de.lmu.ifi.dbs.elki.math.statistics.distribution

Class NormalDistribution

java.lang.Object

de.lmu.ifi.dbs.elki.math.statistics.distribution.AbstractDistribution

de.lmu.ifi.dbs.elki.math.statistics.distribution.NormalDistribution

All Implemented Interfaces:

Distribution

@Alias(value={"GaussianDistribution","normal","gauss"})
public class NormalDistribution
extends AbstractDistribution

Gaussian distribution aka normal distribution

Since:

0.5.0

Author:

Erich Schubert

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
static class	NormalDistribution.Parameterizer Parameterization class

Field Summary

Fields

Modifier and Type	Field and Description
private static double[]	ERF_COEFF1 T.
private static double[]	ERF_COEFF2 T.
private double	mean Mean value for the generator
static double	ONEBYPHIINV075 1 / CDFINV(0.75)
(package private) static double	P_HIGH Treshold for switching nethods for erfinv approximation
(package private) static double	P_LOW Treshold for switching nethods for erfinv approximation
static double	PHIINV075 CDFINV(0.75)
private double	stddev Standard deviation

Fields inherited from class de.lmu.ifi.dbs.elki.math.statistics.distribution.AbstractDistribution

random

Constructor Summary

Constructors

Constructor and Description
NormalDistribution(double mean, double stddev) Constructor for Gaussian distribution
NormalDistribution(double mean, double stddev, java.util.Random random) Constructor for Gaussian distribution
NormalDistribution(double mean, double stddev, RandomFactory random) Constructor for Gaussian distribution

Method Summary

Modifier and Type	Method and Description
double	cdf(double val) Return the cumulative density function at the given value.
static double	cdf(double x, double mu, double sigma) Cumulative probability density function (CDF) of a normal distribution.
static double	erf(double x) Error function for Gaussian distributions = Normal distributions.
static double	erfc(double x) Complementary error function for Gaussian distributions = Normal distributions.
static double	erfcinv(double y) Inverse error function.
double	getMean()
double	getStddev()
double	logpdf(double val) Return the log density of an existing value
static double	logpdf(double x, double mu, double sigma) Log probability density function of the normal distribution.
double	nextRandom() Generate a new random value
double	pdf(double val) Return the density of an existing value
static double	pdf(double x, double mu, double sigma) Probability density function of the normal distribution.
double	quantile(double q) Quantile aka probit (for normal) aka inverse CDF (invcdf, cdf^-1) function.
static double	quantile(double x, double mu, double sigma) Inverse cumulative probability density function (probit) of a normal distribution.
static double	standardNormalCDF(double x) Cumulative probability density function (CDF) of a normal distribution.
static double	standardNormalLogPDF(double x) Log probability density function of the standard normal distribution.
static double	standardNormalPDF(double x) Probability density function of the standard normal distribution.
static double	standardNormalQuantile(double d) Approximate the inverse error function for normal distributions.
java.lang.String	toString() Describe the distribution

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Field Detail

P_LOW

static final double P_LOW

Treshold for switching nethods for erfinv approximation

See Also:

Constant Field Values

P_HIGH

static final double P_HIGH

Treshold for switching nethods for erfinv approximation

See Also:

Constant Field Values

PHIINV075

public static final double PHIINV075

CDFINV(0.75)

See Also:

Constant Field Values

ONEBYPHIINV075

public static final double ONEBYPHIINV075

1 / CDFINV(0.75)

See Also:

Constant Field Values

mean

private double mean

Mean value for the generator

stddev

private double stddev

Standard deviation

ERF_COEFF1

private static final double[] ERF_COEFF1

T. Ooura, http://www.kurims.kyoto-u.ac.jp/~ooura/gamerf.html

ERF_COEFF2

private static final double[] ERF_COEFF2

T. Ooura, http://www.kurims.kyoto-u.ac.jp/~ooura/gamerf.html

Constructor Detail

NormalDistribution

public NormalDistribution(double mean,
                          double stddev,
                          RandomFactory random)

Constructor for Gaussian distribution

Parameters:

mean - Mean

stddev - Standard Deviation

random - Random generator

NormalDistribution

public NormalDistribution(double mean,
                          double stddev,
                          java.util.Random random)

Constructor for Gaussian distribution

Parameters:

mean - Mean

stddev - Standard Deviation

random - Random generator

NormalDistribution

public NormalDistribution(double mean,
                          double stddev)

Constructor for Gaussian distribution

Parameters:

mean - Mean

stddev - Standard Deviation

Method Detail

pdf

public double pdf(double val)

Description copied from interface: Distribution

Return the density of an existing value

Parameters:

val - existing value

Returns:

distribution density

logpdf

public double logpdf(double val)

Description copied from interface: Distribution

Return the log density of an existing value

Parameters:

val - existing value

Returns:

log distribution density

cdf

public double cdf(double val)

Description copied from interface: Distribution

Return the cumulative density function at the given value.

Parameters:

val - existing value

Returns:

cumulative density

quantile

public double quantile(double q)

Description copied from interface: Distribution

Quantile aka probit (for normal) aka inverse CDF (invcdf, cdf^-1) function.

Parameters:

q - Quantile to find

Returns:

Quantile position

nextRandom

public double nextRandom()

Description copied from interface: Distribution

Generate a new random value

Specified by:

nextRandom in interface Distribution

Overrides:

nextRandom in class AbstractDistribution

Returns:

new random value

toString

public java.lang.String toString()

Description copied from interface: Distribution

Describe the distribution

Specified by:

toString in interface Distribution

Overrides:

toString in class java.lang.Object

Returns:

description

getMean

public double getMean()

Returns:

the mean

getStddev

public double getStddev()

Returns:

the standard deviation

erfc

@Reference(authors="T. Ooura",
           title="Gamma / Error Functions",
           booktitle="",
           url="http://www.kurims.kyoto-u.ac.jp/~ooura/gamerf.html",
           bibkey="web/Ooura96")
public static double erfc(double x)

Complementary error function for Gaussian distributions = Normal distributions.

Based on:
Takuya Ooura, http://www.kurims.kyoto-u.ac.jp/~ooura/gamerf.html
Copyright (C) 1996 Takuya OOURA (email: ooura@mmm.t.u-tokyo.ac.jp).
"You may use, copy, modify this code for any purpose and without fee."

Parameters:

x - parameter value

Returns:

erfc(x)

erf

public static double erf(double x)

Error function for Gaussian distributions = Normal distributions.

Parameters:

x - parameter value

Returns:

erf(x)

erfcinv

@Reference(authors="T. Ooura",
           title="Gamma / Error Functions",
           booktitle="",
           url="http://www.kurims.kyoto-u.ac.jp/~ooura/gamerf.html",
           bibkey="web/Ooura96")
public static double erfcinv(double y)

Inverse error function.

Based on:
T. Ooura, http://www.kurims.kyoto-u.ac.jp/~ooura/gamerf.html
Copyright (C) 1996 Takuya OOURA (email: ooura@mmm.t.u-tokyo.ac.jp).
"You may use, copy, modify this code for any purpose and without fee."

Parameters:

y - parameter value

Returns:

erfcinv(y)

pdf

public static double pdf(double x,
                         double mu,
                         double sigma)

Probability density function of the normal distribution.

\[ \frac{1}{\sqrt{2\pi\sigma^2}} \exp(-\frac{(x-\mu)^2}{2\sigma^2}) \]

Parameters:

x - The value.

mu - The mean.

sigma - The standard deviation.

Returns:

PDF of the given normal distribution at x.

logpdf

public static double logpdf(double x,
                            double mu,
                            double sigma)

Log probability density function of the normal distribution.

\[\log\frac{1}{\sqrt{2\pi}} - \log\sigma - \tfrac{(x-\mu)^2}{2\sigma^2}\]

Parameters:

x - The value.

mu - The mean.

sigma - The standard deviation.

Returns:

PDF of the given normal distribution at x.

standardNormalLogPDF

public static double standardNormalLogPDF(double x)

Log probability density function of the standard normal distribution.

\[ \log\frac{1}{\sqrt{2\pi}} -\frac{x^2}{2}) \]

Parameters:

x - The value.

Returns:

PDF of the given normal distribution at x.

standardNormalPDF

public static double standardNormalPDF(double x)

Probability density function of the standard normal distribution.

\[ \frac{1}{\sqrt{2\pi}} \exp(-\frac{x^2}{2}) \]

Parameters:

x - The value.

Returns:

PDF of the given normal distribution at x.

cdf

@Reference(authors="G. Marsaglia",
           title="Evaluating the Normal Distribution",
           booktitle="Journal of Statistical Software 11(4)",
           url="https://doi.org/10.18637/jss.v011.i04",
           bibkey="doi:10.18637/jss.v011.i04")
public static double cdf(double x,
                                                                                                                                                                                                                                                                                        double mu,
                                                                                                                                                                                                                                                                                        double sigma)

Cumulative probability density function (CDF) of a normal distribution.

Reference:

G. Marsaglia
Evaluating the Normal Distribution
Journal of Statistical Software 11(4)

Parameters:

x - value to evaluate CDF at

mu - Mean value

sigma - Standard deviation.

Returns:

The CDF of the given normal distribution at x.

standardNormalCDF

@Reference(authors="G. Marsaglia",
           title="Evaluating the Normal Distribution",
           booktitle="Journal of Statistical Software 11(4)",
           url="https://doi.org/10.18637/jss.v011.i04",
           bibkey="doi:10.18637/jss.v011.i04")
public static double standardNormalCDF(double x)

Cumulative probability density function (CDF) of a normal distribution.

Reference:

G. Marsaglia
Evaluating the Normal Distribution
Journal of Statistical Software 11(4)

Parameters:

x - value to evaluate CDF at

Returns:

The CDF of the given normal distribution at x.

quantile

public static double quantile(double x,
                              double mu,
                              double sigma)

Inverse cumulative probability density function (probit) of a normal distribution.

Parameters:

x - value to evaluate probit function at

mu - Mean value

sigma - Standard deviation.

Returns:

The probit of the given normal distribution at x.

standardNormalQuantile

public static double standardNormalQuantile(double d)

Approximate the inverse error function for normal distributions.

Parameters:

d - Quantile. Must be in [0:1], obviously.

Returns:

Inverse erf.