Gaussian process parameter estimation using zero crossing data from wireless sensors

Abstract

The parameters in a general Gaussian process, including the param- eters in an additive Gaussian noise process, are estimated based on zero crossing data for the total process and arbitrarily filtered ver- sions thereof. A nonlinear weighted least squares estimate is con- sidered and an analysis of the asymptotic covariance matrix of the estimated parameter vector is made. The proposed estimator and the use of zero crossing data are suitable when information of a process is sent from wireless sensors to a node center for further processing due to an efficient use of available bandwidth. data is considered in (13). In (14), the poles of autoregressive mov- ing average processes are estimated based on information of higher order crossings. Time delay estimation based on zero crossing data is considered in (15). Here, estimation of the parameters in general Gaussian pro- cesses based on information of the number of zero crossings for the process itself and for filtered versions of the process. This is done by using a relation between the number of zero crossings and the correlation function of a process. The relation is considered for the process itself as well as for filtered versions thereof. As the number of zero crossings can be registered and the correlation function is a possibly nonlinear function in the unknown parameters, a nonlinear weighted least squares criterion is defined and minimized. There- after, the asymptotic covariance matrix of the estimated parameter vector is analyzed. In the analysis, the covariance matrix of the vector containing the registered number of zero crossings for the process and its filtered versions must be evaluated. Unfortunately, the evaluation of the covariance between the registered number of zero crossings for different processes is nontrivial. As an example of a Gaussian process for illustrating the material in the paper nu- merically, a diffusion process is considered. The rest of the paper is organized as follows. In Section 2, the relation to prior work is given. The parameter estimation is considered in Section 3, the asymptotic normalized covariance matrix of the estimated parame- ter vector is given in Section 4, and second order statistics of zero crossings is considered in Section 5. The numerical examples are presented in Section 6, some discussions are found in Section 7, and conclusions are given in Section 8.