Package org.drip.measure.chisquare

Class R1Central

java.lang.Object

org.drip.measure.continuous.R1Univariate

org.drip.measure.chisquare.R1Central

public class R1Central
extends R1Univariate

R1Central implements the Probability Density Function for the R¹ Central Chi-Square Distribution. The References are:

• Abramowitz, M., and I. A. Stegun (2007): Handbook of Mathematics Functions Dover Book on Mathematics
• Backstrom, T., and J. Fischer (2018): Fast Randomization for Distributed Low Bit-rate Coding of Speech and Audio IEEE/ACM Transactions on Audio, Speech, and Language Processing 26 (1) 19-30
• Chi-Squared Distribution (2019): Chi-Squared Function https://en.wikipedia.org/wiki/Chi-squared_distribution
• Johnson, N. L., S. Kotz, and N. Balakrishnan (1994): Continuous Univariate Distributions 2^nd Edition John Wiley and Sons
• National Institute of Standards and Technology (2019): Chi-Squared Distribution https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm

• Module = Computational Core Module
• Library = Numerical Analysis Library
• Project = R^d Continuous/Discrete Probability Measures
• Package = Chi-Square Distribution Implementation/Properties

Author:

Lakshmi Krishnamurthy

Constructor Summary

Constructors

Constructor	Description
R1Central(double degreesOfFreedom, R1ToR1 gammaEstimator, R1ToR1 digammaEstimator, R2ToR1 lowerIncompleteGammaEstimator)	R1Central Constructor

Method Summary

Modifier and Type	Method	Description
double	cdfScaler()	Retrieve the CDF Scaler
double	chernoffBound(double x)	Compute the Chernoff Upper Bound
R1UnivariateNormal	cltProxy()	Retrieve the Central Limit Theorem Equivalent Normal Distribution Proxy
double	cumulant(int n)	Compute the Cumulant
double	cumulative(double t)	Compute the cumulative under the distribution to the given value
double	degreesOfFreedom()	Retrieve the Degrees of Freedom
double	density(double t)	Compute the Density under the Distribution at the given Variate
double	differentialEntropy()	Retrieve the Differential Entropy of the Distribution
R1ToR1	digammaEstimator()	Retrieve the Digamma Estimator
double	excessKurtosis()	Retrieve the Excess Kurtosis of the Distribution
static R1Central	FromIndependentChiSquared(R1Central[] chiSquaredDistributionArray)	Generate a Consolidated Chi-squared Distribution from Independent Component Distributions
R1ShapeScaleDistribution	gammaDistribution(double scale)	Generate a Gamma-distribution off of the Scaled Chi-Square Distribution
R1ToR1	gammaEstimator()	Retrieve the Gamma Estimator
R2ToR1	lowerIncompleteGammaEstimator()	Retrieve the Lower Incomplete Gamma Estimator
double	mean()	Retrieve the Mean of the Distribution
double	median()	Retrieve the Median of the Distribution
double	mode()	Retrieve the Mode of the Distribution
R1ToR1	momentGeneratingFunction()	Construct the Moment Generating Function
double	nonCentralMoment(int m)	Compute the Non-central Moment about Zero
double	normalizer()	Retrieve the Normalizer
R1ToR1	probabilityGeneratingFunction()	Construct the Probability Generating Function
double	random()	Generate a Random Variable corresponding to the Distribution
double	randomChi()	Generate the Chi Distributed Random Number
double	randomCLTProxy()	Generate CLT Proxy Based Random Number - Proxy to Univariate Normal Distribution
double	randomExponentialHalf()	Generate Exponential (0.5) Distributed Random Number
double	randomFisherProxy()	Generate Fisher Proxy Random Number - Proxy to Univariate Normal Distribution
double	randomGamma(double c)	Generate Gamma Distributed Random Number
double	randomLogProxy()	Generate Logarithm Proxy Based Random Number - Proxy to Univariate Normal Distribution
double	randomMaxwell1()	Generate Maxwell (1) Distributed Random Number
double	randomRayleigh1()	Generate Rayleigh (1) Distributed Random Number
double	randomWilsonHilferty()	Generate Wilson-Hilferty Proxy Random Number - Proxy to Univariate Normal Distribution
double	skewness()	Retrieve the Skewness of the Distribution
double[]	support()	Lay out the Support of the PDF Range
boolean	validCLTProxy()	Indicate if the Current Distribution is a Valid Proxy as a CLT
double	variance()	Retrieve the Variance of the Distribution

Methods inherited from class org.drip.measure.continuous.R1Univariate

bPOE, centralMoment, cvar, expectedShortfall, fisherInformation, histogram, incremental, invCumulative, iqr, kullbackLeiblerDivergence, populationCentralMeasures, quantile, randomArray, supported, tukeyAnomaly, tukeyCriterion

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

R1Central

public R1Central(double degreesOfFreedom,
R1ToR1 gammaEstimator,
R1ToR1 digammaEstimator,
R2ToR1 lowerIncompleteGammaEstimator)
          throws java.lang.Exception

R1Central Constructor

Parameters:

degreesOfFreedom - Degrees of Freedom

gammaEstimator - Gamma Estimator

digammaEstimator - Digamma Estimator

lowerIncompleteGammaEstimator - Lower Incomplete Gamma Estimator

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

Method Details

FromIndependentChiSquared

public static final R1Central FromIndependentChiSquared(R1Central[] chiSquaredDistributionArray)

Generate a Consolidated Chi-squared Distribution from Independent Component Distributions

Parameters:

chiSquaredDistributionArray - Independent Component Distribution Array

Returns:

Consolidated Chi-squared Distribution

degreesOfFreedom

public double degreesOfFreedom()

Retrieve the Degrees of Freedom

Returns:

The Degrees of Freedom

gammaEstimator

public R1ToR1 gammaEstimator()

Retrieve the Gamma Estimator

Returns:

Gamma Estimator

digammaEstimator

public R1ToR1 digammaEstimator()

Retrieve the Digamma Estimator

Returns:

Digamma Estimator

lowerIncompleteGammaEstimator

public R2ToR1 lowerIncompleteGammaEstimator()

Retrieve the Lower Incomplete Gamma Estimator

Returns:

Lower Incomplete Gamma Estimator

support

public double[] support()

Description copied from class: R1Univariate

Lay out the Support of the PDF Range

Specified by:

support in class R1Univariate

Returns:

Support of the PDF Range

density

public double density(double t)
               throws java.lang.Exception

Description copied from class: R1Univariate

Compute the Density under the Distribution at the given Variate

Specified by:

density in class R1Univariate

Parameters:

t - Variate at which the Density needs to be computed

Returns:

The Density

Throws:

java.lang.Exception - Thrown if the input is invalid

cumulative

public double cumulative(double t)
                  throws java.lang.Exception

Description copied from class: R1Univariate

Compute the cumulative under the distribution to the given value

Specified by:

cumulative in class R1Univariate

Parameters:

t - Variate to which the cumulative is to be computed

Returns:

The cumulative

Throws:

java.lang.Exception - Thrown if the inputs are invalid

mean

public double mean()
            throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Mean of the Distribution

Specified by:

mean in class R1Univariate

Returns:

The Mean of the Distribution

Throws:

java.lang.Exception - Thrown if the Mean cannot be estimated

median

public double median()
              throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Median of the Distribution

Overrides:

median in class R1Univariate

Returns:

The Median of the Distribution

Throws:

java.lang.Exception - Thrown if the Median cannot be estimated

mode

public double mode()
            throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Mode of the Distribution

Overrides:

mode in class R1Univariate

Returns:

The Mode of the Distribution

Throws:

java.lang.Exception - Thrown if the Mode cannot be estimated

variance

public double variance()
                throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Variance of the Distribution

Specified by:

variance in class R1Univariate

Returns:

The Variance of the Distribution

Throws:

java.lang.Exception - Thrown if the Variance cannot be estimated

skewness

public double skewness()
                throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Skewness of the Distribution

Overrides:

skewness in class R1Univariate

Returns:

The Skewness of the Distribution

Throws:

java.lang.Exception - Thrown if the Skewness cannot be estimated

excessKurtosis

public double excessKurtosis()
                      throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Excess Kurtosis of the Distribution

Overrides:

excessKurtosis in class R1Univariate

Returns:

The Excess Kurtosis of the Distribution

Throws:

java.lang.Exception - Thrown if the Skewness cannot be estimated

differentialEntropy

public double differentialEntropy()
                           throws java.lang.Exception

Description copied from class: R1Univariate

Retrieve the Differential Entropy of the Distribution

Overrides:

differentialEntropy in class R1Univariate

Returns:

The Differential Entropy of the Distribution

Throws:

java.lang.Exception - Thrown if the Entropy cannot be estimated

momentGeneratingFunction

public R1ToR1 momentGeneratingFunction()

Description copied from class: R1Univariate

Construct the Moment Generating Function

Overrides:

momentGeneratingFunction in class R1Univariate

Returns:

The Moment Generating Function

probabilityGeneratingFunction

public R1ToR1 probabilityGeneratingFunction()

Description copied from class: R1Univariate

Construct the Probability Generating Function

Overrides:

probabilityGeneratingFunction in class R1Univariate

Returns:

The Probability Generating Function

random

public double random()
              throws java.lang.Exception

Description copied from class: R1Univariate

Generate a Random Variable corresponding to the Distribution

Overrides:

random in class R1Univariate

Returns:

Random Variable corresponding to the Distribution

Throws:

java.lang.Exception - Thrown if the Random Instance cannot be estimated

normalizer

public double normalizer()

Retrieve the Normalizer

Returns:

Normalizer

cdfScaler

public double cdfScaler()

Retrieve the CDF Scaler

Returns:

CDF Scaler

chernoffBound

public double chernoffBound(double x)
                     throws java.lang.Exception

Compute the Chernoff Upper Bound

Parameters:

x - A

Returns:

The Chernoff Upper Bound

Throws:

java.lang.Exception - Thrown if the Chernoff Upper Bound cannot be calculated

nonCentralMoment

public double nonCentralMoment(int m)
                        throws java.lang.Exception

Compute the Non-central Moment about Zero

Overrides:

nonCentralMoment in class R1Univariate

Parameters:

m - Non-central Moment Index

Returns:

The Non-central Moment about Zero

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

cumulant

public double cumulant(int n)
                throws java.lang.Exception

Compute the Cumulant

Parameters:

n - Cumulant Index

Returns:

The Cumulant

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

cltProxy

public R1UnivariateNormal cltProxy()

Retrieve the Central Limit Theorem Equivalent Normal Distribution Proxy

Returns:

The Central Limit Theorem Equivalent Normal Distribution Proxy

validCLTProxy

public boolean validCLTProxy()

Indicate if the Current Distribution is a Valid Proxy as a CLT

Returns:

TRUE - The Current Distribution is a Valid Proxy as a CLT

gammaDistribution

public R1ShapeScaleDistribution gammaDistribution(double scale)

Generate a Gamma-distribution off of the Scaled Chi-Square Distribution

Parameters:

scale - The Scale

Returns:

The Gamma Distribution

randomLogProxy

public double randomLogProxy()
                      throws java.lang.Exception

Generate Logarithm Proxy Based Random Number - Proxy to Univariate Normal Distribution

Returns:

Logarithm Proxy Based Random Number - Proxy to Univariate Normal Distribution

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

randomCLTProxy

public double randomCLTProxy()
                      throws java.lang.Exception

Generate CLT Proxy Based Random Number - Proxy to Univariate Normal Distribution

Returns:

CLT Proxy Based Random Number - Proxy to Univariate Normal Distribution

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

randomFisherProxy

public double randomFisherProxy()
                         throws java.lang.Exception

Generate Fisher Proxy Random Number - Proxy to Univariate Normal Distribution

Returns:

Fisher Proxy Random Number - Proxy to Univariate Normal Distribution

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

randomWilsonHilferty

public double randomWilsonHilferty()
                            throws java.lang.Exception

Generate Wilson-Hilferty Proxy Random Number - Proxy to Univariate Normal Distribution

Returns:

Wilson-Hilferty Proxy Random Number - Proxy to Univariate Normal Distribution

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

randomGamma

public double randomGamma(double c)
                   throws java.lang.Exception

Generate Gamma Distributed Random Number

Parameters:

c - The Scale Parameter

Returns:

Gamma Distributed Random Number

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

randomChi

public double randomChi()
                 throws java.lang.Exception

Generate the Chi Distributed Random Number

Returns:

Chi Distributed Random Number

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

randomExponentialHalf

public double randomExponentialHalf()
                             throws java.lang.Exception

Generate Exponential (0.5) Distributed Random Number

Returns:

Exponential (0.5) Distributed Random Number

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

randomRayleigh1

public double randomRayleigh1()
                       throws java.lang.Exception

Generate Rayleigh (1) Distributed Random Number

Returns:

Rayleigh (1) Distributed Random Number

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

randomMaxwell1

public double randomMaxwell1()
                      throws java.lang.Exception

Generate Maxwell (1) Distributed Random Number

Returns:

Maxwell (1) Distributed Random Number

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid