R1NonCentral.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>R1NonCentral</i> implements the Distribution Table for the R<sup>1</sup> Non-central Chi-Square
* Distribution. The References are:
*
* <br><br>
* <ul>
* <li>
* Johnson, N. L., S. Kotz, and N. Balakrishnan (1995): <i>Continuous Univariate Distributions
* 2<sup>nd</sup> Edition</i> <b>John Wiley and Sons</b>
* </li>
* <li>
* Muirhead, R. (2005): <i>Aspects of Multivariate Statistical Theory 2<sup>nd</sup> Edition</i>
* <b>Wiley</b>
* </li>
* <li>
* Non-central Chi-Squared Distribution (2019): Chi-Squared Function
* https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution
* </li>
* <li>
* Sankaran, M. (1963): Approximations to the Non-Central Chi-Square Distribution <i>Biometrika</i>
* <b>50 (1-2)</b> 199-204
* </li>
* <li>
* Young, D. S. (2010): tolerance: An R Package for Estimating Tolerance Intervals <i>Journal of
* Statistical Software</i> <b>36 (5)</b> 1-39
* </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 R1NonCentral
extends org.drip.measure.continuous.R1Univariate
{
private double _cdfScaler = 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;
private org.drip.measure.chisquare.R1NonCentralParameters _r1NonCentralParameters = null;
private org.drip.specialfunction.definition.ModifiedBesselFirstKindEstimator
_modifiedBesselFirstKindEstimator = null;
private double nonCentralMoment (
final int n,
final double[] fourLeadingRawMoments)
throws java.lang.Exception
{
if (n <= 3)
{
return fourLeadingRawMoments[n - 1];
}
double nonCentralMoment = cumulant (
n
);
double degreesOfFreedom = _r1NonCentralParameters.degreesOfFreedom();
double nonCentralityParameter = _r1NonCentralParameters.nonCentralityParameter();
for (int j = 1;
j < n;
++j)
{
nonCentralMoment = nonCentralMoment +
_gammaEstimator.evaluate (
n
) * java.lang.Math.pow (
2.,
j - 1.
) * (
degreesOfFreedom + j * nonCentralityParameter
) * nonCentralMoment (
n - j,
fourLeadingRawMoments
) / _gammaEstimator.evaluate (
n - j + 1
);
}
return nonCentralMoment;
}
/**
* Construct the Standard Instance of R1NonCentral
*
* @param degreesOfFreedom Degrees of Freedom
* @param nonCentralityParameter Non-centrality Parameter
* @param gammaEstimator Gamma Estimator
* @param digammaEstimator Digamma Estimator
* @param lowerIncompleteGammaEstimator Lower Incomplete Gamma Estimator
* @param modifiedBesselFirstKindEstimator Modified Bessel First Kind Estimator
*
* @return The Standard Instance of R1NonCentral
*/
public static final R1NonCentral Standard (
final double degreesOfFreedom,
final double nonCentralityParameter,
final org.drip.function.definition.R1ToR1 gammaEstimator,
final org.drip.function.definition.R1ToR1 digammaEstimator,
final org.drip.function.definition.R2ToR1 lowerIncompleteGammaEstimator,
final org.drip.specialfunction.definition.ModifiedBesselFirstKindEstimator
modifiedBesselFirstKindEstimator)
{
try
{
return new R1NonCentral (
new org.drip.measure.chisquare.R1NonCentralParameters (
degreesOfFreedom,
nonCentralityParameter
),
gammaEstimator,
digammaEstimator,
lowerIncompleteGammaEstimator,
modifiedBesselFirstKindEstimator
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* R1NonCentral Constructor
*
* @param r1NonCentralParameters R<sup>1</sup> Non-central Parameters
* @param gammaEstimator Gamma Estimator
* @param digammaEstimator Digamma Estimator
* @param lowerIncompleteGammaEstimator Lower Incomplete Gamma Estimator
* @param modifiedBesselFirstKindEstimator Modified Bessel First Kind Estimator
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public R1NonCentral (
final org.drip.measure.chisquare.R1NonCentralParameters r1NonCentralParameters,
final org.drip.function.definition.R1ToR1 gammaEstimator,
final org.drip.function.definition.R1ToR1 digammaEstimator,
final org.drip.function.definition.R2ToR1 lowerIncompleteGammaEstimator,
final org.drip.specialfunction.definition.ModifiedBesselFirstKindEstimator
modifiedBesselFirstKindEstimator)
throws java.lang.Exception
{
if (null == (_r1NonCentralParameters = r1NonCentralParameters) ||
null == (_gammaEstimator = gammaEstimator) ||
null == (_digammaEstimator = digammaEstimator) ||
null == (_lowerIncompleteGammaEstimator = lowerIncompleteGammaEstimator) ||
null == (_modifiedBesselFirstKindEstimator = modifiedBesselFirstKindEstimator)
)
{
throw new java.lang.Exception (
"R1NonCentral Constructor => Invalid Inputs"
);
}
_cdfScaler = java.lang.Math.exp (
-0.5 * _r1NonCentralParameters.nonCentralityParameter()
);
}
/**
* Retrieve the R<sup>1</sup> Non-Central Parameters
*
* @return The R<sup>1</sup> Non-Central Parameters
*/
public org.drip.measure.chisquare.R1NonCentralParameters parameters()
{
return _r1NonCentralParameters;
}
/**
* 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;
}
/**
* Retrieve the Modified Bessel First Kind Estimator
*
* @return Modified Bessel First Kind Estimator
*/
public org.drip.specialfunction.definition.ModifiedBesselFirstKindEstimator
modifiedBesselFirstKindEstimator()
{
return _modifiedBesselFirstKindEstimator;
}
@Override public double[] support()
{
return new double[]
{
0.,
java.lang.Double.POSITIVE_INFINITY
};
}
@Override public double density (
final double x)
throws java.lang.Exception
{
if (!supported (
x
))
{
throw new java.lang.Exception (
"R1NonCentral::density => Variate not in Range"
);
}
double degreesOfFreedom = _r1NonCentralParameters.degreesOfFreedom();
double nonCentralityParameter = _r1NonCentralParameters.nonCentralityParameter();
return 0.5 * java.lang.Math.pow (
x / nonCentralityParameter,
0.25 * degreesOfFreedom - 0.5
) * java.lang.Math.exp (
-0.5 * (x + nonCentralityParameter)
) * _modifiedBesselFirstKindEstimator.bigI (
0.5 * degreesOfFreedom - 1.,
java.lang.Math.sqrt (
x * nonCentralityParameter
)
);
}
@Override public double cumulative (
final double t)
throws java.lang.Exception
{
if (!supported (
t
))
{
throw new java.lang.Exception (
"R1NonCentral::cumulative => Invalid Inputs"
);
}
int termCount = 10;
double cumulative = 0.;
double degreesOfFreedom = _r1NonCentralParameters.degreesOfFreedom();
double nonCentralityParameter = _r1NonCentralParameters.nonCentralityParameter();
for (int termIndex = 0;
termIndex < termCount;
++termIndex)
{
cumulative = cumulative +
java.lang.Math.pow (
0.5 * nonCentralityParameter,
termIndex
) * new org.drip.measure.chisquare.R1Central (
degreesOfFreedom + 2 * termIndex,
_gammaEstimator,
_digammaEstimator,
_lowerIncompleteGammaEstimator
).cumulative (
t
) / org.drip.numerical.common.NumberUtil.Factorial (
termIndex
);
}
return _cdfScaler * cumulative;
}
@Override public double mean()
throws java.lang.Exception
{
return _r1NonCentralParameters.degreesOfFreedom() + _r1NonCentralParameters.nonCentralityParameter();
}
@Override public double variance()
throws java.lang.Exception
{
return 2. * _r1NonCentralParameters.degreesOfFreedom() +
4. * _r1NonCentralParameters.nonCentralityParameter();
}
@Override public double skewness()
throws java.lang.Exception
{
double degreesOfFreedom = _r1NonCentralParameters.degreesOfFreedom();
double nonCentralityParameter = _r1NonCentralParameters.nonCentralityParameter();
return (degreesOfFreedom + 3. * nonCentralityParameter) * java.lang.Math.pow (
2.,
degreesOfFreedom + 2. * nonCentralityParameter
);
}
@Override public double excessKurtosis()
throws java.lang.Exception
{
double degreesOfFreedom = _r1NonCentralParameters.degreesOfFreedom();
double nonCentralityParameter = _r1NonCentralParameters.nonCentralityParameter();
double kPlusTwoLambda = degreesOfFreedom + 2. * nonCentralityParameter;
return 12. * (degreesOfFreedom + 4. * nonCentralityParameter) /
(kPlusTwoLambda* kPlusTwoLambda);
}
@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 (
"R1NonCentral::momentGeneratingFunction::evaluate => Invalid Input"
);
}
double oneMinusTwoT = 1. - 2. * t;
return java.lang.Math.exp (
_r1NonCentralParameters.nonCentralityParameter() * t / oneMinusTwoT
) / java.lang.Math.pow (
oneMinusTwoT,
0.5 * _r1NonCentralParameters.degreesOfFreedom()
);
}
};
}
/**
* 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 (
"R1NonCentral::cumulant => Invalid Inputs"
);
}
return (
_r1NonCentralParameters.degreesOfFreedom() + n * _r1NonCentralParameters.nonCentralityParameter()
) * java.lang.Math.pow (
2.,
n - 1.
) * _gammaEstimator.evaluate (
n
);
}
/**
* Compute the Leading Non-central Moments
*
* @return Leading Non-central Moments
*/
public double[] leadingRawMoments()
{
double degreesOfFreedom = _r1NonCentralParameters.degreesOfFreedom();
double nonCentralityParameter = _r1NonCentralParameters.nonCentralityParameter();
double[] fourLeadingRawMomentArray = new double[4];
fourLeadingRawMomentArray[0] = nonCentralityParameter + degreesOfFreedom;
fourLeadingRawMomentArray[1] =
fourLeadingRawMomentArray[0] * fourLeadingRawMomentArray[0] +
2. * (nonCentralityParameter + 2. * degreesOfFreedom);
fourLeadingRawMomentArray[2] =
fourLeadingRawMomentArray[0] * fourLeadingRawMomentArray[0] * fourLeadingRawMomentArray[0] +
6. * fourLeadingRawMomentArray[0] * (nonCentralityParameter + 2. * degreesOfFreedom) +
8. * (nonCentralityParameter + 3. * degreesOfFreedom);
fourLeadingRawMomentArray[3] =
fourLeadingRawMomentArray[0] * fourLeadingRawMomentArray[0] * fourLeadingRawMomentArray[0] *
fourLeadingRawMomentArray[0] +
12. * fourLeadingRawMomentArray[0] * fourLeadingRawMomentArray[0] *
(nonCentralityParameter + 2. * degreesOfFreedom) +
4. * (
11. * degreesOfFreedom * degreesOfFreedom +
44. * degreesOfFreedom * nonCentralityParameter +
36. * nonCentralityParameter * nonCentralityParameter
) +
48. * (nonCentralityParameter + 4. * degreesOfFreedom);
return fourLeadingRawMomentArray;
}
/**
* Compute the Leading central Moments
*
* @return Leading central Moments
*/
public double[] leadingCentralMoments()
{
double degreesOfFreedom = _r1NonCentralParameters.degreesOfFreedom();
double nonCentralityParameter = _r1NonCentralParameters.nonCentralityParameter();
double[] fourLeadingCentralMomentArray = new double[4];
fourLeadingCentralMomentArray[0] = 0.;
fourLeadingCentralMomentArray[1] =
2. * (nonCentralityParameter + 2. * degreesOfFreedom);
fourLeadingCentralMomentArray[2] =
8. * (nonCentralityParameter + 3. * degreesOfFreedom);
fourLeadingCentralMomentArray[3] =
12. * (nonCentralityParameter + 2. * degreesOfFreedom) *
(nonCentralityParameter + 2. * degreesOfFreedom) +
48. * (nonCentralityParameter + 4. * degreesOfFreedom);
return fourLeadingCentralMomentArray;
}
/**
* Compute the Non-central Moment
*
* @param n Moment Index
*
* @return The Non-central Moment
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double nonCentralMoment (
final int n)
throws java.lang.Exception
{
if (0 >= n)
{
throw new java.lang.Exception (
"R1NonCentral::nonCentralMoment => Invalid Inputs"
);
}
return nonCentralMoment (
n,
leadingRawMoments()
);
}
}