R1Central.java
package org.drip.measure.chisquare;
/*
* -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
*/
/*!
* Copyright (C) 2020 Lakshmi Krishnamurthy
* Copyright (C) 2019 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>R1Central</i> implements the Probability Density Function for the R<sup>1</sup> Central Chi-Square
* Distribution. The References are:
*
* <br><br>
* <ul>
* <li>
* Abramowitz, M., and I. A. Stegun (2007): <i>Handbook of Mathematics Functions</i> <b>Dover Book
* on Mathematics</b>
* </li>
* <li>
* Backstrom, T., and J. Fischer (2018): Fast Randomization for Distributed Low Bit-rate Coding of
* Speech and Audio <i>IEEE/ACM Transactions on Audio, Speech, and Language Processing</i> <b>26
* (1)</b> 19-30
* </li>
* <li>
* Chi-Squared Distribution (2019): Chi-Squared Function
* https://en.wikipedia.org/wiki/Chi-squared_distribution
* </li>
* <li>
* Johnson, N. L., S. Kotz, and N. Balakrishnan (1994): <i>Continuous Univariate Distributions
* 2<sup>nd</sup> Edition</i> <b>John Wiley and Sons</b>
* </li>
* <li>
* National Institute of Standards and Technology (2019): Chi-Squared Distribution
* https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm
* </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/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/chisquare/README.md">Chi-Square Distribution Implementation/Properties</a></li>
* </ul>
*
* @author Lakshmi Krishnamurthy
*/
public class R1Central extends org.drip.measure.continuous.R1Univariate
{
private double _degreesOfFreedom = -1;
private double _cdfScaler = java.lang.Double.NaN;
private double _normalizer = java.lang.Double.NaN;
private org.drip.function.definition.R1ToR1 _gammaEstimator = null;
private org.drip.function.definition.R1ToR1 _digammaEstimator = null;
private org.drip.function.definition.R2ToR1 _lowerIncompleteGammaEstimator = null;
/**
* Generate a Consolidated Chi-squared Distribution from Independent Component Distributions
*
* @param chiSquaredDistributionArray Independent Component Distribution Array
*
* @return Consolidated Chi-squared Distribution
*/
public static final R1Central FromIndependentChiSquared (
final org.drip.measure.chisquare.R1Central[] chiSquaredDistributionArray)
{
if (null == chiSquaredDistributionArray || 0 == chiSquaredDistributionArray.length)
{
return null;
}
double degreesOfFreedom = 0;
for (org.drip.measure.chisquare.R1Central chiSquaredDistribution :
chiSquaredDistributionArray)
{
if (null == chiSquaredDistribution)
{
return null;
}
degreesOfFreedom = degreesOfFreedom + chiSquaredDistribution.degreesOfFreedom();
}
try
{
return new R1Central (
degreesOfFreedom,
chiSquaredDistributionArray[0].gammaEstimator(),
chiSquaredDistributionArray[0].digammaEstimator(),
chiSquaredDistributionArray[0].lowerIncompleteGammaEstimator()
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* R1Central Constructor
*
* @param degreesOfFreedom Degrees of Freedom
* @param gammaEstimator Gamma Estimator
* @param digammaEstimator Digamma Estimator
* @param lowerIncompleteGammaEstimator Lower Incomplete Gamma Estimator
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public R1Central (
final double degreesOfFreedom,
final org.drip.function.definition.R1ToR1 gammaEstimator,
final org.drip.function.definition.R1ToR1 digammaEstimator,
final org.drip.function.definition.R2ToR1 lowerIncompleteGammaEstimator)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (
_degreesOfFreedom = degreesOfFreedom
) || 0. >= _degreesOfFreedom ||
null == (_gammaEstimator = gammaEstimator) ||
null == (_digammaEstimator = digammaEstimator) ||
null == (_lowerIncompleteGammaEstimator = lowerIncompleteGammaEstimator))
{
throw new java.lang.Exception ("R1Central Constructor => Invalid Inputs");
}
double halfDOF = 0.5 * _degreesOfFreedom;
_normalizer = (_cdfScaler = 1. / _gammaEstimator.evaluate (halfDOF)) * java.lang.Math.pow (
2.,
-1. * halfDOF
);
}
/**
* Retrieve the Degrees of Freedom
*
* @return The Degrees of Freedom
*/
public double degreesOfFreedom()
{
return _degreesOfFreedom;
}
/**
* Retrieve the Gamma Estimator
*
* @return Gamma Estimator
*/
public org.drip.function.definition.R1ToR1 gammaEstimator()
{
return _gammaEstimator;
}
/**
* Retrieve the Digamma Estimator
*
* @return Digamma Estimator
*/
public org.drip.function.definition.R1ToR1 digammaEstimator()
{
return _digammaEstimator;
}
/**
* Retrieve the Lower Incomplete Gamma Estimator
*
* @return Lower Incomplete Gamma Estimator
*/
public org.drip.function.definition.R2ToR1 lowerIncompleteGammaEstimator()
{
return _lowerIncompleteGammaEstimator;
}
@Override public double[] support()
{
return new double[]
{
0.,
java.lang.Double.POSITIVE_INFINITY
};
}
@Override public double density (
final double t)
throws java.lang.Exception
{
if (!supported (t))
{
throw new java.lang.Exception ("R1Central::density => Variate not in Range");
}
return _normalizer * java.lang.Math.pow (
t,
0.5 * _degreesOfFreedom - 1.
) * java.lang.Math.exp (-0.5 * t);
}
@Override public double cumulative (
final double t)
throws java.lang.Exception
{
if (!supported (t))
{
throw new java.lang.Exception ("R1Central::cumulative => Invalid Inputs");
}
return _cdfScaler * _lowerIncompleteGammaEstimator.evaluate (
0.5 * _degreesOfFreedom,
0.5 * t
);
}
@Override public double mean()
throws java.lang.Exception
{
return _degreesOfFreedom;
}
@Override public double median()
throws java.lang.Exception
{
double oneMinus_twoOver_9dof__ = 1. - (2. / (9. * _degreesOfFreedom));
return _degreesOfFreedom * oneMinus_twoOver_9dof__ * oneMinus_twoOver_9dof__ *
oneMinus_twoOver_9dof__;
}
@Override public double mode()
throws java.lang.Exception
{
return java.lang.Math.max (
_degreesOfFreedom - 2.,
0.
);
}
@Override public double variance()
throws java.lang.Exception
{
return 2. * _degreesOfFreedom;
}
@Override public double skewness()
throws java.lang.Exception
{
return java.lang.Math.sqrt (8. / _degreesOfFreedom);
}
@Override public double excessKurtosis()
throws java.lang.Exception
{
return 12. / _degreesOfFreedom;
}
@Override public double differentialEntropy()
throws java.lang.Exception
{
double halfDOF = 0.5 * _degreesOfFreedom;
return halfDOF + java.lang.Math.log (2. * _gammaEstimator.evaluate (halfDOF)) +
(1. - halfDOF) * _digammaEstimator.evaluate (halfDOF);
}
@Override public org.drip.function.definition.R1ToR1 momentGeneratingFunction()
{
return new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double t)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (t) || t > 0.5)
{
throw new java.lang.Exception
("R1Central::momentGeneratingFunction::evaluate => Invalid Input");
}
return java.lang.Math.pow (
1. - 2. * t,
-0.5 * _degreesOfFreedom
);
}
};
}
@Override public org.drip.function.definition.R1ToR1 probabilityGeneratingFunction()
{
return new org.drip.function.definition.R1ToR1 (null)
{
@Override public double evaluate (
final double t)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (t) ||
t <= 0. || t > java.lang.Math.sqrt (java.lang.Math.E))
{
throw new java.lang.Exception
("R1Central::probabilityGeneratingFunction::evaluate => Invalid Input");
}
return java.lang.Math.pow (
1. - 2. * java.lang.Math.log (t),
-0.5 * _degreesOfFreedom
);
}
};
}
@Override public double random()
throws java.lang.Exception
{
double sumOfStandardNormalSquares = 0.;
for (int drawIndex = 0; drawIndex < _degreesOfFreedom; ++drawIndex)
{
double randomStandardNormal = org.drip.measure.gaussian.NormalQuadrature.InverseCDF
(java.lang.Math.random());
sumOfStandardNormalSquares = sumOfStandardNormalSquares +
randomStandardNormal * randomStandardNormal;
}
return sumOfStandardNormalSquares;
}
/**
* Retrieve the Normalizer
*
* @return Normalizer
*/
public double normalizer()
{
return _normalizer;
}
/**
* Retrieve the CDF Scaler
*
* @return CDF Scaler
*/
public double cdfScaler()
{
return _cdfScaler;
}
/**
* Compute the Chernoff Upper Bound
*
* @param x A
*
* @return The Chernoff Upper Bound
*
* @throws java.lang.Exception Thrown if the Chernoff Upper Bound cannot be calculated
*/
public double chernoffBound (
final double x)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (x) || 0. >= x)
{
throw new java.lang.Exception ("R1Central::chernoffBound => Invalid Inputs");
}
double z = x / _degreesOfFreedom;
if (1. == z)
{
throw new java.lang.Exception ("R1Central::chernoffBound => Invalid Inputs");
}
double _zExponent_OneMinusZ__powerHalfDegreesOfFreedom = java.lang.Math.pow (
z * java.lang.Math.exp (1. - z),
0.5 * _degreesOfFreedom
);
return 1. > z ? _zExponent_OneMinusZ__powerHalfDegreesOfFreedom : 1. -
_zExponent_OneMinusZ__powerHalfDegreesOfFreedom;
}
/**
* Compute the Non-central Moment about Zero
*
* @param m Non-central Moment Index
*
* @return The Non-central Moment about Zero
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double nonCentralMoment (
final int m)
throws java.lang.Exception
{
if (0 > m)
{
throw new java.lang.Exception ("R1Central::nonCentralMoment => Invalid Inputs");
}
double halfDOF = 0.5 * _degreesOfFreedom;
return java.lang.Math.pow (
2.,
m
) * _gammaEstimator.evaluate (m + halfDOF) / _gammaEstimator.evaluate (halfDOF);
}
/**
* Compute the Cumulant
*
* @param n Cumulant Index
*
* @return The Cumulant
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double cumulant (
final int n)
throws java.lang.Exception
{
if (0 > n)
{
throw new java.lang.Exception ("R1Central::cumulant => Invalid Inputs");
}
return _degreesOfFreedom * java.lang.Math.pow (
2.,
n - 1.
) * _gammaEstimator.evaluate (n);
}
/**
* Retrieve the Central Limit Theorem Equivalent Normal Distribution Proxy
*
* @return The Central Limit Theorem Equivalent Normal Distribution Proxy
*/
public org.drip.measure.gaussian.R1UnivariateNormal cltProxy()
{
try
{
return new org.drip.measure.gaussian.R1UnivariateNormal (
_degreesOfFreedom,
2. * _degreesOfFreedom
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Indicate if the Current Distribution is a Valid Proxy as a CLT
*
* @return TRUE - The Current Distribution is a Valid Proxy as a CLT
*/
public boolean validCLTProxy()
{
return 50. <= _degreesOfFreedom;
}
/**
* Generate a Gamma-distribution off of the Scaled Chi-Square Distribution
*
* @param scale The Scale
*
* @return The Gamma Distribution
*/
public org.drip.measure.gamma.R1ShapeScaleDistribution gammaDistribution (
final double scale)
{
try
{
return new org.drip.measure.gamma.R1ShapeScaleDistribution (
new org.drip.measure.gamma.ShapeScaleParameters (
0.5 * _degreesOfFreedom,
2. * scale
),
_gammaEstimator,
_digammaEstimator,
_lowerIncompleteGammaEstimator
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Generate Logarithm Proxy Based Random Number - Proxy to Univariate Normal Distribution
*
* @return Logarithm Proxy Based Random Number - Proxy to Univariate Normal Distribution
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double randomLogProxy()
throws java.lang.Exception
{
return java.lang.Math.log (_degreesOfFreedom);
}
/**
* Generate CLT Proxy Based Random Number - Proxy to Univariate Normal Distribution
*
* @return CLT Proxy Based Random Number - Proxy to Univariate Normal Distribution
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double randomCLTProxy()
throws java.lang.Exception
{
return (random() - _degreesOfFreedom) / java.lang.Math.sqrt (2. * _degreesOfFreedom);
}
/**
* Generate Fisher Proxy Random Number - Proxy to Univariate Normal Distribution
*
* @return Fisher Proxy Random Number - Proxy to Univariate Normal Distribution
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double randomFisherProxy()
throws java.lang.Exception
{
return java.lang.Math.sqrt (2. * random());
}
/**
* Generate Wilson-Hilferty Proxy Random Number - Proxy to Univariate Normal Distribution
*
* @return Wilson-Hilferty Proxy Random Number - Proxy to Univariate Normal Distribution
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double randomWilsonHilferty()
throws java.lang.Exception
{
return java.lang.Math.pow (
random() / _degreesOfFreedom,
1. / 3.
);
}
/**
* Generate Gamma Distributed Random Number
*
* @param c The Scale Parameter
*
* @return Gamma Distributed Random Number
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double randomGamma (
final double c)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (c) || 0. >= c)
{
throw new java.lang.Exception ("R1Central::randomGamma => Invalid Inputs");
}
return random() * c;
}
/**
* Generate the Chi Distributed Random Number
*
* @return Chi Distributed Random Number
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double randomChi()
throws java.lang.Exception
{
return java.lang.Math.sqrt (random());
}
/**
* Generate Exponential (0.5) Distributed Random Number
*
* @return Exponential (0.5) Distributed Random Number
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double randomExponentialHalf()
throws java.lang.Exception
{
if (2. != _degreesOfFreedom)
{
throw new java.lang.Exception ("R1Central::randomExponentialHalf => Invalid Inputs");
}
return random();
}
/**
* Generate Rayleigh (1) Distributed Random Number
*
* @return Rayleigh (1) Distributed Random Number
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double randomRayleigh1()
throws java.lang.Exception
{
if (2. != _degreesOfFreedom)
{
throw new java.lang.Exception ("R1Central::randomRayleigh1 => Invalid Inputs");
}
return random();
}
/**
* Generate Maxwell (1) Distributed Random Number
*
* @return Maxwell (1) Distributed Random Number
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double randomMaxwell1()
throws java.lang.Exception
{
if (3. != _degreesOfFreedom)
{
throw new java.lang.Exception ("R1Central::randomMaxwell1 => Invalid Inputs");
}
return random();
}
}