org.drip.spaces.cover

Class ScaleSensitiveCoveringBounds

java.lang.Object

org.drip.spaces.cover.ScaleSensitiveCoveringBounds

All Implemented Interfaces:

FunctionClassCoveringBounds

public class ScaleSensitiveCoveringBounds
extends java.lang.Object
implements FunctionClassCoveringBounds

ScaleSensitiveCoveringBounds implements the Lower/Upper Bounds for the General Class of Functions in terms of their scale-sensitive dimensions (i.e., the fat shattering coefficients). The References are: 1) D. Pollard (1984): Convergence of Stochastic Processes, Springer, New York. 2) N. Alon, S. Ben-David, N. Cesa-Bianchi, and D. Haussler (1993): Scale-sensitive Dimensions, Uniform- Convergence, and Learnability, Proceedings of the ACM Symposium on the Foundations of Computer Science. 3) P. L. Bartlett, S. R. Kulkarni, and S. E. Posner (1997): Covering Numbers for Real-valued Function Classes, IEEE Transactions on Information Theory 43 (5) 1721-1724.

Author:

Lakshmi Krishnamurthy

Constructor Summary

Constructors

Constructor and Description
ScaleSensitiveCoveringBounds(R1ToR1 r1r1FatShatter, int iSampleSize) ScaleSensitiveCoveringBounds Constructor

Method Summary

Modifier and Type	Method and Description
R1ToR1	fatShatteringFunction() Retrieve the Fat Shattering Coefficient Function
double	logLowerBound(double dblCover) Log of the Lower Bound of the Function Covering Number
double	logUpperBound(double dblCover) Log of the Upper Bound of the Function Covering Number
double	restrictedSubsetCardinality(double dblCover) Compute the Cardinality for the Subset T (|x) that possesses the Specified Cover for the Restriction of the Input Function Class Family F (|x).
int	sampleSize() Retrieve the Sample Size
double	sampleSizeLowerBound(double dblCover) Compute the Minimum Sample Size required to Estimate the Cardinality corresponding to the Specified Cover
double	upperProbabilityBoundWeight(double dblCover) Compute the Log of the Weight Loading Coefficient for the Maximum Cover Term in: {Probability that the Empirical Error .gt.

Methods inherited from class java.lang.Object

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

Constructor Detail

ScaleSensitiveCoveringBounds

public ScaleSensitiveCoveringBounds(R1ToR1 r1r1FatShatter,
                                    int iSampleSize)
                             throws java.lang.Exception

ScaleSensitiveCoveringBounds Constructor

Parameters:

r1r1FatShatter - The Cover Fat Shattering Coefficient Function

iSampleSize - Sample Size

Throws:

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

Method Detail

fatShatteringFunction

public R1ToR1 fatShatteringFunction()

Retrieve the Fat Shattering Coefficient Function

Returns:

The Fat Shattering Coefficient Function

sampleSize

public int sampleSize()

Retrieve the Sample Size

Returns:

The Sample Size

sampleSizeLowerBound

public double sampleSizeLowerBound(double dblCover)
                            throws java.lang.Exception

Compute the Minimum Sample Size required to Estimate the Cardinality corresponding to the Specified Cover

Parameters:

dblCover - The Cover

Returns:

The Minimum Sample Size

Throws:

java.lang.Exception - Thrown if the Minimum Sample Size Cannot be computed

restrictedSubsetCardinality

public double restrictedSubsetCardinality(double dblCover)
                                   throws java.lang.Exception

Compute the Cardinality for the Subset T (|x) that possesses the Specified Cover for the Restriction of the Input Function Class Family F (|x).

Parameters:

dblCover - The Specified Cover

Returns:

The Restricted Subset Cardinality

Throws:

java.lang.Exception - Thrown if the Restricted Subset Cardinality cannot be computed

upperProbabilityBoundWeight

public double upperProbabilityBoundWeight(double dblCover)
                                   throws java.lang.Exception

Compute the Log of the Weight Loading Coefficient for the Maximum Cover Term in: {Probability that the Empirical Error .gt. Cover} .lte. 4 * exp (-m * Cover^2 / 128) * [[Max Covering Number Over the Specified Sample]] Reference is: - D. Haussler (1995): Sphere Packing Numbers for Subsets of the Boolean n-Cube with Bounded Vapnik-Chervonenkis Dimension, Journal of the COmbinatorial Theory A 69 (2) 217.

Parameters:

dblCover - The Specified Cover

Returns:

Log of the Weight Loading Coefficient for the Maximum Cover Term

Throws:

java.lang.Exception - Thrown if the Log of the Weight Loading Coefficient cannot be computed

logLowerBound

public double logLowerBound(double dblCover)
                     throws java.lang.Exception

Description copied from interface: FunctionClassCoveringBounds

Log of the Lower Bound of the Function Covering Number

Specified by:

logLowerBound in interface FunctionClassCoveringBounds

Parameters:

dblCover - The Size of the Cover

Returns:

Log of the Lower Bound of the Function Covering Number

Throws:

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

logUpperBound

public double logUpperBound(double dblCover)
                     throws java.lang.Exception

Description copied from interface: FunctionClassCoveringBounds

Log of the Upper Bound of the Function Covering Number

Specified by:

logUpperBound in interface FunctionClassCoveringBounds

Parameters:

dblCover - The Size of the Cover

Returns:

Log of the Upper Bound of the Function Covering Number

Throws:

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