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;
    }
}