# SuanShu, a Java numerical and statistical library

com.numericalmethod.suanshu.stats.factor.pca

## Class PCAbySVD

• All Implemented Interfaces:
PCA

public class PCAbySVD
extends Object
This class performs Principal Component Analysis (PCA) on a data matrix, using the preferred Singular Value Decomposition (SVD) method.

PCA essentially rotates the set of points around their mean in order to align with the principal components. This moves as much of the variance as possible (using an orthogonal transformation) into the first few dimensions. The values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information.

The R equivalent function is prcomp.

• K. V. Mardia, J. T. Kent and J. M. Bibby, "Multivariate Analysis," London, Academic Press, 1979.
• W. N. Venables and B. D. Ripley, "Modern Applied Statistics with S," New York, Springer-Verlag, 2002.
• Wikipedia: Principal component analysis
• ### Constructor Summary

Constructors
Constructor and Description
PCAbySVD(Matrix data)
Performs Principal Component Analysis, using the preferred SVD method, on a centered and scaled data matrix.
PCAbySVD(Matrix data, boolean centered, boolean scaled)
Performs Principal Component Analysis, using the preferred SVD method, on a data matrix (possibly centered and/or scaled).
PCAbySVD(Matrix data, Vector mean, Vector scale)
Performs Principal Component Analysis, using the preferred SVD method, on a data matrix with (optional) mean vector and scaling vector provided.
• ### Method Summary

All Methods
Modifier and Type Method and Description
DenseVector cumulativeProportionVar()
Gets the cumulative proportion of overall variance explained by the principal components
ImmutableMatrix data()
Gets the original data matrix.
Vector loading(int i)
Matrix loadings()
Vector mean()
Gets the sample means that were subtracted.
int nFactors()
Gets the number of variables in the original data.
int nObs()
Gets the number of observations in the original data; sample size.
Vector proportionVar()
Gets the proportion of overall variance explained by each of the principal components.
double proportionVar(int i)
Gets the proportion of overall variance explained by the i-th principal component.
Vector scale()
Gets the scalings applied to each variable.
Matrix scores()
Gets the scores of supplied data on the principal components.
double sdPrincipalComponent(int i)
Gets the standard deviation of the i-th principal component.
DenseVector sdPrincipalComponents()
Gets the standard deviations of the principal components (i.e., the square roots of the eigenvalues of the correlation (or covariance) matrix, though the calculation is actually done with the singular values of the data matrix)
SVD svd()
Gets the Singular Value Decomposition (SVD) of matrix X.
Matrix X()
Gets the (possibly centered and/or scaled) data matrix X used for the PCA.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### PCAbySVD

public PCAbySVD(Matrix data,
Vector mean,
Vector scale)
Performs Principal Component Analysis, using the preferred SVD method, on a data matrix with (optional) mean vector and scaling vector provided.
Parameters:
data - a matrix that represents the original data
mean - an optional mean vector (of length equal to nFactors) to be subtracted regardless of the flag centered
scale - an optional scaling vector (of length equal to nFactors) to be divided regardless of the flag scaled
• #### PCAbySVD

public PCAbySVD(Matrix data,
boolean centered,
boolean scaled)
Performs Principal Component Analysis, using the preferred SVD method, on a data matrix (possibly centered and/or scaled).
Parameters:
data - a matrix that represents the original data
centered - a logical value indicating whether the variables should be shifted to be zero centered
scaled - a logical value indicating whether the variables should be scaled to have unit variance before the analysis takes place (N.B. in general scaling is advisable; however, it should only be used if there is no constant variable)
• #### PCAbySVD

public PCAbySVD(Matrix data)
Performs Principal Component Analysis, using the preferred SVD method, on a centered and scaled data matrix.
Parameters:
data - a matrix that represents the original data
• ### Method Detail

• #### mean

public Vector mean()
Description copied from interface: PCA
Gets the sample means that were subtracted.
Specified by:
mean in interface PCA
Returns:
the sample means of each variable in the original data
• #### scale

public Vector scale()
Description copied from interface: PCA
Gets the scalings applied to each variable.
Specified by:
scale in interface PCA
Returns:
the scalings applied to each variable in the original data
• #### svd

public SVD svd()
Gets the Singular Value Decomposition (SVD) of matrix X.
Returns:
the Singular Value Decomposition (SVD) of matrix X
• #### sdPrincipalComponents

public DenseVector sdPrincipalComponents()
Gets the standard deviations of the principal components (i.e., the square roots of the eigenvalues of the correlation (or covariance) matrix, though the calculation is actually done with the singular values of the data matrix)
Returns:
the standard deviations of the principal components

public Matrix loadings()
Description copied from interface: PCA
Returns:
• #### data

public ImmutableMatrix data()
Gets the original data matrix.
Returns:
the original data matrix
• #### nObs

public int nObs()
Description copied from interface: PCA
Gets the number of observations in the original data; sample size.
Specified by:
nObs in interface PCA
Returns:
nObs, the number of observations in the original data
• #### nFactors

public int nFactors()
Description copied from interface: PCA
Gets the number of variables in the original data.
Specified by:
nFactors in interface PCA
Returns:
nFactors, the number of variables in the original data
• #### X

public Matrix X()
Description copied from interface: PCA
Gets the (possibly centered and/or scaled) data matrix X used for the PCA.
Specified by:
X in interface PCA
Returns:
the (possibly centered and/or scaled) data matrix X
• #### sdPrincipalComponent

public double sdPrincipalComponent(int i)
Description copied from interface: PCA
Gets the standard deviation of the i-th principal component.
Specified by:
sdPrincipalComponent in interface PCA
Parameters:
i - an index, counting from 1
Returns:
the standard deviation of the i-th principal component.

public Vector loading(int i)
Description copied from interface: PCA
Specified by:
loading in interface PCA
Parameters:
i - an index, counting from 1
Returns:
• #### proportionVar

public Vector proportionVar()
Description copied from interface: PCA
Gets the proportion of overall variance explained by each of the principal components.
Specified by:
proportionVar in interface PCA
Returns:
the proportion of overall variance explained by each of the principal components
• #### proportionVar

public double proportionVar(int i)
Description copied from interface: PCA
Gets the proportion of overall variance explained by the i-th principal component.
Specified by:
proportionVar in interface PCA
Parameters:
i - an index, counting from 1
Returns:
the proportion of overall variance explained by the i-th principal component
• #### cumulativeProportionVar

public DenseVector cumulativeProportionVar()
Description copied from interface: PCA
Gets the cumulative proportion of overall variance explained by the principal components
Specified by:
cumulativeProportionVar in interface PCA
Returns:
the cumulative proportion of overall variance explained by the principal components
• #### scores

public Matrix scores()
Description copied from interface: PCA
Gets the scores of supplied data on the principal components. The signs of the columns of the scores are arbitrary.
Specified by:
scores in interface PCA
Returns:
the scores of supplied data on the principal components