Multivariate Centered Data Matrix
Standardizes raw observation vectors into centered matrix X̃ = X - 1 x̄^T with zero empirical column means.
Statistics & Probability / Principal Component Analysis
FREE TO EXPLORELABORATORY 43 · STATISTICS & PROBABILITY
Principal Component Analysis (PCA), Covariance Decomposition, and Dimensionality Reduction
Investigate multivariate dimensionality reduction, compute sample covariance matrices, execute Singular Value Decomposition (SVD) and eigendecomposition, project observations onto principal axes, and verify variance scaling. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
FIRST PRINCIPAL COMPONENT EIGENVALUE Λ_1—
SAMPLING TELEMETRY
Ready to sample with Scale k = 1.0.
Adjust Scale k, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
PCA is an unsupervised linear method; it cannot detect complex non-linear manifold structures (e.g., Swiss roll). The empirical regression of dominant eigenvalues against total variance scale confirms principal component scaling with slope m = 0.4000 and R² = 1.000000.
Statistical tools & sampling modules.
Sample Covariance Matrix Calculator
Computes sample covariance matrix S = (1/(n-1)) X̃^T X̃ across multi-attribute observations.
SVD & Eigendecomposition Engine
Computes eigenvalues λ_j and orthogonal eigenvectors v_j satisfying S v_j = λ_j v_j via QR iteration.
Scree Plot Variance Ratio Analyzer
Calculates proportion of variance explained by each principal component: PV_j = λ_j / ∑ λ_i.
Orthogonal Projection Transformer
Projects observations onto principal axes Z = X̃ V to construct uncorrelated score coordinates.
Principal Variance Regression Unit
Performs linear regression of dominant eigenvalue λ_1 versus scale factor k to extract slope m = 0.400.
FOLLOW THE METHOD
A reason behind every step.
Investigate multivariate dimensionality reduction, compute sample covariance matrices, execute Singular Value Decomposition (SVD) and eigendecomposition, project observations onto principal axes, and verify variance scaling. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 factor.
01
Configure Bivariate Correlated Data Structure
Set sample size n = 10,000 observations with bivariate covariance matrix Σ having correlation r = 0.80 on the PCA dashboard.
02
Center and Standardize Input Coordinates
Subtract column means to form centered matrix X̃. Verify that empirical means equal zero across all attributes.
03
Execute Eigendecomposition on Sample Covariance
Compute covariance matrix S and solve for eigenvalues (λ_1, λ_2) and orthogonal eigenvectors (v_1, v_2). Verify v_1^T v_2 = 0.
04
Measure Proportion of Variance Explained
Evaluate the scree ratio PV_1 = λ_1 / (λ_1 + λ_2). For r = 0.80 and normalized variance, verify dominant component captures 40% of baseline scale.
05
Step Total Variance Scale Across k = 1 to 6
Increment total variance scale across standardized steps k ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the dominant eigenvalue λ_1.
06
Conduct Linear Regression on PCA Eigenvalue Scaling
Fit a linear regression line y = mx to coordinate pairs (x = k, y = λ_1). Confirm that empirical slope m = 0.4000 validates principal variance scaling.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Principal Component Analysis mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 0.4000 · x
| # | Total Variance Scale Factor k (factor) | First Principal Component Eigenvalue λ_1 (Eigenvalue λ_1) | Total Variance Scale Factor k | First Principal Component Eigenvalue λ_1 | Remove |
|---|
No readings yet. Execute a statistical trial and record it.
Calculate it yourself.
CHECK YOUR UNDERSTANDING