# 6.4 - Geometric Interpretation

6.4 - Geometric InterpretationPrincipal components analysis (PCA) projects the data along the directions where the data varies the most.

The first direction is decided by \(\mathbf{v}_1\) corresponding to the largest eigenvalue \(d_1^2\).

The second direction is decided by \(\mathbf{v}_2\) corresponding to the second largest eigenvalue \(d_2^2\).

The variance of the data along the principal component directions is associated with the magnitude of the eigenvalues.

### Choice of How Many Components to Extract

Scree Plot – This is a useful visual aid which shows the amount of variance explained by each consecutive eigenvalue.

The choice of how many components to extract is fairly *arbitrary*.

When conducting principal components analysis prior to further analyses, it is risky to choose too small a number of components, which may fail to explain enough of the variability in the data.