R1ShapeScaleDistribution.java
package org.drip.measure.gamma;
/*
* -*- 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>R1ShapeScaleDistribution</i> implements the Shape and Scale Parameterization of the R<sup>1</sup> Gamma
* Distribution. The References are:
*
* <br><br>
* <ul>
* <li>
* Devroye, L. (1986): <i>Non-Uniform Random Variate Generation</i> <b>Springer-Verlag</b> New York
* </li>
* <li>
* Gamma Distribution (2019): Gamma Distribution
* https://en.wikipedia.org/wiki/Chi-squared_distribution
* </li>
* <li>
* Louzada, F., P. L. Ramos, and E. Ramos (2019): A Note on Bias of Closed-Form Estimators for the
* Gamma Distribution Derived From Likelihood Equations <i>The American Statistician</i> <b>73
* (2)</b> 195-199
* </li>
* <li>
* Minka, T. (2002): Estimating a Gamma distribution https://tminka.github.io/papers/minka-gamma.pdf
* </li>
* <li>
* Ye, Z. S., and N. Chen (2017): Closed-Form Estimators for the Gamma Distribution Derived from
* Likelihood Equations <i>The American Statistician</i> <b>71 (2)</b> 177-181
* </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/gamma/README.md">R<sup>1</sup> Gamma Distribution Implementation/Properties</a></li>
* </ul>
*
* @author Lakshmi Krishnamurthy
*/
public class R1ShapeScaleDistribution
extends org.drip.measure.continuous.R1Univariate
{
private double _cdfScaler = java.lang.Double.NaN;
private double _pdfScaler = java.lang.Double.NaN;
private org.drip.function.definition.R1ToR1 _gammaEstimator = null;
private org.drip.function.definition.R1ToR1 _digammaEstimator = null;
private org.drip.measure.gamma.ShapeScaleParameters _shapeScaleParameters = null;
private org.drip.function.definition.R2ToR1 _lowerIncompleteGammaEstimator = null;
/**
* Construct a Gamma Distribution from Shape and Rate Parameters
*
* @param shapeParameter Shape Parameter
* @param rateParameter Rate Parameter
* @param gammaEstimator Gamma Estimator
* @param digammaEstimator Digamma Estimator
* @param lowerIncompleteGammaEstimator Lower Incomplete Gamma Estimator
*
* @return Gamma Distribution from Shape Alpha and Rate Beta Parameters
*/
public static final R1ShapeScaleDistribution ShapeRate (
final double shapeParameter,
final double rateParameter,
final org.drip.function.definition.R1ToR1 gammaEstimator,
final org.drip.function.definition.R1ToR1 digammaEstimator,
final org.drip.function.definition.R2ToR1 lowerIncompleteGammaEstimator)
{
return R1ShapeScaleDistribution.Standard (
shapeParameter,
1. / rateParameter,
gammaEstimator,
digammaEstimator,
lowerIncompleteGammaEstimator
);
}
/**
* Shape Summation Based ShapeScaleDistribution
*
* @param shapeParameterArray Shape Parameter Array
* @param scaleParameter Scale Parameter
* @param gammaEstimator Gamma Estimator
* @param digammaEstimator Digamma Estimator
* @param lowerIncompleteGammaEstimator Lower Incomplete Gamma Estimator
*
* @return Shape Summation Based ShapeScaleDistribution
*/
public static final R1ShapeScaleDistribution ShapeSummation (
final double[] shapeParameterArray,
final double scaleParameter,
final org.drip.function.definition.R1ToR1 gammaEstimator,
final org.drip.function.definition.R1ToR1 digammaEstimator,
final org.drip.function.definition.R2ToR1 lowerIncompleteGammaEstimator)
{
if (null == shapeParameterArray)
{
return null;
}
double shapeParameter = 0.;
int shapeParameterArraySize = shapeParameterArray.length;
if (0 == shapeParameterArraySize)
{
return null;
}
for (int shapeParameterIndex = 0;
shapeParameterIndex < shapeParameterArraySize;
++shapeParameterIndex)
{
if (!org.drip.numerical.common.NumberUtil.IsValid (
shapeParameterArray[shapeParameterIndex]
))
{
return null;
}
shapeParameter += shapeParameterArray[shapeParameterIndex];
}
return R1ShapeScaleDistribution.Standard (
shapeParameter,
scaleParameter,
gammaEstimator,
digammaEstimator,
lowerIncompleteGammaEstimator
);
}
/**
* Construct the Standard R1ShapeScaleDistribution Instance
*
* @param shapeParameter Shape Parameter
* @param scaleParameter Scale Parameter
* @param gammaEstimator Gamma Estimator
* @param digammaEstimator Digamma Estimator
* @param lowerIncompleteGammaEstimator Lower Incomplete Gamma Estimator
*
* @return The R1ShapeScaleDistribution Instance
*/
public static final R1ShapeScaleDistribution Standard (
final double shapeParameter,
final double scaleParameter,
final org.drip.function.definition.R1ToR1 gammaEstimator,
final org.drip.function.definition.R1ToR1 digammaEstimator,
final org.drip.function.definition.R2ToR1 lowerIncompleteGammaEstimator)
{
try
{
return new R1ShapeScaleDistribution (
new org.drip.measure.gamma.ShapeScaleParameters (
shapeParameter,
scaleParameter
),
gammaEstimator,
digammaEstimator,
lowerIncompleteGammaEstimator
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
private double randomMarsaglia1977 (
final double shapeParameterIn)
throws java.lang.Exception
{
double shapeParameter = shapeParameterIn < 1. ? shapeParameterIn + 1. : shapeParameterIn;
double d = shapeParameter - 1. / 3.;
double v = 0.;
double u = 0.;
double c = 1. / java.lang.Math.sqrt (
9. * d
);
while (true)
{
double x = org.drip.measure.gaussian.NormalQuadrature.Random();
u = java.lang.Math.random();
v = 1. + c * x;
v = v * v * v;
if (v > 0. &&
0.5 * x * x + d - d * v + d * java.lang.Math.log (
v
) > java.lang.Math.log (
u
)
)
{
double marsagliaRandom =_shapeScaleParameters.scale() * d * v;
return shapeParameter != shapeParameterIn ?
marsagliaRandom * java.lang.Math.pow (
u,
1. / shapeParameterIn
) : marsagliaRandom;
}
}
}
/**
* R1ShapeScaleDistribution Constructor
*
* @param shapeScaleParameters Shape-Scale Parameters
* @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 R1ShapeScaleDistribution (
final org.drip.measure.gamma.ShapeScaleParameters shapeScaleParameters,
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 (null == (_shapeScaleParameters = shapeScaleParameters) ||
null == (_gammaEstimator = gammaEstimator) ||
null == (_digammaEstimator = digammaEstimator) ||
null == (_lowerIncompleteGammaEstimator = lowerIncompleteGammaEstimator)
)
{
throw new java.lang.Exception (
"R1ShapeScaleDistribution Constructor => Invalid Inputs"
);
}
double shape = _shapeScaleParameters.shape();
_pdfScaler = (
_cdfScaler = 1. / _gammaEstimator.evaluate (
shape
)
) * java.lang.Math.pow (
_shapeScaleParameters.scale(),
-1. * shape
);
}
/**
* Retrieve the Shape-Scale Parameters
*
* @return The Shape-Scale Parameters
*/
public org.drip.measure.gamma.ShapeScaleParameters shapeScaleParameters()
{
return _shapeScaleParameters;
}
/**
* 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 (
"ShapeScaleDistribution::density => Variate not in Range"
);
}
return _pdfScaler * java.lang.Math.pow (
t,
_shapeScaleParameters.shape() - 1.
) * java.lang.Math.exp (
-1. * t / _shapeScaleParameters.scale()
);
}
@Override public double cumulative (
final double t)
throws java.lang.Exception
{
if (!supported (
t
))
{
throw new java.lang.Exception (
"ShapeScaleDistribution::cumulative => Invalid Inputs"
);
}
return _cdfScaler * _lowerIncompleteGammaEstimator.evaluate (
_shapeScaleParameters.shape(),
t / _shapeScaleParameters.scale()
);
}
@Override public double mean()
throws java.lang.Exception
{
return _shapeScaleParameters.shape() * _shapeScaleParameters.scale();
}
@Override public double mode()
throws java.lang.Exception
{
double shape = _shapeScaleParameters.shape();
if (shape < 1.)
{
throw new java.lang.Exception (
"ShapeScaleDistribution::mode => No Closed Form Available"
);
}
return (shape - 1.) * _shapeScaleParameters.scale();
}
@Override public double variance()
throws java.lang.Exception
{
double scale = _shapeScaleParameters.scale();
return _shapeScaleParameters.shape() * scale * scale;
}
@Override public double skewness()
throws java.lang.Exception
{
return 2. * java.lang.Math.sqrt (1. / _shapeScaleParameters.shape());
}
@Override public double excessKurtosis()
throws java.lang.Exception
{
return 6. / _shapeScaleParameters.shape();
}
@Override public double differentialEntropy()
throws java.lang.Exception
{
double shape = _shapeScaleParameters.shape();
return shape + java.lang.Math.log (
_shapeScaleParameters.scale() / _cdfScaler
) + (1. - shape) * _digammaEstimator.evaluate (
shape
);
}
@Override public org.drip.function.definition.R1ToR1 momentGeneratingFunction()
{
final double scale = _shapeScaleParameters.scale();
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 >= 1. / scale
)
{
throw new java.lang.Exception (
"ShapeScaleDistribution::momentGeneratingFunction::evaluate => Invalid Input"
);
}
return java.lang.Math.pow (
1. - scale * t,
-1. * _shapeScaleParameters.shape()
);
}
};
}
/**
* 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()
{
double scale = _shapeScaleParameters.scale();
double shape = _shapeScaleParameters.shape();
try
{
return new org.drip.measure.gaussian.R1UnivariateNormal (
shape * scale,
scale * java.lang.Math.sqrt (
shape
)
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Compute the Logarithmic Expectation
*
* @return The Logarithmic Expectation
*
* @throws java.lang.Exception Thrown if the Logarithmic Expectation cannot be computed
*/
public double logarithmicExpectation()
throws java.lang.Exception
{
return _digammaEstimator.evaluate (
_shapeScaleParameters.shape()
) - java.lang.Math.log (
_shapeScaleParameters.scale()
);
}
/**
* Compute the Banneheke-Ekayanake Approximation for the Median when k gte 1
*
* @return The Banneheke-Ekayanake Approximation for the Median
*
* @throws java.lang.Exception Thrown if the Median cannot be computed
*/
public double bannehekeEkayanakeMedianApproximation()
throws java.lang.Exception
{
double shape = _shapeScaleParameters.shape();
if (1. > shape)
{
throw new java.lang.Exception (
"ShapeScaleDistribution::bannehekeEkayanakeMedianApproximation => Invalid Shape Parameter"
);
}
return (3. * shape - 0.8) / (3. * shape - 0.2) * mean();
}
/**
* Compute the Ramanujan-Choi Approximation for the Median
*
* @return The Ramanujan-Choi Approximation for the Median
*/
public double ramanujanChoiMedianApproximation()
{
double shape = _shapeScaleParameters.shape();
double inverseShapeParameter = 1. / shape;
return shape - 1. / 3. +
8. * inverseShapeParameter / 405. +
184. * inverseShapeParameter * inverseShapeParameter / 25515. +
2248. * inverseShapeParameter * inverseShapeParameter * inverseShapeParameter / 3444525.;
}
/**
* Compute the Chen-Rubin Median Lower Bound
*
* @return The Chen-Rubin Median Lower Bound
*/
public double chenRubinMedianLowerBound()
{
return _shapeScaleParameters.shape() - 1. / 3.;
}
/**
* Compute the Chen-Rubin Median Upper Bound
*
* @return The Chen-Rubin Median Upper Bound
*/
public double chenRubinMedianUpperBound()
{
return _shapeScaleParameters.shape();
}
/**
* Generate a Scaled Gamma Distribution
*
* @param scaleFactor The Gamma Distribution Scale Factor
*
* @return Scaled Gamma Distribution
*/
public R1ShapeScaleDistribution scale (
final double scaleFactor)
{
try
{
return new R1ShapeScaleDistribution (
new org.drip.measure.gamma.ShapeScaleParameters (
_shapeScaleParameters.shape(),
_shapeScaleParameters.scale() * scaleFactor
),
_gammaEstimator,
_digammaEstimator,
_lowerIncompleteGammaEstimator
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Retrieve the Array of Natural Parameters
*
* @return Array of Natural Parameters
*/
public double[] naturalParameters()
{
return new double[]
{
_shapeScaleParameters.shape() - 1,
-1. / _shapeScaleParameters.scale()
};
}
/**
* Retrieve the Array of Natural Statistics
*
* @param x X
*
* @return Array of Natural Statistics
*/
public double[] naturalStatistics (
final double x)
{
return org.drip.numerical.common.NumberUtil.IsValid (
x
) ? new double[]
{
x,
java.lang.Math.log (
x
)
} : null;
}
/**
* Generate the Exponential Family Representation
*
* @param x X
*
* @return Exponential Family Representation
*/
public org.drip.measure.gamma.ExponentialFamilyRepresentation exponentialFamilyRepresentation (
final double x)
{
try
{
return new org.drip.measure.gamma.ExponentialFamilyRepresentation (
naturalParameters(),
naturalStatistics (
x
)
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Compute the Laplacian
*
* @param s S
*
* @return The Laplacian
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double laplacian (
final double s)
throws java.lang.Exception
{
if (0. > s)
{
throw new java.lang.Exception (
"ShapeScaleDistribution::laplacian => Invalid Shape Parameter"
);
}
return java.lang.Math.pow (
1. + s * _shapeScaleParameters.scale(),
-1. * _shapeScaleParameters.shape()
);
}
/**
* Generate a Random Variable using the Ahrens-Dieter (1982) Scheme
*
* @return Random Variable using the Ahrens-Dieter (1982) Scheme
*
* @throws java.lang.Exception Thrown if the Random Instance cannot be estimated
*/
public double randomAhrensDieter1982()
throws java.lang.Exception
{
double shape = _shapeScaleParameters.shape();
double eta = 0.;
double random = 0.;
double epsilon = 0.;
int k = (int) shape;
double delta = shape - k;
for (int index = 0;
index < k;
++index)
{
random = random - java.lang.Math.log (
java.lang.Math.random()
);
}
if (0. == delta)
{
return random;
}
while (true)
{
double u = java.lang.Math.random();
double v = java.lang.Math.random();
double w = java.lang.Math.random();
if (u <= java.lang.Math.E / (java.lang.Math.E + delta))
{
epsilon = java.lang.Math.pow (
v,
1. / delta
);
eta = w * java.lang.Math.pow (
epsilon,
delta - 1.
);
}
else
{
epsilon = 1. - java.lang.Math.log (
v
);
eta = w * java.lang.Math.exp (
-1. * epsilon
);
}
if (eta <= java.lang.Math.pow (
epsilon,
delta - 1.
) * java.lang.Math.exp (
-1. * epsilon
)
)
{
break;
}
}
return _shapeScaleParameters.scale() * (random + epsilon);
}
/**
* Generate a Random Variable using the Marsaglia (1977) Scheme
*
* @return Random Variable using the Marsaglia (1977) Scheme
*
* @throws java.lang.Exception Thrown if the Random Instance cannot be estimated
*/
public double randomMarsaglia1977()
throws java.lang.Exception
{
return randomMarsaglia1977 (
_shapeScaleParameters.shape()
);
}
}