Principal Component Analysis (PCA)

The first principal component (PC1) captures the direction of greatest variance in the data. Each subsequent component captures the maximum remaining variance under the constraint that it is orthogonal to all previous components. This property ensures that the first few PCs retain the most informative signal, making PCA ideal for feature reduction in genomic datasets where thousands of genes can be compressed into a handful of latent variables while preserving biological signal.

All principal components are mutually orthogonal—mathematically uncorrelated with one another. This decorrelation eliminates multicollinearity, a critical advantage when preparing data for downstream models like logistic regression or survival analysis. In patient stratification, orthogonal components ensure that each axis represents an independent source of biological variation, preventing redundant encoding of the same molecular signal across multiple features.

PCA is computed via eigendecomposition of the covariance matrix. Each eigenvector defines a principal component direction, while its corresponding eigenvalue quantifies the variance explained by that component. The proportion of total variance explained by the k-th component is calculated as:

• λₖ / Σλᵢ This mathematical foundation allows precise quantification of information retention when reducing dimensionality for multi-omics integration workflows.

PCA assumes that the underlying data structure is linear. It cannot capture complex non-linear manifolds present in single-cell RNA-seq or spatial transcriptomics data. When patient subgroups are separated by non-linear boundaries, techniques like t-SNE, UMAP, or kernel PCA should be considered. Despite this limitation, PCA remains the preferred first-pass method for exploratory data analysis due to its computational efficiency and deterministic output.

PCA is highly sensitive to feature scaling. Variables with larger numeric ranges dominate the principal components, even if they carry less biological relevance. Standard practice requires z-score normalization (mean=0, standard deviation=1) before applying PCA to heterogeneous clinical datasets. Without standardization, features like systolic blood pressure (range 80-200) would overwhelm binary features like mutation status (0/1), distorting the variance structure.

Each principal component is a linear combination of original features, with weights called loadings. High-magnitude loadings reveal which genes or clinical variables drive a component. In biomarker discovery, inspecting PC1 loadings can identify the most influential molecular features separating patient subgroups. This interpretability distinguishes PCA from black-box non-linear methods and supports regulatory submissions requiring explainable feature attribution.

A non-linear dimensionality reduction algorithm that excels at preserving local structure in high-dimensional data. Unlike PCA's linear transformation, t-SNE uses a Student's t-distribution to model pairwise similarities in a low-dimensional space, making it ideal for visualizing single-cell RNA sequencing data and identifying rare cell populations. However, it is computationally intensive and does not preserve global distances, meaning cluster sizes and distances between clusters are not interpretable.

A manifold learning technique that constructs a fuzzy topological representation of the data. UMAP assumes the data is uniformly distributed on a Riemannian manifold and approximates this structure using a k-nearest neighbor graph. It is significantly faster than t-SNE and better preserves the global structure, making it the preferred tool for visualizing large patient cohorts and multi-omics integration results.

A resampling-based methodology that aggregates results from multiple clustering runs to identify robust and stable patient subgroups. By applying a clustering algorithm (often K-Means on PCA-reduced data) to subsamples of the dataset and calculating a consensus matrix, this method quantifies the stability of cluster assignments. It provides a visual consensus heatmap and metrics like the PAC score to determine the optimal number of clusters.

A method that simultaneously learns a low-dimensional feature representation with an autoencoder and assigns clusters in the latent space. DEC first pre-trains a deep autoencoder to map high-dimensional patient data to a compressed bottleneck, then iteratively refines cluster centroids using a KL-divergence loss. This overcomes PCA's linearity by learning complex, non-linear manifolds optimized specifically for patient stratification.

A method for studying the shape of complex patient data using persistent homology. Instead of reducing dimensions, TDA constructs a simplicial complex from the data and tracks topological features like connected components and loops as a filtration parameter varies. The resulting persistence diagram reveals high-dimensional voids and connectivity patterns that linear methods like PCA cannot detect, identifying disease subtypes with unique topological signatures.