PCA variance vs direction: datasets + projection + wᵀCw

Choose a dataset

All datasets are standardised so x and y have mean 0 and variance 1.

Direction angle θ (degrees)

θ = 45.0°

w = (0.707, 0.707)

Show orthogonal direction w⊥ (dotted)

Whiten/Decorrelate (apply Σ^{-1/2} after centring)

Sampling

n = 800

Options

2D covariance matrix

C = (1/n) Σ (xᵢ − μ)(xᵢ − μ)ᵀ

Numeric

Pretty print

[ - -; - - ]

Variance computations

Var along w (raw)

Var along w (wᵀCw)

|difference|

Var along w⊥ (raw)

Overall variance (trace(C))

Mean (x, y)

Std (x, y)

w = (cos θ, sin θ), w⊥ = (−sin θ, cos θ), Var(wᵀX) = wᵀCw, trace(C) = C₁₁ + C₂₂

Instructions

Select a dataset with the buttons at the top.
Move the slider to change θ and watch the projection and the variance curve update.
Toggle “Show orthogonal direction” to display w⊥ on the scatter and its variance curve on the plot.
Use “Centre” to illustrate when the identity Var(wᵀX) = wᵀCw holds for the covariance shown.
Toggle “Whiten/Decorrelate” to apply whitening (decorrelate and unit-variance) after centring.
The scatter plot uses equal axis scaling (square aspect) to avoid geometric distortion.

Scatter: equal scaling on x and y, direction w through μ, projected points on the line, dotted connectors

Variance vs θ (y-axis shows range), with current θ marked

Variance along linear projections (wᵀx) and Principal Component Analysis Demo