Abstract
We introduce an index based on information theory to quantify the stationarity of a stochastic process. The index compares on the one hand the information contained in the increment at the time scale of the process at time t with, on the other hand, the extra information in the variable at time t that is not present at time . By varying the scale , the index can explore a full range of scales. We thus obtain a multi-scale quantity that is not restricted to the first two moments of the density distribution, nor to the covariance, but that probes the complete dependences in the process. This index indeed provides a measure of the regularity of the process at a given scale. Not only is this index able to indicate whether a realization of the process is stationary, but its evolution across scales also indicates how rough and non-stationary it is. We show how the index behaves for various synthetic processes proposed to model fluid turbulence, as well as on experimental fluid turbulence measurements.
Highlights
Many if not most real-world phenomena are intrinsically non-stationary, while most if not all classical tools such as Fourier analysis and power spectrum, correlation function, wavelet transforms, etc., are only defined for—and supposed to operate on—signals which are stationary
The weak-sense stationarity assumption most commonly used in practice requires the first two moments of the probability distribution to exist and to be time-invariant, as well as the auto-covariance function that is required to be time-translation invariant, which leads to the definition of the correlation function
We introduce an index based on information theory to quantify the stationarity of a signal
Summary
Many if not most real-world phenomena are intrinsically non-stationary, while most if not all classical tools such as Fourier analysis and power spectrum, correlation function, wavelet transforms, etc., are only defined for—and supposed to operate on—signals which are stationary. Other approaches have suggested using the roughness of the process, computed in sliding windows, to quantify the order of its non-stationarity [14] We proposed following such an approach, but generalizing it on the full range of scales, without restricting it to an appropriate time window. We introduce an index based on information theory to quantify the stationarity of a signal Is this index able to indicate whether a realization of the process is stationary at a given scale—typically the size of the realization—but its evolution across scales indicates how rough and non-stationary the process is. We use these Gaussian scale-invariant processes with long-range dependence structures as a set Entropy 2021, 23, 1609 of benchmarks where numerical estimations can be compared with analytical results.
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