Online estimation algorithm for the unknown probability density functions of random parameters in auto-regression and exogenous stochastic systems

The authors present a new method to estimate the unknown probability density functions (PDFs) of random parameters for non-Gaussian dynamic stochastic systems. The system is represented by an auto-regression and exogenous model, where the parameters and the system noise term are random processes that are characterised by their unknown PDFs. Under the assumption that each random parameter and the noise term are independent and are an identically distributed sequence, a simple mathematical relationship is established between the measured output PDF of the system and the unknown PDFs of the random parameters and noise term. The moment generating function in probability theory has been used to transfer the multiple convolution integration into a simple algebraic operation. An identification algorithm is then established that estimates these unknown PDFs of the parameters and the noise term by using the measured output PDFs and the system input. A simulated example is given to show the effectiveness of the proposed method.