What does the notation $D = 2$ -> D' = 1 mean in this PCA introduction?

In the PCA setup shown here, the data matrix X is organized such that its rows are data points and its columns are features. Each row is treated as a D-dimensional vector.

Conditions: Rows correspond to data points.; Columns correspond to features.; Each row lies in $\mathbb{R}^D$.

The video uses an ellipse to explain principal components because it provides an intuitive geometric model for the data cloud. The main axes of the ellipse are identified as the directions PCA wants to find, which are called principal components.

Conditions: The data are approximated by an ellipse.; The ellipse is described verbally as 'basically a two-dimensional Gaussian.'

The PCA algorithm subtracts the mean from the data to centralize it, meaning that the mean of all points should be zero. This step prepares the data for the variance-based step that follows, making the formula for the scatter matrix simpler.

Conditions: The data must first be represented as a matrix X.; The subtraction is by the mean $\overline{X}$.; After this step, the video uses $\overline{X} = 0$ in the scatter-matrix discussion.

The clip explains that PCA chooses which directions to keep based on the spread of the data. It uses a 2D scatter plot to show that projecting along the smaller-spread direction leaves the points separated, while projecting along the larger-spread direction collapses them.

Conditions: Demonstrated on a 2D scatter plot.; The explanation is geometric and intuitive rather than algebraic.

Projecting onto the blue line loses less information because the spread along that direction is smaller. Clamping all points onto one line in that direction would not lose that much information.

Conditions: Demonstrated on a 2D scatter plot.; The explanation is geometric and intuitive rather than algebraic.