10  Unsupervised Learning

10.1 K-Means Clustering

So far, we have explored various supervised learning algorithms such as Decision Trees and Random Forests, which rely on labeled data with known outcomes. In contrast, unsupervised learning techniques analyze unlabeled data to identify patterns, making them particularly useful for clustering and association problems. Among these, K-means clustering stands out as one of the simplest and most widely used algorithms.

K-means clustering aims to divide a dataset into non-overlapping groups based on similarity. Given a set of data points, each represented as a vector in a multi-dimensional space, the algorithm assigns each point to one of \(k\) clusters in a way that minimizes the variation within each cluster. This is done by reducing the sum of squared distances between each point and its assigned cluster center. Mathematically, we seek to minimize:

\[\begin{equation*} \sum_{i=1}^{k}\sum_{\boldsymbol{x}\in S_i} \left\|\boldsymbol{x}-\boldsymbol{\mu}_i\right\|^2 \end{equation*}\]

where \(S_i\) represents each cluster and \(\boldsymbol{\mu}_i\) is the mean of the points within that cluster.

10.1.1 Lloyd’s Algorithm

K-means clustering is typically solved using Lloyd’s algorithm, which operates iteratively as follows:

1. Initialization: Select \(k\) initial cluster centroids \(\boldsymbol{\mu}_i\) randomly.
2. Iteration:

   • Assignment step: Assign each point \(\boldsymbol{x}\) to the cluster whose centroid is closest based on the squared Euclidean distance.

   • Update step: Recompute the centroids as the mean of all points assigned to each cluster:

     \[\begin{equation*} \boldsymbol{\mu}_i \leftarrow \frac{1}{|S_i|} \sum_{\boldsymbol{x}_j \in S_i} \boldsymbol{x}_j \end{equation*}\]

3. Termination: The process stops when either the assignments no longer change or a predefined number of iterations is reached.

10.1.2 Example: Iris Data

K-means clustering can be implemented using the scikit-learn library. Below, we apply it to the Iris dataset.

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets
from sklearn.cluster import KMeans

# Load the Iris dataset
iris = datasets.load_iris()
X = iris.data[:, :2]  # Using only two features
y = iris.target

We visualize the observations based on their true species labels.

# Scatter plot of true species labels
fig, ax = plt.subplots()
scatter = ax.scatter(X[:, 0], X[:, 1], c=y,
                      cmap='viridis', edgecolors='k')
ax.legend(*scatter.legend_elements(), loc="upper left",
          title="Species")
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")
plt.title("True Species Distribution")
plt.show()

Now, we apply K-means clustering to the data.

# Train K-means model
Kmean = KMeans(n_clusters=3, init='k-means++',
               n_init=10, random_state=42)
Kmean.fit(X)

KMeans(n_clusters=3, n_init=10, random_state=42)

In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.

Several parameters can be adjusted for better performance. See: <https://scikit-learn.org/stable/modules/generated/ sklearn.cluster.KMeans.html>

K-means provides cluster centroids, representing the center of each cluster.

# Print predicted cluster centers
print("Cluster Centers:")
print(Kmean.cluster_centers_)

Cluster Centers:
[[6.81276596 3.07446809]
 [5.77358491 2.69245283]
 [5.006      3.428     ]]

We plot the centroids along with clustered points.

# Plot centroids on the scatter plot
fig, ax = plt.subplots()
ax.scatter(X[:, 0], X[:, 1], c=Kmean.labels_,
           cmap='viridis', edgecolors='k', alpha=0.5)
ax.scatter(Kmean.cluster_centers_[:, 0],
           Kmean.cluster_centers_[:, 1],
           c="black", s=200, marker='s',
           label="Centroids")
ax.legend()
plt.xlabel("Feature 1")
plt.ylabel("Feature 2")
plt.title("K-means Clustering Results")
plt.show()

10.1.2.1 Comparing True and Predicted Labels

By plotting the results side by side, we can see how well K-means clustering approximates the true labels.

# Compare true vs. predicted labels
fig, axs = plt.subplots(ncols=2, figsize=(12, 5),
                        constrained_layout=True)

# True labels plot
axs[0].scatter(X[:, 0], X[:, 1], c=y,
               cmap='viridis', alpha=0.5,
               edgecolors='k')
axs[0].set_title("True Labels")
axs[0].set_xlabel("Feature 1")
axs[0].set_ylabel("Feature 2")

# Predicted clusters plot
axs[1].scatter(X[:, 0], X[:, 1], c=Kmean.labels_,
               cmap='viridis', alpha=0.5,
               edgecolors='k')
axs[1].scatter(Kmean.cluster_centers_[:, 0],
               Kmean.cluster_centers_[:, 1],
               marker="s", c="black", s=200,
               alpha=1, label="Centroids")
axs[1].set_title("Predicted Clusters")
axs[1].set_xlabel("Feature 1")
axs[1].set_ylabel("Feature 2")
axs[1].legend()
plt.show()

10.1.3 Making Predictions on New Data

Once trained, the model can classify new data points.

# Sample test data points
sample_test = np.array([[3, 4], [7, 4]])

# Predict cluster assignment
print("Predicted Clusters:", Kmean.predict(sample_test))

Predicted Clusters: [2 0]

10.1.4 Discussion

K-means is intuitive but has limitations:

• Sensitivity to initialization: Poor initialization can yield suboptimal results. k-means++ mitigates this issue.
• Choosing the number of clusters: The choice of \(k\) is critical. The elbow method helps determine an optimal value.
• Assumption of spherical clusters: K-means struggles when clusters have irregular shapes. Alternative methods such as kernel-based clustering may be more effective.

Despite its limitations, K-means is a fundamental tool in exploratory data analysis and practical applications.

10.2 K-Prototypes Clustering

This section was prepared by Mario Tomaino, a Senior majoring in Mathematics/Statistics. This section explores K-Prototypes clustering, an unsupervised machine learning algorithm used to cluster data that contains both numerical and categorical variables.

We will be generating a dataset containing individuals of certain ages and incomes, and clustering them based on their technology product preferences. We will then create some visualizations to further understand the clusters and characteristics of the individuals.

10.2.1 What is K-Prototypes Clustering?

• Clustering is the unsupervised classification of patterns into groups
• It helps group similar observations into distinct, interpretable clusters
• Traditional algorithms like k-means only work with numerical data
• Real world data often includes both numerical and categorical features, which makes K-Prototypes necessary

10.2.2 K-Means Clustering

• Works with numerical data only
• Implemented using the scikit-learn library
• Uses Euclidean distance to measure similarity (distance between two points)
• Here, cluster centroids are the mean of all points in the cluster
• Common in applications with quantitative features (incident zip, latitude, longitude)

10.2.3 K-Modes Clustering

• Designed for categorical data only
• Uses dissimilarity (counts how many attributes differ)
• Here, centroids are the mode (most common category)
• Useful when data contains strings or categories (borough, complaint type)

10.2.4 Why K-Prototypes?

• Many datasets include both numbers (latitude, longitude) and categories (borough, complaint type)
• K-Means: numerical only
• K-Modes: categorical only
• K-Prototypes: combines both types into a single clustering algorithm
• Here, the centroid is a mix:
  • Mean for numeric features
  • Mode for categorical features

10.2.5 Why is the centroid important?

The algorithm uses centroids to:

1. Measure how close a data point is to a cluster
2. Reassign points to the closest cluster
3. Update the cluster’s center as new points are assigned

This process repeats until centroids stop changing much (convergence).

10.2.6 How K-Prototypes Works

• K-Prototypes combines K-Means and K-Modes
  • Minimizes cost function by combining:
    • Euclidean distance for numeric features (numerical distance from cluster average)
    • Dissimilarity for categorical features
• A hyperparameter γ (gamma) balances weight of numerical and categorical variables

10.2.7 Similarity Measure (Distance Function)

distance = (euclidian distance) + γ * (categorical dissimilarity).

Precise numerical formula can be found at (Huang, 1997) page two.

• Measures how different each data point is from the cluster centroids
• Helps assign each point to the most appropriate cluster
• γ (gamma) balances numeric and categorical importance

10.2.8 Python Example

Basic example of a dataset containing customers that we will cluster based on Age, Income, and Preferred Product.

Here, we choose γ = 0.5 because it gives us an equal trade-off between the numeric and categorical parts of the distance function.

This is the code we will use to generate our dataset.

import pandas as pd
import numpy as np
from kmodes.kprototypes import KPrototypes

np.random.seed(42)
# generated 25 random ages between 20 and 59
ages = np.random.randint(20, 60, size=25)
# generated 25 random incomes between 30 and 119 
# (thousands of dollars annually)
incomes = np.random.randint(30, 120, size=25)
# generate 25 random product categories, each with their own probability
products = np.random.choice(['Phone', 'Laptop', 'Tablet', 'Accessory'], size=25,
                            p=[0.4, 0.3, 0.2, 0.1])

df = pd.DataFrame({
    'Age': ages,
    'Income': incomes,
    'Product': products
})

# fitting k-prototypes model with several different gamma values
results = {}
for gamma in [0.1, 0.5, 1.0]:
    kproto = KPrototypes(
        n_clusters=2,
        init='Huang',
        random_state=42,
        gamma=gamma
    )
    clusters = kproto.fit_predict(df.to_numpy(), categorical=[2])
    results[gamma] = clusters

# choose gamma value and add to dataframe
chosen_gamma = 0.5
df['Cluster'] = results[chosen_gamma]

print(f"Using gamma = {chosen_gamma}")
print(df.head(10))

import pandas as pd
import numpy as np
from kmodes.kprototypes import KPrototypes

np.random.seed(42)
# generated 25 random ages between 20 and 59
ages = np.random.randint(20, 60, size=25)
# generated 25 random incomes between 30 and 119 
# (thousands of dollars annually)
incomes = np.random.randint(30, 120, size=25)
# generate 25 random product categories, each with their own probability
products = np.random.choice(['Phone', 'Laptop', 'Tablet', 'Accessory'], size=25,
                            p=[0.4, 0.3, 0.2, 0.1])

df = pd.DataFrame({
    'Age': ages,
    'Income': incomes,
    'Product': products
})

# fitting k-prototypes model with several different gamma values
results = {}
for gamma in [0.1, 0.5, 1.0]:
    kproto = KPrototypes(
        n_clusters=2,
        init='Huang',
        random_state=42,
        gamma=gamma
    )
    clusters = kproto.fit_predict(df.to_numpy(), categorical=[2])
    results[gamma] = clusters

# choose gamma value and add to dataframe
chosen_gamma = 0.5
df['Cluster'] = results[chosen_gamma]

print(f"Using gamma = {chosen_gamma}")
print(df.head(10))

Using gamma = 0.5
   Age  Income    Product  Cluster
0   58     118     Tablet        0
1   48      78      Phone        0
2   34      88      Phone        0
3   27      71     Laptop        0
4   40      89     Laptop        0
5   58     109      Phone        0
6   38      44     Laptop        1
7   42      91      Phone        0
8   30      91  Accessory        0
9   30      76      Phone        0

Above, we see the first 10 rows of our dataset of 25 random customers.

10.2.9 Scatter Plot

This code will allow us to create a scatterplot. This scatterplot will vizualize our K-Prototypes clustering, and split our customers into two clusters.

# vizualizing cluster assignments for gamma value chosen
import matplotlib.pyplot as plt

plt.figure(figsize=(8, 6))

# scatter plot by cluster
for cluster in df['Cluster'].unique():
    subset = df[df['Cluster'] == cluster]
    plt.scatter(
        subset['Age'],
        subset['Income'],
        label=f'Cluster {cluster}',
        s=100,
        edgecolor='black'
    )

plt.title(f'K-Prototypes Clustering (γ = {chosen_gamma})')
plt.xlabel('Age')
plt.ylabel('Income')
plt.legend()
plt.tight_layout()
plt.show()

# vizualizing cluster assignments for gamma value chosen
import matplotlib.pyplot as plt

plt.figure(figsize=(8, 6))

# scatter plot by cluster
for cluster in df['Cluster'].unique():
    subset = df[df['Cluster'] == cluster]
    plt.scatter(
        subset['Age'],
        subset['Income'],
        label=f'Cluster {cluster}',
        s=100,
        edgecolor='black'
    )

plt.title(f'K-Prototypes Clustering (γ = {chosen_gamma})')
plt.xlabel('Age')
plt.ylabel('Income')
plt.legend()
plt.tight_layout()
plt.show()

• Our model created two clusters:
  • Cluster 0 (Blue):
    • Contains roughly 19/25 observations
    • Wide age range (late 20s through late 50s)
    • Mostly moderate to high incomes (roughly 70-120)
  • Cluster 1 (Orange):
    • Contains roughly 6/25 observations, smaller group
    • Younger age range (20s through early 40s)
    • Mostly lower income (30-50)

10.2.10 Bar Chart For Product Counts

The following code will create a bar chart for product counts. Specifically, the raw counts of each product for each cluster, or the amount of customers in each cluster who prefer each product.

The following code will also give us the proportions of each product for each cluster, or the percentage of customers per cluster that prefer a given product.

# computing counts of products in each cluster
counts = pd.crosstab(df['Cluster'], df['Product'])
print("Raw counts:\n", counts)

# normalize counts to proportions per cluster
# divide each row by its total so values sum to 1
props = counts.div(counts.sum(axis=1), axis=0)
print("\nProportions:\n", props)

# plotting a grouped bar chart of product proportions
import matplotlib.pyplot as plt

props.plot(kind='bar', figsize=(8,5))
plt.title(f'Product Preference by Cluster (γ={chosen_gamma})')
plt.xlabel('Cluster')
plt.ylabel('Proportion of Products')
plt.legend(title='Product', bbox_to_anchor=(1.02, 1))
plt.tight_layout()
plt.show()

# computing counts of products in each cluster
counts = pd.crosstab(df['Cluster'], df['Product'])
print("Raw counts:\n", counts)

# normalize counts to proportions per cluster
# divide each row by its total so values sum to 1
props = counts.div(counts.sum(axis=1), axis=0)
print("\nProportions:\n", props)

# plotting a grouped bar chart of product proportions
import matplotlib.pyplot as plt

props.plot(kind='bar', figsize=(8,5))
plt.title(f'Product Preference by Cluster (γ={chosen_gamma})')
plt.xlabel('Cluster')
plt.ylabel('Proportion of Products')
plt.legend(title='Product', bbox_to_anchor=(1.02, 1))
plt.tight_layout()
plt.show()

Raw counts:
 Product  Accessory  Laptop  Phone  Tablet
Cluster                                  
0                2       6      8       2
1                2       1      3       1

Proportions:
 Product  Accessory    Laptop     Phone    Tablet
Cluster                                         
0         0.111111  0.333333  0.444444  0.111111
1         0.285714  0.142857  0.428571  0.142857

Bar Chart Analysis

	Accessory	Laptop	Phone	Tablet
Cluster 0	2 (11.1%)	6 (33.3%)	8 (44.4%)	2 (11.1%)
Cluster 1	2 (28.6%)	1 (14.3%)	3 (42.9%)	1 (14.3%)

• Cluster 0: (larger, higher-income & mixed-age cluster)
  • Phones are the most popular (44.4% of purchases)
  • Laptops follow (33.3% of purchases)
  • Very few Accessory and Tablet purchases
• Cluster 1: (smaller, younger & lower income cluster)
  • Phone is still the most popular (42.9%)
  • Accessory increases (28.6%)
  • Laptops drop considerably (14.3%), tablets slightly higher (14%)

We can see that both groups favor phones, although Cluster 0 purchases more laptops, and Cluster 1 purchases more accessories.

These results suggest to us that the younger/lower-income cluster is less likely to purchase the higher-priced laptops and more likely to pick smaller items, like accessories.

10.2.11 Real-World Use Cases

• Patient Profiling in Healthcare
  • Age, BMI, Cholesterol level (Numeric)
  • Smoking status, Medical history (Categorical)
• Insurance Customer Risk Grouping
  • Age, Annual Premium (Numeric)
  • Car type, Marital status (Categorical)
• Socioeconomic Grouping
  • Household income, Number of dependents (Numeric)
  • Home ownership status, Education level (categorical)

10.2.12 Conclusion

• K-Prototypes combines numerical and categorical variables into one clustering algorithm
• It is ideal for real-world datasets where not all features are numbers
• It is also customizeable, as the γ (gamma) parameter allows you to balance numerical and categorical importance

10.2.13 Further Readings

• kmodes GitHub repo
• K-Means API Reference
• Clustering Large Data Sets with Mixed Numeric and Categorical Values
• Extensions to the k-Means Algorithm for Clustering Large Data Sets with Categorical Values

10.3 Stochastic Neighbor Embedding

Stochastic Neighbor Embedding (SNE) is a dimensionality reduction technique used to project high-dimensional data into a lower-dimensional space (often 2D or 3D) while preserving local neighborhoods of points. It is particularly popular for visualization tasks, helping to reveal clusters or groupings among similar points. Key characteristics include:

• Unsupervised: It does not require labels, relying on similarity or distance metrics among data points.
• Probabilistic framework: Pairwise distances in the original space are interpreted as conditional probabilities, which SNE attempts to replicate in the lower-dimensional space.
• Common for exploratory data analysis: Especially useful for high-dimensional datasets such as images, text embeddings, or genetic data.

10.3.1 Statistical Rationale

The core idea behind SNE is to preserve local neighborhoods of each point in the data:

1. For each point \(x_i\) in the high-dimensional space, SNE defines a conditional probability \(p_{j|i}\) that represents how likely \(x_j\) is a neighbor of \(x_i\).

2. The probability \(p_{j|i}\) is modeled using a Gaussian distribution centered on \(x_i\):

   \[ p_{j|i} = \frac{\exp\left(- \| x_i - x_j \|^2 / 2 \sigma_i^2\right)}{\sum_{k \neq i} \exp\left(- \| x_i - x_k \|^2 / 2 \sigma_i^2\right)}, \] where \(\sigma_i\) is a variance parameter controlling the neighborhood size.

3. Each point \(x_i\) is mapped to a lower-dimensional counterpart \(y_i\), and a corresponding probability \(q_{j|i}\) is defined similarly in that space.

4. The objective function minimizes the Kullback–Leibler (KL) divergence between the high-dimensional and low-dimensional conditional probabilities, encouraging a faithful representation of local neighborhoods.

10.3.2 t-SNE Variation

The t-SNE (t-distributed Stochastic Neighbor Embedding) addresses two main issues in the original formulation of SNE:

• The crowding problem: In high dimensions, pairwise distances tend to spread out; in 2D or 3D, they can crowd together. t-SNE uses a Student t-distribution (with one degree of freedom) in the low-dimensional space, which has heavier tails than a Gaussian.
• Symmetric probabilities: t-SNE symmetrizes probabilities \(p_{ij} = (p_{j|i} + p_{i|j}) / (2N)\), simplifying computation.

The Student t-distribution for low-dimensional similarity is given by: \[ q_{ij} = \frac{\bigl(1 + \| y_i - y_j \|^2 \bigr)^{-1}}{\sum_{k \neq l} \bigl(1 + \| y_k - y_l \|^2 \bigr)^{-1}}. \] This heavier tail ensures that distant points are not forced too close, thus reducing the crowding effect.

10.3.3 Supervised Variation

Although SNE and t-SNE are fundamentally unsupervised, it is possible to integrate label information. In a supervised variant, distances between similarly labeled points may be reduced (or differently weighted), and additional constraints can be imposed to promote class separation in the lower-dimensional embedding. These approaches can help when partial label information is available and you want to blend supervised and unsupervised insights.

10.3.4 Demonstration with a Subset of the NIST Digits Data

Below is a brief example in Python using t-SNE on a small subset of the MNIST digits (which is itself a curated subset of the original NIST data).

import numpy as np
from sklearn.datasets import fetch_openml
from sklearn.manifold import TSNE
import matplotlib.pyplot as plt

mnist = fetch_openml('mnist_784', version=1)
X = mnist.data[:2000]
y = mnist.target[:2000]

tsne = TSNE(n_components=2, perplexity=30, learning_rate='auto', 
            init='random', random_state=42)
X_embedded = tsne.fit_transform(X)

# Create a separate scatter plot for each digit to show a legend
plt.figure()
digits = np.unique(y)
for digit in digits:
    idx = (y == digit)
    plt.scatter(
        X_embedded[idx, 0],
        X_embedded[idx, 1],
        label=f"Digit {digit}",
        alpha=0.5
    )
plt.title("t-SNE on a Subset of MNIST Digits (by class)")
plt.xlabel("Dimension 1")
plt.ylabel("Dimension 2")
plt.legend()
plt.show()

In the visualization:

• Points belonging to the same digit typically cluster together.
• Ambiguous or poorly written digits often end up bridging two clusters.
• Some digits, such as 3 and 5, may be visually similar and can appear partially overlapping in the 2D space.

10.4 Principal Component Analysis (PCA)

The following section is written by Mezmur Edo, a PhD student in the physics department. This section will focus on the motivation, intuition and theory behind PCA. It will also demonstrate the importance of scaling for proper implementation of PCA.

10.4.1 Motivation

Some of the motivations behind PCA are:

• Computation Efficiency

• Feature Extraction

• Visualization

• Curse of dimensionality

10.4.1.1 Curse of Dimensionality

The Euclidean distance between data points, which we represent as vectors, shrinks with the number of dimensions. To demonstrate this, let’s generate 10,000 vectors of n dimensions each, where n ranges from 2 to 50, with integer entries ranging from -100 to 100. By selecting a random vector, Q, of the same dimension, we can calculate the Euclidean distance of Q to each of these 10,000 vectors. The plot below shows the logarithm, to the base 10, of difference between the maximum and minimum distances divided by the minimum distance as a function of the number of dimensions.

#import libraries
import numpy as np
import matplotlib.pyplot as plt
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler, scale, normalize
import os
import math
from matplotlib.ticker import AutoMinorLocator

#define a list to store delta values
#delta is the logarithm, to the base 10, of difference between
#the maximum and minimum Euclidean distances divided 
#by the minimum distance
deltas = []

#loop through dimensions from 2 to 49
for N in range(2, 50):
  #generate 10,000 random N-dimensional vectors, P, and
  #a single random N-dimensional vector, Q
  P = [np.random.randint(-100, 100, N) for _ in range(10000)]
  Q = np.random.randint(-100, 100, N)
  
  #calculate the Euclidean distances between each point in P and Q
  diffs = [np.linalg.norm(p - Q) for p in P]
  
  #find the maximum and minimum Euclidean distances
  mxd = max(diffs)
  mnd = min(diffs)
  
  #calculate delta
  delta = math.log10(mxd - mnd) / mnd
  deltas.append(delta)

#plot delta versus N, the number of dimensions
plt.plot(range(2, 50), deltas)
plt.xlabel('Number of dimension', loc='right', fontsize=10)
plt.ylabel('Euclidean Distance', loc='top', fontsize=10)
ax = plt.gca()

#add minor locators to the axes
ax.xaxis.set_minor_locator(AutoMinorLocator())
ax.yaxis.set_minor_locator(AutoMinorLocator())

plt.show()

10.4.2 Intuition

We aim to find orthogonal directions of maximum variance in data. Directions with sufficiently low variance in the data can be removed.

rng = np.random.RandomState(0)
n_samples = 200

#generate a 2D dataset with 200 entries from 
#a multivariate normal distribution
#with covariances [[3, 3], [3, 4]]
#and mean [0, 0]
X = rng.multivariate_normal(mean=[0,0], \
cov=[[3, 3], [3, 4]], size=n_samples)

#perform PCA on the generated data to find 
#the two principal components
pca = PCA(n_components=2).fit(X)

#plot the generated data wih label 'Data'
plt.scatter(X[:,0], X[:,1], label = 'Data')

#plot the first principal component scaled by 
#its explained variance
#set color, linewidth and label
first_principal_cpt_explained_var = pca.explained_variance_[0]
first_principal_cpt = [[0, pca.components_[0][0]*first_principal_cpt_explained_var] \
, [0, pca.components_[0][1]*first_principal_cpt_explained_var]]

plt.plot(first_principal_cpt[0], first_principal_cpt[1] \
, color='green', linewidth=5 \
, label = r'First Principal Component ($p_1$)')

#plot the second principal component scaled by 
#its explained variance
#set color, linewidth and label
second_principal_cpt_explained_var = pca.explained_variance_[1]
second_principal_cpt = [[0, pca.components_[1][0]*second_principal_cpt_explained_var] \
, [0, pca.components_[1][1]*second_principal_cpt_explained_var]]

plt.plot(second_principal_cpt[0],  second_principal_cpt[1] \
, color='red', linewidth=5 \
, label = r'Second Principal Component ($p_2$)')

plt.title("")
plt.xlabel("First Feature", loc = 'right', fontsize = 10)
plt.ylabel("Second Feature", loc = 'top', fontsize = 10)
plt.legend()
plt.show()

We can then project the data onto the first principal component direction, \(p_1\).

10.4.3 Theory

Let \(x\) be a data point with features \(f_1\), \(f_2\), \(f_3\), …, \(f_n\),

\[x = \begin{pmatrix} f_1\\ f_2\\ f_3\\ .\\ .\\ .\\ f_n \end{pmatrix}. \]

The projection of x onto p is then,

\[x^{T} \frac{p}{||p||}.\]

Hence, the projection of all data points onto the principal component direction, p, can be written as,

\[\begin{pmatrix} x_1^{T} \frac{p}{||p||}\\ x_2^{T} \frac{p}{||p||}\\ x_3^{T} \frac{p}{||p||}\\ .\\ .\\ .\\ x_m^{T} \frac{p}{||p||} \end{pmatrix} = X\frac{p}{||p||},\]

where:

• X is the design matrix consisting m datapoints.

10.4.3.1 The Optimization Problem

Let \(\bar{x}\) be the sample mean vector such that,

\[\bar{x} = \frac{1}{m}\sum_{i=1}^{m}x^{(i)}.\]

The sample covariance matrix is then given by,

\[S = \frac{1}{m} X^TX - \bar{x}\bar{x}^T,\]

where:

• \(S_{ij}\) is the covarance of feature i and feature j.

For a sample mean of the projected data, \(\bar{a}\),

\[\bar{a} = \frac{1}{m}\sum_{i=1}^{m}x^{(i)T}p = \bar{x}^Tp,\]

the sample variance of the projected data can be written as,

\[\sigma^{2}= \frac{1}{m}\sum_{i=1}^{m}(x^{(i)T}p)^2 - \bar{a}^{2} = p^{T}Sp.\]

Then, our optimization problem simplifies to maximizing the sample variance,

\[\max_p \space p^{T}Sp \space s.t. ||p||=1,\]

which has the following solution,

\[Sp = \lambda p.\]

10.4.3.2 Scikit-learn Implementation

Computation can be done using the single value decomposition of X,

\[X = U \Sigma V^T.\]

If the data is mean-centered (the default option in scikit-learn), the sample covariance matrix is given by,

\[S = \frac{1}{m} X^TX = \frac{1}{m} V\Sigma U^T U \Sigma V^T = V\frac{1}{m}\Sigma^2V^T,\]

which is the eigenvalue decomposition of S, with its eigenvectors as the columns of \(V\) and the corresponding eigenvalues as diagonal entries of \(\frac{1}{m}\Sigma^2\).

The variance explained by the j-th principal component, \(p_j\), is \(\lambda_{j}\) and the total variance explained is the sum of all the eigenvalues, which is also equal to the trace of S. The total variance explained by the first k principal componentsis then given by,

\[\frac{\sum_{j=1}^{k} \lambda_j}{trace(s)}.\]

10.4.4 PCA With and Without Scaling

For proper implementation of PCA, data must be scaled. To demonstrate this, we generate a dataset with the first 4 features selected from a normal distribution with mean 0 and standard deviation 1. We then append a fifth feature drawn from a uniform distribution with integer entries ranging from 1 to 10. The plot of the projection of the data onto first principal component versus the projection onto the second principal component does not show the expected noise structure unless the data is scaled.

np.random.seed(42)

#generate a feature of size 10,000 with integer entries 
#ranging from 1 to 10
feature = np.random.randint(1, 10, 10000)
N = 10000
P = 4

#generate a 4D dataset drawn from a normal distribution of 10,000 entries
#then append the feature to X, making it a 5D dataset
X = np.random.normal(size=[N,P])
X = np.append(X, feature.reshape(10000,1), axis = 1)

#perform PCA with 2 components on the dataset without scaling
pca = PCA(2)
pca_no_scale = pca.fit_transform(X)

#plot the projection of the data onto the first principal
#component versus the projection onto 
#the second principal component
plt.scatter(pca_no_scale[:,0], pca_no_scale[:,1])
plt.title("PCA without Scaling")
plt.xlabel("Principal Component 1", loc = 'right', fontsize = 10)
plt.ylabel("Principal Component 2", loc = 'top', fontsize = 10)
plt.show()

#scale data, mean-center and divide by the standard deviation
Xn = scale(X)

#perform PCA with 2 components on the scaled data
pca = PCA(2)
pca_scale = pca.fit_transform(Xn)

#plot the projection of the data onto the first principal 
#component versus the projection onto
#the second principal component
plt.scatter(pca_scale[:,0], pca_scale[:,1])
plt.title("PCA with Scaling")
plt.xlabel("Principal Component 1", loc = 'right', fontsize = 10)
plt.ylabel("Principal Component 2", loc = 'top', fontsize = 10)
plt.show()

10.4.5 Summary

• PCA is a dimensionality reduction technique that projects data onto directions which explain the most variance in the data.

• The principal component directions are the eigenvectors of the sample covariance matrix and the corresponding eigenvalues represent the variances explained.

• For proper implementation of PCA, data must be mean-centered, scikit-learn default, and scaled.

10.4.6 Further Readings

• Principal component analysis: a review and recent developments

• Principal Components Analysis

10.5 Choosing the Optimal Number of Clusters

This section was contributed by Nicholas Pfeifer, a junior majoring in Statistics and minoring in Real Estate and Computer Science.

This section will cover the following:

• Why use clustering? What are its applications?

• K-means Clustering and Hierarchical Clustering algorithms

• How to determine the optimal number of clusters

10.5.1 Why Clustering? What is it?

Clustering is an exploratory approach looking to identify natural categories in the data. The overall goal is to Place observations into groups (“clusters”) based on similarities or patterns. It can be viewed as an Unsupervised Learning technique since the algorithm does not use a target variable to discover patterns and make groups. This is in contrast to regression, for instance, where the target variable is used in the process of generating a model. Clustering can be effective at identifying trends, patterns, or outliers in a dataset.

• Clustering is useful when…
  • the true number of clusters is not known in advance
  • working with large unlabeled data
  • looking to detect anomolies/outliers

10.5.1.1 Applications

Clustering has a plethora of applications. Some of the most popular ones are outlined below.

• Market Reasearch
  • Customer Segmentation - grouping customers by demographics or behaviors
  • Sales Analysis - based on the clusters, which groups purchase the product/service and which groups do not
• Anomaly Detection
  • Banks - combat fraud by distinguishing characteristics that stand out
• Image Segmentation
  • Identifying sections, objects, or regions of interest
  • Classify land using satellite imagery - vegetation, industrial use, etc.

10.5.2 How to measure the quality of clustering outcome

When assigning data points to clusters, there are two aspects to consider when judging the quality of the resulting clusters:

1. Intra-cluster Distance: The distance between data points within a cluster (can also be referred to as within-cluster distance)
   • The smaller the distance/variation within clusters, the better the clustering result
   • Ideally similar data points are clustered together
2. Inter-cluster Distance: The distance between data points in separate clusters (can also be referred to as between-cluster distance)
   • The larger the distance/variation between clusters, the better the clustering result
   • Ideally dissimilar data points are in different clusters

In essence, the objective is for points within a cluster to be as similar to each other as possible, and for points belonging to different clusters to be as distinct as possible.

The following code outputs two possible ways to cluster 10 observations from the MNIST handwritten digits dataset introduced in the Unsupervised Learning chapter of these class notes. The dimensionally of the observations has been reduced to 2 dimensions using t-SNE in order to make visualization easier.

from sklearn.datasets import fetch_openml
import numpy as np
import pandas as pd
from sklearn.manifold import TSNE
import matplotlib.pyplot as plt

mnist = fetch_openml('mnist_784', version=1)
mnist_example_df = pd.DataFrame(mnist.data)
mnist_example_df = mnist_example_df[:10]

tsne = TSNE(n_components=2, perplexity=5,
            learning_rate='auto',
            init='random', random_state=416)

mnist_example_df = tsne.fit_transform(mnist_example_df)

mnist_example_df = pd.DataFrame(mnist_example_df)
mnist_example_df.columns = ['dimension_1', 'dimension_2']

mnist_example_df['clustering_1'] = [1, 1, 3, 2, 3, 3, 2, 1, 2, 3]
mnist_example_df['clustering_2'] = [1, 1, 2, 2, 3, 3, 1, 3, 2, 2]

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 5))

ax1.scatter(mnist_example_df['dimension_1'],
            mnist_example_df['dimension_2'],
            c = mnist_example_df['clustering_1'],
            cmap = 'rainbow')
ax1.set_xlabel('Dimension 1')
ax1.set_ylabel('Dimension 2')
ax1.set_title('Clustering 1')

ax2.scatter(mnist_example_df['dimension_1'],
            mnist_example_df['dimension_2'],
            c = mnist_example_df['clustering_2'],
            cmap = 'rainbow')
ax2.set_xlabel('Dimension 1')
ax2.set_ylabel('Dimension 2')
ax2.set_title('Clustering 2')

plt.tight_layout();

Here are two different clusterings. Hopefully it is apparent which clustering is preferred. Clustering 1 is better than clustering 2 since points in the same cluster are closer to each other, and the clusters themselves are further apart. Some points in clustering 2 are more similar to points of other clusters than points within their own cluster. Ideally a clustering more closely resembles the result seen in clustering 1.

10.5.3 Clustering Algorithms

They are many different clustering algorithms out there, but for simplicity this section will focus on the K-means and Hierarchical clustering algorithms.

• K-means
  • Top-down approach
  • Centroid based
• Hierarchical (Agglomerative)
  • Bottom-up approach
  • Tree-like structure
• Others include:
  • K-mediods, DBSCAN, Gaussian Mixture Model, etc.

10.5.3.1 K-means Algorithm

The K-means algorithm has already been introduced in the unsupervised learning chapter, so this will serve as a brief refresher. The steps of the algorithm are as follows:

1. Must specify a number of clusters k
2. Data points are randomly assigned to k intial clusters
3. The centroid of each cluster is calculated
4. Data points are reassigned to the cluster with the closest centroid according to euclidean distance
5. Iterate the previous 2 steps until cluster assignments no longer change or a set number of iterations have been completed

from sklearn.cluster import KMeans

mnist_example_df = mnist_example_df.drop(['clustering_1', 'clustering_2'],
axis = 1)

kmeans = KMeans(n_clusters = 3, random_state = 416, 
n_init = 16).fit(mnist_example_df)

mnist_example_df['labels'] = kmeans.labels_

plt.figure(figsize=(10, 7))
plt.scatter(mnist_example_df['dimension_1'],
        mnist_example_df['dimension_2'],
        c = mnist_example_df['labels'],
        cmap = 'rainbow')
plt.scatter(kmeans.cluster_centers_[:, 0],
        kmeans.cluster_centers_[:, 1],
        marker = '*', c = 'y', label = 'Centroids',
        s = 100)

plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2')
plt.title('K-means with k = 3')
plt.legend();

Here is an example clustering result of K-means clustering with k = 3 (clusters) on the same 10 MNIST observations. The final centroids are included in the plot.

10.5.3.2 Hierarchical Clustering Algorithm

Hierarchical clustering is another algorithm which differs substantially from K-means. Something particularly of note is that the hierarchical approach does not require the number of clusters to be specified in advance. This can be seen as a drawback of K-means. The decision on the number of clusters can be made by based on the resulting tree-like structure called a Dendrogram. The steps of this algorithm are shown below:

1. Each data point is initailly assigned to its own cluster
2. Check the distance between every possible pair of clusters
3. Merge the closest pair of clusters into one cluster
4. Iterate the previous 2 steps until all of the data points are in one cluster
5. Cut the resulting Dendrogram

from sklearn.cluster import AgglomerativeClustering
from scipy.cluster.hierarchy import dendrogram, linkage

mnist_example_df = mnist_example_df.drop(['labels'], axis = 1)

fig, axs = plt.subplots(2, 2, figsize=(10, 8))

H_Clust = AgglomerativeClustering(n_clusters = None, distance_threshold = 0,
linkage = 'ward')
clusters = H_Clust.fit_predict(mnist_example_df)

clust_linkage = linkage(mnist_example_df, method = 'ward',
metric = 'euclidean')

#plt.figure(figsize=(10, 7))
dendrogram(clust_linkage, ax = axs[0, 1])
axs[0, 1].set_title('Dendrogram')
axs[0, 1].set_xlabel('Sample Index')
axs[0, 1].set_ylabel('Distance')

axs[0, 0].scatter(mnist_example_df['dimension_1'],
            mnist_example_df['dimension_2'], c=clusters, cmap='rainbow')
for i, label in enumerate(range(0, 10)):
    axs[0, 0].text(mnist_example_df['dimension_1'][i] - 3,
    mnist_example_df['dimension_2'][i], str(label),
    fontsize = 16, ha = 'right')
axs[0, 0].set_title('Hierarchical Clustering')
axs[0, 0].set_xlabel('Dimension 1')
axs[0, 0].set_ylabel('Dimension 2')


H_Clust = AgglomerativeClustering(n_clusters = 4, distance_threshold = None,
linkage = 'ward')
clusters = H_Clust.fit_predict(mnist_example_df)

clust_linkage = linkage(mnist_example_df, method = 'ward',
metric = 'euclidean')

#plt.figure(figsize=(10, 7))
dendrogram(clust_linkage, color_threshold = 70, ax = axs[1, 1])
axs[1, 1].set_title('Dendrogram')
axs[1, 1].set_xlabel('Sample Index')
axs[1, 1].set_ylabel('Distance')


axs[1, 0].scatter(mnist_example_df['dimension_1'],
            mnist_example_df['dimension_2'], c=clusters, cmap='rainbow')
for i, label in enumerate(range(0, 10)):
    axs[1, 0].text(mnist_example_df['dimension_1'][i] - 3,
    mnist_example_df['dimension_2'][i], str(label),
    fontsize = 16, ha = 'right')
axs[1, 0].set_title('Hierarchical Clustering')
axs[1, 0].set_xlabel('Dimension 1')
axs[1, 0].set_ylabel('Dimension 2')

plt.tight_layout()

plt.show()

Here is an example of Hierarchical clustering on the same 10 MNIST observations. The top row is the result when the number of clusters has not been specified. In the top left plot each data point is its own cluster. The indices of the data points can be seen in the dendrogram to the right. For this plot (top right) the colors are irrelevant. Potential clusterings of data points can be seen as the clusters are merged from bottom to top. The smaller the vertical distance the closer those clusters are to each other (and vice-versa). In the bottom row, the algorithm has been instructed to generate 4 clusters. The colors in the dendrogram do not align with those shown in the plot, so it is better to refer to the indices. Here, the dendrogram has been cut such that the closest clusters are merged together until there are 4 clusters. How to choose the height to cut the dendrogram will be discussed later on in the section.

10.5.4 Methods for selecting the optimal number of clusters

Selecting the optimal number of clusters is important since the results can be misleading if our clustering differs greatly from the true number of clusters. There are many different methods for selecting the optimal number of clusters, but for now we will delve into 4 of the most popular methods. It is important to note that no method works well in every scenario and that different methods can give differing results.

Here are the methods covered in this section:

• Inspect a Dendrogram
• Elbow Method
• Silhouette Method
• Gap Statistic

10.5.4.1 Hierarchical Clustering Example

In this example we will continue to use the MNIST dataset, however this time 2000 observations will be selected at random to be clustered.

from sklearn.datasets import fetch_openml
import numpy as np
import pandas as pd
from sklearn.utils import resample

# Fetching NIST dataset
mnist = fetch_openml('mnist_784', version=1)

mnist_df = pd.DataFrame(mnist.data)

# Taking a random sample of 2000 images
mnist_rand = resample(mnist_df, n_samples = 2000, random_state = 416)

mnist_rand = mnist_rand.reset_index().drop('index', axis = 1)

# Keeping track of the target values
mnist_target_df = pd.DataFrame(mnist.target)
mnist_target_rand = resample(mnist_target_df,
                             n_samples = 2000,
                             random_state = 416)
mnist_target_rand = mnist_target_rand.reset_index().drop('index', axis = 1)

# Distribution is fairly even
mnist_target_rand['class'].value_counts()

class
1    211
3    209
2    208
5    202
9    202
8    201
6    198
7    197
0    189
4    183
Name: count, dtype: int64

The distribution of the 2000 randomly sampled handwritten digits is shown above. The distribution of the digitsappears to be fairly evenly distributed.

Once again, the dimensionality of these images is reduced to 2 dimensions using t-SNE.

from sklearn.manifold import TSNE
import matplotlib.pyplot as plt

# t-SNE dimensionality reduction
tsne = TSNE(n_components=2, perplexity=30,
            learning_rate='auto',
            init='random', random_state=416)

mnist_embedded = tsne.fit_transform(mnist_rand)

mnist_embedded_df = pd.DataFrame(mnist_embedded)
mnist_embedded_df.columns = ['dimension_1', 'dimension_2']

plt.figure(figsize=(10, 7))
plt.scatter(mnist_embedded_df['dimension_1'],
            mnist_embedded_df['dimension_2'])
plt.title('Random Sample of 2000 MNIST Digits')
plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2');

Here is a scatterplot of the 2000 randomly sampled images above without looking at their actual labels.

from sklearn.cluster import AgglomerativeClustering
from scipy.cluster.hierarchy import dendrogram, linkage

H_Clust =  AgglomerativeClustering(n_clusters = None, distance_threshold = 0,
linkage = 'ward')
clusters = H_Clust.fit_predict(mnist_embedded_df)

clust_linkage = linkage(mnist_embedded_df, method = 'ward')

plt.figure(figsize=(10, 7))
dendrogram(clust_linkage)
plt.title('Dendrogram')
plt.xlabel('Sample Index')
plt.ylabel('Distance')
plt.show()

After conducting Hierarchical clustering without specifying the number of clusters, we have a dendrogram. Now comes the decision of where to make a horizontal cut. There is a paper about “dynamic cuts” that are flexible and do not cut at a constant height, but that is outside of the current scope (Langfelder et al. (2008)). When looking at the dendrogram above, suppose we do not know the true number of clusters. Generally, when cutting the tree, we want the resulting clusters to be around the same height. Vertical distance represents dissimilarity, so we do not want clusters of high disimilarity to be merged together. Remember that good clustering involves small distances within clusters and large distances between clusters. This is a subjective approach and sometimes it may be difficult to find the best height to cut the dendrogram. Perhaps with domain knowledge a predefined threshold could be a good height at which to cut. For this example I chose to cut the tree at a height of 200. That resulted in 11 clusters which will be analyzed below.

from scipy.cluster.hierarchy import cut_tree

# cut the tree
new_clusters = cut_tree(clust_linkage, height = 200)

mnist_embedded_df['cluster'] = new_clusters

# Plot the new clusters
plt.figure(figsize=(10, 7))
plt.scatter(mnist_embedded_df['dimension_1'],
            mnist_embedded_df['dimension_2'], c=mnist_embedded_df['cluster'], 
            cmap='rainbow')
plt.title('Hierarchical Clustering (11 clusters)')
plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2')
plt.show()

Here are the 11 clusters obtained after cutting the tree. What do these clusters signify? Maybe by adding some labels to the clusters, that will become more clear.

# Plot clusters with labels (cluster labels not actual!)

plt.figure(figsize=(10, 7))
scatter = plt.scatter(mnist_embedded_df['dimension_1'],
             mnist_embedded_df['dimension_2'], c=mnist_embedded_df['cluster'], 
             cmap='rainbow')
plt.title('Hierarchical Clustering (11 clusters)')
plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2')
legend1 = plt.legend(*scatter.legend_elements(), title="Cluster")
plt.gca().add_artist(legend1)
plt.show()

Now the clusters have been associated with their cluster label, but this does not represent the actual handwritten digits.

In this case it is hard to determine what the clusters signify if the target values are unknown. However we do know the target value (actual handwritten digit) for each image. This information can help to label the clusters and make them more interpretable.

mnist_embedded_df['actual'] = mnist_target_rand['class']

# calculating mode and proportion of observations in the cluster that are the 
# mode in each cluster
modes = mnist_embedded_df.groupby('cluster').agg(
            {'actual': [lambda x: x.mode().iloc[0],
             lambda y: (y == y.mode().iloc[0]).sum()/len(y)]})

modes.columns = ['mode', 'proportion']
modes

	mode	proportion
cluster
0	4	0.482394
1	8	0.680180
2	3	0.726923
3	0	0.942708
4	7	0.490196
5	6	0.936893
6	7	0.732394
7	5	0.897436
8	2	0.853846
9	1	0.892157
10	1	0.519481

This code above calculates the mode digit of each cluster along with the proportion of observations in the cluster that are the mode. Now let’s label the clusters by their mode.

# Plot clusters with (actual) labels (modes)

new_labels = modes['mode']

plt.figure(figsize=(10, 7))
scatter = plt.scatter(mnist_embedded_df['dimension_1'],
             mnist_embedded_df['dimension_2'], c=mnist_embedded_df['cluster'], 
             cmap='rainbow')
plt.title('Hierarchical Clustering (11 clusters)')
plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2')

handles, _ = scatter.legend_elements()
plt.legend(handles, new_labels, title="Mode")

plt.show()

Now we can get a better understanding of the clustering. Although there are 11 clusters in total, you will notice that every digit does not appear as the mode of a cluster. 9 is not the mode of any cluster while 4 and 7 are the modes of multiple clusters. At the very least the clusters with 4 and 7 as the mode are very close to each other. Also intuitively the digits 0, 6, and 8 are written similarly, so it makes sense to see those clusters in the same general area.

Just out of curiosity, let’s look at the actual distribution of the digits.

# Showing the actual distribution of classes

mnist_embedded_df['actual'] = mnist_embedded_df['actual'].astype('int64')

plt.figure(figsize=(10, 7))
scatter = plt.scatter(mnist_embedded_df['dimension_1'],
            mnist_embedded_df['dimension_2'],
            c = mnist_embedded_df['actual'], cmap='rainbow')
plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2')
plt.title('True Distribution of Target')
legend1 = plt.legend(*scatter.legend_elements(), title="Value")
plt.gca().add_artist(legend1);

Analyzing the cluster performance by viewing the actual distribution of the target is becoming de facto supervized learning, but not really since the clustering algorithm does not know or use the information of the target. For the purposes of this section it is just to see how well the clustering found the true clusters. For the most part it looks like the clustering did a moderately good job at identifying the true clusters of the digits in 2 dimensions. The digits 4, 7, and 9 seem to be very similar in 2D and is understandably more difficult for the algorithm to distinguish.

Since the true number of clusters is known, let’s see what it looks like with 10 clusters just out of curiosity again.

# Try cutting with 10 clusters instead
new_clusters = cut_tree(clust_linkage, n_clusters=10)
mnist_embedded_df['cluster'] = new_clusters

modes = mnist_embedded_df.groupby('cluster').agg(
            {'actual': [lambda x: x.mode().iloc[0],
             lambda y: (y == y.mode().iloc[0]).sum()/len(y),
             lambda x: x.value_counts().index[1],
             lambda y: (y == y.value_counts().index[1]).sum()/len(y)]})

modes.columns = ['mode', 'proportion', 'mode_2', 'proportion_2']

# Plot clusters with mode labels

new_labels = modes['mode']

plt.figure(figsize=(10, 7))
scatter = plt.scatter(mnist_embedded_df['dimension_1'],
             mnist_embedded_df['dimension_2'], c=mnist_embedded_df['cluster'],
             cmap='rainbow')
plt.title('Hierarchical Clustering (10 clusters)')
plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2')

handles, _ = scatter.legend_elements()
plt.legend(handles, new_labels, title="Mode")

plt.show()

The difference appears to be that the two clusters with 4 as mode merged into one cluster.

modes

	mode	proportion	mode_2	proportion_2
cluster
0	4	0.482394	9	0.454225
1	8	0.680180	5	0.274775
2	3	0.726923	5	0.192308
3	0	0.942708	2	0.031250
4	7	0.490196	9	0.385621
5	6	0.936893	0	0.029126
6	7	0.732394	4	0.232394
7	5	0.897436	3	0.064103
8	2	0.853846	8	0.084615
9	1	0.633634	2	0.243243

This table contains the mode of each cluster as well as the second most common value in each cluster denoted at mode_2. Interestingly, 9 appears as the second most common value in 3 different clusters.

10.5.4.2 K-means Clustering Example: Elbow Method

For the next 3 methods the K-means algorithm will be used on the same random 2000 MNIST images in 2 dimensions.

The goal of the Elbow method is to minimize the within cluster sum of squares (WSS), which is also refered to as inertia. The optimal number of clusters is K such that adding another cluster does not (significantly) improve WSS. Whenever the number of clusters increases, inertia will decrease since there are fewer points in each cluster that become closer to their cluster’s center. The idea of the Elbow method is that the rate of decrease in WSS changes based on the optimal number of clusters, K. When k < K, (approaching optimal number) inertia decreases rapidly. When k > K, (going past optimal number) inertia decreases slowly. K is found by plotting inertia over a range of k and looking for a bend or “elbow”, hence the name.

# K-means
from sklearn.cluster import KMeans

# removing non-nist columns
mnist_embedded_df = mnist_embedded_df.drop(['cluster', 'actual'], axis = 1)

# elbow method for k between 1 and 20 on same MNIST data

wcss = []

for k in range(1, 21):
     model = KMeans(n_clusters = k, random_state = 416).fit(mnist_embedded_df)
     wcss.append(model.inertia_)

plt.figure(figsize=(10, 7))
plt.plot(range(1, 21), wcss, 'bx-')
plt.xlabel('Number of Clusters (k)')
plt.ylabel('Within-Cluster Sum of Squares')
plt.title('Elbow Method')
plt.show()
# Seems inconclusive, maybe 7?

The code above stores the inertia for values of k between 1 and 20, and creates the plot. In this case it is somewhat inconclusive. It looks like the decrease in inertia starts to slow down at k = 7. Like with the dendrogram, this method is also subjective.

10.5.4.3 K-means Clustering Example: Silhouette Method

Next is the Silhouette method, which is the most objective method of the 4 covered in this section

Silhouette Score

Before delving into the Silhouette method, it is good to get an understanding of Silhouette Score. The silhouette s of a data point is, \[s = (b-a)/\max(a, b).\]

• At each data point, the distance to its cluster’s center = a
• And the distance to the second best cluster center = b
  • Second best suggests closest cluster that is not the current cluster
• s can take any value between -1 and 1

Interpreting Silhouette Score

There are 3 main categories that a data point can fall into:

• If a data point is very close to its own cluster and very far from the second best cluster (a is small, and b is big), then s is close to 1 (close to \(b/b\))
• If a data point is roughly the same distance to its own cluster as the second best cluster (\(a \approx b\)), then s \(\approx\) 0
• If a data point is very far from its own cluster and very close to the second best cluster (a is big, and b is small), then s is close to -1 (close to -a/a)

For optimal clustering, we want most data points to fall into the first category. In other words we want silhouette scores to be as close to 1 as possible.

Silhouette Coefficient

The Silhouette Coefficient is represented by the average silhouette score of the data points. This metric does a good job of summarizing both within-cluster and between-cluster variation. The closer the Silhouette Coefficient is to 1, the better the clustering. Similar to the Elbow method, the optimal K is selected by calculating the Silhouette Coefficient over a range of k’s, and choosing K with the maximum Silhouette Coefficient.

# Silhoutte method
from sklearn.metrics import silhouette_score

silhouette_average_scores = []

for k in range (2, 21):
    kmeans = KMeans(n_clusters = k, random_state = 416)
    cluster_labels = kmeans.fit_predict(mnist_embedded_df)

    silhouette_avg = silhouette_score(mnist_embedded_df, cluster_labels)
    silhouette_average_scores.append(silhouette_avg)

# Plot silhouette scores
plt.figure(figsize=(10, 7))
plt.plot(list(range(2,21)), silhouette_average_scores, marker='o')
plt.title("Silhouette Coefficients")
plt.xlabel("Number of Clusters (k)")
plt.ylabel("Average Silhouette Score")
plt.show()
# k = 7 has the highest average silhouette score

Here the Silhouette Coefficient is calculated for k between 2 and 20. The maximum occurs at k = 7, which is coincidentally the same result as the Elbow method. Let’s visualize how these 7 clusters look on our 2000 MNIST digits.

kmeans = KMeans(n_clusters = 7, random_state = 416)
cluster_labels = kmeans.fit_predict(mnist_embedded_df)
mnist_embedded_df['cluster'] = cluster_labels

# K-means with k= 7 (cluster labels, not actual!)
plt.figure(figsize=(10, 7))
scatter = plt.scatter(mnist_embedded_df['dimension_1'],
             mnist_embedded_df['dimension_2'], c=mnist_embedded_df['cluster'],
             cmap='rainbow')
plt.title('K-means with k = 7 clusters')
plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2')
legend1 = plt.legend(*scatter.legend_elements(), title="Cluster")
plt.gca().add_artist(legend1)
plt.show()

These are just the cluster labels, not the actual digits.

mnist_embedded_df['actual'] = mnist_target_rand['class']

modes = mnist_embedded_df.groupby('cluster').agg(
            {'actual': [lambda x: x.mode().iloc[0],
             lambda y: (y == y.mode().iloc[0]).sum()/len(y),
             lambda x: x.value_counts().index[1],
             lambda y: (y == y.value_counts().index[1]).sum()/len(y)]})

modes.columns = ['mode', 'proportion', 'mode_2', 'proportion_2']
modes

	mode	proportion	mode_2	proportion_2
cluster
0	1	0.655280	2	0.248447
1	8	0.458781	2	0.390681
2	7	0.550769	4	0.215385
3	6	0.726592	5	0.205993
4	3	0.606349	5	0.228571
5	9	0.451957	4	0.380783
6	0	0.862559	5	0.075829

Here are the modes which can be used to label the 7 clusters.

# Plot clusters with (actual) labels (modes)
new_labels = modes['mode']

plt.figure(figsize=(10, 7))
scatter = plt.scatter(mnist_embedded_df['dimension_1'],
             mnist_embedded_df['dimension_2'], c=mnist_embedded_df['cluster'],
             cmap='rainbow')
plt.title('K-means with k = 7 clusters')
plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2')

handles, _ = scatter.legend_elements()
plt.legend(handles, new_labels, title="Mode")

plt.show()

Now we have the actual mode labels of the 7 clusters obtained from K-means. Interestingly, the area that used to have 4 as the label now has 9. Now this digits that do not appear as the mode in any cluster are 4, 5, and 8. Looking back at the modes table we see that these digits frequently appear as the second most common value in a cluster at a high rate. Obviously 7 is not the true number of clusters, but perhaps the 2D representation is obscuring the ability to find disimilarities between some of the digits.

# Showing the actual distribution of classes

mnist_embedded_df['actual'] = mnist_embedded_df['actual'].astype('int64')

plt.figure(figsize=(10, 7))
scatter = plt.scatter(mnist_embedded_df['dimension_1'],
            mnist_embedded_df['dimension_2'],
            c = mnist_embedded_df['actual'], cmap='rainbow')
plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2')
plt.title('True Distribution of Target')
legend1 = plt.legend(*scatter.legend_elements(), title="Value")
plt.gca().add_artist(legend1);

For reference, here is the true distribution of the handwritten digits again.

10.5.4.4 K-means Clustering Example: Gap Statistic

The last method to be covered is the Gap Statistic. The Gap Statistic for a number of clusters k can be written as

\[Gap(k) = \frac{1}{B}\sum_{b=1}^{B} \log(W_{kb}) - \log(W_k).\]

• Compares the total (within) intra-cluster variation for a range of k’s with their expected values
• Calculated by comparing the inertia of a clustered dataset with the inertia of a uniformly distributed random data set (covering the same ranges in the data space)
• A number of random samples (B) are generated that are then clustered over a range of k’s while keeping track of the inertia
• \(W_{kb}\) is the inertia of the b-th random sample with k clusters and \(W_k\) is the inertia of the original data with k clusters

We also need the standard deviation,

\[s_k = \sqrt{1 + \frac{1}{B}}\sqrt{\frac{1}{B}\sum_{b=1}^{B} (\log(W_{kb}) - \overline{W})^2}.\]

Where

\[\overline{W} = \frac{1}{B}\sum_{b=1}^{B} \log(W_{kb}).\]

Choose the smallest k such that the gap statistic is within one standard deviation of the gap at k + 1.

This can be represented by the expression,

\[Gap(k) \geq Gap(k+1) - s_{k+1}.\]

The optimal k may vary over multiple gap statistic simulations since there is randomness involved.

# gap statistic

# removing non-nist columns
mnist_embedded_df = mnist_embedded_df.drop(['cluster', 'actual'], axis = 1)

def calc_gap_statistic(data, max_k, n = 10):
    # Generate reference data from a uniform distribution
    def generate_reference_data(X):
        return np.random.uniform(low = data.min(axis=0),
        high = data.max(axis=0),
        size=X.shape)

    gap_values = []

    # Loop over a range of k values
    for k in range(1, max_k + 1):
        # Fit K-means to the original data
        kmeans = KMeans(n_clusters = k, random_state = 416)
        kmeans.fit(data)
        original_inertia = kmeans.inertia_
    
        # Compute the average inertia for the reference datasets
        reference_inertia = []
        for _ in range(n):
            random_data = generate_reference_data(data)
            kmeans.fit(random_data)
            reference_inertia.append(kmeans.inertia_)
        
        # Calculate the Gap statistic
        gap = np.log(np.mean(reference_inertia)) - np.log(original_inertia)
        gap_values.append(gap)

    return gap_values

gap_values = calc_gap_statistic(mnist_embedded_df, 20, n = 100)

plt.figure(figsize=(10, 7))
plt.plot(range(1, 21), gap_values, marker='o')
plt.title('Gap Statistic vs Number of Clusters')
plt.xlabel('Number of Clusters (k)')
plt.ylabel('Gap Statistic')
plt.grid()
plt.show()
# 2 is the best?

Here a function is defined to calculate the gap statistic. It is calculated for k between 1 and 20 with B = 100 random datasets (the more datasets that are used, the more computationally expensive). In the plot we are looking for the k where the gap statistic is greater than at k + 1 minus standard deviation. In this case we do not even need standard deviation since we observe that Gap(2) is greater than Gap(3). This means that the optimal K is 2 based on this method.

This process can also be conducted using the gapstatistics package.

pip install gapstatistics

from gapstatistics import gapstatistics

gs = gapstatistics.GapStatistics(distance_metric='euclidean')

optimal = gs.fit_predict(K = 20, X = np.array(mnist_embedded_df))

print(f'Optimal: {optimal}')

Optimal: 2

The result is also an optimal K of 2. It appears that this method is not very good for this dataset.

kmeans = KMeans(n_clusters = 2, random_state = 416)
cluster_labels = kmeans.fit_predict(mnist_embedded_df)
mnist_embedded_df['cluster'] = cluster_labels

# Cluster labels!
plt.figure(figsize=(10, 7))
scatter = plt.scatter(mnist_embedded_df['dimension_1'],
             mnist_embedded_df['dimension_2'], c=mnist_embedded_df['cluster'],
             cmap='rainbow')
plt.title('K-means with k = 2 clusters')
plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2')
legend1 = plt.legend(*scatter.legend_elements(), title="Cluster")
plt.gca().add_artist(legend1)
plt.show()

Here we have K-means with k = 2 with default cluster labels.

mnist_embedded_df['actual'] = mnist_target_rand['class']

modes = mnist_embedded_df.groupby('cluster').agg(
            {'actual': [lambda x: x.mode().iloc[0],
             lambda y: (y == y.mode().iloc[0]).sum()/len(y),
             lambda x: x.value_counts().index[1],
             lambda y: (y == y.value_counts().index[1]).sum()/len(y)]})

modes.columns = ['mode', 'proportion', 'mode_2', 'proportion_2']

# Plot clusters with labels (actual)
new_labels = modes['mode']

plt.figure(figsize=(10, 7))
scatter = plt.scatter(mnist_embedded_df['dimension_1'],
             mnist_embedded_df['dimension_2'], c=mnist_embedded_df['cluster'],
             cmap='rainbow')
plt.title('K-means with k = 2 clusters')
plt.xlabel('Dimension 1')
plt.ylabel('Dimension 2')

handles, _ = scatter.legend_elements()
plt.legend(handles, new_labels, title="Mode")

plt.show()

Here are the mode labels but that does not tell us very much.

mnist_embedded_df[mnist_embedded_df['cluster'] == 0]['actual'].value_counts()
# 1, 9, 7, 4, and 2 are similar

actual
1    211
9    197
7    196
4    181
2    109
5     77
8     38
3     11
6      1
0      0
Name: count, dtype: int64

mnist_embedded_df[mnist_embedded_df['cluster'] == 1]['actual'].value_counts()
# 3, 6, 0, 8, and 5 are similar

actual
3    198
6    197
0    189
8    163
5    125
2     99
9      5
4      2
7      1
1      0
Name: count, dtype: int64

At the very least we can see which images of handwritten digits look similar in 2 dimensions.

10.5.5 Conclusions

• Using clustering we can figure out which digits look similar to each other when writing by hand
• The true number of clusters in 2D may be different than in the original dimensions. Maybe the algorithms would be better at identifying the different clusters of the MNIST data in 3D
• Choosing the right number of clusters can be challenging but is very important
• There are many methods for selecting the optimal number of clusters and they can yield different results

10.5.6 Further Readings

Defining clusters from a hierarchical cluster tree

sklearn AgglomerativeClustering Documentation

gapstatistics PyPI Documentation

How many Clusters? - Towards Data Science

10.6 Autoencoders

This section was written by Kyle Reed, a senior at the University of Connecticut double majoring in Applied Data Analysis and Geographic Information Science.

This section will explore:

• What an Autoencoder is

• How an Autoencoder works

• Potential applications of autoencoders

10.6.1 Introduction

Autoencoders are a specialized type of neural network used in unsupervised learning. They are trained to encode input data into a compressed representation and then decode it back to something as close as possible to the original. This process forces the model to learn the most important features or patterns in the data.

Unlike traditional supervised learning models, autoencoders do not require labeled data. Instead, the model learns from the data itself by minimizing the difference between the original input and the reconstructed output.

10.6.2 How it works

An autoencoder consists of two primary components:

• Encoder: Deconstructs the input data into a lower-dimensional representation.

• Decoder: Reconstructs the original input from the compressed encoding.

10.6.2.1 Encoder

The encoder compresses data into smaller lower-dimensional groupings. It learns the data and identifies the most essential features of the data while discarding redundant information.

10.6.2.2 Decoder

The decoder reconstructs the data set from the compressed analyzed data generated by the encoder. The decoder attempts to reproduce the data as closely as possible by reversing the compression process.

10.6.3 Application

Autoencoders can be used for:

• Data Compression Reduce dataset size for storage and transmission while retaining key information.

• Anomaly Detection Identify unusual patterns that differ from the learned norm based on reconstruction error.

• Image/Audio Refining
  Remove noise, fill missing pixels or sound samples, colorize images, and more.

• Data Refining/Denoising
  Improve dataset quality by correcting errors and filling missing values.

10.6.4 Example usage

For the example, I wish show how autoencoders can be used to compress data and refine data/images. The mnist data set will be used which is a collection of various numbers that are drawn out on a small pixel image.

10.6.4.1 Load and Prepare Data

#Import necessary packages
from tensorflow.keras.datasets import mnist
from tensorflow.keras import layers, models
import numpy as np
import matplotlib.pyplot as plt

#Load data from dataset
(X_train, _), (X_test, _) = mnist.load_data()

#Ensure proper format and divide by 255 to normalize data
X_train = X_train.astype("float32") / 255.
X_test = X_test.astype("float32") / 255.

#Reshape the image to make it one-dimensional
X_train = X_train.reshape((len(X_train), -1))
X_test = X_test.reshape((len(X_test), -1))

2025-05-06 14:59:56.288201: I tensorflow/core/platform/cpu_feature_guard.cc:210] This TensorFlow binary is optimized to use available CPU instructions in performance-critical operations.
To enable the following instructions: AVX2 FMA, in other operations, rebuild TensorFlow with the appropriate compiler flags.

Here I used Tensorflow Keras package which is one of the most common and easy to use frameworks people use for building and training autoencoders.

Data is broken into train and test data, and converted to floating-point format. Also, the data is divided by 255 as that is the maximum value for the grey scale.

Reshaping images as seen in the last two lines allows to neural network to compress the data easier when it is one-dimensional.

10.6.4.2 Build The Encoder

#Set the input size to the number of pixels for each image
input_dim = X_train.shape[1]

#Create and define elements of the autoencoder
autoencoder = models.Sequential([
    layers.Input(shape=(input_dim,)),
    layers.Dense(256, activation='relu'),  #Initial encoding
    layers.Dense(128, activation='relu'),   #Compressed version (Bottleneck)
    layers.Dense(256, activation='relu'),  #Reconstruction
    layers.Dense(input_dim, activation='sigmoid')  #Final reconstructed version
])

For each of the activation lines, the number represents the number of neurons for each layer, so ‘256’ means that this layer transforms the input into a 256-dimensional representation

When choosing the number of neurons, you want to pick a number that fits your data well. More neurons are needed for larger complex models, but they are not necessary for smaller, less complex models as this could cause overfitting.

10.6.4.3 Train the encoder

#Prepare the autoencoder with the optomizer 
autoencoder.compile(optimizer='adam', loss='binary_crossentropy')

#Train the autoencoder
autoencoder.fit(X_train, X_train,
                epochs=40,
                batch_size=256,
                shuffle=True,
                validation_data=(X_test, X_test))

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235/235 ━━━━━━━━━━━━━━━━━━━━ 8s 27ms/step - loss: 0.2903 - val_loss: 0.1134

Epoch 2/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 13s 59ms/step - loss: 0.1185

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.1175 

  8/235 ━━━━━━━━━━━━━━━━━━━━ 3s 17ms/step - loss: 0.1170

 11/235 ━━━━━━━━━━━━━━━━━━━━ 4s 19ms/step - loss: 0.1166

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205/235 ━━━━━━━━━━━━━━━━━━━━ 0s 21ms/step - loss: 0.1089

207/235 ━━━━━━━━━━━━━━━━━━━━ 0s 21ms/step - loss: 0.1089

209/235 ━━━━━━━━━━━━━━━━━━━━ 0s 21ms/step - loss: 0.1088

212/235 ━━━━━━━━━━━━━━━━━━━━ 0s 21ms/step - loss: 0.1088

215/235 ━━━━━━━━━━━━━━━━━━━━ 0s 21ms/step - loss: 0.1087

218/235 ━━━━━━━━━━━━━━━━━━━━ 0s 21ms/step - loss: 0.1086

221/235 ━━━━━━━━━━━━━━━━━━━━ 0s 21ms/step - loss: 0.1085

225/235 ━━━━━━━━━━━━━━━━━━━━ 0s 21ms/step - loss: 0.1085

229/235 ━━━━━━━━━━━━━━━━━━━━ 0s 20ms/step - loss: 0.1084

232/235 ━━━━━━━━━━━━━━━━━━━━ 0s 20ms/step - loss: 0.1083

235/235 ━━━━━━━━━━━━━━━━━━━━ 0s 20ms/step - loss: 0.1082

235/235 ━━━━━━━━━━━━━━━━━━━━ 5s 22ms/step - loss: 0.1082 - val_loss: 0.0937

Epoch 3/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 11s 49ms/step - loss: 0.0971

  4/235 ━━━━━━━━━━━━━━━━━━━━ 3s 17ms/step - loss: 0.0960 

  8/235 ━━━━━━━━━━━━━━━━━━━━ 3s 16ms/step - loss: 0.0957

 11/235 ━━━━━━━━━━━━━━━━━━━━ 3s 17ms/step - loss: 0.0955

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235/235 ━━━━━━━━━━━━━━━━━━━━ 5s 19ms/step - loss: 0.0930 - val_loss: 0.0873

Epoch 4/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 9s 40ms/step - loss: 0.0869

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0874

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230/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0866

234/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0866

235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0866 - val_loss: 0.0826

Epoch 5/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 44ms/step - loss: 0.0790

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0804 

  9/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0812

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235/235 ━━━━━━━━━━━━━━━━━━━━ 5s 19ms/step - loss: 0.0824 - val_loss: 0.0795

Epoch 6/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 2:28 634ms/step - loss: 0.0806

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235/235 ━━━━━━━━━━━━━━━━━━━━ 0s 21ms/step - loss: 0.0799

235/235 ━━━━━━━━━━━━━━━━━━━━ 6s 23ms/step - loss: 0.0799 - val_loss: 0.0778

Epoch 7/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 11s 49ms/step - loss: 0.0792

  4/235 ━━━━━━━━━━━━━━━━━━━━ 4s 18ms/step - loss: 0.0784 

  7/235 ━━━━━━━━━━━━━━━━━━━━ 4s 18ms/step - loss: 0.0781

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 19ms/step - loss: 0.0781 - val_loss: 0.0762

Epoch 8/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 43ms/step - loss: 0.0775

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235/235 ━━━━━━━━━━━━━━━━━━━━ 5s 19ms/step - loss: 0.0768 - val_loss: 0.0753

Epoch 9/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 43ms/step - loss: 0.0752

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0762 

  9/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0764

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235/235 ━━━━━━━━━━━━━━━━━━━━ 5s 19ms/step - loss: 0.0757 - val_loss: 0.0744

Epoch 10/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 9s 42ms/step - loss: 0.0755

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224/235 ━━━━━━━━━━━━━━━━━━━━ 0s 16ms/step - loss: 0.0745

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 18ms/step - loss: 0.0745 - val_loss: 0.0735

Epoch 11/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 13s 57ms/step - loss: 0.0757

  4/235 ━━━━━━━━━━━━━━━━━━━━ 3s 17ms/step - loss: 0.0751 

  7/235 ━━━━━━━━━━━━━━━━━━━━ 4s 18ms/step - loss: 0.0748

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 17/235 ━━━━━━━━━━━━━━━━━━━━ 3s 17ms/step - loss: 0.0743

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215/235 ━━━━━━━━━━━━━━━━━━━━ 0s 17ms/step - loss: 0.0738

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 18ms/step - loss: 0.0738 - val_loss: 0.0728

Epoch 12/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 11s 48ms/step - loss: 0.0729

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 17ms/step - loss: 0.0728 

  8/235 ━━━━━━━━━━━━━━━━━━━━ 5s 25ms/step - loss: 0.0730

  9/235 ━━━━━━━━━━━━━━━━━━━━ 7s 34ms/step - loss: 0.0730

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223/235 ━━━━━━━━━━━━━━━━━━━━ 0s 19ms/step - loss: 0.0731

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230/235 ━━━━━━━━━━━━━━━━━━━━ 0s 19ms/step - loss: 0.0731

234/235 ━━━━━━━━━━━━━━━━━━━━ 0s 19ms/step - loss: 0.0731

235/235 ━━━━━━━━━━━━━━━━━━━━ 5s 20ms/step - loss: 0.0731 - val_loss: 0.0723

Epoch 13/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 1:22 354ms/step - loss: 0.0725

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0723   

  9/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0723

 13/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0723

 17/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0724

 21/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0724

 25/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0724

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0723 - val_loss: 0.0733

Epoch 14/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 46ms/step - loss: 0.0740

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0736 

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 19ms/step - loss: 0.0722 - val_loss: 0.0712

Epoch 15/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 13s 58ms/step - loss: 0.0720

  4/235 ━━━━━━━━━━━━━━━━━━━━ 12s 56ms/step - loss: 0.0716

  5/235 ━━━━━━━━━━━━━━━━━━━━ 12s 56ms/step - loss: 0.0716

  9/235 ━━━━━━━━━━━━━━━━━━━━ 8s 36ms/step - loss: 0.0717 

 13/235 ━━━━━━━━━━━━━━━━━━━━ 6s 29ms/step - loss: 0.0717

 17/235 ━━━━━━━━━━━━━━━━━━━━ 5s 26ms/step - loss: 0.0717

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 18ms/step - loss: 0.0714 - val_loss: 0.0708

Epoch 16/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 9s 42ms/step - loss: 0.0707

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0710 - val_loss: 0.0707

Epoch 17/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 9s 40ms/step - loss: 0.0713

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0715

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 16ms/step - loss: 0.0708 - val_loss: 0.0701

Epoch 18/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 46ms/step - loss: 0.0726

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0705 - val_loss: 0.0701

Epoch 19/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 11s 49ms/step - loss: 0.0707

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0706 

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0700 - val_loss: 0.0698

Epoch 20/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 43ms/step - loss: 0.0696

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0702 

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205/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0700

209/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0700

212/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0700

215/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0700

218/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0700

221/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0699

224/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0699

227/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0699

231/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0699

235/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0699

235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0699 - val_loss: 0.0696

Epoch 21/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 11s 48ms/step - loss: 0.0717

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0703 

  9/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0703

 13/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0702

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214/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0697

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230/235 ━━━━━━━━━━━━━━━━━━━━ 0s 15ms/step - loss: 0.0697

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 16ms/step - loss: 0.0697 - val_loss: 0.0692

Epoch 22/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 5:15 1s/step - loss: 0.0693

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 13ms/step - loss: 0.0691

 10/235 ━━━━━━━━━━━━━━━━━━━━ 2s 13ms/step - loss: 0.0689

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235/235 ━━━━━━━━━━━━━━━━━━━━ 6s 18ms/step - loss: 0.0692 - val_loss: 0.0690

Epoch 23/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 12s 55ms/step - loss: 0.0691

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0693 - val_loss: 0.0690

Epoch 24/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 43ms/step - loss: 0.0687

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0686 

  7/235 ━━━━━━━━━━━━━━━━━━━━ 8s 35ms/step - loss: 0.0685

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235/235 ━━━━━━━━━━━━━━━━━━━━ 5s 19ms/step - loss: 0.0690 - val_loss: 0.0687

Epoch 25/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 9s 42ms/step - loss: 0.0687

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 18ms/step - loss: 0.0688 - val_loss: 0.0693

Epoch 26/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 44ms/step - loss: 0.0689

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0693 

  9/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0693

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 18ms/step - loss: 0.0687 - val_loss: 0.0688

Epoch 27/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 44ms/step - loss: 0.0676

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 18ms/step - loss: 0.0685 - val_loss: 0.0683

Epoch 28/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 9s 40ms/step - loss: 0.0698

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0691

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 16ms/step - loss: 0.0683 - val_loss: 0.0688

Epoch 29/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 11s 51ms/step - loss: 0.0704

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0696 

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0684 - val_loss: 0.0681

Epoch 30/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 46ms/step - loss: 0.0654

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0666 

  9/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0672

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 16ms/step - loss: 0.0681 - val_loss: 0.0682

Epoch 31/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 44ms/step - loss: 0.0668

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0681 - val_loss: 0.0678

Epoch 32/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 13s 59ms/step - loss: 0.0686

  4/235 ━━━━━━━━━━━━━━━━━━━━ 5s 22ms/step - loss: 0.0681 

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 16ms/step - loss: 0.0678 - val_loss: 0.0677

Epoch 33/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 44ms/step - loss: 0.0682

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0677 - val_loss: 0.0677

Epoch 34/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 44ms/step - loss: 0.0697

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0688 

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0678 - val_loss: 0.0680

Epoch 35/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 9s 42ms/step - loss: 0.0679

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0682

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 16ms/step - loss: 0.0676 - val_loss: 0.0674

Epoch 36/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 12s 55ms/step - loss: 0.0650

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 16ms/step - loss: 0.0675 - val_loss: 0.0676

Epoch 37/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 43ms/step - loss: 0.0675

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0676 

  9/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0674

 13/235 ━━━━━━━━━━━━━━━━━━━━ 3s 15ms/step - loss: 0.0673

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 17ms/step - loss: 0.0675 - val_loss: 0.0673

Epoch 38/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 44ms/step - loss: 0.0668

  5/235 ━━━━━━━━━━━━━━━━━━━━ 3s 14ms/step - loss: 0.0671 

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235/235 ━━━━━━━━━━━━━━━━━━━━ 4s 18ms/step - loss: 0.0673 - val_loss: 0.0672

Epoch 39/40


  1/235 ━━━━━━━━━━━━━━━━━━━━ 10s 45ms/step - loss: 0.0676

  4/235 ━━━━━━━━━━━━━━━━━━━━ 3s 17ms/step - loss: 0.0672 

  8/235 ━━━━━━━━━━━━━━━━━━━━ 3s 16ms/step - loss: 0.0674

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Epoch 40/40


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<keras.src.callbacks.history.History at 0x132c838c0>

For the model preparation I used “Adam,” which stands for Adaptive Moment Estimation. This is a very popular optimizer for autoencoders that works well with noisy or large data sets. binary_crossentropy calculates the difference between the original and reconstructed image.

For training: ‘epochs’ = number of full passes through the data,‘batch_size’ = number of samples per training step, ‘shuffle’ = True, randomly shuffles the data each epoch to help construct the data.

10.6.4.4 Reconstruct The Data

#Reconstruct the images
reconstruct_images = autoencoder.predict(X_test)

#Set the parameters for the plot with original and reconstructed images
n = 10
plt.figure(figsize=(16, 4))
for i in range(n):
    #Top plot for original images
    ax = plt.subplot(2, n, i + 1)
    plt.imshow(X_test[i].reshape(28, 28), cmap="gray")
    plt.title("Original")
    plt.axis("off")
    
    #Bottom plot for reconstructed images
    ax = plt.subplot(2, n, i + 1 + n)
    plt.imshow(reconstruct_images[i].reshape(28, 28), cmap="gray")
    plt.title("Reconstructed")
    plt.axis("off")
plt.show()

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As we can see above the reconstructed images are slightly more pixelated and not as distict as the original. In order to improve on this the number of epochs or creating more dimensions when building the model.

10.6.5 Conclusion

• Autoencoders compress and reconstruct data, enabling pattern recognition without labeled data.

• Useful in tasks like anomaly detection, data cleaning, and feature extraction.

• Training involves minimizing reconstruction error, using loss functions such as MSE.

10.6.6 Further Readings

• Doersch’s Tutorial on Variational Autoencoders
• Michelucci’s Deep Learning with TensorFlow 2

Huang, Z. (1997). Clustering large data sets with mixed numeric and categorical values. Proceedings of the 1st Pacific-Asia Conference on Knowledge Discovery and Data Mining (PAKDD), 21–34.

Langfelder, P., Zhang, B., & Horvath, S. (2008). Defining clusters from a hierarchical cluster tree: The dynamic tree cut package for r. Bioinformatics, 24(5), 719–720.