Scattering Spectra

The Scattering Spectra [1] are a set of statistics that provide an interpretable low-dimensional representation of multi-scale time-series.

They can be used for time-series analysis and generation, but also to assess self-similarity in the data as well as to detect non-Gaussianity.

For a quick start, see tutorial.ipynb

Generation of S&P log-returns in a Scattering Spectra model.

Installation

From a venv with python>=3.10 run the commands below to install the required packages.

pip install git+https://github.com/RudyMorel/scattering_spectra

1. Analysis

The Scattering Spectra provide a dashboard to analyze time-series.

Standard model of time series can be loaded using load_data from frontend.py. The function analyze computes the Scattering Spectra, they can be visualized using the function plot_dashboard.

from scatspectra import load_data, SPDaily, analyze, plot_dashboard

# DATA
x_brownian = load_data(name='fbm', R=256, T=6063, H=0.5)
x_mrw = load_data(name='mrw', R=256, T=6063, H=0.5, lam=0.2)
x_snp = SPDaily().lnx  # S&P500 daily prices from 2000 to 2024

# ANALYSIS
scat_brownian = analyze(x_brownian)
scat_mrw = analyze(x_mrw)
scat_snp = analyze(x_snp)

# VISUALIZATION
plot_dashboard([scat_brownian, scat_mrw, scat_snp], labels=['Brownian', 'MRW', 'S&P']);

The dashboard consists of 4 spectra that can be interpreted as follows:

• $\Phi_1(x)[j]$ are the sparsity factors. The lower these coefficients the sparser the signal $x$.
• $\Phi_2(x)[j]$ is the standard wavelet spectrum. A steep curve indicates long-range dependencies.
• $\Phi_3(x)[a]$ is the phase-modulus cross-spectrum.
  • $|\Phi_3(x)[a]|$ is zero if the underlying process is invariant to sign change $x \overset{d}{=}-x$.
    These coefficients quantify non-invariance to sign-change often called "skewness".
  • Arg $\Phi_3(x)[a]$ is zero if the underlying process $x$ is time-symmetric: $x(t) \overset{d}{=}x(-t)$. These coefficients quantify time-asymmetry. They are typically non-zero for processes with time-causality.
• $\Phi_4(x)[a,b]$ is the modulus cross-spectrum.
  • $|\Phi_4(x)[a,b]|$ quantify envelope dependencies. Steep curves indicate long-range dependencies.
  • Arg $\Phi_4(x)[a,b]$ quantify envelope time-asymmetry. These coefficients are typically non-zero for processes with time-causality.

For further interpretation see [1].

Self-Similarity

Self-Similar	Not Self-Similar
Self-similarity distance: 2.1	Self-similarity distance: 22.0

Processes encountered in Finance, Seismology, Physics may exhibit a form of regularity called self-similarity. Also refered as scale invariance, it states that the process is similar at different scales. Assessing such property on a single realization is a hard problem.

Similarly to time-stationarity that has a wide-sense definition that can be tested statistically on the covariance of a process $x(t)$, we introduced in [1] a wide-sense definition of self-similarity called wide-sense Self-Similarity. It can be tested on a covariance matrix across times $t,t'$ and scales $j,j'$.

The function self_simi_obstruction_score from frontend.py assesses self-similarity on a time-series.

2. Generation

A model of the process $x$ can be defined from the Scattering Spectra. Such model can be sampled using gradient descent [1].

Function generate from frontend.py takes observed data $x$ as input and return realizations of our model of $x$.

from scatspectra import generate

# DATA
snp_data = SPDaily()

# GENERATION
gen_data = generate(snp_data, cuda=CUDA, tol_optim=6e-3)

# VISUALIZATION
fig, axes = plt.subplots(2, 1, figsize=(10, 4))
axes[0].plot(snp_data.dlnx[0, 0, :], color="lightskyblue", linewidth=0.5)
axes[1].plot(gen_data.dlnx[0, 0, :], color="coral", linewidth=0.5)
axes[0].set_ylim(-0.1, 0.1)
axes[1].set_ylim(-0.1, 0.1)

See tutorial.ipynb and testing.ipynb for more code examples.

[1] "Scale Dependencies and Self-Similar Models with Wavelet Scattering Spectra"

Rudy Morel et al. - https://arxiv.org/abs/2204.10177