PCA Principal Component Analysis (PCA) on raw data.
COEFF = PCA(X) returns the principal component coefficients for the N
by P data matrix X. Rows of X correspond to observations and columns to
variables. Each column of COEFF contains coefficients for one principal
component. The columns are in descending order in terms of component
variance (LATENT). PCA, by default, centers the data and uses the
singular value decomposition algorithm. For the non-default options,
use the name/value pair arguments.
[COEFF, SCORE] = PCA(X) returns the principal component score, which is
the representation of X in the principal component space. Rows of SCORE
correspond to observations, columns to components. The centered data
can be reconstructed by SCORE*COEFF'.
[COEFF, SCORE, LATENT] = PCA(X) returns the principal component
variances, i.e., the eigenvalues of the covariance matrix of X, in
LATENT.
[COEFF, SCORE, LATENT, TSQUARED] = PCA(X) returns Hotelling's T-squared
statistic for each observation in X. PCA uses all principal components
to compute the TSQUARED (computes in the full space) even when fewer
components are requested (see the 'NumComponents' option below). For
TSQUARED in the reduced space, use MAHAL(SCORE,SCORE).
[COEFF, SCORE, LATENT, TSQUARED, EXPLAINED] = PCA(X) returns a vector
containing the percentage of the total variance explained by each
principal component.
[COEFF, SCORE, LATENT, TSQUARED, EXPLAINED, MU] = PCA(X) returns the
estimated mean, MU, when 'Centered' is set to true; and all zeros when
set to false.
[...] = PCA(..., 'PARAM1',val1, 'PARAM2',val2, ...) specifies optional
parameter name/value pairs to control the computation and handling of
special data types. Parameters are:
'Algorithm' - Algorithm that PCA uses to perform the principal
component analysis. Choices are:
'svd' - Singular Value Decomposition of X (the default).
'eig' - Eigenvalue Decomposition of the covariance matrix. It
is faster than SVD when N is greater than P, but less
accurate because the condition number of the covariance
is the square of the condition number of X.
'als' - Alternating Least Squares (ALS) algorithm which finds
the best rank-K approximation by factoring a X into a
N-by-K left factor matrix and a P-by-K right factor
matrix, where K is the number of principal components.
The factorization uses an iterative method starting with
random initial values. ALS algorithm is designed to
better handle missing values. It deals with missing
values without listwise deletion (see {'Rows',
'complete'}).
'Centered' - Indicator for centering the columns of X. Choices are:
true - The default. PCA centers X by subtracting off column
means before computing SVD or EIG. If X contains NaN
missing values, NANMEAN is used to find the mean with
any data available.
false - PCA does not center the data. In this case, the original
data X can be reconstructed by X = SCORE*COEFF'.
'Economy' - Indicator for economy size output, when D the degrees of
freedom is smaller than P. D, is equal to M-1, if data
is centered and M otherwise. M is the number of rows
without any NaNs if you use 'Rows', 'complete'; or the
number of rows without any NaNs in the column pair that
has the maximum number of rows without NaNs if you use
'Rows', 'pairwise'. When D < P, SCORE(:,D+1:P) and
LATENT(D+1:P) are necessarily zero, and the columns of
COEFF(:,D+1:P) define directions that are orthogonal to
X. Choices are:
true - This is the default. PCA returns only the first D
elements of LATENT and the corresponding columns of
COEFF and SCORE. This can be significantly faster when P
is much larger than D. NOTE: PCA always returns economy
size outputs if 'als' algorithm is specifed.
false - PCA returns all elements of LATENT. Columns of COEFF and
SCORE corresponding to zero elements in LATENT are
zeros.
'NumComponents' - The number of components desired, specified as a
scalar integer K satisfying 0 < K <= P. When specified,
PCA returns the first K columns of COEFF and SCORE.
'Rows' - Action to take when the data matrix X contains NaN
values. If 'Algorithm' option is set to 'als, this
option is ignored as ALS algorithm deals with missing
values without removing them. Choices are:
'complete' - The default action. Observations with NaN values
are removed before calculation. Rows of NaNs are
inserted back into SCORE at the corresponding
location.
'pairwise' - If specified, PCA switches 'Algorithm' to 'eig'.
This option only applies when 'eig' method is used.
The (I,J) element of the covariance matrix is
computed using rows with no NaN values in columns I
or J of X. Please note that the resulting covariance
matrix may not be positive definite. In that case,
PCA terminates with an error message.
'all' - X is expected to have no missing values. All data
are used, and execution will be terminated if NaN is
found.
'Weights' - Observation weights, a vector of length N containing all
positive elements.
'VariableWeights' - Variable weights. Choices are:
- a vector of length P containing all positive elements.
- the string 'variance'. The variable weights are the inverse of
sample variance. If 'Centered' is set true at the same time,
the data matrix X is centered and standardized. In this case,
PCA returns the principal components based on the correlation
matrix.
The following parameter name/value pairs specify additional options
when alternating least squares ('als') algorithm is used.
'Coeff0' - Initial value for COEFF, a P-by-K matrix. The default is
a random matrix.
'Score0' - Initial value for SCORE, a N-by-K matrix. The default is
a matrix of random values.
'Options' - An options structure as created by the STATSET function.
PCA uses the following fields:
'Display' - Level of display output. Choices are 'off' (the
default), 'final', and 'iter'.
'MaxIter' - Maximum number of steps allowed. The default is
1000. Unlike in optimization settings, reaching
MaxIter is regarded as convergence.
'TolFun' - Positive number giving the termination tolerance for
the cost function. The default is 1e-6.
'TolX' - Positive number giving the convergence threshold
for relative change in the elements of L and R. The
default is 1e-6.
Example:
load hald;
[coeff, score, latent, tsquared, explained] = pca(ingredients);
See also PPCA, PCACOV, PCARES, BIPLOT, BARTTEST, CANONCORR, FACTORAN,
ROTATEFACTORS.
Documentation for pca
doc pca
Other uses of pca
gpuArray/pca tall/pca
