Statistics: pca

### Function Reference: `pca`

statistics: coeff = pca (x)
statistics: coeff = pca (x, Name, Value)
statistics: [coeff, score, latent] = pca (…)
statistics: [coeff, score, latent, tsquared] = pca (…)
statistics: [coeff, score, latent, tsquared, explained, mu] = pca (…)

Performs a principal component analysis on a data matrix.

A principal component analysis of a data matrix of N observations in a D dimensional space returns a D×D transformation matrix, to perform a change of basis on the data. The first component of the new basis is the direction that maximizes the variance of the projected data.

Input argument:

- x : a N×D data matrix

The following Name, Value pair arguments can be used:

- `"Algorithm"` defines the algorithm to use:
  - `"svd"` (default), for singular value decomposition
  - `"eig"` for eigenvalue decomposition
- `"Centered"` is a boolean indicator for centering the observation data. It is `true` by default.
- `"Economy"` is a boolean indicator for the economy size output. It is `true` by default. Hence, `pca` returns only the elements of latent that are not necessarily zero, and the corresponding columns of coeff and score, that is, when N <= D, only the first N - 1.
- `"NumComponents"` defines the number of components k to return. If k < p, then only the first k columns of coeff and score are returned.
- `"Rows"` defines how to handle missing values:
  - `"complete"` (default), missing values are removed before computation.
  - `"pairwise"` (only valid when `"Algorithm"` is `"eig"`), the covariance of rows with missing data is computed using the available data, but the covariance matrix could be not positive definite, which triggers the termination of `pca`.
  - `"complete"`, missing values are not allowed, `pca` terminates with an error if there are any.
- `"Weights"` defines observation weights as a vector of positive values of length N.
- `"VariableWeights"` defines variable weights:
  - a vector of positive values of length D.
  - the string `"variance"` to use the sample variance as weights.

Return values:

- coeff : the principal component coefficients, a D×D transformation matrix
- score : the principal component scores, the representation of x in the principal component space
- latent : the principal component variances, i.e., the eigenvalues of the covariance matrix of x
- tsquared : Hotelling’s T-squared Statistic for each observation in x
- explained : the percentage of the variance explained by each principal component
- mu : the estimated mean of each variable of x, it is zero if the data are not centered

Matlab compatibility note: the alternating least square method ’als’ and associated options ’Coeff0’, ’Score0’, and ’Options’ are not yet implemented

### References

1. Jolliffe, I. T., Principal Component Analysis, 2nd Edition, Springer, 2002

See also: barttest, factoran, pcacov, pcares

Source Code: pca