Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and here for more.

PCA - Loadings and Scores

If we look at PCA more formally, it turns out that the PCA is based on a decomposition of the data matrix X into two matrices V and U:

The two matrices V and U are orthogonal. The matrix V is usually called the loadings matrix, and the matrix U is called the scores matrix. The loadings can be understood as the weights for each original variable when calculating the principal component. The matrix U contains the original data in a rotated coordinate system.

There are a few common plots which are always used in connection with PCA:

  • the scores/scores plot (left part of the figure below),
  • the corresponding loading/loading plot (right part of the figure below)
  • the plotting of loadings as spectral lines,
  • and the plot of the ordered eigenvalues.

The following  interactive example  shows all of these plots for a data set originating from genuine and forged banknotes.

The word "score" has its roots in old Skandinavian languages. An explanation has been given by W. Clements in the Globe Review, Oct-21, 1999:

"Score" rose from the Old Norse "skor2, meaning notch, and "skera" meaning to cut or shear -- an origin it shares with "shard" and "share" (from the notion of divvying something up). It entered the English language in the 14th century as a verb meaning to "notch with lines" and a noun meaning "twenty" -- a use familiar from the Bible's "threescore and ten" (that is 70) and Abraham Lincoln's "fourscore and seven years ago" (that is 87). What did 20 have to do with cutting? The best available guess is that 20 was a standard reference mark when putting notches in a stick (known as a tally) to keep track of debts owed.

Since then the word has aquired scores of meanings. To score with someone .......".