PCA · Explore together

Find a useful direction

Rotate a projection axis and measure how much variation survives.

One person controls the experiment. The other predicts and questions. Swap roles after a trial.

ti=(xi−xˉ)⋅u,∥u∥=1t_i=(x_i-\bar x)\cdot u,\qquad \lVert u\rVert=1

Scroll across the diagram to compare points and projections.

Rotate a projection axis through centered data Six centered illustrative points project perpendicularly onto a vertical line. Solid circles are the original points; hollow circles are the projections. xy -3-2-1123

Original dataProjectionDashed segments meet the axis at right angles.

Choose a direction together. Which line would keep the most of this cloud’s spread? Predict an angle, then test it.

0° is horizontal; 90° is vertical. Opposite directions represent the same line.

Original variance
5.563
Projected variance
0.897
Spread retained
16.1%

The original points stay fixed. Rotate the axis to compare how much spread a single coordinate keeps.

Illustrative data, centered at (0, 0), with both coordinates in the same units. Variance uses the average squared distance from the mean (divide by 6).

What is PCA trying to keep?

Each point becomes one signed coordinate along the axis. PCA chooses the direction with the greatest variance of those coordinates. Here that also minimizes the total squared lengths of the dashed reconstruction errors.

Discuss: why would a line pointing the opposite way preserve the same variance? If x and y were measured on very different scales, what should you check before using PCA?

Compare your evidence

What can you explain now?

Ask your partner to restate your reasoning. An experiment suggests a pattern; your explanation should say why it holds and when it applies.

  1. Why are the points centered before measuring variance?
  2. What disappears when two different points project to the same place?
  3. Why do an axis and the same axis turned 180° keep the same variance?