ScaleSensitiveCoveringBounds.java

  1. package org.drip.spaces.cover;

  3. /*
  4.  * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
  5.  */

  7. /*!
  8.  * Copyright (C) 2020 Lakshmi Krishnamurthy
  9.  * Copyright (C) 2019 Lakshmi Krishnamurthy
  10.  * Copyright (C) 2018 Lakshmi Krishnamurthy
  11.  * Copyright (C) 2017 Lakshmi Krishnamurthy
  12.  * Copyright (C) 2016 Lakshmi Krishnamurthy
  13.  * Copyright (C) 2015 Lakshmi Krishnamurthy
  14.  * 
  15.  *  This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
  16.  *      asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
  17.  *      analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
  18.  *      equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
  19.  *      numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
  20.  *      and computational support.
  21.  *  
  22.  *      https://lakshmidrip.github.io/DROP/
  23.  *  
  24.  *  DROP is composed of three modules:
  25.  *  
  26.  *  - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
  27.  *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
  28.  *  - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
  29.  * 
  30.  *  DROP Product Core implements libraries for the following:
  31.  *  - Fixed Income Analytics
  32.  *  - Loan Analytics
  33.  *  - Transaction Cost Analytics
  34.  * 
  35.  *  DROP Portfolio Core implements libraries for the following:
  36.  *  - Asset Allocation Analytics
  37.  *  - Asset Liability Management Analytics
  38.  *  - Capital Estimation Analytics
  39.  *  - Exposure Analytics
  40.  *  - Margin Analytics
  41.  *  - XVA Analytics
  42.  * 
  43.  *  DROP Computational Core implements libraries for the following:
  44.  *  - Algorithm Support
  45.  *  - Computation Support
  46.  *  - Function Analysis
  47.  *  - Model Validation
  48.  *  - Numerical Analysis
  49.  *  - Numerical Optimizer
  50.  *  - Spline Builder
  51.  *  - Statistical Learning
  52.  * 
  53.  *  Documentation for DROP is Spread Over:
  54.  * 
  55.  *  - Main                     => https://lakshmidrip.github.io/DROP/
  56.  *  - Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
  57.  *  - GitHub                   => https://github.com/lakshmiDRIP/DROP
  58.  *  - Repo Layout Taxonomy     => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
  59.  *  - Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
  60.  *  - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
  61.  *  - Release Versions         => https://lakshmidrip.github.io/DROP/version.html
  62.  *  - Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
  63.  *  - Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
  64.  *  - JUnit                    => https://lakshmidrip.github.io/DROP/junit/index.html
  65.  *  - Jacoco                   => https://lakshmidrip.github.io/DROP/jacoco/index.html
  66.  * 
  67.  *  Licensed under the Apache License, Version 2.0 (the "License");
  68.  *      you may not use this file except in compliance with the License.
  69.  *   
  70.  *  You may obtain a copy of the License at
  71.  *      http://www.apache.org/licenses/LICENSE-2.0
  72.  *  
  73.  *  Unless required by applicable law or agreed to in writing, software
  74.  *      distributed under the License is distributed on an "AS IS" BASIS,
  75.  *      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  76.  *  
  77.  *  See the License for the specific language governing permissions and
  78.  *      limitations under the License.
  79.  */

  81. /**
  82.  * <i>ScaleSensitiveCoveringBounds</i> implements the Lower/Upper Bounds for the General Class of Functions
  83.  * in terms of their scale-sensitive dimensions (i.e., the fat shattering coefficients). The References are:
  84.  *
  85.  * <br><br>
  86.  *  <ul>
  87.  *      <li>
  88.  *          N. Alon, S. Ben-David, N. Cesa-Bianchi, and D. Haussler (1993): Scale-sensitive Dimensions,
  89.  *              Uniform-Convergence, and Learnability <i>Proceedings of the ACM Symposium on the Foundations
  90.  *                  of Computer Science</i>
  91.  *      </li>
  92.  *      <li>
  93.  *          P. L. Bartlett, S. R. Kulkarni, and S. E. Posner (1997): Covering Numbers for Real-valued
  94.  *              Function Classes <i>IEEE Transactions on Information Theory</i> <b>43 (5)</b> 1721-1724 
  95.  *      </li>
  96.  *      <li>
  97.  *          D. Pollard (1984): <i>Convergence of Stochastic Processes</i> <b>Springer</b> New York
  98.  *      </li>
  99.  *  </ul>
  100.  *
  101.  * <br><br>
  102.  *  <ul>
  103.  *      <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
  104.  *      <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/StatisticalLearningLibrary.md">Statistical Learning Library</a></li>
  105.  *      <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>
  106.  *      <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>
  107.  *  </ul>
  108.  * <br><br>
  109.  *
  110.  * @author Lakshmi Krishnamurthy
  111.  */

  113. public class ScaleSensitiveCoveringBounds implements org.drip.spaces.cover.FunctionClassCoveringBounds {
  114.     private int _iSampleSize = -1;
  115.     private org.drip.function.definition.R1ToR1 _r1r1FatShatter = null;

  117.     /**
  118.      * ScaleSensitiveCoveringBounds Constructor
  119.      * 
  120.      * @param r1r1FatShatter The Cover Fat Shattering Coefficient Function
  121.      * @param iSampleSize Sample Size
  122.      * 
  123.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  124.      */

  126.     public ScaleSensitiveCoveringBounds (
  127.         final org.drip.function.definition.R1ToR1 r1r1FatShatter,
  128.         final int iSampleSize)
  129.         throws java.lang.Exception
  130.     {
  131.         if (null == (_r1r1FatShatter = r1r1FatShatter) || 0 >= (_iSampleSize = iSampleSize))
  132.             throw new java.lang.Exception ("ScaleSensitiveCoveringBounds ctr: Invalid Inputs");
  133.     }

  135.     /**
  136.      * Retrieve the Fat Shattering Coefficient Function
  137.      * 
  138.      * @return The Fat Shattering Coefficient Function
  139.      */

  141.     public org.drip.function.definition.R1ToR1 fatShatteringFunction()
  142.     {
  143.         return _r1r1FatShatter;
  144.     }

  146.     /**
  147.      * Retrieve the Sample Size
  148.      * 
  149.      * @return The Sample Size
  150.      */

  152.     public int sampleSize()
  153.     {
  154.         return _iSampleSize;
  155.     }

  157.     /**
  158.      * Compute the Minimum Sample Size required to Estimate the Cardinality corresponding to the Specified
  159.      *  Cover
  160.      * 
  161.      * @param dblCover The Cover
  162.      * 
  163.      * @return The Minimum Sample Size
  164.      * 
  165.      * @throws java.lang.Exception Thrown if the Minimum Sample Size Cannot be computed
  166.      */

  168.     public double sampleSizeLowerBound (
  169.         final double dblCover)
  170.         throws java.lang.Exception
  171.     {
  172.         if (!org.drip.numerical.common.NumberUtil.IsValid (dblCover) || 0. == dblCover)
  173.             throw new java.lang.Exception
  174.                 ("ScaleSensitiveCoveringBounds::sampleSizeLowerBound => Invalid Inputs");

  176.         double dblLog2 = java.lang.Math.log (2.);

  178.         return 2. * _r1r1FatShatter.evaluate (0.25 * dblCover) * java.lang.Math.log (64. * java.lang.Math.E *
  179.             java.lang.Math.E / (dblCover * dblLog2)) / dblLog2;
  180.     }

  182.     /**
  183.      * Compute the Cardinality for the Subset T (|x) that possesses the Specified Cover for the Restriction
  184.      *  of the Input Function Class Family F (|x).
  185.      *  
  186.      * @param dblCover The Specified Cover
  187.      * 
  188.      * @return The Restricted Subset Cardinality
  189.      * 
  190.      * @throws java.lang.Exception Thrown if the Restricted Subset Cardinality cannot be computed
  191.      */

  193.     public double restrictedSubsetCardinality (
  194.         final double dblCover)
  195.         throws java.lang.Exception
  196.     {
  197.         if (!org.drip.numerical.common.NumberUtil.IsValid (dblCover) || 0. == dblCover)
  198.             throw new java.lang.Exception
  199.                 ("ScaleSensitiveCoveringBounds::restrictedSubsetCardinality => Invalid Inputs");

  201.         double dblLog2 = java.lang.Math.log (2.);

  203.         double dblFatShatteringCoefficient = _r1r1FatShatter.evaluate (0.25 * dblCover);

  205.         if (_iSampleSize < 2. * dblFatShatteringCoefficient * java.lang.Math.log (64. * java.lang.Math.E *
  206.             java.lang.Math.E / (dblCover * dblLog2)) / dblLog2)
  207.             throw new java.lang.Exception
  208.                 ("ScaleSensitiveCoveringBounds::restrictedSubsetCardinality => Invalid Inputs");

  210.         return 6. * dblFatShatteringCoefficient * java.lang.Math.log (16. / dblCover) * java.lang.Math.log
  211.             (32. * java.lang.Math.E * _iSampleSize / (dblFatShatteringCoefficient * dblCover)) / dblLog2 +
  212.                 dblLog2;
  213.     }

  215.     /**
  216.      * Compute the Log of the Weight Loading Coefficient for the Maximum Cover Term in:
  217.      * 
  218.      *  {Probability that the Empirical Error .gt. Cover} .lte. 4 * exp (-m * Cover^2 / 128) *
  219.      *      [[Max Covering Number Over the Specified Sample]]
  220.      * 
  221.      * Reference is:
  222.      *
  223.      *  - D. Haussler (1995): Sphere Packing Numbers for Subsets of the Boolean n-Cube with Bounded
  224.      *      Vapnik-Chervonenkis Dimension, Journal of the COmbinatorial Theory A 69 (2) 217.
  225.      *
  226.      * @param dblCover The Specified Cover
  227.      * 
  228.      * @return Log of the Weight Loading Coefficient for the Maximum Cover Term
  229.      * 
  230.      * @throws java.lang.Exception Thrown if the Log of the Weight Loading Coefficient cannot be computed
  231.      */

  233.     public double upperProbabilityBoundWeight (
  234.         final double dblCover)
  235.         throws java.lang.Exception
  236.     {
  237.         if (!org.drip.numerical.common.NumberUtil.IsValid (dblCover) || 0. == dblCover)
  238.             throw new java.lang.Exception
  239.                 ("ScaleSensitiveCoveringBounds::upperProbabilityBoundWeight => Invalid Inputs");

  241.         return java.lang.Math.log (4.) - (dblCover * dblCover * _iSampleSize / 128.);
  242.     }

  244.     @Override public double logLowerBound (
  245.         final double dblCover)
  246.         throws java.lang.Exception
  247.     {
  248.         return restrictedSubsetCardinality (dblCover);
  249.     }

  251.     @Override public double logUpperBound (
  252.         final double dblCover)
  253.         throws java.lang.Exception
  254.     {
  255.         return _r1r1FatShatter.evaluate (4. * dblCover) / 32.;
  256.     }
  257. }
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