Copulas — Modeling Dependence
Advanced Statistical Methods
Separating Marginal Behavior From Joint Dependence
Copulas model the dependence structure between variables independently of their marginal distributions, thanks to Sklar's theorem. Gaussian, t, and Archimedean copulas capture different tail dependence patterns.
- Finance — Model joint extreme losses across assets for portfolio risk management
- Insurance — Correlate claim amounts across multiple coverage lines for solvency modeling
- Hydrology — Link rainfall and river flow distributions for flood risk assessment
Copulas let you model how variables move together without being constrained by their individual distributions.
Gaussian Copula
Student's t-Copula
Archimedean Copulas
Dependence Measures
Vine Copulas
import numpy as np
from scipy import stats
from scipy.special import gamma
class CopulaModel:
def __init__(self, data):
self.data = np.asarray(data)
self.n, self.d = self.data.shape
def empirical_cdf(self):
ranks = np.apply_along_axis(stats.rankdata, 0, self.data)
return ranks / (self.n + 1)
def gaussian_copula_fit(self):
U = self.empirical_cdf()
Z = stats.norm.ppf(np.clip(U, 1e-10, 1-1e-10))
R = np.corrcoef(Z.T)
return R
def t_copula_fit(self):
U = self.empirical_cdf()
Z = stats.t.ppf(np.clip(U, 1e-10, 1-1e-10), df=5)
R = np.corrcoef(Z.T)
return R, 5.0
def clayton_copula(self, u, v, theta):
return (u**(-theta) + v**(-theta) - 1)**(-1/theta)
def gumbel_copula(self, u, v, theta):
return np.exp(-((-np.log(u))**theta + (-np.log(v))**theta)**(1/theta))
def frank_copula(self, u, v, theta):
num = np.exp(-theta*u) - 1
den = np.exp(-theta) - 1
return -np.log(1 + (num * (np.exp(-theta*v) - 1)) / (den * np.exp(-theta*(u+v)/2))) / theta
def kendalls_tau(self, u, v):
n = len(u)
concordant = 0
discordant = 0
for i in range(n):
for j in range(i+1, n):
d = (u[i] - u[j]) * (v[i] - v[j])
if d > 0:
concordant += 1
elif d < 0:
discordant += 1
return (concordant - discordant) / (concordant + discordant)
def tail_dependence_coefficient(self, u, v, threshold=0.95):
n = len(u)
upper = np.sum((u > threshold) & (v > threshold)) / (n * (1 - threshold))
lower = np.sum((u < 1-threshold) & (v < 1-threshold)) / (n * (1 - threshold))
return upper, lower
def simulate_gaussian(self, R, n_samples):
d = R.shape[0]
L = np.linalg.cholesky(R)
Z = np.random.standard_normal((n_samples, d))
X = Z @ L.T
U = stats.norm.cdf(X)
return U