Visualize the Dictionary of Obscure Words with T-SNE

I recently published on a wrapper around The Dictionary of Obscure Words (originally from this website http://phrontistery.info) for Python and in this post we'll see how to create a visualization to highlight few entries from the dictionary using the dimensionality reduction technique called T-SNE. The dictionary is available on github at this address https://github.com/JustGlowing/obscure_words and can be installed as follows:

pip install git+https://github.com/JustGlowing/obscure_words

We can now import the dictionary and create a vectorial representation of each word:

import matplotlib.pyplot as plt
import numpy as np
from obscure_words import load_obscure_words
from sklearn.feature_extraction.text import TfidfVectorizer, CountVectorizer
from sklearn.manifold import TSNE

obscure_dict = load_obscure_words()
words = np.array(list(obscure_dict.keys()))
definitions = np.array(list(obscure_dict.values()))

vectorizer = TfidfVectorizer(stop_words=None)
X = vectorizer.fit_transform(definitions)

projector = TSNE(random_state=0)
XX = projector.fit_transform(X)

In the snippet above, we compute a Tf-Idf representation using the definition of each word. This gives us a vector for each word in our dictionary, but each of these vectors has many elements as the total number of words used in all the definitions. Since we can't plot all the features extracted, we reduce our data to 2 dimensions we use T-SNE. We have now a mapping that allows us to place each word in a point of a bi-dimensional space. There's one problem remaining, how can we plot the words in a way that we can still read them? Here's a solution:

from sklearn.cluster import KMeans
from sklearn.preprocessing import StandardScaler
from sklearn.metrics import pairwise_distances

def textscatter(x, y, text, k=10):
    X = np.array([x, y]).T
    clustering = KMeans(n_clusters=k)
    scaler = StandardScaler()
    clustering.fit(scaler.fit_transform(X))
    centers = scaler.inverse_transform(clustering.cluster_centers_)
    selected = np.argmin(pairwise_distances(X, centers), axis=0)
    plt.scatter(x, y, s=6, c=clustering.predict(scaler.transform(X)), alpha=.05)
    for i in selected:
        plt.text(x[i], y[i], text[i], fontsize=10)

plt.figure(figsize=(16, 16))
textscatter(XX[:, 0], XX[:, 1], 
            [w+'\n'+d for w, d in zip(words, definitions)], 20)
plt.show()

In the function textscatter we segment all the points created at the previous steps in k clusters using K-Means, then we plot the word related to the center of cluster (and also its definion). Given the properties of K-Means we know that the centers are distant from each other and with the right choice of k we can maximize the number of words we can display. This is the result of the snippet above:

(click on the figure to see the entire chart)

A visual introduction to the Gap Statistics

We have previously seen how to implement KMeans. However, the results of this algorithm strongly rely on the choice of the parameter K. According to statistical folklore the best K is located at the 'elbow' of the clusters inertia while K increases. This heuristic has been translated into a more formalized procedure by the Gap Statistics and in this post we'll see how to pick K in an optimal way using the Gap Statistics. The main idea of the methodology is to compare the clusters inertia on the data to cluster and a reference dataset. The optimal choice of K is given by k for which the gap between the two results is maximum. To illustrate this idea, let’s pick as reference dataset a uniformly distributed set of points and see the result of KMeans increasing K:

import numpy as np
import matplotlib.pyplot as plt

from sklearn.datasets import make_blobs
from sklearn.metrics import pairwise_distances
from sklearn.cluster import KMeans


reference = np.random.rand(100, 2)
plt.figure(figsize=(12, 3))
for k in range(1,6):
    kmeans = KMeans(n_clusters=k)
    a = kmeans.fit_predict(reference)
    plt.subplot(1,5,k)
    plt.scatter(reference[:, 0], reference[:, 1], c=a)
    plt.xlabel('k='+str(k))
plt.tight_layout()
plt.show()

From the figure above we can see that the algorithm evenly splits the points K clusters even if there's no separation between them. Let’s now do the same on a target dataset with 3 natural clusters:

X = make_blobs(n_samples=100, n_features=2,
               centers=3, cluster_std=.8,)[0]

plt.figure(figsize=(12, 3))
for k in range(1,6):
    kmeans = KMeans(n_clusters=k)
    a = kmeans.fit_predict(X)
    plt.subplot(1,5,k)
    plt.scatter(X[:, 0], X[:, 1], c=a)
    plt.xlabel('k='+str(k))
plt.tight_layout()
plt.show()

Here we note that the algorithm, with K=2, correctly isolates one of the clusters grouping the other two together. Then, with K=3, correctly identifies the natural clusters. But, with K=4 and K=5 some of the natural clusters are split in two. If we plot the inertia in both cases we'll see something interesting:

def compute_inertia(a, X):
    W = [np.mean(pairwise_distances(X[a == c, :])) for c in np.unique(a)]
    return np.mean(W)

def compute_gap(clustering, data, k_max=5, n_references=5):
    if len(data.shape) == 1:
        data = data.reshape(-1, 1)
    reference = np.random.rand(*data.shape)
    reference_inertia = []
    for k in range(1, k_max+1):
        local_inertia = []
        for _ in range(n_references):
            clustering.n_clusters = k
            assignments = clustering.fit_predict(reference)
            local_inertia.append(compute_inertia(assignments, reference))
        reference_inertia.append(np.mean(local_inertia))
    
    ondata_inertia = []
    for k in range(1, k_max+1):
        clustering.n_clusters = k
        assignments = clustering.fit_predict(data)
        ondata_inertia.append(compute_inertia(assignments, data))
        
    gap = np.log(reference_inertia)-np.log(ondata_inertia)
    return gap, np.log(reference_inertia), np.log(ondata_inertia)

k_max = 5
gap, reference_inertia, ondata_inertia = compute_gap(KMeans(), X, k_max)


plt.plot(range(1, k_max+1), reference_inertia,
         '-o', label='reference')
plt.plot(range(1, k_max+1), ondata_inertia,
         '-o', label='data')
plt.xlabel('k')
plt.ylabel('log(inertia)')
plt.show()

On the reference dataset the inertia goes down’ very slowly while on the target dataset it assumes the shape of an elbow! We can now compute the Gap Statistics for each K computing the difference of the two curves showed above:

plt.plot(range(1, k_max+1), gap, '-o')
plt.ylabel('gap')
plt.xlabel('k')

It’s easy to see that the Gap is maximum for K=3, just the right choice for our target dataset.

For a more formal introduction you can check out the following paper: Tibshirani R, Walther G, Hastie T. Estimating the number of clusters in a dataset via the gap statistic. Journal of the Royal Statistics Society 2001.