Sequential Bayesian Modeling of non-stationary signals

are involved until the development of Sequential Monte Carlo techniques which are also known as the particle filters. In particle filtering, the problem is expressed in terms of state-space equations where the linearity and Gaussianity requirements of the Kalman filtering are generalized. Therefore, we need information about the functional form of the state variations. In this thesis, we bring a general solution for the cases where these variations are unknown and the process distributions cannot be expressed by any closed form probability density function. Here, we propose a novel modeling scheme which is as unified as possible to cover all these problems. Therefore we study the performance analysis of our unifying particle filtering methodology on non-stationary Alpha Stable process modeling. It is well known that the probability density functions of these processes cannot be expressed in closed form, except for some limited number of cases. Moreover, this distribution family presents a direct generalization from Gaussian to non-Gaussian distributions, since they have common properties, such as the stability property and the Central Limit Theorem. To model time structures of these processes, linear autoregressions are utilized, which are widely used in the literature. We propose three novel techniques to model non-stationary alpha stable processes. These include the modeling of time-varying autoregressive processes with known, unknown and constant, unknown and time-varying distribution parameters, respectively. Successful performances of these techniques have been shown by empirical analysis. It has also been demonstrated that the empirical results approach to their posterior ii Cramer Rao Lower Bound values and time-varying alpha stable processes can be modeled in their most general form succesfully. Next, to extend our unifying approach to model non-stationary cross-correlated vector autoregressive processes which are widely encountered in biomedical applications, mobile communications and chemical process modeling. Here, we extend our particle filtering scheme to multivariate cases so that the relationships between different processes can also be modeled. By means of our novel methodology, relationships between non- Gaussian vector autoregressive processes can also be modeled. Successful simulation results show that this extension can be used as a building block to model more challenging problems which are discussed below. Finally, to provide a solution to model non-stationary mixtures of cross-correlated processes, our methodology is expanded to its most unifying form. This modeling scheme can also be interpreted as a Dependent Component Analysis where both the mixing matrix and the latent processes (sources) are modeled by only observing their mixtures. Here, we propose two novel techniques. First method is used to model non-stationary mixtures of cross-correlated processes which do not possess time structures, while the second one is utilized for modeling non-stationary mixtures of cross-correlated autoregressive processes. Successful simulation results verify that our particle filtering methodology is very flexible and provides a unifying solution for the modeling of non-stationary processes in all cases described above.