# PCA、特征向量与协方差椭圆可视化

通过几何解释理解降维技术：协方差矩阵、特征分解、主成分分析与降维演示

> 标准页面: https://elysiatools.com/zh/visualizations/pca-eigenvector

- **分类:** Math

## 概述

Comprehensive interactive visualization of Principal Component Analysis (PCA) demonstrating dimensionality reduction through geometric interpretation. Features bivariate Gaussian data generation with adjustable correlation coefficient ρ (-1 to 1), noise level σ (0 to 2), and sample size n (100 to 1000). Covariance matrix computation and eigendecomposition showing eigenvalues (variance) and eigenvectors (principal directions). Visual elements include scatter plot with data points, covariance ellipses at 1σ, 2σ, 3σ levels, eigenvector arrows for PC1 and PC2 with color coding (PC1: green #22c55e, PC2: orange #f97316), mean point indicator, and projected/reconstructed points. Analysis panel displays covariance matrix Σ, eigenvalues λ₁ and λ₂, eigenvectors v₁ and v₂, explained variance ratio bars, and reconstruction error (MSE). Interactive features include preset scenarios (uncorrelated, strong positive/negative correlation, high noise), data centering toggle, projection visualization, dimensionality reduction slider (k components), and view mode switching between original and PC-transformed space. Educational content covers covariance matrix formula Σ = (1/n)XᵀX, eigendecomposition Σv = λv, PCA transformation z = Qᵀ(x-μ), reconstruction x̂ = Q_k z_k + μ, covariance ellipse parametric equation, step-by-step PCA process (centering, covariance computation, eigendecomposition, projection, optional dimensionality reduction and reconstruction), and practical applications in data visualization, feature extraction (Eigenfaces), noise reduction, image compression, anomaly detection, and handling multicollinearity in regression. Uses KaTeX for mathematical formula rendering with formulas for covariance matrix, eigendecomposition, PCA transform, reconstruction, explained variance ratio, and covariance ellipse. Multi-language support (zh, en, es, fr, de, ru, pt).

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