Examples

2D Visualization

The visualization feature can support 2D and 3D plot of copulas.

     import numpy as np
     import matplotlib.pyplot as plt
     from matplotlib import cm
     from pycopula.copula import ArchimedeanCopula
     from mpl_toolkits.mplot3d import Axes3D
     from pycopula.visualization import pdf_2d, cdf_2d

     # The Clayton copula
     clayton = ArchimedeanCopula(family="clayton", dim=2)

     # Visualization of CDF and PDF
     u, v, C = cdf_2d(clayton)
     u, v, c = pdf_2d(clayton)

     # Plotting
     fig = plt.figure()
     ax = fig.add_subplot(121, projection='3d', title="Clayton copula CDF")
     X, Y = np.meshgrid(u, v)

     ax.set_zlim(0, 5)
     ax.plot_surface(X, Y, c, cmap=cm.Blues)
     ax.plot_wireframe(X, Y, c, color='black', alpha=0.3)

     ax = fig.add_subplot(122, title="Clayton copula PDF")
     ax.contour(X, Y, c, levels = np.arange(0,5,0.15))

     plt.show()

3D Visualization

This sample allows you to plot the CDF of a two-dimensional copula thanks to Matplotlib3D.

     import numpy as np
     import matplotlib.pyplot as plt
     from matplotlib import cm
     from pycopula.copula import *
     from mpl_toolkits.mplot3d import Axes3D
     from pycopula.visualization import cdf_2d

     # Storing the copulas in array
     copulas = []
     copulas.append(ArchimedeanCopula(family="clayton", dim=2))
     copulas.append(Copula(dim=2, name='frechet_up'))
     copulas.append(Copula(dim=2, name='frechet_down'))
     copulas.append(GaussianCopula(dim=2))

     names = [ 'Clayton copula', 'Fréchet-Hoeffding upper bound', 'Fréchet-Hoeffding lower bound', 'Gaussian copula' ]

     fig = plt.figure()
     index = 1

     # For each copula
     for c, i in zip(copulas, range(1,5)):
             # We get the CDF values
             u, v, C = cdf_2d(c)

             # Subplotting the current copula's CDF
             ax = fig.add_subplot(220 + i, projection='3d', title=names[i-1])
             X, Y = np.meshgrid(u, v)
             ax.set_zlim(0, 1)
             ax.plot_surface(X, Y, C, cmap=cm.Blues)
             ax.plot_wireframe(X, Y, C, color='black', alpha=0.3)
             ax.view_init(30, 260)

     plt.show()

Sampling

     import numpy as np
     import matplotlib.pyplot as plt
     from matplotlib import cm
     from pycopula.copula import *
     from mpl_toolkits.mplot3d import Axes3D
     from pycopula.visualization import pdf_2d
     from pycopula.simulation import simulate

     # The Gaussian copula with specified covariance matrix
     gaussian = GaussianCopula(dim=2, sigma=[[1, 0.8], [0.8, 1]])
     # Visualization of CDF and PDF
     u, v, c = pdf_2d(gaussian)
     X, Y = np.meshgrid(u, v)

     # Sampling of size 1000 with Gaussian copula
     sim = simulate(gaussian, 1000)

     # Plotting
     fig = plt.figure()
     ax = fig.add_subplot(111, title="Gaussian copula PDF")
     ax.contour(X, Y, c, levels = np.arange(0,5,0.15), alpha=0.5)
     plt.scatter([ s[0] for s in sim ], [s[1] for s in sim ], alpha=0.6, edgecolors='none')
     plt.title("Simulation of 1000 points with Gaussian copula")
     plt.xlim(0, 1)
     plt.ylim(0, 1)

     plt.show()

Concentration Functions

     import matplotlib.pyplot as plt
     from pycopula.copula import ArchimedeanCopula
     from pycopula.visualization import concentrationFunction
     from pycopula.simulation import simulate

     # The Clayton copula
     clayton = ArchimedeanCopula(dim=2, family='clayton')

     # Sampling of size 1000 with Clayton copula
     sim = simulate(clayton, 1000)

     # Computing theoritical and empirical concentrations functions
     downI, upI, tailDown, tailUp = concentrationFunction(sim)
     ClaytonDown = [ clayton.concentrationDown(x) for x in downI ]
     ClaytonUp = [ clayton.concentrationUp(x) for x in upI ]

     # Plotting
     plt.figure()
     plt.plot(downI, tailDown, color='red', linewidth=3, label="Empirical concentration")
     plt.plot(upI, tailUp, color='red', linewidth=3)
     plt.plot(downI, ClaytonDown, color='blue', linewidth=1, label="Clayton concentration")
     plt.plot(upI, ClaytonUp, color='blue', linewidth=1)
     plt.plot([0.5, 0.5], [0, 1], color='gray', alpha=0.6, linestyle='--', linewidth=1)
     plt.title("Lower-Upper tail dependence Coefficients")
     plt.xlabel("Lower Tail      Upper Tail")
     plt.legend(loc=0)

     plt.show()