Statistical analysis of DWT coefficients of fGn processes using ARFIMA(p,d,q) models

Abstract

Fractional Gaussian noise (fGn) provides an important parametric representation for the data recorded from long-memory processes. Also it has been well established in literature that the orthogonal wavelet transforms prove to be the optimal bases to represent the data as fGn or fBm (fractional Brownian motion) models. This paper highlights the statistical properties of discrete wavelet transform (DWT) coefficients in the wavelet expansion of fGn. Statistical analysis was carried out by analyzing the inter-scale and intra-scale correlations of the DWT coefficients for wavelets with varying vanishing moments. Two types of auto-regressive moving average (ARMA) models were fit to the wavelet coefficients of fGn, namely, (i) ARMA(p,q) and (ii) ARFIMA(p,d,q) models. The latter represents the ARMA models with fractional differencing. Using the Akaike information criteria (AIC) and the Bayesian information criteria (BIC), it has been shown that ARFIMA models best represent the wavelet coefficients of fGn. The above observation holds good, when wavelets with increasing number of vanishing moments are used for obtaining DWT coefficients. After estimating the optimal model and its parameters, different properties pertaining to the inter-scale and intra-scale correlations were verified using these models.