A wavelet-based approach to revealing the Tweedie distribution type in sparse data

Abstract

We propose two approaches to the analysis of sparse stochastic data, which exhibit a power-law dependence between their first and second moments (Taylor’s law), and determining the respective power index, when it has a value between 1 and 2. They are based on the analysis of components of the Haar wavelet expansion. The first method uses a dependence between the first iterations of averaging and wavelet coefficients with the subsequent studying the statistics of zero paddings on the line, which corresponds the linear dependence between two first moments of the analysed distributions. The second method refers to Taylor’s plot formed by the full set of wavelet coefficients. It is discussed that such representations provide also an opportunity to check time–scale stability of analysed data and distinguish between particular cases of the Tweedie probabilistic distribution. Both simulated series and real marine species abundance data for five spatial regions of the Pacific are used as example illustrating an applicability of these approaches.