de.unihalle.informatik.MiToBo.math.statistics

Class PCA

java.lang.Object

de.unihalle.informatik.Alida.operator.ALDOperator

de.unihalle.informatik.MiToBo.core.operator.MTBOperator

de.unihalle.informatik.MiToBo.math.statistics.PCA

All Implemented Interfaces:

de.unihalle.informatik.Alida.datatypes.ALDConfigurationValidator, de.unihalle.informatik.Alida.operator.events.ALDOperatorExecutionProgressEventListener, EventListener

@ALDAOperator(genericExecutionMode=ALL,
              level=STANDARD,
              allowBatchMode=false,
              shortDescription="This class implements the Karhunen-Loeve transformation, also known as PCA.")
public class PCA
extends MTBOperator

This class implements the Karhunen-Loeve transformation, also known as PCA.

Given a data matrix A where each column contains a data vector, first the covariance matrix of the data, i.e., $A\cdot A^T$, is calculated. Then the eigenvalues and eigenvectors of this matrix are computed according to

\begin{equation} A \cdot A^T \cdot \vec{v} = \lambda \cdot \vec{v} \end{equation}

A subset of the eigenvectors corresponding to the largest eigenvalues is then selected according to the given dimension reduction mode. These are finally used to form the basis of a new vector space with reduced dimensionality. The (mean-free) input data is projected into this space for dimension reduction and yields the result of the operator.

In case that the dimensionality of the data is larger than the available number of samples, i.e., the input data matrix has more row than columns, the calculations are simplified by using the matrix $A^T\cdot A$ instead of the covariance matrix which is larger in this case. For the eigenvectors and values of this matrix the following equation holds:

\begin{equation} A^T \cdot A \cdot \vec{w} = \lambda \cdot \vec{w} \;\; \Longleftrightarrow \;\; A \cdot (A^T \cdot A ) \cdot \vec{w} = \lambda \cdot A \cdot \vec{w} \;\; \Longleftrightarrow \;\; (A \cdot A^T ) \cdot A \cdot \vec{w} = \lambda \cdot A \cdot \vec{w} \end{equation}

In detail, the eigenvectors of this matrix, denoted by $\vec{w}$ in the above equation, can be used to calculate the eigenvectors $\vec{v} = A \cdot \vec{w}$ of the covariance matrix without need for explicitly solving the problem for the larger matrix.

Note that if no proper sub-space can be determined, e.g., because only a single data item is provided, no transformation is applied and the output data is identical to the input data.

Author:

moeller

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	PCA.ReductionMode Available modes for determining the sub-space dimensionality.

Nested classes/interfaces inherited from class de.unihalle.informatik.Alida.operator.ALDOperator

de.unihalle.informatik.Alida.operator.ALDOperator.HidingMode, de.unihalle.informatik.Alida.operator.ALDOperator.HistoryConstructionMode

Field Summary

Fields

Modifier and Type	Field and Description
protected Jama.Matrix	C Covariance matrix calculated from mean-free data.
private int	componentNum Number of sub-space components in mode ReductionMode.NUMBER_COMPONENTS.
protected int	dataDim Dimensionality of the input data.
private double[][]	dataset Input data with each column containing a data vector.
protected double[]	eigenVals Set of computed eigenvalues.
protected Jama.Matrix	eigenVects Matrix of eigenvectors, each column containing a vector.
private boolean	isMeanFree Flag for indicating if input data is already mean-free.
protected double[]	mean Average vector of input dataset.
protected double[][]	meanfreeData Normalized, i.e., mean-free, dataset.
protected Jama.Matrix	meanfreeDataMatrix Normalized, i.e., mean-free, data matrix.
private PCA.ReductionMode	mode Mode for dimension reduction, i.e., how to determine the sub-space dimensionality.
protected Jama.Matrix	P_t The final transformation matrix to be used for dimension reduction.
private double	percentageVar Variance fraction for automatic dimension selection in mode ReductionMode.PERCENTAGE_VARIANCE.
private double[][]	resultData Resulting data set with each column containing a data vector.
protected int	sampleCount Number of data samples in input data.
protected int	subDim Dimensionality of the sub-space as either specified by the user or automatically determined based on the percentage of variance.

Fields inherited from class de.unihalle.informatik.Alida.operator.ALDOperator

completeDAG, name, operatorExecutionEventlistenerList, portHashAccess, verbose, versionProvider

Constructor Summary

Constructors

Constructor and Description
PCA() Default constructor.

Method Summary

Modifier and Type	Method and Description
protected void	calculateCovarianceMatrixAndEigenstuff() Calculates covariance matrix and eigenvalues and -vectors.
protected void	calculateMeanFreeData() Computes the average data vector and makes data mean-free.
protected void	determineSubspaceDimension() Determines desired sub-space dimensionality according to selected mode.
protected void	doDimensionReduction() Does the actual dimension reduction by data projection into sub-space.
protected void	examineDataset() Extracts number of samples and their dimension from dataset.
String	getDocumentation()
double[]	getEigenvalues() Get calculated eigenvalues in ascending order.
double[][]	getEigenvects() Get calculated eigenvectors, one vector per column, in ascending order.
double[][]	getResultData() Get the transformed dataset.
protected void	operate() This method does the actual work.
void	setDataset(double[][] ds) Specify an input dataset.
void	setMeanFreeData(boolean b) Set flag to indicate if data is already mean-free.
void	setNumberOfComponents(int compNum) Number of sub-space components if reduction mode is NUMBER_COMPONENTS.
void	setPercentageOfVariance(double p) Fraction of variance to be represented in the sub-space if the reduction mode is PERCENTAGE_VARIANCE.
void	setReductionMode(PCA.ReductionMode rm) Specify the mode for selecting the sub-space dimensionality.

Methods inherited from class de.unihalle.informatik.MiToBo.core.operator.MTBOperator

readResolve

Methods inherited from class de.unihalle.informatik.Alida.operator.ALDOperator

addOperatorExecutionProgressEventListener, addParameter, addParameter, addParameterUnconditioned, fieldContained, fireOperatorExecutionProgressEvent, getALDPortHashAccessKey, getConstructionMode, getHidingMode, getInactiveParameterNames, getInInoutNames, getInInoutNames, getInNames, getInOutNames, getMissingRequiredInputs, getName, getNumParameters, getOutInoutNames, getOutNames, getParameter, getParameterDescriptor, getParameterDescriptorUnconditioned, getParameterNames, getParameterUnconditioned, getSupplementalNames, getVerbose, getVersion, handleOperatorExecutionProgressEvent, hasInOutParameters, hasParameter, isAnnotatedParameter, isConfigured, print, print, print, printInterface, printInterface, readHistory, reinitializeParameterDescriptors, removeOperatorExecutionProgressEventListener, removeParameter, runOp, runOp, runOp, setConstructionMode, setConstructionMode, setConstructionMode, setHidingMode, setName, setParameter, setParameterUnconditioned, setVerbose, toStringVerbose, unconfiguredItems, validate, validateCustom, validateGeneric, writeHistory, writeHistory, writeHistory

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

dataset

@Parameter(label="Dataset",
           required=true,
           dataIOOrder=-1,
           direction=IN,
           description="Dataset.")
private double[][] dataset

Input data with each column containing a data vector.

isMeanFree

@Parameter(label="Is data mean-free?",
           required=true,
           direction=IN,
           dataIOOrder=1,
           mode=ADVANCED,
           description="Set to true, if data is already mean-free.")
private boolean isMeanFree

Flag for indicating if input data is already mean-free.

mode

@Parameter(label="Reduction Mode",
           required=true,
           dataIOOrder=2,
           direction=IN,
           description="Mode.")
private PCA.ReductionMode mode

Mode for dimension reduction, i.e., how to determine the sub-space dimensionality.

componentNum

@Parameter(label="Number of Components",
           required=true,
           direction=IN,
           dataIOOrder=3,
           description="Number of components, i.e., sub-space dimensionality.")
private int componentNum

Number of sub-space components in mode ReductionMode.NUMBER_COMPONENTS.

percentageVar

@Parameter(label="Variance fraction",
           required=true,
           direction=IN,
           dataIOOrder=4,
           description="Percentage of data variance to be contained in sub-space.")
private double percentageVar

Variance fraction for automatic dimension selection in mode ReductionMode.PERCENTAGE_VARIANCE.

resultData

@Parameter(label="Result Dataset",
           required=true,
           direction=OUT,
           description="Result dataset.")
private transient double[][] resultData

Resulting data set with each column containing a data vector.

dataDim

protected int dataDim

Dimensionality of the input data.

sampleCount

protected int sampleCount

Number of data samples in input data.

mean

protected transient double[] mean

Average vector of input dataset.

meanfreeData

protected transient double[][] meanfreeData

Normalized, i.e., mean-free, dataset.

meanfreeDataMatrix

protected transient Jama.Matrix meanfreeDataMatrix

Normalized, i.e., mean-free, data matrix.

C

protected transient Jama.Matrix C

Covariance matrix calculated from mean-free data.

The scaling by the number of samples is omitted here as this is just a constant factor in eigenvalue and -vector calculations.

eigenVals

protected transient double[] eigenVals

Set of computed eigenvalues.

Note that the values are in ascending order.

eigenVects

protected transient Jama.Matrix eigenVects

Matrix of eigenvectors, each column containing a vector.

The vectors are sorted according to their eigenvalues, i.e., the vector corresponding to the largest eigenvalue can be found in the last column.

subDim

protected transient int subDim

Dimensionality of the sub-space as either specified by the user or automatically determined based on the percentage of variance.

P_t

protected transient Jama.Matrix P_t

The final transformation matrix to be used for dimension reduction.

This matrix is already transposed, i.e., each row contains a sub-space basis vector and the number of rows is equal to the dimension of the sub-space.

Constructor Detail

PCA

public PCA()
    throws de.unihalle.informatik.Alida.exceptions.ALDOperatorException

Default constructor.

Throws:

de.unihalle.informatik.Alida.exceptions.ALDOperatorException

Method Detail

setDataset

public void setDataset(double[][] ds)

Specify an input dataset.

Parameters:

ds - Dataset to process.

setMeanFreeData

public void setMeanFreeData(boolean b)

Set flag to indicate if data is already mean-free.

Parameters:

b - If true, the input data is assumed to be mean-free already.

setReductionMode

public void setReductionMode(PCA.ReductionMode rm)

Specify the mode for selecting the sub-space dimensionality.

Parameters:

rm - Mode for dimension reduction.

setNumberOfComponents

public void setNumberOfComponents(int compNum)

Number of sub-space components if reduction mode is NUMBER_COMPONENTS.

Parameters:

compNum - Number of components, i.e., eigenvectors, to use.

setPercentageOfVariance

public void setPercentageOfVariance(double p)

Fraction of variance to be represented in the sub-space if the reduction mode is PERCENTAGE_VARIANCE.

Parameters:

p - Fraction of variance to represent in sub-space.

getResultData

public double[][] getResultData()

Get the transformed dataset.

Returns:

Resulting dataset.

getEigenvalues

public double[] getEigenvalues()

Get calculated eigenvalues in ascending order.

Returns:

Set of eigenvalues, null if calculations are not yet completed.

getEigenvects

public double[][] getEigenvects()

Get calculated eigenvectors, one vector per column, in ascending order.

Returns:

Set of eigenvectors, null if calculations are not yet completed.

operate

protected void operate()

This method does the actual work.

Specified by:

operate in class de.unihalle.informatik.Alida.operator.ALDOperator

examineDataset

protected void examineDataset()

Extracts number of samples and their dimension from dataset.

calculateMeanFreeData

protected void calculateMeanFreeData()

Computes the average data vector and makes data mean-free.

calculateCovarianceMatrixAndEigenstuff

protected void calculateCovarianceMatrixAndEigenstuff()

Calculates covariance matrix and eigenvalues and -vectors.

determineSubspaceDimension

protected void determineSubspaceDimension()

Determines desired sub-space dimensionality according to selected mode.

doDimensionReduction

protected void doDimensionReduction()

Does the actual dimension reduction by data projection into sub-space.

getDocumentation

public String getDocumentation()

Overrides:

getDocumentation in class de.unihalle.informatik.Alida.operator.ALDOperator