Agglomerative Clustering Metrics: Hierarchical Clustering Techniques

Introduction

Agglomerative clustering is a hierarchical clustering method used to group similar objects together. It starts with each object as its own cluster, and then iteratively merges the most similar clusters together until a stopping criterion is met. In this lab, we will demonstrate the effect of different metrics on the hierarchical clustering using agglomerative clustering algorithm.

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Import libraries and generate waveform data

First, we import the necessary libraries and generate waveform data that will be used in this lab.

import matplotlib.pyplot as plt
import matplotlib.patheffects as PathEffects
import numpy as np
from sklearn.cluster import AgglomerativeClustering
from sklearn.metrics import pairwise_distances

np.random.seed(0)

# Generate waveform data
n_features = 2000
t = np.pi * np.linspace(0, 1, n_features)

def sqr(x):
    return np.sign(np.cos(x))

X = list()
y = list()
for i, (phi, a) in enumerate([(0.5, 0.15), (0.5, 0.6), (0.3, 0.2)]):
    for _ in range(30):
        phase_noise = 0.01 * np.random.normal()
        amplitude_noise = 0.04 * np.random.normal()
        additional_noise = 1 - 2 * np.random.rand(n_features)
        # Make the noise sparse
        additional_noise[np.abs(additional_noise) < 0.997] = 0

        X.append(
            12
            * (
                (a + amplitude_noise) * (sqr(6 * (t + phi + phase_noise)))
                + additional_noise
            )
        )
        y.append(i)

X = np.array(X)
y = np.array(y)

Plot the ground-truth labeling

We plot the ground-truth labeling of the waveform data.

n_clusters = 3

labels = ("Waveform 1", "Waveform 2", "Waveform 3")

colors = ["#f7bd01", "#377eb8", "#f781bf"]

# Plot the ground-truth labelling
plt.figure()
plt.axes([0, 0, 1, 1])
for l, color, n in zip(range(n_clusters), colors, labels):
    lines = plt.plot(X[y == l].T, c=color, alpha=0.5)
    lines[0].set_label(n)

plt.legend(loc="best")

plt.axis("tight")
plt.axis("off")
plt.suptitle("Ground truth", size=20, y=1)

Plot the distances

We plot the interclass distances for different metrics.

for index, metric in enumerate(["cosine", "euclidean", "cityblock"]):
    avg_dist = np.zeros((n_clusters, n_clusters))
    plt.figure(figsize=(5, 4.5))
    for i in range(n_clusters):
        for j in range(n_clusters):
            avg_dist[i, j] = pairwise_distances(
                X[y == i], X[y == j], metric=metric
            ).mean()
    avg_dist /= avg_dist.max()
    for i in range(n_clusters):
        for j in range(n_clusters):
            t = plt.text(
                i,
                j,
                "%5.3f" % avg_dist[i, j],
                verticalalignment="center",
                horizontalalignment="center",
            )
            t.set_path_effects(
                [PathEffects.withStroke(linewidth=5, foreground="w", alpha=0.5)]
            )

    plt.imshow(avg_dist, interpolation="nearest", cmap="cividis", vmin=0)
    plt.xticks(range(n_clusters), labels, rotation=45)
    plt.yticks(range(n_clusters), labels)
    plt.colorbar()
    plt.suptitle("Interclass %s distances" % metric, size=18, y=1)
    plt.tight_layout()

Plot clustering results

We plot the clustering results for different metrics.

for index, metric in enumerate(["cosine", "euclidean", "cityblock"]):
    model = AgglomerativeClustering(
        n_clusters=n_clusters, linkage="average", metric=metric
    )
    model.fit(X)
    plt.figure()
    plt.axes([0, 0, 1, 1])
    for l, color in zip(np.arange(model.n_clusters), colors):
        plt.plot(X[model.labels_ == l].T, c=color, alpha=0.5)
    plt.axis("tight")
    plt.axis("off")
    plt.suptitle("AgglomerativeClustering(metric=%s)" % metric, size=20, y=1)

Summary

In this lab, we demonstrated the effect of different metrics on the hierarchical clustering using agglomerative clustering algorithm. We generated waveform data and plotted the ground-truth labeling, interclass distances, and clustering results for different metrics. We observed that the clustering results varied with the choice of metric and that the cityblock distance performed the best in separating the waveforms.

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