from generate_datasets import make_point_clouds
import matplotlib.pyplot as plt
n_samples_per_class = 1
point_clouds, labels = make_point_clouds(n_samples_per_class, 15, 0.1)
point_clouds.shape
print(f"There are {point_clouds.shape[0]} point clouds in {point_clouds.shape[2]} dimensions, "
      f"each with {point_clouds.shape[1]} points.")

There are 3 point clouds in 3 dimensions, each with 225 points.

point_clouds.shape

(3, 225, 3)

def plot_in_axis(ax, point_cloud):
    xs, ys, zs = zip(*point_cloud)
    ax.scatter(list(xs),list(ys),list(zs))
    return ax

def plot_3d_data(point_cloud):
    fig = plt.figure()
    ax = fig.add_subplot(projection='3d')
    ax = plot_in_axis(ax)
    plt.show()

def plot_3_figures(point_cloud_1, point_cloud_2, point_cloud_3): 
    # Set up a figure three times as wide as it is tall
    fig = plt.figure(figsize=plt.figaspect(0.3))
    point_clouds = [point_cloud_1, point_cloud_2, point_cloud_3]
    for i in range(len(point_clouds)):
        axis = fig.add_subplot(1, 3, i+1, projection='3d')
        plot_in_axis(axis, point_clouds[i])
    plt.show()

Let's take one point cloud for each of the three categories

circumference = point_clouds[0]
sphere = point_clouds[n_samples_per_class]
torus = point_clouds[n_samples_per_class*2]
plot_3_figures(circumference, sphere, torus)

We will compute persistence diagrams for the Vietoris Rips and for the Cech filtrations.

Vietoris-Rips filtration

from gtda.homology import VietorisRipsPersistence
VR = VietorisRipsPersistence(homology_dimensions=[0, 1, 2])
diagrams_VR = VR.fit_transform(point_clouds)
diagrams_VR.shape

(3, 402, 3)

Cech filtration

import numpy as np

def filter_Cech_diagram(CH_diagram):
    cleaned_diagram = []
    for point in CH_diagram[0]:
        if not np.isclose(point[0], point[1], atol=.01):
            cleaned_diagram.append([point[0], point[1], point[2]])
    return np.array([cleaned_diagram])

from gtda.homology import EuclideanCechPersistence
CH = EuclideanCechPersistence(homology_dimensions=[0, 1])
diagrams_CH = CH.fit_transform([point_clouds[n_samples_per_class*2]]) # Currently it only works with a list of ONE diagram
# We need to remove points from the diagonal: the algorithm returns points in the diagonal needed to compute
# persistence diagrams
diagrams_CH = list(map(lambda point_cloud: 
                   filter_Cech_diagram(CH.fit_transform([point_clouds[n_samples_per_class*2]])), 
                   point_clouds))
diagrams_CH = np.vstack(diagrams_CH)
diagrams_CH.shape

(3, 952, 3)

Now let's plot the diagrams for the Cech and the Vietoris-Rips filtration. We can do it either with the plot_diagram function as well as with the plot of the VR method.

from gtda.plotting import plot_diagram

plot_diagram(diagrams_VR[0])

or

VR.plot(diagrams_VR, sample=0)

Now let's plot both Cech and Vietoris-Rips persistence diagrams.

import chart_studio.plotly as py
from ipywidgets import widgets
import plotly.graph_objects as go

def plot_VR_and_Cech(ppdd_VR, ppdd_Cech):
    fig_VR = plot_diagram(ppdd_VR)
    fig_VR.update_layout(title="Vietoris-Rips persistence diagram")
    fig_Cech = plot_diagram(ppdd_Cech)
    fig_Cech.update_layout(title="Cech persistence diagram")
    fig_both = widgets.HBox(children=[go.FigureWidget(fig_VR), go.FigureWidget(fig_Cech)])
    return fig_both

point_cloud_torus_id = n_samples_per_class*2
plot_VR_and_Cech(diagrams_VR[point_cloud_torus_id], diagrams_CH[point_cloud_torus_id]) #diagrams from sampled torus 

Now let's compute some features from our persistence diagram.

from gtda.diagrams import PersistenceEntropy, PersistenceImage

PE = PersistenceEntropy()
features = PE.fit_transform(diagrams_VR)

features[0][1]

array([0.        , 0.        , 0.        , 0.        , 0.00758692,
       0.02731114, 0.04703537, 0.06675959, 0.08648382, 0.10620804,
       0.12593226, 0.14565649, 0.16538071, 0.18510494, 0.20482916,
       0.22455338, 0.24427761, 0.26400183, 0.28372606, 0.30345028,
       0.3231745 , 0.34289873, 0.36262295, 0.38234718, 0.4020714 ,
       0.42179562, 0.44151985, 0.46124407, 0.4809683 , 0.50069252,
       0.52041674, 0.54014097, 0.55986519, 0.57958942, 0.59931364,
       0.61903786, 0.63876209, 0.65848631, 0.67821054, 0.69793476,
       0.71765898, 0.73738321, 0.75710743, 0.74068368, 0.72095945,
       0.70123523, 0.68151101, 0.66178678, 0.64206256, 0.62233833,
       0.60261411, 0.58288989, 0.56316566, 0.54344144, 0.52371721,
       0.50399299, 0.48426877, 0.46454454, 0.44482032, 0.42509609,
       0.40537187, 0.38564765, 0.36592342, 0.3461992 , 0.32647497,
       0.30675075, 0.28702653, 0.2673023 , 0.24757808, 0.22785385,
       0.20812963, 0.18840541, 0.16868118, 0.14895696, 0.12923273,
       0.10950851, 0.08978429, 0.07006006, 0.05033584, 0.03061162,
       0.01088739, 0.        , 0.        , 0.        , 0.        ,
       0.        , 0.        , 0.        , 0.        , 0.        ,
       0.        , 0.        , 0.        , 0.        , 0.        ,
       0.        , 0.        , 0.        , 0.        , 0.        ])

import matplotlib.image as mpimg

PI = PersistenceImage()
features = PI.fit_transform(diagrams_VR)

plt.imshow(features[0][1])

<matplotlib.image.AxesImage at 0x197a0e66070>

from gtda.diagrams import Silhouette
PS = Silhouette()
features = PS.fit_transform_plot(diagrams_VR)

features

array([[[0.00000000e+00, 7.39900115e-03, 1.37935752e-02, 1.79439340e-02,
         1.86358909e-02, 1.64152840e-02, 1.25623199e-02, 8.57831872e-03,
         5.43301115e-03, 3.15642730e-03, 1.57422868e-03, 5.80019867e-04,
         2.54822622e-04, 9.57567790e-05, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00],
        [0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         7.58692007e-03, 2.73111441e-02, 4.70353680e-02, 6.67595920e-02,
         8.64838160e-02, 1.06208040e-01, 1.25932264e-01, 1.45656488e-01,
         1.65380712e-01, 1.85104936e-01, 2.04829160e-01, 2.24553384e-01,
         2.44277608e-01, 2.64001832e-01, 2.83726056e-01, 3.03450280e-01,
         3.23174504e-01, 3.42898728e-01, 3.62622952e-01, 3.82347176e-01,
         4.02071400e-01, 4.21795624e-01, 4.41519848e-01, 4.61244072e-01,
         4.80968296e-01, 5.00692520e-01, 5.20416744e-01, 5.40140968e-01,
         5.59865192e-01, 5.79589416e-01, 5.99313640e-01, 6.19037864e-01,
         6.38762088e-01, 6.58486312e-01, 6.78210536e-01, 6.97934760e-01,
         7.17658984e-01, 7.37383208e-01, 7.57107432e-01, 7.40683679e-01,
         7.20959455e-01, 7.01235231e-01, 6.81511007e-01, 6.61786783e-01,
         6.42062559e-01, 6.22338335e-01, 6.02614111e-01, 5.82889887e-01,
         5.63165663e-01, 5.43441439e-01, 5.23717215e-01, 5.03992991e-01,
         4.84268767e-01, 4.64544543e-01, 4.44820319e-01, 4.25096095e-01,
         4.05371871e-01, 3.85647647e-01, 3.65923423e-01, 3.46199199e-01,
         3.26474975e-01, 3.06750751e-01, 2.87026527e-01, 2.67302303e-01,
         2.47578079e-01, 2.27853855e-01, 2.08129631e-01, 1.88405407e-01,
         1.68681183e-01, 1.48956959e-01, 1.29232735e-01, 1.09508511e-01,
         8.97842870e-02, 7.00600630e-02, 5.03358390e-02, 3.06116150e-02,
         1.08873911e-02, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00],
        [0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 3.77615735e-03, 6.07607452e-03,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00,
         0.00000000e+00, 0.00000000e+00, 0.00000000e+00, 0.00000000e+00]]])