Wavelet Covariance Matrix Structure and Bayesian-Wavelet Estimation of Autoregressive Process Parameters with Long-Term Memory

The data obtained from observing a phenomenon over time is very common. One of the most popular models in time series and signal processing is the Autoregressive moving average model (ARMA). If the investigated time series has long memory, autoregressive fractional moving average model or in other words (ARFIMA), would be appropriate. The ARFIMA (p, d, q) model was first introduced by [1]. Classic methods for modeling, inference and estimating these processes lead to complex calculations of covariance structure and likelihood functions that make data processing difficult.Wavelet transform is one of the most powerful tools in analyzing such functions and best performs these functions from different time and location  perspectives as well as high and low frequencies. Wavelet transform due to the decreasing correlation property is a very efficient method in analysis and inference for estimating long-term memory processes.The values ​​obtained from wavelet transforms for long-term memory processes, in spite of the complex covariance structure of these processes, the wavelet coefficients are almost uncorrelated and thus much easier to handle [2]. The dense covariance structure of such processes makes it difficult to accurately calculate the maximum likelihood function of data sets [3]. In these cases, the Bayesian method can be easily used to calculate wavelet coefficients. In this paper, while briefly introducing wavelet transform in section two, the ARFIMA model and covariance matrix structure of this model is investigated in section three.  In section four, theBayesian estimation of ARFIMA parameters based on wavelets are calculated. At the end section, ‎we survey the theoretical outcomes with numerical computation by using simulation to described purpose estimation.

In this scheme, first we explain wavelet transformation, ARFIMA model and covariance matrix structure of this model. By using wavelet decomposition, Bayesian estimation of ARFIMA model parameters are calculated. The performance of purpose estimation is assessed with simulated data for comparing with respect to another estimators.

We discuss in detail wavelet transformation and autoregressive fractional  moving average model with long memory. The structure of covariance matrices of wavelet coefficients and Bayesian wavelet estimation of parameters are investigated.
At the end we used simulation study to examine our proposed estimation. Notice that, obtained results confirm that proposed estimation is better than another.

The following conclusions were drawn from this research.Wavelet transform due to the decreasing correlation property is very efficient method in estimating long-term memory processes.  The main purpose of this paper is to provide a new wavelet estimation of ARFIMA model parameters via Bayesian method.The main characteristic of this method is that it can be easily used and therefore many calculations are reduced.The proposed method can be applied for estimating of parameters in the simulation.According to the figures, we conclude   that Bayesian wavelet estimation of autoregressive process parameters is appropriate and better than with respect to other estimations.According to the table, by increasing the sample size, standard error of proposed estimator is decreased, so it was shown that the new proposed method is better with respect to others.