"fat-tailed" pseudo-random number generation algorithm?

Started by Ron N. April 25, 2007
In previous threads, it was mentioned that noise with
a Gaussian distribution could be quickly approximated
by summing 12 uniform RNG's.

Is there an low overhead algorithm to generate samples
whose distribution approximates a fat-tailed, extreme
kurtosis, Pareto, or power-law decay distributions?

This would seem to be useful when simulating or
testing filters against noise sources thought to be
more turbulent or chaotic than uniform or Gaussian.


Thanks,

rhn A.T nicholson d.0.t C-o-M

Ron N wrote:
> In previous threads, it was mentioned that noise with > a Gaussian distribution could be quickly approximated > by summing 12 uniform RNG's. > > Is there an low overhead algorithm to generate samples > whose distribution approximates a fat-tailed, extreme > kurtosis, Pareto, or power-law decay distributions?
The extremely heavy-tailed Cauchy distribution can be generated by using the ratio of two zero-mean Gaussian rv's. Other ratios can be used as well, generally resulting in heavy-tailed distributions: http://en.wikipedia.org/wiki/Ratio_distribution
> > This would seem to be useful when simulating or > testing filters against noise sources thought to be > more turbulent or chaotic than uniform or Gaussian.
Turbulence and chaos are not modeled well using iid processes, regardless of the distribution. The characteristic behaviour of certain chaotic "time" series (stocks, sunspots, climatological phenomena, DNA patterns, music, serial chemical and physical measurements, humain gait patterns, you name it) is modeled using long- range correlations. Long-range correlated stochastic processes have auto-correlation functions which are not summable (resulting in a pole at DC in the PSD). The long ago past influences the far future. "long range correlation" seems to be a good starting point at Google. Regards, Andor
Ron N. wrote:
> In previous threads, it was mentioned that noise with > a Gaussian distribution could be quickly approximated > by summing 12 uniform RNG's. > > Is there an low overhead algorithm to generate samples > whose distribution approximates a fat-tailed, extreme > kurtosis, Pareto, or power-law decay distributions? > > This would seem to be useful when simulating or > testing filters against noise sources thought to be > more turbulent or chaotic than uniform or Gaussian. > > > Thanks, > > rhn A.T nicholson d.0.t C-o-M >
Look for the Matlab/Octave function called 'starnd'. John
Ron N. wrote:
> In previous threads, it was mentioned that noise with > a Gaussian distribution could be quickly approximated > by summing 12 uniform RNG's. > > Is there an low overhead algorithm to generate samples > whose distribution approximates a fat-tailed, extreme > kurtosis, Pareto, or power-law decay distributions? > > This would seem to be useful when simulating or > testing filters against noise sources thought to be > more turbulent or chaotic than uniform or Gaussian. > > > Thanks, > > rhn A.T nicholson d.0.t C-o-M >
Look for the Matlab/Octave function called 'starnd'. John
Ron N wrote:
> In previous threads, it was mentioned that noise with > a Gaussian distribution could be quickly approximated > by summing 12 uniform RNG's. > > Is there an low overhead algorithm to generate samples > whose distribution approximates a fat-tailed, extreme > kurtosis, Pareto, or power-law decay distributions?
The extremely heavy-tailed Cauchy distribution can be generated by using the ratio of two zero-mean Gaussian rv's. Other ratios can be used as well, generally resulting in heavy-tailed distributions: http://en.wikipedia.org/wiki/Ratio_distribution
> > This would seem to be useful when simulating or > testing filters against noise sources thought to be > more turbulent or chaotic than uniform or Gaussian.
Turbulence and chaos are not modeled well using iid processes, regardless of the distribution. The characteristic behaviour of certain chaotic "time" series (stocks, sunspots, climatological phenomena, DNA patterns, music, serial chemical and physical measurements, humain gait patterns, you name it) is modeled using long- range correlations. Long-range correlated stochastic processes have auto-correlation functions which are not summable (resulting in a pole at DC in the PSD). The long ago past influences the far future. "long range correlation" seems to be a good starting point at Google. Regards, Andor
In previous threads, it was mentioned that noise with
a Gaussian distribution could be quickly approximated
by summing 12 uniform RNG's.

Is there an low overhead algorithm to generate samples
whose distribution approximates a fat-tailed, extreme
kurtosis, Pareto, or power-law decay distributions?

This would seem to be useful when simulating or
testing filters against noise sources thought to be
more turbulent or chaotic than uniform or Gaussian.


Thanks,

rhn A.T nicholson d.0.t C-o-M