Can We Characterize Variabilities of Variabilities in Real Time?

Variabilities are inherent in complex natural and man-made systems including complex big data in industrial processes. Characterization of variabilities becomes a foundation of industrial AI(IAI). Traditional methods to measure variability such as variance, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Z$</tex>-score etc. do not work for complex data series such as random processes with heavy-tailedness in distribution or long-range dependence (LRD). Both types are of fractal nature with a self-similarity (SS) property that follows an inverse power law (IPL) trend. The Hurst parameter is a dimensionless factor used to characterize the presence and magnitude of the self-similarity property. Many existing Hurst parameter estimators are for offline or batch processing, when, in real applications, stream processing or online real-time processing is needed. A moving window method is used for online estimation of Hurst parameters in our previous paper with a detailed investigation on which estimator performs best for online processing, the effects of window sizes, and computational cost evaluation. In this work, we extend the previous study to time-varying Hurst parameters where the Hurst parameter may have its own variability. The key contributions in this paper are 1) two signal generators in Matlab that generate a random process with given desirable Hurst curve <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H$</tex><inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf>(t) with numerical verification; 2) moving window Aggregate Variance method implemented in Simulink for stream processing to estimate Ĥ (t) with various <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$H$</tex><inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf>(t) shapes such as ramp, square, sawtooth and sine waves. To demonstrate the variabilities of variabilities characterization ability of this work, a special example is given for a case when H<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf>(t) itself is a fractional Gaussian noise (fGn) with a constant Hurst parameter H<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> where we can use the estimated constant Ĥ<inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> in the obtained Ĥ(t). It is interesting to see that the possibility of characterizing the variabilities in variability is enabled using the tools presented in this work.