PCA Dimensionality Lab

Principal Component Analysis (PCA) simplifies complex datasets by finding new directions (components) that capture the maximum variance in clinical data.

The Math of PCA

1. Centre data (mean = 0) 2. Calc Covariance Matrix: Σ = (1/n)XᵀX 3. Find Eigenvectors (v) & Eigenvalues (λ): Σv = λv 4. Project data onto top K eigenvectors: z = Xv₁..ₖ λᵢ represents variance captured by PCᵢ.

Interpreting PCs

PC1 follows the direction of greatest spread. PC2 is orthogonal to PC1 and captures the next most variance. Often, 2-3 PCs can describe 90%+ of clinical information from dozens of biomarkers.

⚖️ Ethical Reality

PCA ignores labels (Unsupervised). If a small minority group has distinct biological features but low "variance" in the overall cohort, PCA may compress their unique signal into noise, leading to invisible healthcare disparities in the reduced feature space.

① PCA Configuration

Biomarker Correlation0.8

linear relationship between metrics

Patient Count (N)60

size of clinical cohort

② Variance Formula

        Total Var = Σ λᵢ

        Explained Ratioᵢ = λᵢ / Σ λⱼ

        Loadings = Eigenvectors scaled by √λ

        PC1 Variance: —%

        PC2 Variance: —%

③ Component Statistics

Explained Cumulative Variance

—

Eigenvalue 1

—

Dim Reduction

④ Original Feature Space — Glucose vs BMI · line = PC1 vector · width = PC2 spread

⑤ Principal Component Projection — transformed space (PC1 vs PC2) · axes are decorrelated