How to recognise whether an observed person walks or runs? We consider a dynamic environment where observations (e.g. the posture of a person) are caused by different dynamic processes (walking or running) which are active one at a time and which may transition from one to another at any time. For this setup, switching dynamic models have been suggested previously, mostly, for linear and nonlinear dynamics in discrete time. Motivated by basic principles of computations in the brain (dynamic, internal models) we suggest a model for switching nonlinear differential equations. The switching process in the model is implemented by a Hopfield network and we use parametric dynamic movement primitives to represent arbitrary rhythmic motions. The model generates observed dynamics by linearly interpolating the primitives weighted by the switching variables and it is constructed such that standard filtering algorithms can be applied. In two experiments with synthetic planar motion and a human motion capture data set we show that inference with the unscented Kalman filter can successfully discriminate several dynamic processes online.
In @cite_22 , the authors derive a smoothing algorithm based on variational inference for an Ornstein-Uhlenbeck (OU) process which is switched by a random telegraph process. Thus, this model can only switch between two constant drifts. Similarly, @cite_0 @cite_20 propose Markov chain Monte Carlo inference for a switched OU process where the number of different parameter sets as well as the parameter values are automatically determined from the data. However, these parameters are limited to the constant drift and diffusion parameters of the OU process which cannot implement generic nonlinear dynamics. In contrast to these models, we approximate a change point process using our continuous-valued switching dynamics which allows us to maintain a coherent continuous framework and apply standard filtering algorithms.
@cite_0 @cite_22 @cite_20
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We develop Markov chain Monte Carlo methodology for Bayesian inference for non-Gaussian Ornstein-Uhlenbeck stochastic volatility processes. The approach introduced involves expressing the unobserved stochastic volatility process in terms of a suitable marked Poisson process. We introduce two specific classes of Metropolis-Hastings algorithms which correspond to different ways of jointly parameterizing the marked point process and the model parameters. The performance of the methods is investigated for different types of simulated data. The approach is extended to consider the case where the volatility process is expressed as a superposition of Ornstein-Uhlenbeck processes. We apply our methodology to the US dollar-Deutschmark exchange rate. Copyright 2004 Royal Statistical Society. In this paper, we introduce a new Markov chain Monte Carlo approach to Bayesian analysis of discretely observed diffusion processes. We treat the paths between any two data points as missing data. As such, we show that, because of full dependence between the missing paths and the volatility of the diffusion, the rate of convergence of basic algorithms can be arbitrarily slow if the amount of the augmentation is large. We offer a transformation of the diffusion which breaks down dependency between the transformed missing paths and the volatility of the diffusion. We then propose two efficient Markov chain Monte Carlo algorithms to sample from the posterior-distribution of the transformed missing observations and the parameters of the diffusion. We apply our results to examples involving simulated data and also to Eurodollar short-rate data. This paper provides methods for carrying out likelihood based inference for diffusion driven models, for example discretely observed multivariate diffusions, continuous time stochastic volatility models and counting process models. The diffusions can potentially be non-stationary. Although our methods are sampling based, making use of Markov chain Monte Carlo methods to sample the posterior distribution of the relevant unknowns, our general strategies and details are different from previous work along these lines. The methods we develop are simple to implement and simulation efficient. Importantly, unlike previous methods, the performance of our technique is not worsened, in fact it improves, as the degree of latent augmentation is increased to reduce the bias of the Euler approximation. In addition, our method is not subject to a degeneracy that afflicts previous techniques when the degree of latent augmentation is increased. We also discuss issues of model choice, model checking and filtering. The techniques and ideas are applied to both simulated and real data. In this paper, a novel algorithm to blindly separate an instantaneous linear underdetermined mixture of nonstationary sources is proposed. It means that the number of sources exceeds the number of channels of the available data. The separation is based on the working assumption that the sources are piecewise stationary with a different variance in each block. It proceeds in two steps: 1) estimating the mixing matrix, and 2) computing the optimum beamformer in each block to maximize the signal-to-interference ratio of each separated signal with respect to the remaining signals. Estimating the mixing matrix is accomplished through a specialized tensor decomposition of the set of sample covariance matrices of the received mixture in each block. It utilizes optimum weighting, which allows statistically efficient (CRB attaining) estimation provided that the data obey the assumed Gaussian piecewise stationary model. In simulations, performance of the algorithm is successfully tested on blind separation of 16 speech signals from nine linear instantaneous mixtures of these signals.