ScaleSensitiveCoveringBounds.java
package org.drip.spaces.cover;
/*
* -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
*/
/*!
* Copyright (C) 2020 Lakshmi Krishnamurthy
* Copyright (C) 2019 Lakshmi Krishnamurthy
* Copyright (C) 2018 Lakshmi Krishnamurthy
* Copyright (C) 2017 Lakshmi Krishnamurthy
* Copyright (C) 2016 Lakshmi Krishnamurthy
* Copyright (C) 2015 Lakshmi Krishnamurthy
*
* This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
* asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
* analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
* equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
* numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
* and computational support.
*
* https://lakshmidrip.github.io/DROP/
*
* DROP is composed of three modules:
*
* - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
* - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
* - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
*
* DROP Product Core implements libraries for the following:
* - Fixed Income Analytics
* - Loan Analytics
* - Transaction Cost Analytics
*
* DROP Portfolio Core implements libraries for the following:
* - Asset Allocation Analytics
* - Asset Liability Management Analytics
* - Capital Estimation Analytics
* - Exposure Analytics
* - Margin Analytics
* - XVA Analytics
*
* DROP Computational Core implements libraries for the following:
* - Algorithm Support
* - Computation Support
* - Function Analysis
* - Model Validation
* - Numerical Analysis
* - Numerical Optimizer
* - Spline Builder
* - Statistical Learning
*
* Documentation for DROP is Spread Over:
*
* - Main => https://lakshmidrip.github.io/DROP/
* - Wiki => https://github.com/lakshmiDRIP/DROP/wiki
* - GitHub => https://github.com/lakshmiDRIP/DROP
* - Repo Layout Taxonomy => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
* - Javadoc => https://lakshmidrip.github.io/DROP/Javadoc/index.html
* - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
* - Release Versions => https://lakshmidrip.github.io/DROP/version.html
* - Community Credits => https://lakshmidrip.github.io/DROP/credits.html
* - Issues Catalog => https://github.com/lakshmiDRIP/DROP/issues
* - JUnit => https://lakshmidrip.github.io/DROP/junit/index.html
* - Jacoco => https://lakshmidrip.github.io/DROP/jacoco/index.html
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
*
* You may obtain a copy of the License at
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
*
* See the License for the specific language governing permissions and
* limitations under the License.
*/
/**
* <i>ScaleSensitiveCoveringBounds</i> 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:
*
* <br><br>
* <ul>
* <li>
* N. Alon, S. Ben-David, N. Cesa-Bianchi, and D. Haussler (1993): Scale-sensitive Dimensions,
* Uniform-Convergence, and Learnability <i>Proceedings of the ACM Symposium on the Foundations
* of Computer Science</i>
* </li>
* <li>
* P. L. Bartlett, S. R. Kulkarni, and S. E. Posner (1997): Covering Numbers for Real-valued
* Function Classes <i>IEEE Transactions on Information Theory</i> <b>43 (5)</b> 1721-1724
* </li>
* <li>
* D. Pollard (1984): <i>Convergence of Stochastic Processes</i> <b>Springer</b> New York
* </li>
* </ul>
*
* <br><br>
* <ul>
* <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
* <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/StatisticalLearningLibrary.md">Statistical Learning Library</a></li>
* <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/spaces/README.md">R<sup>1</sup> and R<sup>d</sup> Vector/Tensor Spaces (Validated and/or Normed), and Function Classes</a></li>
* <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/spaces/cover/README.md">Vector Spaces Covering Number Estimator</a></li>
* </ul>
* <br><br>
*
* @author Lakshmi Krishnamurthy
*/
public class ScaleSensitiveCoveringBounds implements org.drip.spaces.cover.FunctionClassCoveringBounds {
private int _iSampleSize = -1;
private org.drip.function.definition.R1ToR1 _r1r1FatShatter = null;
/**
* ScaleSensitiveCoveringBounds Constructor
*
* @param r1r1FatShatter The Cover Fat Shattering Coefficient Function
* @param iSampleSize Sample Size
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public ScaleSensitiveCoveringBounds (
final org.drip.function.definition.R1ToR1 r1r1FatShatter,
final int iSampleSize)
throws java.lang.Exception
{
if (null == (_r1r1FatShatter = r1r1FatShatter) || 0 >= (_iSampleSize = iSampleSize))
throw new java.lang.Exception ("ScaleSensitiveCoveringBounds ctr: Invalid Inputs");
}
/**
* Retrieve the Fat Shattering Coefficient Function
*
* @return The Fat Shattering Coefficient Function
*/
public org.drip.function.definition.R1ToR1 fatShatteringFunction()
{
return _r1r1FatShatter;
}
/**
* Retrieve the Sample Size
*
* @return The Sample Size
*/
public int sampleSize()
{
return _iSampleSize;
}
/**
* Compute the Minimum Sample Size required to Estimate the Cardinality corresponding to the Specified
* Cover
*
* @param dblCover The Cover
*
* @return The Minimum Sample Size
*
* @throws java.lang.Exception Thrown if the Minimum Sample Size Cannot be computed
*/
public double sampleSizeLowerBound (
final double dblCover)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (dblCover) || 0. == dblCover)
throw new java.lang.Exception
("ScaleSensitiveCoveringBounds::sampleSizeLowerBound => Invalid Inputs");
double dblLog2 = java.lang.Math.log (2.);
return 2. * _r1r1FatShatter.evaluate (0.25 * dblCover) * java.lang.Math.log (64. * java.lang.Math.E *
java.lang.Math.E / (dblCover * dblLog2)) / dblLog2;
}
/**
* Compute the Cardinality for the Subset T (|x) that possesses the Specified Cover for the Restriction
* of the Input Function Class Family F (|x).
*
* @param dblCover The Specified Cover
*
* @return The Restricted Subset Cardinality
*
* @throws java.lang.Exception Thrown if the Restricted Subset Cardinality cannot be computed
*/
public double restrictedSubsetCardinality (
final double dblCover)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (dblCover) || 0. == dblCover)
throw new java.lang.Exception
("ScaleSensitiveCoveringBounds::restrictedSubsetCardinality => Invalid Inputs");
double dblLog2 = java.lang.Math.log (2.);
double dblFatShatteringCoefficient = _r1r1FatShatter.evaluate (0.25 * dblCover);
if (_iSampleSize < 2. * dblFatShatteringCoefficient * java.lang.Math.log (64. * java.lang.Math.E *
java.lang.Math.E / (dblCover * dblLog2)) / dblLog2)
throw new java.lang.Exception
("ScaleSensitiveCoveringBounds::restrictedSubsetCardinality => Invalid Inputs");
return 6. * dblFatShatteringCoefficient * java.lang.Math.log (16. / dblCover) * java.lang.Math.log
(32. * java.lang.Math.E * _iSampleSize / (dblFatShatteringCoefficient * dblCover)) / dblLog2 +
dblLog2;
}
/**
* 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.
*
* @param dblCover The Specified Cover
*
* @return 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
*/
public double upperProbabilityBoundWeight (
final double dblCover)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (dblCover) || 0. == dblCover)
throw new java.lang.Exception
("ScaleSensitiveCoveringBounds::upperProbabilityBoundWeight => Invalid Inputs");
return java.lang.Math.log (4.) - (dblCover * dblCover * _iSampleSize / 128.);
}
@Override public double logLowerBound (
final double dblCover)
throws java.lang.Exception
{
return restrictedSubsetCardinality (dblCover);
}
@Override public double logUpperBound (
final double dblCover)
throws java.lang.Exception
{
return _r1r1FatShatter.evaluate (4. * dblCover) / 32.;
}
}