P-model multifractal time series

p-model time series for various p values
P-model time series for various p values. The blue line is the normal p-model time series, the red line its fractionally integrated counter part with a power spectral slope of -2.

The p-model is used to generate time series and fields with a fractal structure. These time series can be made more smooth by fractional integration. Such time series are occasionally used to model fractal cloud time series. This page illustrates the range of time series that can be made with this model, and gives a Matlab code to generated 1D time series.
The p-model itself can only produce stationary time series, i.e time series where the variance is finite if you would extrapolate its power spectrum to infinite large scales. Optionally you can also filter the result from the p-model in Fourier space to give it another fractal slope, e.g. to make it continuous and nonstationary. This is also called fractional integration.
The parameter of the p-model is p. With p values close to 1 or 0 the time series is very peaked. With values close to 0.5 the p-model is much calmer; p=0.5 results in a constant unity vector.
The parameter for the fractal integration is the slope of the power spectrum. Davis et al. calls slopes flatter than -1 stationary, and slopes between -1 and -3 nonstationary, with stationary increments. These nonstationary cases are at least continuous, but not differentiable. Slopes steeper than -3 are nonstionary and differentiable.


Below you can find Matlab programs to generate p-model time series (and some more links). These programs are free for scientific work, and are distributed under the GNU public license. Please, inform me of any bugs you find. The programs have been kept as simple as possible; all add-ons have been removed for clarity.

More information

Last update: 25 January 2004
URL: http://www2.meteo.uni-bonn.de/victor/themes/surrogates/pmodel/