Copulas
=======

The copulas module provides copula functions for modeling dependencies between stochastic variables.

.. automodule:: pal.copulas
   :members:
   :undoc-members:
   :show-inheritance:

Available Copulas
-----------------

The following copulas are available for modeling dependencies:

- **GaussianCopula**: Gaussian (Normal) copula for modeling symmetric dependencies
- **StudentsTCopula**: Student's t copula for modeling symmetric dependencies with tail dependence
- **GumbelCopula**: Gumbel copula for modeling upper tail dependence
- **ClaytonCopula**: Clayton copula for modeling lower tail dependence
- **FrankCopula**: Frank copula for symmetric dependence
- **JoeCopula**: Joe copula for modeling upper tail dependence
- **PlackettCopula**: Plackett copula for modeling symmetric dependencies
- **GalambosCopula**: Galambos copula for modeling upper tail dependence
- **HuslerReissCopula**: Hüsler-Reiss copula for flexible upper tail dependence structures
- **MM1Copula**: Mixture of max-id copulas for flexible upper tail dependence 
- **ExtremalTCopula**: Extremal-t copula, also known as the t-EV copula, for flexible upper tail dependence

Usage Example
-------------

.. code-block:: python

   from pal import distributions, copulas

   # Create independent variables
   var1 = distributions.Gamma(alpha=2.5, theta=2).generate()
   var2 = distributions.LogNormal(mu=1, sigma=0.5).generate()

   # Apply Gumbel copula to create dependency
   copulas.GumbelCopula(theta=1.2).apply([var1, var2])