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.