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CauchyDistribution Class

Cauchy-Lorentz distribution.
Inheritance Hierarchy
SystemObject
  Accord.Statistics.DistributionsDistributionBase
    Accord.Statistics.Distributions.UnivariateUnivariateContinuousDistribution
      Accord.Statistics.Distributions.UnivariateCauchyDistribution

Namespace:  Accord.Statistics.Distributions.Univariate
Assembly:  Accord.Statistics (in Accord.Statistics.dll) Version: 3.7.0
Syntax
[SerializableAttribute]
public class CauchyDistribution : UnivariateContinuousDistribution, 
	IFittableDistribution<double, CauchyOptions>, IFittable<double, CauchyOptions>, 
	IFittable<double>, IFittableDistribution<double>, IDistribution<double>, 
	IDistribution, ICloneable, ISampleableDistribution<double>, IRandomNumberGenerator<double>, 
	IFormattable
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The CauchyDistribution type exposes the following members.

Constructors
  NameDescription
Public methodCauchyDistribution
Constructs a Cauchy-Lorentz distribution with location parameter 0 and scale 1.
Public methodCauchyDistribution(Double, Double)
Constructs a Cauchy-Lorentz distribution with given location and scale parameters.
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Properties
  NameDescription
Public propertyEntropy
Gets the entropy for this distribution.
(Overrides UnivariateContinuousDistributionEntropy.)
Public propertyLocation
Gets the distribution's location parameter x0.
Public propertyMean
Cauchy's mean is undefined.
(Overrides UnivariateContinuousDistributionMean.)
Public propertyMedian
Gets the median for this distribution.
(Overrides UnivariateContinuousDistributionMedian.)
Public propertyMode
Gets the mode for this distribution.
(Overrides UnivariateContinuousDistributionMode.)
Public propertyQuartiles
Gets the Quartiles for this distribution.
(Inherited from UnivariateContinuousDistribution.)
Public propertyScale
Gets the distribution's scale parameter gamma.
Public propertyStatic memberStandard
Gets the Standard Cauchy Distribution, with zero location and unitary shape.
Public propertyStandardDeviation
Gets the Standard Deviation (the square root of the variance) for the current distribution.
(Inherited from UnivariateContinuousDistribution.)
Public propertySupport
Gets the support interval for this distribution.
(Overrides UnivariateContinuousDistributionSupport.)
Public propertyVariance
Cauchy's variance is undefined.
(Overrides UnivariateContinuousDistributionVariance.)
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Methods
  NameDescription
Public methodClone
Creates a new object that is a copy of the current instance.
(Overrides DistributionBaseClone.)
Public methodComplementaryDistributionFunction
Gets the complementary cumulative distribution function (ccdf) for this distribution evaluated at point x. This function is also known as the Survival function.
(Inherited from UnivariateContinuousDistribution.)
Public methodCumulativeHazardFunction
Gets the cumulative hazard function for this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.)
Public methodDistributionFunction(Double)
Gets the cumulative distribution function (cdf) for this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.)
Public methodDistributionFunction(Double, Double)
Gets the cumulative distribution function (cdf) for this distribution in the semi-closed interval (a; b] given as P(a < X ≤ b).
(Inherited from UnivariateContinuousDistribution.)
Public methodEquals
Determines whether the specified object is equal to the current object.
(Inherited from Object.)
Protected methodFinalize
Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection.
(Inherited from Object.)
Public methodFit(Double)
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)
Public methodFit(Double, IFittingOptions)
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)
Public methodFit(Double, Double)
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)
Public methodFit(Double, Int32)
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)
Public methodCode exampleFit(Double, Double, CauchyOptions)
Fits the underlying distribution to a given set of observations.
Public methodCode exampleFit(Double, Double, IFittingOptions)
Fits the underlying distribution to a given set of observations.
(Overrides UnivariateContinuousDistributionFit(Double, Double, IFittingOptions).)
Public methodFit(Double, Int32, IFittingOptions)
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.)
Public methodGenerate
Generates a random observation from the current distribution.
(Inherited from UnivariateContinuousDistribution.)
Public methodGenerate(Random)
Generates a random observation from the current distribution.
(Overrides UnivariateContinuousDistributionGenerate(Random).)
Public methodGenerate(Int32)
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.)
Public methodGenerate(Int32, Double)
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.)
Public methodGenerate(Int32, Random)
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.)
Public methodGenerate(Int32, Double, Random)
Generates a random vector of observations from the current distribution.
(Overrides UnivariateContinuousDistributionGenerate(Int32, Double, Random).)
Public methodGetHashCode
Serves as the default hash function.
(Inherited from Object.)
Public methodGetRange
Gets the distribution range within a given percentile.
(Inherited from UnivariateContinuousDistribution.)
Public methodGetType
Gets the Type of the current instance.
(Inherited from Object.)
Public methodHazardFunction
Gets the hazard function, also known as the failure rate or the conditional failure density function for this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.)
Protected methodInnerComplementaryDistributionFunction
Gets the complementary cumulative distribution function (ccdf) for this distribution evaluated at point x. This function is also known as the Survival function.
(Inherited from UnivariateContinuousDistribution.)
Protected methodInnerDistributionFunction
Gets the cumulative distribution function (cdf) for this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionInnerDistributionFunction(Double).)
Protected methodInnerInverseDistributionFunction
Gets the inverse of the cumulative distribution function (icdf) for this distribution evaluated at probability p. This function is also known as the Quantile function.
(Inherited from UnivariateContinuousDistribution.)
Protected methodInnerLogProbabilityDensityFunction
Gets the log-probability density function (pdf) for this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionInnerLogProbabilityDensityFunction(Double).)
Protected methodInnerProbabilityDensityFunction
Gets the probability density function (pdf) for this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionInnerProbabilityDensityFunction(Double).)
Public methodInverseDistributionFunction
Gets the inverse of the cumulative distribution function (icdf) for this distribution evaluated at probability p. This function is also known as the Quantile function.
(Inherited from UnivariateContinuousDistribution.)
Public methodLogCumulativeHazardFunction
Gets the log of the cumulative hazard function for this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.)
Public methodLogProbabilityDensityFunction
Gets the log-probability density function (pdf) for this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.)
Protected methodMemberwiseClone
Creates a shallow copy of the current Object.
(Inherited from Object.)
Public methodProbabilityDensityFunction
Gets the probability density function (pdf) for this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.)
Public methodQuantileDensityFunction
Gets the first derivative of the inverse distribution function (icdf) for this distribution evaluated at probability p.
(Inherited from UnivariateContinuousDistribution.)
Public methodStatic memberRandom(Double, Double)
Generates a random observation from the Cauchy distribution with the given parameters.
Public methodStatic memberRandom(Double, Double, Int32)
Generates a random vector of observations from the Cauchy distribution with the given parameters.
Public methodStatic memberRandom(Double, Double, Random)
Generates a random observation from the Cauchy distribution with the given parameters.
Public methodStatic memberRandom(Double, Double, Int32, Double)
Generates a random vector of observations from the Cauchy distribution with the given parameters.
Public methodStatic memberRandom(Double, Double, Int32, Random)
Generates a random vector of observations from the Cauchy distribution with the given parameters.
Public methodStatic memberRandom(Double, Double, Int32, Double, Random)
Generates a random vector of observations from the Cauchy distribution with the given parameters.
Public methodToString
Returns a String that represents this instance.
(Inherited from DistributionBase.)
Public methodToString(IFormatProvider)
Returns a String that represents this instance.
(Inherited from DistributionBase.)
Public methodToString(String)
Returns a String that represents this instance.
(Inherited from DistributionBase.)
Public methodToString(String, IFormatProvider)
Returns a String that represents this instance.
(Overrides DistributionBaseToString(String, IFormatProvider).)
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Extension Methods
  NameDescription
Public Extension MethodHasMethod
Checks whether an object implements a method with the given name.
(Defined by ExtensionMethods.)
Public Extension MethodIsEqual
Compares two objects for equality, performing an elementwise comparison if the elements are vectors or matrices.
(Defined by Matrix.)
Public Extension MethodToT
Converts an object into another type, irrespective of whether the conversion can be done at compile time or not. This can be used to convert generic types to numeric types during runtime.
(Defined by ExtensionMethods.)
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Remarks

The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The simplest Cauchy distribution is called the standard Cauchy distribution. It has the distribution of a random variable that is the ratio of two independent standard normal random variables.

References:

Examples

The following example demonstrates how to instantiate a Cauchy distribution with a given location parameter x0 and scale parameter γ (gamma), calculating its main properties and characteristics:

double location = 0.42;
double scale = 1.57;

// Create a new Cauchy distribution with x0 = 0.42 and γ = 1.57 
CauchyDistribution cauchy = new CauchyDistribution(location, scale);

// Common measures
double mean = cauchy.Mean;     // NaN - Cauchy's mean is undefined.
double var = cauchy.Variance;  // NaN - Cauchy's variance is undefined.
double median = cauchy.Median; // 0.42

// Cumulative distribution functions
double cdf = cauchy.DistributionFunction(x: 0.27);           // 0.46968025841608563
double ccdf = cauchy.ComplementaryDistributionFunction(x: 0.27);          // 0.53031974158391437
double icdf = cauchy.InverseDistributionFunction(p: 0.69358638272337991); // 1.5130304686978195

// Probability density functions
double pdf = cauchy.ProbabilityDensityFunction(x: 0.27);     // 0.2009112009763413
double lpdf = cauchy.LogProbabilityDensityFunction(x: 0.27); // -1.6048922547266871

// Hazard (failure rate) functions
double hf = cauchy.HazardFunction(x: 0.27); // 0.3788491832800277
double chf = cauchy.CumulativeHazardFunction(x: 0.27); // 0.63427516833243092

// String representation
string str = cauchy.ToString(CultureInfo.InvariantCulture); // "Cauchy(x; x0 = 0.42, γ = 1.57)

The following example shows how to fit a Cauchy distribution (estimate its location and shape parameters) given a set of observation values.

// Create an initial distribution
CauchyDistribution cauchy = new CauchyDistribution();

// Consider a vector of univariate observations
double[] observations = { 0.25, 0.12, 0.72, 0.21, 0.62, 0.12, 0.62, 0.12 };

// Fit to the observations
cauchy.Fit(observations);

// Check estimated values
double location = cauchy.Location; //  0.18383
double gamma    = cauchy.Scale;    // -0.10530

It is also possible to estimate only some of the Cauchy parameters at a time. For this, you can specify a CauchyOptions object and pass it alongside the observations:

// Create options to estimate location only
CauchyOptions options = new CauchyOptions()
{
    EstimateLocation = true,
    EstimateScale = false
};

// Create an initial distribution with a pre-defined scale
CauchyDistribution cauchy = new CauchyDistribution(location: 0, scale: 4.2);

// Fit to the observations
cauchy.Fit(observations, options);

// Check estimated values
double location = cauchy.Location; //  0.3471218110202
double gamma    = cauchy.Scale;    //  4.2 (unchanged)
See Also