Simulate a Multifractal Time Series Using the p-Model

Generates a multiplicative binomial cascade ("p-model") time series (Meneveau & Sreenivasan 1987), optionally fractionally integrated in Fourier space to a prescribed power-spectrum slope (Davis et al. 1997) to produce a continuous, nonstationary series superposed on the multifractal cascade.

Usage

pmodel(n_values = 256L, p = 0.375, slope = NULL, seed = 42L)

Arguments

n_values

Integer. Desired length of the output series. Internally rounded up to the next power of two for the cascade construction, then truncated back to n_values. Default 256.

p

Numeric in (0, 1). The p-model parameter: values near 0 or 1 produce a very peaked (strongly multifractal) series; values near 0.5 produce a calm series (p = 0.5 gives an exactly constant unit vector – no multifractality at all). Default 0.375.

slope

Numeric or NULL. If supplied, the p-model series is fractionally integrated in Fourier space to have this power-spectrum slope, producing a continuous, nonstationary series (slopes between -1 and -3 give stationary increments; steeper than -3 is differentiable) – see Davis et al. (1997) for the regimes. If NULL (default), no fractional integration is applied.

seed

Integer or NULL. Seeds the random sign draws at every cascade level. The original MATLAB implementation reseeds to a fixed seed (42) at every level unconditionally, making the cascade fully deterministic by design (each level's random signs are literally a prefix of the next level's, since the RNG stream restarts from the same point every time). This port preserves that same reseed-every-level structure, with seed (default 42, matching the original) controlling it. Pass seed = NULL to draw fresh randomness at every level instead (no reseeding) – unlike the original, which has no way to disable this.

Value

If slope is NULL: a numeric vector of length n_values, the p-model series itself. If slope is supplied: a list with

x

The fractionally integrated series.

y

The underlying p-model series before integration.

Details

Clean-room R reimplementation of pmodel.m, written by the late Victor Venema (no license header present in the source file; see inst/COPYRIGHTS). The underlying random sign draws use R's own RNG rather than attempting to replicate MATLAB's specific generator bit-for-bit (not feasible across languages, and not meaningful for a synthetic-data generator characterised by its cascade rule rather than particular RNG bytes) – see the seed argument below for how the original's reseed-every-level behaviour is preserved. Validated against two RNG-independent structural properties that hold for any random sign sequence: at p = 0.5 the cascade is provably an exact constant vector of 1s, and total mass is conserved exactly at every cascade level (sum(y) doubles per level) for any p – see tests/testthat/test-pmodel.R.

References

Meneveau C, Sreenivasan KR. Simple multifractal cascade model for fully developed turbulence. Phys Rev Lett 1987;59:1424-1427.

Davis A, Marshak A, Cahalan R, Wiscombe W. The Landsat scale break in stratocumulus as a three-dimensional radiative transfer effect: implications for cloud remote sensing. J Atmos Sci 1997;54(2):241-260.

Venema V, Bachner S, Rust HW, Simmer C. Statistical characteristics of surrogate data based on geophysical measurements. Nonlin Processes Geophys 2006;13:449-466. (Requested citation for this code by its original author, Victor Venema.)

Examples

y <- pmodel(1024, p = 0.3)
range(y)
#> [1]  0.006046618 28.925465498

# p = 0.5 always gives a constant series, regardless of seed:
all(pmodel(64, p = 0.5) == 1)
#> [1] TRUE