Abstract: Bayesian approach can be used for parameter identification and extraction in state space models and its ability for analyzing sequence of data in dynamical system is proved in different literatures. In this paper, adaptive Kalman filter with Bayesian approach for identification of variances in measurement parameter noise is developed. Next, it is applied for estimation of the dynamical state and measurement data in discrete linear dynamical system. This algorithm at each step time estimates noise variance in measurement noise and state of system with Kalman filter. Next, approximation is designed at each step separately and consequently sufficient statistics of the state and noise variances are computed with a fixed-point iteration of an adaptive Kalman filter. Different simulations are applied for showing the influence of noise variance in measurement data on algorithm. Firstly, the effect of noise variance and its distribution on detection and identification performance is simulated in Kalman filter without Bayesian formulation. Then, simulation is applied to adaptive Kalman filter with the ability of noise variance tracking in measurement data. In these simulations, the influence of noise distribution of measurement data in each step is estimated, and true variance of data is obtained by algorithm and is compared in different scenarios. Afterwards, one typical modeling of nonlinear state space model with inducing noise measurement is simulated by this approach. Finally, the performance and the important limitations of this algorithm in these simulations are explained.
Abstract: This paper presents a prediction performance of
feedforward Multilayer Perceptron (MLP) and Echo State Networks
(ESN) trained with extended Kalman filter. Feedforward neural
networks and ESN are powerful neural networks which can track and
predict nonlinear signals. However, their tracking performance
depends on the specific signals or data sets, having the risk of
instability accompanied by large error. In this study we explore this
process by applying different network size and leaking rate for
prediction of nonlinear or chaotic signals in MLP neural networks.
Major problems of ESN training such as the problem of initialization
of the network and improvement in the prediction performance are
tackled. The influence of coefficient of activation function in the
hidden layer and other key parameters are investigated by simulation
results. Extended Kalman filter is employed in order to improve the
sequential and regulation learning rate of the feedforward neural
networks. This training approach has vital features in the training of
the network when signals have chaotic or non-stationary sequential
pattern. Minimization of the variance in each step of the computation
and hence smoothing of tracking were obtained by examining the
results, indicating satisfactory tracking characteristics for certain
conditions. In addition, simulation results confirmed satisfactory
performance of both of the two neural networks with modified
parameterization in tracking of the nonlinear signals.