Tempered Fractionally Differenced White Noise

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Tempered fractionally differenced (TFD) white noise () can be defined using the backshift operator in the relation , where is Gaussian white noise, , and is the series length. In this Demonstration, we consider the ranges of values: , , and . This Demonstration explores the dependence on and .


The time series plot is shown on an arbitrary scale to emphasize the general shape, independent of scale and location. The initial parameter settings, and , provide an appropriate model for turbulent flow. You can generate a wide variety of random patterns using the "random seed" slider. You can display the associated spectrum and autocorrelation using the "plot" buttons. See Details.


Contributed by: Ian McLeod, Mark Meerschaert and Farzad Sabzikar (September 2016)
(Western University, Michigan State University, Iowa State University)
Open content licensed under CC BY-NC-SA



The selected range for the tempering parameter , , and the differencing parameter , , were chosen to exhibit a wide variety of time series behavior. When is large or is close to zero, they are almost independent. When is not too large and is not too small, there is a moderately strong dependence that can serve to model turbulent flow.

Snapshot 1: The sample and theoretical autocorrelations reveal the true structure and its estimate from the data. When the true dependence is strong, as in the case where is small and is not small, the sample estimates are not accurate and have large biases even in quite large samples. Due to this strong dependence, many spurious patterns in the sample autocorrelations may be generated. For example, setting "random seed" to 185 with and generates a spurious apparently periodic autocorrelation plot.

Snapshot 2: Log spectrum is plotted with the sample periodogram. The sample periodogram points scatter about the underlying theoretical value more randomly, illustrating that there is less bias than with the sample autocorrelation.

Snapshot 3: Log spectrum versus log frequency is plotted. The lowest frequency is chosen to illustrate that the TFD process has a bounded spectrum near the origin but obeys a power-law decay for intermediate frequencies.

The TFD model was suggested for turbulent flow in [1] and its extension to a more general family of models, denoted by is discussed in [1] and [2]. This model is an extension to the FARIMA process.


[1] M. M. Meerschaert, F. Sabzikar, M. S. Phanikumar and A. Zeleke, "Tempered Fractional Time Series Model for Turbulence in Geophysical Flows," Journal of Statistical Mechanics: Theory and Experiment, 9, 2014 P09023. stacks.iop.org/1742-5468/2014/i=9/a=P09023.

[2] A. I. McLeod, M. M. Meerschaert and F. Sabzikar, "Tempered Fractional Time Series," working paper, 2016.

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