12.2 Hierarchical Clustering
4 min read•august 7, 2024
Hierarchical clustering organizes data into a tree-like structure, revealing relationships between points. It's a versatile method that can uncover hidden patterns and group similar items together, making it useful for various fields like biology and marketing.
This approach offers two main types: agglomerative (bottom-up) and divisive (top-down). By using different linkage methods, it can adapt to various data structures and reveal insights about the underlying relationships in your dataset.
Types of Hierarchical Clustering
Agglomerative and Divisive Clustering
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- starts with each data point as a separate cluster and iteratively merges the closest until all points belong to a single cluster
- Also known as a "bottom-up" approach since it begins with individual data points and builds up to a single cluster
- At each step, the two closest clusters are combined into a new cluster
- The process continues until a desired number of clusters is reached or all data points are in one cluster
- begins with all data points in a single cluster and recursively splits the clusters until each data point is in its own cluster
- Follows a "top-down" approach, starting with a single cluster containing all data and dividing it into smaller clusters
- At each step, the largest cluster is split into two smaller clusters based on a chosen criterion
- The splitting process continues until each data point is in its own cluster or a desired number of clusters is achieved
Dendrograms
- A is a tree-like diagram used to visualize the arrangement of clusters produced by hierarchical clustering
- The x-axis represents the data points, while the y-axis represents the distance or dissimilarity between clusters
- Each merge or split is represented by a horizontal line connecting the clusters
- The height of the horizontal line indicates the distance between the merged or split clusters
- Dendrograms allow for easy interpretation of the clustering results
- The closer two data points or clusters are connected in the dendrogram, the more similar they are
- Cutting the dendrogram at a specific height (distance threshold) determines the final number of clusters
- Example: In a dendrogram of animal species, closely related species (cats and tigers) will be connected at a lower height compared to more distantly related species (cats and elephants)
Linkage Methods
Distance-based Linkage Methods
- determines the distance between two clusters as the minimum distance between any two points in the clusters
- Also known as the nearest neighbor method
- Tends to create long, chain-like clusters and can be sensitive to noise and outliers
- Example: In a dataset of cities, single linkage would consider the distance between two clusters of cities as the shortest distance between any pair of cities from each cluster
- calculates the distance between two clusters as the maximum distance between any two points in the clusters
- Also referred to as the farthest neighbor method
- Tends to create compact, tightly-bound clusters and is less susceptible to noise and outliers compared to single linkage
- Example: In a dataset of animal species, complete linkage would consider the distance between two clusters of species as the largest distance between any pair of species from each cluster
- computes the distance between two clusters as the average distance between all pairs of points in the clusters
- Strikes a balance between single and complete linkage methods
- Less affected by noise and outliers compared to single linkage, but may not create clusters as compact as complete linkage
- Example: In a dataset of customer preferences, average linkage would calculate the distance between two clusters of customers by taking the mean of all pairwise distances between customers from each cluster
Variance-based Linkage Methods
- aims to minimize the total within-cluster variance when merging clusters
- At each step, the merge that results in the smallest increase in total within-cluster variance is chosen
- Tends to create clusters of similar sizes and shapes
- Example: In a dataset of stock prices, Ward's method would merge clusters of stocks in a way that minimizes the overall variance within each cluster
Evaluating Linkage Methods
- measures the correlation between the original pairwise distances and the distances obtained from the dendrogram
- A high cophenetic correlation (close to 1) indicates that the dendrogram accurately represents the original distances between data points
- Helps in assessing the quality of the clustering results and comparing different linkage methods
- Example: If the cophenetic correlation is 0.9, it suggests that the dendrogram preserves 90% of the original pairwise distances, indicating a good fit
Key Terms to Review (19)
Agglomerative clustering: Agglomerative clustering is a type of hierarchical clustering method that builds a hierarchy of clusters by successively merging smaller clusters into larger ones. This bottom-up approach starts with each data point as its own cluster and repeatedly combines them based on a similarity measure until a single cluster encompasses all the data points or a specified number of clusters is achieved. This method is widely used for its intuitive nature and the ability to visualize the results through dendrograms.
Average linkage: Average linkage is a method used in hierarchical clustering to determine the distance between clusters by calculating the average distance between all pairs of objects in the two clusters. This technique helps create a balanced representation of the overall similarity between clusters, allowing for a more stable clustering structure. Average linkage is particularly useful in producing clusters that are more evenly sized and can help mitigate the influence of outliers.
Clusters: Clusters refer to groups of data points that are similar to each other within a dataset, often identified through clustering algorithms. These algorithms aim to partition data into distinct groups based on feature similarities, allowing for the identification of patterns and structures in complex datasets. By organizing data into clusters, it becomes easier to analyze relationships and make predictions based on shared characteristics among the grouped observations.
Complete Linkage: Complete linkage is a clustering method used in hierarchical clustering where the distance between two clusters is defined as the maximum distance between any single pair of points in the two clusters. This approach emphasizes the furthest points in each cluster, leading to tighter and more compact clusters compared to other methods.
Cophenetic correlation: Cophenetic correlation is a statistical measure that evaluates how well the distances between clusters in a hierarchical clustering dendrogram match the original distances between the data points. This metric helps to assess the quality of the clustering solution, indicating how closely the clustering structure reflects the relationships between individual observations. A higher cophenetic correlation value suggests a more accurate representation of the data's structure in the hierarchy.
Cutting the tree: Cutting the tree refers to the process of determining the optimal number of clusters in hierarchical clustering by selecting a specific height on a dendrogram. This action essentially 'cuts' the dendrogram into distinct clusters, allowing for the identification of groups within the data that share similar characteristics. The height at which the cut is made directly influences the granularity of the clusters formed, which can significantly affect the interpretation and usefulness of the clustering results.
Dendrogram: A dendrogram is a tree-like diagram that visually represents the arrangement of clusters resulting from hierarchical clustering. It showcases how data points are grouped together at various levels of similarity or dissimilarity, allowing for an intuitive understanding of the relationships among the data. Dendrograms help to illustrate both the structure of the data and the results of clustering algorithms, providing insight into the hierarchical nature of the data relationships.
Divisive Clustering: Divisive clustering is a top-down hierarchical clustering technique that starts with a single cluster containing all data points and recursively splits this cluster into smaller sub-clusters. This method contrasts with agglomerative clustering, where clusters are formed from individual points that are merged together. The process continues until a stopping criterion is met, such as reaching a specified number of clusters or achieving a desired level of homogeneity within clusters.
Euclidean Distance: Euclidean distance is a measure of the straight-line distance between two points in Euclidean space. It serves as a key component in clustering algorithms, determining how similar or different data points are to one another based on their coordinates in multi-dimensional space. The ability to quantify distance is crucial for grouping similar items together and forming meaningful clusters, which enhances the understanding and interpretation of data patterns.
Gene expression analysis: Gene expression analysis is the process of measuring the activity levels of genes to understand their function and regulation within a biological context. This analysis helps researchers identify which genes are turned on or off in various conditions, such as disease states or developmental stages, providing insights into cellular processes and potential therapeutic targets.
Manhattan Distance: Manhattan distance is a metric used to measure the distance between two points in a grid-based system, calculated as the sum of the absolute differences of their Cartesian coordinates. It reflects the total distance traveled along axes at right angles, similar to navigating through a city grid where only horizontal and vertical paths are available. This measure is particularly useful in various algorithms, especially in clustering methods like hierarchical clustering, where it helps determine the similarity between data points based on their position in a multidimensional space.
Market segmentation: Market segmentation is the process of dividing a broad consumer or business market into smaller, more defined categories based on shared characteristics. This approach allows businesses to tailor their marketing strategies, products, and services to meet the specific needs and preferences of different segments, ultimately enhancing customer satisfaction and driving sales.
Python's scikit-learn: Python's scikit-learn is a powerful open-source machine learning library designed for data analysis and predictive modeling. It provides a range of tools for implementing various machine learning algorithms, including classification, regression, and clustering techniques, making it an essential resource for data scientists. Scikit-learn integrates well with other Python libraries like NumPy and pandas, allowing users to preprocess data efficiently and visualize results easily.
R: In statistics, 'r' typically represents the correlation coefficient, a measure that quantifies the degree of relationship between two variables. It can range from -1 to 1, where -1 indicates a perfect negative correlation, 0 indicates no correlation, and 1 indicates a perfect positive correlation. This concept is essential in various statistical methods for understanding relationships within data sets, making predictions, and assessing the strength and direction of associations.
Silhouette score: The silhouette score is a metric used to evaluate the quality of clustering in data science. It provides a way to measure how similar an object is to its own cluster compared to other clusters, with a higher silhouette score indicating better-defined and separated clusters. This score helps in assessing the effectiveness of clustering algorithms like K-means, hierarchical, and density-based clustering, as well as understanding the impact of dimensionality reduction methods on clustering results.
Single linkage: Single linkage is a method used in hierarchical clustering that determines the distance between two clusters based on the shortest distance between any two points in the clusters. This approach can significantly influence the shape of the resulting dendrogram, as it tends to create elongated, chain-like clusters. The method is particularly useful when dealing with clusters of varying shapes and sizes, and its efficiency helps identify natural groupings in datasets.
Subclusters: Subclusters are smaller, distinct groups that emerge within larger clusters during the hierarchical clustering process. They represent more refined categorizations of data points that share similar characteristics, allowing for a deeper understanding of the overall dataset. Identifying subclusters helps in revealing patterns that may not be apparent at the broader cluster level.
Ward's Method: Ward's Method is a hierarchical clustering algorithm that aims to minimize the variance within each cluster while maximizing the variance between clusters. It accomplishes this by calculating the sum of squared differences from the mean for all observations in a cluster, making it particularly effective for producing compact and well-separated clusters in the context of data analysis.
Within-cluster sum of squares: The within-cluster sum of squares (WCSS) measures the total variance within each cluster in a clustering algorithm. It quantifies how close the data points in a cluster are to each other, where lower values indicate that data points are tightly packed around the centroid. This term is essential in evaluating the compactness and coherence of clusters, especially when determining the optimal number of clusters in hierarchical clustering methods.