Principal Components and Cluster Analysis

A correlation matrix or set of standardized variables is given and you must compute the first few principal components and interpret them.

PCA finds eigenvectors of the covariance (or correlation) matrix sorted by eigenvalue. Each principal component is a linear combination of the original variables; the first explains the most variance. Use a scree plot or cumulative variance threshold (e.g., 80%) to pick the number of components. Interpret each PC by its dominant loadings.

PCs are eigenvectors of Σ; variance explained = eigenvalues / trace(Σ).

A dataset is given and you must apply k-means and determine the optimal number of clusters.

Initialize k centroids randomly, assign each observation to the nearest centroid, recompute centroids as cluster means, repeat until assignments stabilize. To select k, use the elbow method on within-cluster sum of squares or the gap statistic. Standardize variables before clustering so distance is meaningful.

WSS = Σ_k Σ_{i in C_k} ||x_i - μ_k||^2; choose k at the elbow.

A pairwise distance matrix or dataset is given, and you must build an agglomerative hierarchical clustering and read the dendrogram.

Start with each observation as its own cluster. At each step, merge the two closest clusters using a linkage rule: complete (max distance), single (min), average, or Ward (variance). Visualize as a dendrogram; cut horizontally at a chosen height to define clusters. Different linkages produce qualitatively different cluster shapes.

Linkage choices: complete (compact), single (chained), average, Ward (variance-minimizing).