🤖Statistical Prediction Unit 10 – Unsupervised Learning: PCA & Clustering

Key Concepts

• Unsupervised learning extracts patterns and insights from unlabeled data without predefined target variables
• Dimensionality reduction techniques compress high-dimensional data into lower-dimensional representations while preserving important information
  • Helps mitigate the curse of dimensionality and improves computational efficiency
• Clustering algorithms group similar data points together based on their inherent characteristics and patterns
  • Enables the discovery of hidden structures and relationships within the data
• Principal Component Analysis (PCA) is a linear dimensionality reduction technique that identifies the directions of maximum variance in the data
• K-means and hierarchical clustering are popular partitioning and hierarchical clustering algorithms, respectively
• Evaluation metrics such as silhouette score and Davies-Bouldin index assess the quality and validity of clustering results
• Real-world applications of unsupervised learning include customer segmentation, anomaly detection, and image compression

Dimensionality Reduction Techniques

• Dimensionality reduction maps high-dimensional data to a lower-dimensional space while retaining the most important information
• Reduces the number of features or variables in the dataset, making it more manageable and computationally efficient
• Helps alleviate the curse of dimensionality, which refers to the challenges posed by high-dimensional data (sparsity, increased computational complexity)
• Linear techniques, such as PCA, project the data onto a lower-dimensional linear subspace
  • Identifies the directions of maximum variance in the data and constructs new features (principal components) along those directions
• Non-linear techniques, like t-SNE and UMAP, capture complex non-linear relationships in the data
  • Preserve the local structure and neighborhood relationships of the data points in the lower-dimensional space
• Dimensionality reduction facilitates data visualization, noise reduction, and feature extraction

Principal Component Analysis (PCA)

• PCA is a linear dimensionality reduction technique that transforms the data into a new coordinate system
• Identifies the directions of maximum variance in the data, known as principal components
  • The first principal component captures the most variance, followed by the second, and so on
• Projects the data onto the principal components to obtain a lower-dimensional representation
• Preserves the global structure of the data while minimizing the reconstruction error
• The number of principal components can be selected based on the desired level of variance retention (explained variance ratio)
• PCA is sensitive to the scale of the features, so data standardization is often performed beforehand
• Limitations of PCA include its linearity assumption and inability to capture non-linear relationships

Clustering Methods

• Clustering is an unsupervised learning task that involves grouping similar data points together based on their inherent characteristics
• Partitioning clustering algorithms, such as K-means, assign each data point to a single cluster
  • Aim to minimize the within-cluster variation and maximize the between-cluster separation
• Hierarchical clustering algorithms, like agglomerative and divisive clustering, create a hierarchy of clusters
  • Agglomerative clustering starts with each data point as a separate cluster and iteratively merges the closest clusters
  • Divisive clustering starts with all data points in a single cluster and recursively splits the clusters
• Density-based clustering, such as DBSCAN, identifies clusters as dense regions separated by areas of lower density
• Model-based clustering, like Gaussian Mixture Models (GMM), assumes that the data is generated from a mixture of probability distributions
• The choice of clustering algorithm depends on the nature of the data, desired cluster shape, and prior knowledge about the number of clusters

K-means Algorithm

• K-means is a popular partitioning clustering algorithm that aims to partition the data into K clusters
• Initializes K cluster centroids randomly or using techniques like K-means++
• Iteratively assigns each data point to the nearest centroid based on a distance metric (Euclidean distance)
  • Updates the cluster centroids by computing the mean of the data points assigned to each cluster
• Repeats the assignment and update steps until convergence (centroids no longer change significantly) or a maximum number of iterations is reached
• The objective is to minimize the sum of squared distances between data points and their assigned centroids (within-cluster sum of squares)
• The choice of K, the number of clusters, is a hyperparameter that needs to be specified in advance
  • Techniques like the elbow method or silhouette analysis can help determine an appropriate value for K
• K-means is computationally efficient and scales well to large datasets but is sensitive to the initial centroid positions

Hierarchical Clustering

• Hierarchical clustering creates a tree-like structure (dendrogram) that represents the hierarchical relationships between clusters
• Agglomerative clustering is a bottom-up approach that starts with each data point as a separate cluster
  • Iteratively merges the closest clusters based on a linkage criterion (single, complete, average, or Ward's linkage)
  • Continues merging clusters until a desired number of clusters is reached or all data points belong to a single cluster
• Divisive clustering is a top-down approach that starts with all data points in a single cluster
  • Recursively splits the clusters into smaller subsets based on a splitting criterion
  • Continues splitting until a desired number of clusters is reached or each data point forms its own cluster
• The dendrogram visualizes the merging or splitting process and allows for the selection of the number of clusters by cutting the dendrogram at a specific height
• Hierarchical clustering does not require specifying the number of clusters in advance but can be computationally expensive for large datasets

Evaluation Metrics

• Evaluation metrics assess the quality and validity of clustering results in the absence of ground truth labels
• Internal evaluation metrics measure the compactness and separation of clusters based on the data itself
  • Silhouette coefficient computes the average silhouette score for each data point, indicating how well it fits into its assigned cluster compared to other clusters
  • Davies-Bouldin index measures the ratio of within-cluster distances to between-cluster distances, with lower values indicating better clustering
• External evaluation metrics compare the clustering results to external ground truth labels, if available
  • Adjusted Rand Index (ARI) measures the similarity between the clustering and the ground truth, accounting for chance agreements
  • Normalized Mutual Information (NMI) quantifies the mutual information between the clustering and the ground truth, normalized by the entropy of both partitions
• Visualization techniques, such as scatter plots or t-SNE embeddings, can provide qualitative insights into the clustering results
• The choice of evaluation metric depends on the specific goals and characteristics of the clustering task

Real-world Applications

• Customer segmentation in marketing divides customers into distinct groups based on their behavior, preferences, or demographics
  • Enables targeted marketing strategies and personalized recommendations
• Anomaly detection identifies unusual or abnormal instances in the data that deviate from the normal patterns
  • Detects fraudulent transactions, network intrusions, or manufacturing defects
• Image compression reduces the size of images by identifying and preserving the most important visual information
  • PCA can be used to compress images by projecting them onto a lower-dimensional space
• Document clustering organizes a large collection of text documents into coherent groups based on their content similarity
  • Facilitates information retrieval, topic modeling, and content recommendation systems
• Bioinformatics applies clustering techniques to analyze gene expression data, identify co-expressed genes, and discover biological pathways
• Social network analysis uses clustering to identify communities or groups of closely connected individuals within a social network
• Recommender systems employ clustering to group users or items with similar preferences and generate personalized recommendations