Package org.drip.spaces.metric

Class RdContinuousBanach

java.lang.Object

org.drip.spaces.tensor.RdAggregate

org.drip.spaces.tensor.RdContinuousVector

org.drip.spaces.metric.RdContinuousBanach

All Implemented Interfaces:

GeneralizedMetricVectorSpace, RdNormed, GeneralizedVector, RdGeneralizedVector

Direct Known Subclasses:

RdContinuousBall, RdContinuousHilbert

public class RdContinuousBanach
extends RdContinuousVector
implements RdNormed

RdContinuousBanach implements the Normed, Bounded/Unbounded Continuous l_p R^d Spaces. The Reference we've used is:

• Carl, B., and I. Stephani (1990): Entropy, Compactness, and the Approximation of Operators Cambridge University Press Cambridge UK

It provides the following Functionality:

• Construct the Standard l^p R^d Continuous Banach Space Instance
• Construct the Supremum (i.e., l^Infinity) R^d Continuous Banach Space Instance
• RdContinuousBanach Space Constructor

Module	Computational Core Module
Library	Statistical Learning Library
Project	R1 and Rd Vector/Tensor Spaces (Validated and/or Normed), and Function Classes
Package	Hilbert/Banach Normed Metric Spaces

Author:

Lakshmi Krishnamurthy

Field Summary

Fields inherited from interface org.drip.spaces.metric.GeneralizedMetricVectorSpace

EUCLIDEAN_NORM, L2_NORM, LINFINITITY_NORM

Constructor Summary

Constructors

Constructor	Description
RdContinuousBanach(R1ContinuousVector[] r1ContinuousVectorArray, Rd rdDistribution, int pNorm)	RdContinuousBanach Space Constructor

Method Summary

Modifier and Type	Method	Description
double	borelMeasureSpaceExpectation(RdToR1 rdToR1Function)	Compute the Borel Measure Expectation for the specified Rd To R1 Function over the full Input Space
Rd	borelSigmaMeasure()	Retrieve the Borel Sigma Rd Probability Measure
int	pNorm()	Retrieve the P-Norm Index of the Metric Space
double	populationMetricNorm()	Retrieve the Population Metric Norm
double[]	populationMode()	Retrieve the Population Mode
double	populationSupremumNorm()	Compute the Population Supremum Norm of the Sample
double	sampleMetricNorm(double[] xArray)	Compute the Metric Norm of the Sample
double	sampleSupremumNorm(double[] xArray)	Compute the Supremum Norm of the Sample
static RdContinuousBanach	StandardBanach(int iDimension, Rd distRd, int iPNorm)	Construct the Standard lp Rd Continuous Banach Space Instance
static RdContinuousBanach	SupremumBanach(int iDimension, Rd distRd)	Construct the Supremum (i.e., lInfinity) Rd Continuous Banach Space Instance

Methods inherited from class org.drip.spaces.tensor.RdContinuousVector

cardinality, hyperVolume, leftDimensionEdge, leftEdge, rightDimensionEdge, rightEdge, Standard

Methods inherited from class org.drip.spaces.tensor.RdAggregate

dimension, isPredictorBounded, match, subset, validateInstance, vectorSpaces

Methods inherited from class java.lang.Object

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

Methods inherited from interface org.drip.spaces.tensor.GeneralizedVector

cardinality, hyperVolume, isPredictorBounded, leftEdge, match, rightEdge, subset

Methods inherited from interface org.drip.spaces.tensor.RdGeneralizedVector

dimension, leftDimensionEdge, rightDimensionEdge, validateInstance, vectorSpaces

Constructor Details

RdContinuousBanach

public RdContinuousBanach(R1ContinuousVector[] r1ContinuousVectorArray,
Rd rdDistribution,
int pNorm)
                   throws java.lang.Exception

RdContinuousBanach Space Constructor

Parameters:

r1ContinuousVectorArray - Array of R¹ Continuous Vector

rdDistribution - The R^d Borel Sigma Measure

pNorm - The p-norm of the Space

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

Method Details

StandardBanach

public static final RdContinuousBanach StandardBanach(int iDimension,
Rd distRd,
int iPNorm)

Construct the Standard l^p R^d Continuous Banach Space Instance

Parameters:

iDimension - The Space Dimension

distRd - The R^d Borel Sigma Measure

iPNorm - The p-norm of the Space

Returns:

The Standard l^p R^d Continuous Banach Space Instance

SupremumBanach

public static final RdContinuousBanach SupremumBanach(int iDimension,
Rd distRd)

Construct the Supremum (i.e., l^Infinity) R^d Continuous Banach Space Instance

Parameters:

iDimension - The Space Dimension

distRd - The R^d Borel Sigma Measure

Returns:

The Supremum (i.e., l^Infinity) R^d Continuous Banach Space Instance

pNorm

public int pNorm()

Description copied from interface: GeneralizedMetricVectorSpace

Retrieve the P-Norm Index of the Metric Space

Specified by:

pNorm in interface GeneralizedMetricVectorSpace

Returns:

The P-Norm Index of the Metric Space

borelSigmaMeasure

public Rd borelSigmaMeasure()

Description copied from interface: RdNormed

Retrieve the Borel Sigma R^d Probability Measure

Specified by:

borelSigmaMeasure in interface RdNormed

Returns:

The Borel Sigma R^d Probability Measure

sampleSupremumNorm

public double sampleSupremumNorm(double[] xArray)
                          throws java.lang.Exception

Description copied from interface: RdNormed

Compute the Supremum Norm of the Sample

Specified by:

sampleSupremumNorm in interface RdNormed

Parameters:

xArray - The Sample

Returns:

The Supremum Norm of the Sample

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

sampleMetricNorm

public double sampleMetricNorm(double[] xArray)
                        throws java.lang.Exception

Description copied from interface: RdNormed

Compute the Metric Norm of the Sample

Specified by:

sampleMetricNorm in interface RdNormed

Parameters:

xArray - The Sample

Returns:

The Metric Norm of the Sample

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

populationMode

public double[] populationMode()

Description copied from interface: RdNormed

Retrieve the Population Mode

Specified by:

populationMode in interface RdNormed

Returns:

The Population Mode

populationSupremumNorm

public double populationSupremumNorm()
                              throws java.lang.Exception

Description copied from interface: RdNormed

Compute the Population Supremum Norm of the Sample

Specified by:

populationSupremumNorm in interface RdNormed

Returns:

The Population Supremum Norm of the Sample

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

populationMetricNorm

public double populationMetricNorm()
                            throws java.lang.Exception

Description copied from interface: GeneralizedMetricVectorSpace

Retrieve the Population Metric Norm

Specified by:

populationMetricNorm in interface GeneralizedMetricVectorSpace

Returns:

The Population Metric Norm

Throws:

java.lang.Exception - The Population Metric Norm cannot be computed

borelMeasureSpaceExpectation

public double borelMeasureSpaceExpectation(RdToR1 rdToR1Function)
                                    throws java.lang.Exception

Description copied from interface: RdNormed

Compute the Borel Measure Expectation for the specified R^d To R¹ Function over the full Input Space

Specified by:

borelMeasureSpaceExpectation in interface RdNormed

Parameters:

rdToR1Function - R^d To R¹ Function

Returns:

The Borel Measure Expectation for the specified R^d To R¹ Function over the full Input Space

Throws:

java.lang.Exception - Thrown if the Population Mode cannot be calculated