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This example shows how to use the UQpy DiffusionMaps class to reveal the embedded structure of noisy data.

Import the necessary libraries. Here we import standard libraries such as numpy and matplotlib, but also need to import the DiffusionMaps class from UQpy implemented in the DimensionReduction module.

import numpy as np
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
from UQpy.utilities.kernels.GaussianKernel import GaussianKernel
from UQpy.dimension_reduction.diffusion_maps.DiffusionMaps import DiffusionMaps

Sample points randomly following a parametric curve and plot the 3D graphic.

a = 6
b = 1
k = 10
u = np.linspace(0, 2 * np.pi, 1000)

v = k * u

x0 = (a + b * np.cos(0.8 * v)) * (np.cos(u))
y0 = (a + b * np.cos(0.8 * v)) * (np.sin(u))
z0 = b * np.sin(0.8 * v)

rox = 0.2
roy = 0.2
roz = 0.2
x = x0 + rox * np.random.normal(0, 1, len(x0))
y = y0 + roy * np.random.normal(0, 1, len(y0))
z = z0 + roz * np.random.normal(0, 1, len(z0))

X = np.array([x, y, z]).transpose()

fig = plt.figure()
ax = fig.gca(projection='3d')
ax.scatter(x, y, z, c='b', cmap=plt.cm.Spectral, s=8)
ax.plot(x0, y0, z0, 'r', label='parametric curve')
plt.show()

Instantiate the class DiffusionMaps using alpha=1; n_eigenvectors=3, because the first eigenvector is non-informative. Moreover, a Gaussian is used in the kernel construction.

dmaps = DiffusionMaps(data=X, alpha=1, n_eigenvectors=5,
                      kernel=GaussianKernel(epsilon=0.3))

Plot the second and third diffusion coordinates to reveal the embedded structure of the data.

color = dmaps.eigenvectors[:, 1]
plt.scatter(dmaps.diffusion_coordinates[:, 1], dmaps.diffusion_coordinates[:, 2], c=color, cmap=plt.cm.Spectral, s=8)
plt.axis('equal')
plt.show()

Use the colormap to observe how the embedded structure is distributed in the original set.

fig = plt.figure()
ax = fig.gca(projection='3d')
ax.scatter(x, y, z, c=color, cmap=plt.cm.Spectral, s=8)
plt.show()