Distributions associated with general runs and patterns in hidden Markov models

This paper gives a method for computing distributions associated with patterns in the state sequence of a hidden Markov model, conditional on observing all or part of the observation sequence. Probabilities are computed for very general classes of patterns (competing patterns and generalized later patterns), and thus, the theory includes as special cases results for a large class of problems that have wide application. The unobserved state sequence is assumed to be Markovian with a general order of dependence. An auxiliary Markov chain is associated with the state sequence and is used to simplify the computations. Two examples are given to illustrate the use of the methodology. Whereas the first application is more to illustrate the basic steps in applying the theory, the second is a more detailed application to DNA sequences, and shows that the methods can be adapted to include restrictions related to biological knowledge.Comment: Published in at http://dx.doi.org/10.1214/07-AOAS125 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

The State Space Models Toolbox for MATLAB

State Space Models (SSM) is a MATLAB toolbox for time series analysis by state space methods. The software features fully interactive construction and combination of models, with support for univariate and multivariate models, complex time-varying (dy- namic) models, non-Gaussian models, and various standard models such as ARIMA and structural time-series models. The software includes standard functions for Kalman fil- tering and smoothing, simulation smoothing, likelihood evaluation, parameter estimation, signal extraction and forecasting, with incorporation of exact initialization for filters and smoothers, and support for missing observations and multiple time series input with com- mon analysis structure. The software also includes implementations of TRAMO model selection and Hillmer-Tiao decomposition for ARIMA models. The software will provide a general toolbox for time series analysis on the MATLAB platform, allowing users to take advantage of its readily available graph plotting and general matrix computation capabilities.

Bayesian model comparison for compartmental models with applications in positron emission tomography

We develop strategies for Bayesian modelling as well as model comparison, averaging and selection for compartmental models with particular emphasis on those that occur in the analysis of positron emission tomography (PET) data. Both modelling and computational issues are considered. Biophysically inspired informative priors are developed for the problem at hand, and by comparison with default vague priors it is shown that the proposed modelling is not overly sensitive to prior specification. It is also shown that an additive normal error structure does not describe measured PET data well, despite being very widely used, and that within a simple Bayesian framework simultaneous parameter estimation and model comparison can be performed with a more general noise model. The proposed approach is compared with standard techniques using both simulated and real data. In addition to good, robust estimation performance, the proposed technique provides, automatically, a characterisation of the uncertainty in the resulting estimates which can be considerable in applications such as PET

Smoothing dynamic positron emission tomography time courses using functional principal components

A functional smoothing approach to the analysis of PET time course data is presented. By borrowing information across space and accounting for this pooling through the use of a nonparametric covariate adjustment, it is possible to smooth the PET time course data thus reducing the noise. A new model for functional data analysis, the Multiplicative Nonparametric Random Effects Model, is introduced to more accurately account for the variation in the data. A locally adaptive bandwidth choice helps to determine the correct amount of smoothing at each time point. This preprocessing step to smooth the data then allows Subsequent analysis by methods Such as Spectral Analysis to be substantially improved in terms of their mean squared error