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
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  and its extension to a more general family of models, denoted by
is discussed in  and . This model is an extension to the FARIMA process.
 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
, 2014 P09023. stacks.iop.org/1742-5468/2014/i=9/a=P09023
 A. I. McLeod, M. M. Meerschaert and F. Sabzikar, "Tempered Fractional Time Series," working paper, 2016.