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
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_spectra1. 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, analyze
# 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=0.1)
x_snp = load_data(name='snp') # S&P500 daily prices from 2000 to 2024
# ANALYSIS
scat_brownian = analyze(x_brownian)
scat_mrw = analyze(x_mrw)
scat_snp = analyze(np.log10(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
x = load_data(name='mrw', R=1, T=4096, H=0.5, lam=0.3)
# GENERATION
x_gen = generate(x, cuda=True, tol_optim=1e-2)
# VISUALIZATION
fig, axes = plt.subplots(2, 1, figsize=(10,5))
axes[0].plot(np.diff(x)[0,0,:], color='lightskyblue', linewidth=0.5)
axes[1].plot(np.diff(x_gen)[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