NormalQuadrature.java
package org.drip.measure.gaussian;
/*
* -*- 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
* Copyright (C) 2014 Lakshmi Krishnamurthy
* Copyright (C) 2011 Robert Sedgewick
*
* 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>NormalQuadrature</i> implements the Quadrature Metrics behind the Univariate Normal Distribution. It
* implements the Incremental, the Cumulative, and the Inverse Cumulative Distribution Densities.
*
* <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/NumericalAnalysisLibrary.md">Numerical Analysis Library</a></li>
* <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/measure/README.md">R<sup>d</sup> Continuous/Discrete Probability Measures</a></li>
* <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/measure/gaussian/README.md">R<sup>1</sup> R<sup>d</sup> Covariant Gaussian Quadrature</a></li>
* </ul>
*
* @author Robert Sedgewick
* @author Lakshmi Krishnamurthy
*/
public class NormalQuadrature {
private static final double InverseCDF (
final double dblY,
final double dblTolerance,
final double dblLowCutoff,
final double dblHighCutoff)
throws java.lang.Exception
{
double dblMid = 0.5 * (dblHighCutoff + dblLowCutoff);
if (dblHighCutoff - dblLowCutoff < dblTolerance) return dblMid;
return CDF (dblMid) > dblY ? InverseCDF (dblY, dblTolerance, dblLowCutoff, dblMid) : InverseCDF
(dblY, dblTolerance, dblMid, dblHighCutoff);
}
/**
* Retrieve the Density at the specified Point using Zero Mean and Unit Variance
*
* @param dblX The Ordinate
*
* @return The Density at the specified Point Zero Mean and Unit Variance
*
* @throws java.lang.Exception Thrown if Inputs are Invalid
*/
public static final double Density (
final double dblX)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (dblX))
throw new java.lang.Exception ("NormalQuadrature::Density => Invalid Inputs");
return java.lang.Math.exp (-0.5 * dblX * dblX) / java.lang.Math.sqrt (2 * java.lang.Math.PI);
}
/**
* Compute the Cumulative Distribution Function up to the specified Variate
*
* @param dblX The Variate
*
* @return The Cumulative Distribution Function up to the specified Variate
*
* @throws java.lang.Exception thrown if the Inputs are Invalid
*/
public static final double CDF (
final double dblX)
throws java.lang.Exception
{
if (java.lang.Double.isNaN (dblX))
throw new java.lang.Exception ("NormalQuadrature::CDF => Invalid Inputs");
if (dblX < -8.) return 0.;
if (dblX > 8.) return 1.;
double dblSum = 0.;
double dblTerm = dblX;
for (int i = 3; dblSum + dblTerm != dblSum; i += 2) {
dblSum = dblSum + dblTerm;
dblTerm = dblTerm * dblX * dblX / i;
}
return 0.5 + dblSum * Density (dblX);
}
/**
* Compute the Inverse CDF of the Distribution up to the specified Y
*
* @param dblY Y
*
* @return The Inverse CDF of the Distribution up to the specified Y
*
* @throws java.lang.Exception Thrown if Inputs are Invalid
*/
public static final double InverseCDF (
final double dblY)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (dblY))
throw new java.lang.Exception ("NormalQuadrature::InverseCDF => Invalid Inputs");
return InverseCDF (dblY, .00000001, -8., 8.);
}
/**
* Compute the Probit of the Distribution up to the specified p
*
* @param p p
*
* @return The Probit of the Distribution up to the specified p
*
* @throws java.lang.Exception Thrown if Inputs are Invalid
*/
public static final double Probit (
final double p)
throws java.lang.Exception
{
return InverseCDF (p);
}
/**
* Generate a Random Univariate Number following a Gaussian Distribution
*
* @return The Random Univariate Number
*
* @throws java.lang.Exception Thrown the Random Number cannot be generated
*/
public static final double Random()
throws java.lang.Exception
{
return InverseCDF (java.lang.Math.random());
}
/**
* Compute the Error Function of x
*
* @param x x
*
* @return The Error Function of x
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public static final double ERF (
final double x)
throws java.lang.Exception
{
return 2. * CDF (x * java.lang.Math.sqrt (2.)) - 1.;
}
/**
* Compute the Error Function Complement of x
*
* @param x x
*
* @return The Error Function Complement of x
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public static final double ERFC (
final double x)
throws java.lang.Exception
{
return 2. * CDF (1. - x * java.lang.Math.sqrt (2.));
}
}