Fast Identification via Risk-Sensitive Methods

In this thesis we primarily consider the development and applications of risk-sensitive identification schemes for fast convergence of parameter estimates. Initially we consider the risk-sensitive identification problem for hidden Markov models (HMMs). Finite dimensional filters are derived, for the risk-sensitive identification of HMMs, by combining the techniques of maximum likelihood (ML) identification and risk-sensitive state estimation. Simulations show that the risk-sensitive filters may offer the possibility of faster convergence of the HMM parameter estimates (hence the phrase fast identification) under certain “rule of thumb” conditions. In addition, we consider the application of the risk-sensitive filters to the blind adaptive equalization problem, for both finite impulse response (FIR) and infinite impulse response (IIR) channels. The blind adaptive equalization studies demonstrate that the application of risk-sensitive filters to this problem offer the possibility of improved performance over a number of existing HMM based schemes, in line with the “rule of thumb” conditions established in earlier studies. The successful derivation and application of the risk-sensitive filters for the identification of HMMs inspired the development of risk-sensitive filters for the identification of partially observed discrete-time linear systems. The derivation of the risk-sensitive filters is a combination of the techniques of ML identification and risk-sensitive state estimation. Simulation studies show that under certain conditions risk-sensitive filtering offer the possibility of improved rates of convergence of the parameter estimates for partially observed discrete-time linear systems. Additional work considered the application of risk-sensitive filters to the identification of auto-regressive, moving average, exogenous (ARMAX) processes. The aim was to show that risk-sensitive filtering could deliver improved convergence of the estimates of the exogenous, or colored noise, parameters. However, the opposite was true, and we show that the risk-sensitive filters fail to identify the exogenous parameters. The failure of the risk-sensitive identification scheme is due to the identification being poorly posed, so that necessary and sufficient conditions of a particular theorem are not met. We then present an identification scheme which takes an instrumental variable (IV) approach to the identification of HMMs. This approach is suggested as a method for identification if the state of the HMM is correlated with the process noise. Simulation studies are presented which demonstrate, at least under the conditions of the studies, that the IV filters can deliver improved parameter estimates when the system noise is colored. Finally, this thesis considers the proof of an existence theorem and the development of practical algorithms for an antenna array problem.