Abstract
The stochastic analysis in the scale domain (instead of the traditional lag or frequency domains) is introduced as a robust means to identify, model and simulate the Hurst–Kolmogorov (HK) dynamics, ranging from small (fractal) to large scales exhibiting the clustering behavior (else known as the Hurst phenomenon or long-range dependence). The HK clustering is an attribute of a multidimensional (1D, 2D, etc.) spatio-temporal stationary stochastic process with an arbitrary marginal distribution function, and a fractal behavior on small spatio-temporal scales of the dependence structure and a power-type on large scales, yielding a high probability of low- or high-magnitude events to group together in space and time. This behavior is preferably analyzed through the second-order statistics, and in the scale domain, by the stochastic metric of the climacogram, i.e., the variance of the averaged spatio-temporal process vs. spatio-temporal scale.
Highlights
The main themes of the encyclopedia entry are the Hurst–Kolmogorov (HK) clustering behavior and its stochastic analysis in the scale domain.encyclopedia10400771.1
The HK clustering is an attribute of a multidimensional (1D, 2D, etc.) spatio-temporal stationary stochastic process with an arbitrary marginal distribution function, and a fractal behavior on small spatio-temporal scales of the dependence structure and a power-type on large scales, yielding a high probability of low- or high-magnitude events to group together in space and time
− c(0), and power spectrum, where it canthat be the seenvariogram that the variogram and the the climacogram exhibit theexhibit smallest at the long-range scales and, in particuclimacogram theuncertainty smallest uncertainty at the long-range scales and, in particular, on lar, on the double logarithmic slope slope that isthat attributed to the strength of the long-range the double logarithmic is attributed to the strength of the long-range dependence dependence
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Hurst discovered a tendency of highdischarge years to cluster into high-flow periods, and low-discharge years to cluster into low-flow periods This behavior, known as the Hurst phenomenon or Joseph effect [2], is characterized by long-term persistence (LTP; called long-range dependence), which leads to high unpredictability on large scales due to the clustering of events as compared to the purely random process (i.e., white noise) or other short-range dependence models (e.g., Markovian). It is worth noting that the stochastic simulation of the HK dynamics is still a mathematical challenge [9] since it requires the explicit preservation of high-order moments in a vast range of scales [10,11], affecting both the intermittent (fractal) behavior in small scales [12] and the dependence in extremes [13]. Climate, and aspresent compared to annual aannual roulette timeseries resembling a white noise process
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