Editing Principal component analysis (section)

== Intuition ==
[[File:SCREE_plot.jpg|thumb|A scree plot that is meant to help interpret the PCA and decide how many components to retain. The start of the bend in the line (point of inflexion or "knee") should indicate how many components are retained, hence in this example, three factors should be retained.]]

PCA can be thought of as fitting a ''p''-dimensional [[ellipsoid]] to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small.

To find the axes of the ellipsoid, we must first center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we compute the [[covariance matrix]] of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must [[Normalization (statistics)|normalize]] each of the orthogonal eigenvectors to turn them into unit vectors. Once this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues.

[[Biplot]]s and [[scree plot]]s (degree of [[explained variance]]) are used to interpret findings of the PCA.