PCAbyEigen (SuanShuExtension API)

java.lang.Object
- com.numericalmethod.suanshu.stats.pca.PCAbyEigen

All Implemented Interfaces:

PCA, java.io.Serializable
```
public class PCAbyEigen
extends java.lang.Object
```
This class performs a principal component analysis (PCA) on the given data matrix. The calculation is done using eigen on the correlation or covariance matrix. A preferred method of calculation is to use singular value decomposition on the data matrix, as is done in PrincipalComponentsAnalysis.
Each eigenvalue is proportional to the portion of the "variance" (more correctly of the sum of the squared distances of the points from their multidimensional mean) that is correlated with each eigenvector. The sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean. 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 princomp. The main difference is that we use divisor (nObs - 1) instead of nObs for the sample covariance matrix.
See Also:
- 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
, Serialized Form

Constructor Summary

Constructors
Constructor and Description
`PCAbyEigen(Matrix data)` Perform a principal component analysis, using the eigen method and the preferred correlation matrix, on a given data matrix.
`PCAbyEigen(Matrix data, boolean correlation)` Perform a principal component analysis, using the eigen method, on a given data matrix.
`PCAbyEigen(Matrix data, boolean correlation, Matrix V)` Perform a principal component analysis, using the eigen method, on a given data matrix with an optional correlation (or covariance) matrix provided.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`Vector`	`cumulativeProportionVar()` Get the cumulative proportion of overall variance explained by the principal components
`ImmutableMatrix`	`data()` Get the original data matrix.
`Eigen`	`eigen()` Get the eigenvalue decomposition of the correlation (or covariance) matrix.
`Matrix`	`loadings()` Get the matrix of variable loadings.
`Vector`	`loadings(int i)` Get the loading vector of the i-th principal component.
`Vector`	`mean()` Get the sample means that were subtracted.
`int`	`nFactors()` Get the number of variables in the original data.
`int`	`nObs()` Get the number of observations in the original data; sample size.
`Vector`	`proportionVar()` Get the proportion of overall variance explained by each of the principal components.
`double`	`proportionVar(int i)` Get the proportion of overall variance explained by the i-th principal component.
`Vector`	`scale()` Get the scalings applied to each variable.
`Matrix`	`scores()` Get the scores of supplied data on the principal components.
`Vector`	`sdPrincipalComponent()` Get the standard deviations of the principal components (i.e., the square roots of the eigenvalues of the covariance or correlation matrix).
`double`	`sdPrincipalComponent(int i)` Get the standard deviation of the i-th principal component.
`Matrix`	`V()` Get the correlation (or covariance) matrix used for the PCA.
`Matrix`	`X()` Get 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
  - PCAbyEigen
```
public PCAbyEigen(Matrix data,
                  boolean correlation,
                  Matrix V)
```
    Perform a principal component analysis, using the eigen method, on a given data matrix with an optional correlation (or covariance) matrix provided.
    
    Parameters:
    
    data - an nObs*nFactors numeric matrix which provides the original data for the principal component analysis
    
    correlation - a logical value indicating whether the correlation matrix (preferred) or the covariance matrix should be used (N.B. the correlation matrix can only be used if there is no constant variable)
    
    V - an optional correlation (or covariance) matrix; if supplied, this is used rather than the correlation (or covariance) matrix of the centered (and possibly scaled) data
  - PCAbyEigen
```
public PCAbyEigen(Matrix data,
                  boolean correlation)
```
    Perform a principal component analysis, using the eigen method, on a given data matrix.
    
    Parameters:
    
    data - a matrix which provides the original data for the principal component analysis
    
    correlation - a logical value indicating whether the correlation matrix (preferred) or the covariance matrix should be used (N.B. the correlation matrix can only be used if there is no constant variable)
  - PCAbyEigen
```
public PCAbyEigen(Matrix data)
```
    Perform a principal component analysis, using the eigen method and the preferred correlation matrix, on a given data matrix.
    
    Parameters:
    
    data - matrix which provides the original data for the principal component analysis
- Method Detail
  - scale
```
public Vector scale()
```
    Get the scalings applied to each variable. If covariance matrix is used instead of the (preferred) correlation matrix, no scaling is performed.
    
    Specified by:
    
    scale in interface PCA
    
    Returns:
    
    the scalings applied to each variable in the original data
  - V
```
public Matrix V()
```
    Get the correlation (or covariance) matrix used for the PCA.
    
    Returns:
    
    the correlation (or covariance) matrix
  - eigen
```
public Eigen eigen()
```
    Get the eigenvalue decomposition of the correlation (or covariance) matrix.
    
    Returns:
    
    the eigenvalue decomposition of the correlation (or covariance) matrix
  - sdPrincipalComponent
```
public Vector sdPrincipalComponent()
```
    Get the standard deviations of the principal components (i.e., the square roots of the eigenvalues of the covariance or correlation matrix).
    
    Returns:
    
    the standard deviations of the principal components
  - loadings
```
public Matrix loadings()
```
    Get the matrix of variable loadings. The signs of the columns of the loading are arbitrary.
    
    Returns:
    
    the matrix of variable loadings
  - proportionVar
```
public Vector proportionVar()
```
    Get 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
  - data
```
public ImmutableMatrix data()
```
    Get the original data matrix.
    
    Returns:
    
    the original data matrix
  - nObs
```
public int nObs()
```
    Description copied from interface: PCA
    
    Get 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
    
    Get the number of variables in the original data.
    
    Specified by:
    
    nFactors in interface PCA
    
    Returns:
    
    nFactors, the number of variables in the original data
  - mean
```
public Vector mean()
```
    Description copied from interface: PCA
    
    Get the sample means that were subtracted.
    
    Specified by:
    
    mean in interface PCA
    
    Returns:
    
    the sample means of each variable in the original data
  - X
```
public Matrix X()
```
    Description copied from interface: PCA
    
    Get 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
    
    Get 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.
  - loadings
```
public Vector loadings(int i)
```
    Description copied from interface: PCA
    
    Get the loading vector of the i-th principal component.
    
    Specified by:
    
    loadings in interface PCA
    
    Parameters:
    
    i - an index, counting from 1
    
    Returns:
    
    the loading vector of the i-th principal component
  - proportionVar
```
public double proportionVar(int i)
```
    Description copied from interface: PCA
    
    Get 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 Vector cumulativeProportionVar()
```
    Description copied from interface: PCA
    
    Get 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
    
    Get 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

Class PCAbyEigen

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

PCAbyEigen

PCAbyEigen

PCAbyEigen

Method Detail

scale

V

eigen

sdPrincipalComponent

loadings

proportionVar

data

nObs

nFactors

mean

X

sdPrincipalComponent

loadings

proportionVar

cumulativeProportionVar

scores