PCAbySVD (SuanShuExtension API)

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

All Implemented Interfaces:

PCA, java.io.Serializable
```
public class PCAbySVD
extends java.lang.Object
```
This class performs a Principal Component Analysis (PCA) on the given data matrix using the preferred singular value decomposition (SVD) method. The R equivalent function is prcomp.
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
`PCAbySVD(Matrix data)` Perform a principal component analysis, using the preferred SVD method, on a centered and scaled data matrix.
`PCAbySVD(Matrix data, boolean centered, boolean scaled)` Perform a principal component analysis, using the preferred SVD method, on a given data matrix (possibly centered and/or scaled).
`PCAbySVD(Matrix data, boolean centered, boolean scaled, Vector mean, Vector scale)` Perform a Principal Component Analysis, using the preferred SVD method, on a given data matrix with (optional) mean vector and scaling vector 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.
`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, the scaling vector to be divided.
`Matrix`	`scores()` Get the scores of supplied data on the principal components.
`DenseVector`	`sdPrincipalComponent()` Get 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)
`double`	`sdPrincipalComponent(int i)` Get the standard deviation of the i-th principal component.
`SVD`	`svd()` Get the singular value decomposition (SVD) of matrix X.
`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
  - PCAbySVD
```
public PCAbySVD(Matrix data,
                boolean centered,
                boolean scaled,
                Vector mean,
                Vector scale)
```
    Perform a Principal Component Analysis, using the preferred SVD method, on a given data matrix with (optional) mean vector and scaling vector provided.
    
    Parameters:
    
    data - a matrix which provides the original data for the principal component analysis
    
    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)
    
    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)
```
    Perform a principal component analysis, using the preferred SVD method, on a given data matrix (possibly centered and/or scaled).
    
    Parameters:
    
    data - a matrix which provides the original data for the principal component analysis
    
    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)
```
    Perform a principal component analysis, using the preferred SVD method, on a centered and scaled data matrix.
    
    Parameters:
    
    data - a matrix which provides the original data for the principal component analysis
- Method Detail
  - mean
```
public Vector mean()
```
    Get 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()
```
    Get the scalings applied to each variable, the scaling vector to be divided.
    
    Specified by:
    
    scale in interface PCA
    
    Returns:
    
    the scalings applied to each variable in the original data
  - svd
```
public SVD svd()
```
    Get the singular value decomposition (SVD) of matrix X.
    
    Returns:
    
    the singular value decomposition (SVD) of matrix X
  - sdPrincipalComponent
```
public DenseVector sdPrincipalComponent()
```
    Get 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
  - 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
  - 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
  - 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 Vector proportionVar()
```
    Description copied from interface: PCA
    
    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
  - 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 PCAbySVD

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

PCAbySVD

PCAbySVD

PCAbySVD

Method Detail

mean

scale

svd

sdPrincipalComponent

loadings

data

nObs

nFactors

X

sdPrincipalComponent

loadings

proportionVar

proportionVar

cumulativeProportionVar

scores