CoveringBoundsHelper.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>CoveringBoundsHelper</i> contains the assortment of Utilities used in the Computation of Upper Bounds
* for Normed Single Function Spaces and Function Space Products. The References are:
*
* <br><br>
* <ul>
* <li>
* Carl, B. (1985): Inequalities of the Bernstein-Jackson type and the Degree of Compactness of
* Operators in Banach Spaces <i>Annals of the Fourier Institute</i> <b>35 (3)</b> 79-118
* </li>
* <li>
* Carl, B., and I. Stephani (1990): <i>Entropy, Compactness, and the Approximation of Operators</i>
* <b>Cambridge University Press</b> Cambridge UK
* </li>
* <li>
* Williamson, R. C., A. J. Smola, and B. Scholkopf (2000): Entropy Numbers of Linear Function
* Classes, in: <i>Proceedings of the 13th Annual Conference on Computational Learning
* Theory</i> <b>ACM</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 CoveringBoundsHelper {
/**
* Compute the Dyadic Entropy Number from the nth Entropy Number
*
* @param dblLogNEntropyNumber Log of the nth Entropy Number
*
* @return The Dyadic Entropy Number
*
* @throws java.lang.Exception Thrown if the Dyadic Entropy Number cannot be calculated
*/
public static final double DyadicEntropyNumber (
final double dblLogNEntropyNumber)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (dblLogNEntropyNumber))
throw new java.lang.Exception ("CoveringBoundsHelper::DyadicEntropyNumber => Invalid Inputs");
return 1. + (dblLogNEntropyNumber / java.lang.Math.log (2.));
}
/**
* Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product
* across both the Function Classes
*
* @param mocbA The Maurey Operator Covering Bounds for Class A
* @param mocbB The Maurey Operator Covering Bounds for Class B
* @param iEntropyNumberIndexA Entropy Number Index for Class A
* @param iEntropyNumberIndexB Entropy Number Index for Class B
*
* @return The Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product
* across both the Function Classes
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public static final double CarlStephaniProductBound (
final org.drip.spaces.cover.MaureyOperatorCoveringBounds mocbA,
final org.drip.spaces.cover.MaureyOperatorCoveringBounds mocbB,
final int iEntropyNumberIndexA,
final int iEntropyNumberIndexB)
throws java.lang.Exception
{
if (null == mocbA || null == mocbB)
throw new java.lang.Exception
("CoveringBoundsHelper::CarlStephaniProductBound => Invalid Maurey Bounds for the Function Class");
return mocbA.entropyNumberUpperBound (iEntropyNumberIndexA) * mocbB.entropyNumberUpperBound
(iEntropyNumberIndexB);
}
/**
* Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product
* across both the Function Classes using the Function Class Norm
*
* @param mocbA The Maurey Operator Covering Bounds for Class A
* @param mocbB The Maurey Operator Covering Bounds for Class B
* @param dblNormA The Function Class A Norm
* @param dblNormB The Function Class B Norm
* @param iEntropyNumberIndex Entropy Number Index for either Class
*
* @return The Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric
* Product across both the Function Classes using the Function Norm
*/
public static final org.drip.spaces.cover.CarlStephaniNormedBounds CarlStephaniProductNorm (
final org.drip.spaces.cover.MaureyOperatorCoveringBounds mocbA,
final org.drip.spaces.cover.MaureyOperatorCoveringBounds mocbB,
final double dblNormA,
final double dblNormB,
final int iEntropyNumberIndex)
{
if (null == mocbA || null == mocbB) return null;
try {
return new org.drip.spaces.cover.CarlStephaniNormedBounds (mocbA.entropyNumberUpperBound
(iEntropyNumberIndex) * dblNormB, mocbB.entropyNumberUpperBound (iEntropyNumberIndex) *
dblNormA);
} catch (java.lang.Exception e) {
e.printStackTrace();
}
return null;
}
}