IntegralOperator.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>IntegralOperator</i> implements the R<sup>x</sup> L<sub>2</sub> To R<sup>x</sup> L<sub>2</sub> Mercer
* Kernel 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 abstract class IntegralOperator {
private org.drip.measure.continuous.Rd _distRd = null;
private org.drip.function.definition.RdToR1 _funcRdToR1 = null;
private org.drip.spaces.metric.R1Normed _r1OperatorOutput = null;
private org.drip.learning.kernel.SymmetricRdToNormedR1Kernel _kernel = null;
/**
* IntegralOperator Constructor
*
* @param kernel The Symmetric Mercer Kernel - this should be R^x L2 X R^x L2 To R^1
* @param funcRdToR1 The R^d To R^1 Operator Function
* @param r1OperatorOutput The Kernel Integral Operator Output Space - this is R^1 L2
*
* @throws java.lang.Exception Thrown if the Inputs are invalid
*/
public IntegralOperator (
final org.drip.learning.kernel.SymmetricRdToNormedR1Kernel kernel,
final org.drip.function.definition.RdToR1 funcRdToR1,
final org.drip.spaces.metric.R1Normed r1OperatorOutput)
throws java.lang.Exception
{
if (null == (_kernel = kernel) || null == (_funcRdToR1 = funcRdToR1) || null == (_r1OperatorOutput =
r1OperatorOutput) || null == (_distRd = _kernel.inputMetricVectorSpace().borelSigmaMeasure()))
throw new java.lang.Exception ("IntegralOperator ctr: Invalid Inputs");
}
/**
* Retrieve the Symmetric R^d To R^1 Kernel
*
* @return The Symmetric R^d To R^1 Kernel
*/
public org.drip.learning.kernel.SymmetricRdToNormedR1Kernel kernel()
{
return _kernel;
}
/**
* Retrieve the R^d To R^1 Kernel Operator Function
*
* @return The R^d To R^1 Kernel Operator Function
*/
public org.drip.function.definition.RdToR1 kernelOperatorFunction()
{
return _funcRdToR1;
}
/**
* Retrieve the Input Space Borel Sigma Measure
*
* @return The Input Space Borel Sigma Measure
*/
public org.drip.measure.continuous.Rd inputSpaceBorelMeasure()
{
return _distRd;
}
/**
* Retrieve the Kernel Integral Operator Output Space
*
* @return The Kernel Integral Operator Output Space
*/
public org.drip.spaces.metric.R1Normed outputVectorMetricSpace()
{
return _r1OperatorOutput;
}
/**
* Compute the Operator's Kernel Integral across the specified X Variate Instance
*
* @param adblX Validated Vector Instance X
*
* @return The Operator's Kernel Integral across the specified X Variate Instance
*
* @throws java.lang.Exception Thrown if the Inputs are invalid
*/
public double computeOperatorIntegral (
final double[] adblX)
throws java.lang.Exception
{
org.drip.function.definition.RdToR1 funcRdToR1 = new org.drip.function.definition.RdToR1 (null) {
@Override public int dimension()
{
return null == adblX ? 0 : adblX.length;
}
@Override public double evaluate (
final double[] adblY)
throws java.lang.Exception
{
return _kernel.evaluate (adblX, adblY) * _funcRdToR1.evaluate (adblY);
}
};
return _kernel.inputMetricVectorSpace().borelMeasureSpaceExpectation (funcRdToR1);
}
/**
* Indicate the Kernel Operator Integral's Positive-definiteness across the specified X Variate Instance
*
* @param adblX Validated Vector Instance X
*
* @return TRUE - The Kernel Operator Integral is Positive Definite across the specified X Variate
* Instance
*/
public boolean isPositiveDefinite (
final double[] adblX)
{
try {
return 0 < computeOperatorIntegral (adblX);
} catch (java.lang.Exception e) {
e.printStackTrace();
}
return false;
}
/**
* Eigenize the Kernel Integral Operator
*
* @return The Eigenization Output
*/
public abstract org.drip.learning.kernel.IntegralOperatorEigenContainer eigenize();
}