scaleinvariance 0.9.1

Simulation and analysis of multifractal fields

scaleinvariance

Simulation and analysis tools for scale-invariant processes and multifractal fields.

Documentation

View example FIF simulation output in the Multifractal Explorer

Current Features

Analysis

Hurst exponent estimation

• Haar fluctuation method: haar_fluctuation_hurst()
• Structure function method: structure_function_hurst()
• Spectral method: spectral_hurst()

All methods support multi-dimensional arrays, averaging over dimensions that are orthogonal to the specified dimension along which spectra are calculated (specified by axis. Data and fit line for plotting may be returned with return_fit=True.

Simulation

• 1D fractionally integrated flux (FIF): FIF_1D() - Multifractal cascade simulation; causal/acausal
• N-D fractionally integrated flux (FIF): FIF_ND() - Isotropic N-D multifractals for arbitrary dimensions (Example shown above)
• 1D fractional Brownian motion: fBm_1D_circulant() - Fast spectral synthesis
• N-D fractional Brownian motion: fBm_ND_circulant() - Isotropic N-D (2D, 3D, 4D, etc.) fBm fields
• 1D fBm (fractional integration): fBm_1D() - Extended Hurst range (-0.5, 1.5) with causal/acausal kernels

For FIF simulation methods (FIF_1D, FIF_ND), the currently supported range is 0.5 <= alpha <= 2 with alpha != 1. The kernel construction method 'LS2010' (Lovejoy & Schertzer 2010) is used for flux kernels. For observable kernels, both 'LS2010' and 'spectral' are supported.

Installation

pip install scaleinvariance

Agent Skill (Highly Recommended for Agents)

An agent skill is included in this repository. For Claude Code:

mkdir -p ~/.claude/skills/scaleinvariance
cp agent-skills/scaleinvariance/SKILL.md ~/.claude/skills/scaleinvariance/

Codex:

mkdir -p ~/.codex/skills/scaleinvariance
cp agent-skills/scaleinvariance/SKILL.md ~/.codex/skills/scaleinvariance/

Performance

By default, scaleinvariance uses NumPy for all computations. However, if PyTorch is installed, the package automatically detects it and uses PyTorch for simulation functions, providing potentially huge performance gains (depending on your machine):

# NumPy-only installation (minimal dependencies)
pip install scaleinvariance

# With PyTorch for faster simulations
pip install scaleinvariance[torch]

Control backend explicitly:

import scaleinvariance

# Check current backend
print(scaleinvariance.get_backend())  # 'torch' if available, else 'numpy'

# Force specific backend
scaleinvariance.set_backend('numpy')  # Always use NumPy
scaleinvariance.set_backend('torch')  # Use torch (raises error if not installed)

# Configure threading (defaults to 90% CPU count)
scaleinvariance.set_num_threads(8)

Basic Usage

from scaleinvariance import fBm_1D_circulant, fBm_ND_circulant, FIF_1D, haar_fluctuation_hurst

# Generate 1D fractional Brownian motion
fBm_1d = fBm_1D_circulant(1024, H=0.7)

# Generate 2D fractional Brownian motion
fBm_2d = fBm_ND_circulant((512, 512), H=0.7)

# Generate 3D fractional Brownian motion
fBm_3d = fBm_ND_circulant((256, 256, 128), H=0.7)

# Generate multifractal FIF timeseries
fif = FIF_1D(2**16, alpha=1.8, C1=0.1, H=0.3)

# Override FIF kernels explicitly if needed
fif_custom = FIF_1D(
    2**16,
    alpha=1.8,
    C1=0.1,
    H=0.3,
    causal=False,
    kernel_construction_method_flux='LS2010',
    kernel_construction_method_observable='spectral',
)

# Estimate Hurst exponent
H_est, H_err = haar_fluctuation_hurst(fBm_1d)
print(f"Estimated H = {H_est:.3f} ± {H_err:.3f}")

Kernel Selection

For FIF simulations:

• kernel_construction_method_flux controls the cascade kernel.
• kernel_construction_method_observable controls the final observable kernel.
• Flux kernels support 'LS2010' (and deprecated 'naive' for 1D only).
• Observable kernels support 'spectral' (default) and 'LS2010'. FIF_1D also supports 'spectral_odd' and deprecated 'naive'.
• The old kernel_construction_method= argument is no longer accepted by FIF_1D or FIF_ND.
• naive FIF kernels remain available only for comparison and debugging and emit a warning because they are not remotely accurate.

For fBm:

• fBm_ND_circulant() is recommended. It uses a spectral synthesis method and produces essentially perfectly scale invariant fields.
• fBm_1D() is included for comparsion or for causal simulations.

Examples

Work in progress. See Multifractal Explorer for visuals.

Run examples:

python examples/fif_comparison_demo.py

Data source for LGMR: https://www.ncei.noaa.gov/access/paleo-search/study/33112

Testing

Tests are organised into three directories under tests/:

Directory	Purpose	How to run
tests/functional/	Sanity checks and gross-accuracy tests. Should always pass.	pytest
tests/numerical/	Theory-validation scripts run directly as Python programs. Some produce failures due to finite-size/discretization effects.	python tests/numerical/<script>.py
tests/visual/	Scripts that generate plots for manual inspection.	python tests/visual/<script>.py

FIF validation scripts, which test scaling over multiple ranges of scale, live in tests/numerical/. They are designed to be run as standalone Python programs, not via pytest, and they generate many large FIF realizations to reach statistical convergence. These scripts are also known to produce some failures, especially near grid scales, because finite-size effects are not fully mitigated by the LS2010 corrections or their spectral alternatives.

Run the functional suite:

pytest

Run a specific visual test:

python tests/visual/test_1D_fif_structure_function.py 0.3 0.1 1.8

Requirements

• Python ≥ 3.8
• NumPy, SciPy
• PyTorch (optional, for simulation speedup)