pyscarcopula 0.17.1

Python library for bivariate, multivariate, vine, and stochastic copula models

pyscarcopula

A Python library for copula modelling: bivariate, multivariate, vine, and stochastic copula models for financial time series and risk analytics.

• About
• Install
• Features
• Mathematical background
• Examples and docs
• License

About

pyscarcopula fits bivariate and multivariate dependence models using copulas in Python. Alongside classical constant-parameter copulas, it supports stochastic copula autoregressive (SCAR) models where the copula parameter is driven by a latent Ornstein-Uhlenbeck process or Kendall's tau follows a bounded Jacobi diffusion.

The package is aimed at financial time series, risk modelling, and experiments with dynamic dependence. It provides bivariate copulas, C-vines, R-vines, conditional sampling, prediction, goodness-of-fit diagnostics, and risk metrics.

Supported estimation methods:

Method	Key	Description
Maximum likelihood	mle	Constant copula parameter
SCAR transfer matrix	scar-tm-ou	Deterministic OU latent-state likelihood
SCAR Jacobi transfer matrix	scar-tm-jacobi	Deterministic Kendall-tau diffusion likelihood
SCAR Monte Carlo	scar-p-ou, scar-m-ou	Monte Carlo alternatives
GAS	gas	Observation-driven score model

Install

pip install pyscarcopula

The package includes a required pybind11 C++ extension. It provides built-in copula kernels, static likelihoods, GAS, and SCAR-TM-OU numerical evaluation. Official wheels bundle the extension and do not need a local compiler. Source and editable installs require a C++17 compiler:

• Windows: Microsoft C++ Build Tools / Visual Studio Build Tools
• Linux: GCC or Clang with the usual Python development headers
• macOS: Xcode Command Line Tools

For local development:

git clone https://github.com/AANovokhatskiy/pyscarcopula
cd pyscarcopula
pip install -e ".[test]"

To run the full test suite from the source tree, build the C++ extension in place first:

python setup.py build_ext --inplace
pytest --run-validation

pytest --run-validation enables optional validation tests. A source checkout without a successfully built extension is incomplete for the default bivariate GAS workflow.

Optional benchmark checks are disabled by default. Enable them explicitly:

PYSCA_RUN_BENCHMARKS=1 pytest -m benchmark

Core dependencies: numpy, numba, scipy, joblib, tqdm.

Verify a native installation with:

python -m pyscarcopula._native_smoke

Features

Copula families

• Archimedean: Gumbel, Frank, Clayton, Joe, including rotations where supported
• Elliptical: Gaussian and Student-t
• Independence copula for null models and vine pruning
• Multivariate Gaussian, Student-t, equicorrelation, and stochastic Student models
• Explicit CopulaBase / BivariateCopula / MultivariateCopula hierarchy with capability-based strategy validation

Vine copulas

• C-vine pair-copula construction with fixed star structure
• R-vine pair-copula construction with Dissmann-style structure selection
• Automatic family and rotation selection per edge using AIC/BIC
• Tree-level and edge-level truncation
• Mixed MLE, SCAR, GAS, and independence edges within one vine

Sampling and prediction

• Unconditional sampling from fitted bivariate and vine models
• Conditional sampling for R-vines, including exact suffix/rebuild paths and runtime-DAG plus MCMC fallback for arbitrary conditioning sets
• PredictConfig for explicit prediction options
• Reproducible random generation via rng
• JSON persistence through model.save() and ModelClass.load()

Diagnostics and risk

• Rosenblatt-transform based goodness-of-fit tests
• Mixture Rosenblatt transform for stochastic models
• Predictive time-varying copula parameter paths
• VaR and CVaR utilities in pyscarcopula.contrib

Mathematical background

By Sklar's theorem, a joint distribution can be represented as

F(x_1, \ldots, x_d) = C(F_1(x_1), \ldots, F_d(x_d)),

where C is a copula and F_i are marginal distributions. This separates marginal modelling from dependence modelling.

For a one-parameter Archimedean copula with generator phi,

C(u_1, \ldots, u_d; \theta)
  = \phi^{-1}(\phi(u_1; \theta) + \cdots + \phi(u_d; \theta)).

In SCAR models the copula parameter is time-varying:

\theta_t = \Psi(x_t),
\qquad
dx_t = \kappa(\mu - x_t)dt + \nu dW_t,

where x_t is a latent Ornstein-Uhlenbeck process and Psi maps the latent state to the valid parameter domain. scar-tm-jacobi instead evolves Kendall's tau directly with a bounded Jacobi diffusion and maps tau back to the copula parameter for families that implement tau_to_param.

The transfer matrix method evaluates the latent-state likelihood by exploiting the Markov structure of the latent process. The path integral is computed as a sequence of matrix-vector products on a discretized grid or spectral basis, avoiding Monte Carlo variance at the cost of numerical approximation.

For SCAR-TM-OU, transition_method='auto' uses a Hermite spectral likelihood except in narrow-kernel OU regimes, where it uses local Gauss-Hermite. In standardized stationary coordinates, the OU transition is diagonal in the probabilists-Hermite basis with eigenvalues rho**n, rho = exp(-kappa * dt). The observation densities are projected back to this truncated basis by Gauss-Hermite quadrature. This turns the latent path integral into repeated small matrix multiplications and diagonal scalings. See docs/guide/performance.md for the details and the available transition_method values, including the matrix/local numerical fallbacks used when a spectral evaluation is not accepted.

SCAR-TM-OU uses the bundled C++ extension as its only production numerical engine. No backend argument is needed.

result = fit(copula, u, method="scar-tm-ou")

GAS uses the compiled numerical evaluator for likelihood, score recursion, state updates, prediction, and Rosenblatt paths. There is no backend selector or silent Python fallback:

result = fit(copula, u, method="gas")

Use the default scaling="unit" for production. scaling="fisher" remains an experimental, numerically sensitive mode.

See docs/guide/performance.md for supported C++ families and numerical options.

Custom Python copulas remain useful with custom Python strategies, sampling, and diagnostics. Built-in native production strategies do not execute arbitrary Python copula kernels: unsupported classes fail before optimization.

Vine copulas decompose a d-dimensional dependence model into bivariate copulas arranged in a sequence of trees. R-vines choose the tree structure from data subject to the proximity condition; C-vines use a fixed star structure.

Examples and docs

Worked notebooks are available in examples/:

• 01_basic_api.ipynb
• 02_bivariate.ipynb
• 03_multivariate.ipynb
• 04_vine.ipynb
• 05_risk_metrics.ipynb
• 06_pyvinecopulib_comparison.ipynb

Additional documentation is in docs/. Method semantics are described in docs/guide/estimation-methods.md, and performance-related details are kept in docs/guide/performance.md. Migration notes for the native-core and multivariate namespace changes are in docs/release-notes/native-core-migration.md.

License

MIT License. See LICENSE.txt.