Pycop
Python library for multivariate dependence modeling with Copulas
Install / Use
/learn @maximenc/PycopREADME
How to cite
If you use pycop in a scientific publication
@article{nicolas2022pycop,
title={pycop: a Python package for dependence modeling with copulas},
author={Nicolas, Maxime LD},
journal={Zenodo Software Package},
volume={70},
pages={7030034},
year={2022}
}
Overview
Pycop is the most complete tool for modeling multivariate dependence with Python. The package provides methods such as estimation, random sample generation, and graphical representation for commonly used copula functions. The package supports the use of mixture models defined as convex combinations of copulas. Other methods based on the empirical copula such as the non-parametric Tail Dependence Coefficient are given.
Some of the features covered:
- Elliptical copulas (Gaussian & Student) and common Archimedean Copulas functions
- Mixture model of multiple copula functions (up to 3 copula functions)
- Multivariate random sample generation
- Empirical copula method
- Parametric and Non-parametric Tail Dependence Coefficient (TDC)
Available copula function
<p align="center">| Copula | Bivariate <br /> Graph & Estimation | Multivariate <br /> Simulation | |--- | :-: | :-: | | Mixture | ✓ | ✓ | | Gaussian | ✓ | ✓ | | Student | ✓ | ✓ | | Clayton | ✓ | ✓ | | Rotated Clayton | ✓ | ✓ | | Gumbel | ✓ | ✓ | | Rotated Gumbel | ✓ | ✓ | | Frank | ✓ | ✓ | | Joe | ✓ | ✓ | | Rotated Joe | ✓ | ✓ | | Galambos | ✓ | ✗ | | Rotated Galambos | ✓ | ✗ | | BB1 | ✓ | ✗ | | BB2 | ✓ | ✗ | | FGM | ✓ | ✗ | | Plackett | ✓ | ✗ | | AMH | ✗ | ✓ |
</p>Usage
Install pycop using pip
pip install pycop
Examples
Table of Contents
Graphical Representation
We first create a copula object by specifying the copula familly
from pycop import archimedean
cop = archimedean(family="clayton")
Plot the cdf and pdf of the copula.
3d plot
cop = archimedean(family="gumbel")
cop.plot_cdf([2], plot_type="3d", Nsplit=100 )
cop.plot_pdf([2], plot_type="3d", Nsplit=100, cmap="cividis" )
Contour plot
plot the contour
cop = archimedean(family="plackett")
cop.plot_cdf([2], plot_type="contour", Nsplit=100 )
cop.plot_pdf([2], plot_type="contour", Nsplit=100, )
It is also possible to add specific marginals
cop = archimedean.archimedean(family="clayton")
from scipy.stats import norm
marginals = [
{
"distribution": norm, "loc" : 0, "scale" : 0.8,
},
{
"distribution": norm, "loc" : 0, "scale": 0.6,
}]
cop.plot_mpdf([2], marginals, plot_type="3d",Nsplit=100,
rstride=1, cstride=1,
antialiased=True,
cmap="cividis",
edgecolor='black',
linewidth=0.1,
zorder=1,
alpha=1)
lvls = [0.02, 0.05, 0.1, 0.2, 0.3]
cop.plot_mpdf([2], marginals, plot_type="contour", Nsplit=100, levels=lvls)
Mixture plot
mixture of 2 copulas
from pycop import mixture
cop = mixture(["clayton", "gumbel"])
cop.plot_pdf([0.2, 2, 2], plot_type="contour", Nsplit=40, levels=[0.1,0.4,0.8,1.3,1.6] )
# plot with defined marginals
cop.plot_mpdf([0.2, 2, 2], marginals, plot_type="contour", Nsplit=50)
cop = mixture(["clayton", "gaussian", "gumbel"])
cop.plot_pdf([1/3, 1/3, 1/3, 2, 0.5, 4], plot_type="contour", Nsplit=40, levels=[0.1,0.4,0.8,1.3,1.6] )
cop.plot_mpdf([1/3, 1/3, 1/3, 2, 0.5, 2], marginals, plot_type="contour", Nsplit=50)
Simulation
Gaussian
from scipy.stats import norm
from pycop import simulation
n = 2 # dimension
m = 1000 # sample size
corrMatrix = np.array([[1, 0.8], [0.8, 1]])
u1, u2 = simulation.simu_gaussian(n, m, corrMatrix)
Adding gaussian marginals, (using distribution.ppf from scipy.statsto transform uniform margin to the desired distribution)
u1 = norm.ppf(u1)
u2 = norm.ppf(u2)
Student
u1, u2 = simulation.simu_tstudent(n, m, corrMatrix, nu=1)
Archimedean
List of archimedean cop available
u1, u2 = simulation.simu_archimedean("gumbel", n, m, theta=2)
u1, u2 = 1 - u1, 1 - u2
Rotated
u1, u2 = 1 - u1, 1 - u2
High dimension
n = 3 # Dimension
m = 1000 # Sample size
corrMatrix = np.array([[1, 0.9, 0], [0.9, 1, 0], [0, 0, 1]])
u = simulation.simu_gaussian(n, m, corrMatrix)
u = norm.ppf(u)
u = simulation.simu_archimedean("clayton", n, m, theta=2)
u = norm.ppf(u)
Mixture simulation
Simulation from a mixture of 2 copulas
n = 3
m = 2000
combination = [
{"type": "clayton", "weight": 1/3, "theta": 2},
{"type": "gumbel", "weight": 1/3, "theta": 3}
]
u = simulation.simu_mixture(n, m, combination)
u = norm.ppf(u)
Simulation from a mixture of 3 copulas
corrMatrix = np.array([[1, 0.8, 0], [0.8, 1, 0], [0, 0, 1]])
combination = [
{"type": "clayton", "weight": 1/3, "theta": 2},
{"type": "student", "weight": 1/3, "corrMatrix": corrMatrix, "nu":2},
{"type": "gumbel", "weight": 1/3, "theta":3}
]
u = simulation.simu_mixture(n, m, combination)
u = norm.ppf(u)
Estimation
Estimation available : CMLE
Canonical Maximum Likelihood Estimation (CMLE)
Import a sample with pandas
import pandas as pd
import numpy as np
df = pd.read_csv("data/msci.csv")
df.index = pd.to_datetime(df["Date"], format="%m/%d/%Y
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