IntegralOperatorEigenComponent.java

package org.drip.learning.kernel;

/*
 * -*- 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>IntegralOperatorEigenComponent</i> holds the Eigen-Function Space and the Eigenvalue Functions/Spaces
 * of the R<sup>x</sup> L<sub>2</sub> To R<sup>x</sup> L<sub>2</sub> Kernel Linear Integral Operator defined
 * by:
 * 
 *      T_k [f(.)] := Integral Over Input Space {k (., y) * f(y) * d[Prob(y)]}
 *  
 * <br><br>
 *  The References are:
 * <br><br>
 * <ul>
 *  <li>
 *      Ash, R. (1965): <i>Information Theory</i> <b>Inter-science</b> New York
 *  </li>
 *  <li>
 *      Konig, H. (1986): <i>Eigenvalue Distribution of Compact Operators</i> <b>Birkhauser</b> Basel,
 *          Switzerland
 *  </li>
 *  <li>
 *      Smola, A. J., A. Elisseff, B. Scholkopf, and R. C. Williamson (2000): Entropy Numbers for Convex
 *          Combinations and mlps, in: <i>Advances in Large Margin Classifiers, A. Smola, P. Bartlett, B.
 *          Scholkopf, and D. Schuurmans - editors</i> <b>MIT Press</b> Cambridge, MA
 *  </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</a></li>
 *      <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/learning">Agnostic Learning Bounds under Empirical Loss Minimization Schemes</a></li>
 *      <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/learning/kernel">Statistical Learning Banach Mercer Kernels</a></li>
 *  </ul>
 *
 * @author Lakshmi Krishnamurthy
 */

public class IntegralOperatorEigenComponent {
    private double _dblEigenValue = java.lang.Double.NaN;
    private org.drip.learning.kernel.EigenFunctionRdToR1 _efRdToR1 = null;
    private org.drip.spaces.rxtor1.NormedRdToNormedR1 _rkhsFeatureMap = null;

    /**
     * IntegralOperatorEigenComponent Constructor
     * 
     * @param efRdToR1 Normed R^d To Normed R^1 Eigen-Function
     * @param dblEigenValue The Eigenvalue
     * 
     * @throws java.lang.Exception Thrown if the Inputs are Invalid
     */

    public IntegralOperatorEigenComponent (
        final org.drip.learning.kernel.EigenFunctionRdToR1 efRdToR1,
        final double dblEigenValue)
        throws java.lang.Exception
    {
        if (null == (_efRdToR1 = efRdToR1) || !org.drip.numerical.common.NumberUtil.IsValid (_dblEigenValue =
            dblEigenValue))
            throw new java.lang.Exception ("IntegralOperatorEigenComponent ctr: Invalid Inputs");

        final org.drip.function.definition.RdToR1 eigenFuncRdToR1 = _efRdToR1.function();

        if (null != eigenFuncRdToR1) {
            org.drip.function.definition.RdToR1 rkhsFeatureMapRdToR1 = new
                org.drip.function.definition.RdToR1 (null) {
                @Override public int dimension()
                {
                    return org.drip.function.definition.RdToR1.DIMENSION_NOT_FIXED;
                }

                @Override public double evaluate (
                    final double[] adblX)
                    throws java.lang.Exception
                {
                    return java.lang.Math.sqrt (_dblEigenValue) * eigenFuncRdToR1.evaluate (adblX);
                }
            };

            org.drip.spaces.metric.RdNormed rdContinuousInput = efRdToR1.inputMetricVectorSpace();

            org.drip.spaces.metric.R1Normed r1ContinuousOutput = efRdToR1.outputMetricVectorSpace();

            org.drip.spaces.metric.R1Continuous r1Continuous = org.drip.spaces.metric.R1Continuous.Standard
                (r1ContinuousOutput.leftEdge(), r1ContinuousOutput.rightEdge(),
                    r1ContinuousOutput.borelSigmaMeasure(), 2);

            _rkhsFeatureMap = rdContinuousInput instanceof org.drip.spaces.metric.RdCombinatorialBanach ? new
                org.drip.spaces.rxtor1.NormedRdCombinatorialToR1Continuous
                    ((org.drip.spaces.metric.RdCombinatorialBanach) rdContinuousInput, r1Continuous,
                        rkhsFeatureMapRdToR1) : new org.drip.spaces.rxtor1.NormedRdContinuousToR1Continuous
                            ((org.drip.spaces.metric.RdContinuousBanach) rdContinuousInput, r1Continuous,
                                rkhsFeatureMapRdToR1);
        }
    }

    /**
     * Retrieve the Eigen-Function
     * 
     * @return The Eigen-Function
     */

    public org.drip.learning.kernel.EigenFunctionRdToR1 eigenFunction()
    {
        return _efRdToR1;
    }

    /**
     * Retrieve the Eigenvalue
     * 
     * @return The Eigenvalue
     */

    public double eigenvalue()
    {
        return _dblEigenValue;
    }

    /**
     * Retrieve the Feature Map Space represented via the Reproducing Kernel Hilbert Space
     * 
     * @return The Feature Map Space representation using the Reproducing Kernel Hilbert Space
     */

    public org.drip.spaces.rxtor1.NormedRdToNormedR1 rkhsFeatureMap()
    {
        return _rkhsFeatureMap;
    }

    /**
     * Retrieve the RKHS Feature Map Parallelepiped Agnostic Upper Bound Length
     * 
     * @return The RKHS Feature Map Parallelepiped Agnostic Upper Bound Length
     */

    public double rkhsFeatureParallelepipedLength()
    {
        return 2. * _efRdToR1.agnosticUpperBound() * java.lang.Math.sqrt (_dblEigenValue);
    }

    /**
     * Compute the Eigen-Component Contribution to the Kernel Value
     * 
     * @param adblX The X Variate Array
     * @param adblY The Y Variate Array
     * 
     * @return The Eigen-Component Contribution to the Kernel Value
     * 
     * @throws java.lang.Exception Thrown if the Inputs are invalid
     */

    public double evaluate (
        final double[] adblX,
        final double[] adblY)
        throws java.lang.Exception
    {
        org.drip.function.definition.RdToR1 eigenFuncRdToR1 = _efRdToR1.function();

        return eigenFuncRdToR1.evaluate (adblX) * eigenFuncRdToR1.evaluate (adblY) * _dblEigenValue;
    }
}