LOOK INSIDE THE CLOUD
Learn with your cloud.
How much survives in three dimensions?
Each bar is a component’s share of total variance. The line adds those shares together. The first three bars are the dimensions of your cloud.
▥ Individual variance― Cumulative variance┄ 90% / 95%
The math behind the plot
EVRⱼ = λⱼ / Σ λℓ
λⱼ is the variance along component j, ordered from largest to smallest. Retained variance is the sum of the retained components’ ratios.
Z = X̃Vₖ · X̂ = ZVₖᵀ
‖X̃ − X̂‖² / ‖X̃‖² = 1 − Σⱼ₌₁ᵏ EVRⱼ
X̃ is centered data, optionally standardized. Vₖ contains k orthonormal component directions. PCA maximizes retained variance and minimizes squared reconstruction error with the same projection. Error is measured in the preprocessed space.
Which measurements shape the cloud?
Loadings are the weights used to combine your features. The three largest absolute weights are highlighted; positive and negative weights point in opposite directions.
zᵢⱼ = Σℓ x̃ᵢℓ vℓⱼ
A point’s component coordinate is a weighted sum of its centered, optionally standardized measurements. Flipping all weights and scores for a component mirrors the axis without changing the PCA solution. These weights describe the projection, not causal importance.
Try it with four points.
From Assignment 2, Part 1.1: A = (1, 3), B = (3, 1), C = (5, 3), D = (7, 5). Change the units of the second feature and watch PC1 rotate.
See the flat projection
● Centered points ○ Reconstructed from PC1 ― PC1 ┄ Discarded distance
PCA follows variance, so changing units can change its answer. Standardizing removes that dependence on units. It is a modeling choice: shared scales and meaningful low variance may justify leaving it off.
Why the covariance convention matters
This lesson uses population covariance and population standard deviations, dividing by n = 4, to match the assignment. The main cloud uses sample covariance and sample standard deviations, dividing by n − 1. With consistent preprocessing, component directions and explained-variance ratios agree; eigenvalues and standardized scores depend on the convention.
For numerical stability, unstandardized cloud inputs are also divided by one shared scale. This preserves directions and variance ratios. Display coordinates are uniformly resized to fit the scene; the lesson reports its own calculation units.
Variance measures information retained in this representation. It does not guarantee that groups or classes remain distinguishable. Soft particles and cloud density are visual effects, not statistical density estimates.