique employed to implement it. The sequential method has been tested on a simulated data set based on marked point processes widely used in the statistical analysis of seismic sequences: the Poisson model, the stress release model and the Etas model.

General smoothing formulas for Markov-modulated Poisson observations

In this paper, we compute general smoothing dynamics for partially observed dynamical systems generating Poisson observations. We consider two model classes, each Markov modulated Poisson processes, whose stochastic intensities depend upon the state of an unobserved Markov process. In one model class, the hidden state process is a continuously-valued Ito/spl circ/ process, which gives rise to a continuous sample-path stochastic intensity. In the other model class, the hidden state process is a continuous-time Markov chain, giving rise to a pure jump stochastic intensity. To compute filtered estimates of state process, we establish dynamics, whose solutions are unnormalized marginal probabilities; however, these dynamics include Lebesgue-Stieltjes stochastic integrals. By adapting the transformation techniques introduced by J. M. C. Clark, we compute filter dynamics which do not include these stochastic integrals. To construct smoothers, we exploit a duality between our forward and backward transformed dynamics and thereby completely avoid the technical complexities of backward evolving stochastic integral equations. The general smoother dynamics we present can readily be applied to specific smoothing algorithms, referred to in the literature as: Fixed point smoothing, fixed lag smoothing and fixed interval smoothing. It is shown that there is a clear motivation to compute smoothers via transformation techniques similar to those presented by J. M. C. Clark, that is, our smoothers are easily obtained without recourse to two sided stochastic integration. A computer simulation is included.

Hidden Markov model approach to spectral analysis for hyperspectral imagery

The hidden Markov model (HMM) has been widely used in speech recognition where it models a speech signal as a doubly sto- chastic process with a hidden state process that can be observed only through a sequence of observations. We present a new application of the HMM in hyperspectral image analysis inspired by the analogy be- tween the temporal variability of a speech signal and the spectral vari- ability of a remote sensing image pixel vector. The idea is to model a hyperspectral spectral vector as a stochastic process where the spectral correlation and band-to-band variability are modeled by a hidden Markov process with parameters determined by the spectrum of the vector that forms a sequence of observations. With this interpretation, a new HMM- based spectral measure, referred to as the HMM information divergence (HMMID), is derived to characterize spectral properties. To evaluate the performance of this new measure, it is further compared to two com- monly used spectral measures, Euclidean distance (ED) and the spectral angle mapper (SAM), and the recently proposed spectral information divergence (SID). The experimental results show that the HMMID per- forms better than the other three measures in characterizing spectral information at the expense of computational complexity. © 2001 Society of