GammaDistribution Class |
Namespace: Accord.Statistics.Distributions.Univariate
[SerializableAttribute] public class GammaDistribution : UnivariateContinuousDistribution, IFittableDistribution<double, GammaOptions>, IFittable<double, GammaOptions>, IFittable<double>, IFittableDistribution<double>, IDistribution<double>, IDistribution, ICloneable, ISampleableDistribution<double>, IRandomNumberGenerator<double>
The GammaDistribution type exposes the following members.
Name | Description | |
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GammaDistribution |
Constructs a Gamma distribution.
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GammaDistribution(Double, Double) |
Constructs a Gamma distribution.
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Name | Description | |
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Entropy |
Gets the entropy for this distribution.
(Overrides UnivariateContinuousDistributionEntropy.) | |
Mean |
Gets the mean for this distribution.
(Overrides UnivariateContinuousDistributionMean.) | |
Median |
Gets the median for this distribution.
(Inherited from UnivariateContinuousDistribution.) | |
Mode |
Gets the mode for this distribution.
(Overrides UnivariateContinuousDistributionMode.) | |
Quartiles |
Gets the Quartiles for this distribution.
(Inherited from UnivariateContinuousDistribution.) | |
Rate |
Gets the inverse scale parameter β = 1/θ.
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Scale |
Gets the distribution's scale
parameter θ (theta).
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Shape |
Gets the distribution's
shape parameter k.
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Standard |
Gets the standard Gamma distribution,
with scale θ = 1 and location k = 1.
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StandardDeviation |
Gets the Standard Deviation (the square root of
the variance) for the current distribution.
(Inherited from UnivariateContinuousDistribution.) | |
Support |
Gets the support interval for this distribution.
(Overrides UnivariateContinuousDistributionSupport.) | |
Variance |
Gets the variance for this distribution.
(Overrides UnivariateContinuousDistributionVariance.) |
Name | Description | |
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Clone |
Creates a new object that is a copy of the current instance.
(Overrides DistributionBaseClone.) | |
ComplementaryDistributionFunction |
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.) | |
CumulativeHazardFunction |
Gets the cumulative hazard function for this
distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.) | |
DistributionFunction(Double) |
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.) | |
DistributionFunction(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.) | |
Equals | Determines whether the specified object is equal to the current object. (Inherited from Object.) | |
Estimate |
Estimates a new Gamma distribution from a given set of observations.
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Finalize | Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.) | |
Fit(Double) |
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.) | |
Fit(Double, IFittingOptions) |
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.) | |
Fit(Double, Double) |
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.) | |
Fit(Double, Int32) |
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.) | |
Fit(Double, Double, GammaOptions) |
Fits the underlying distribution to a given set of observations.
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Fit(Double, Double, IFittingOptions) |
Fits the underlying distribution to a given set of observations.
(Overrides UnivariateContinuousDistributionFit(Double, Double, IFittingOptions).) | |
Fit(Double, Int32, IFittingOptions) |
Fits the underlying distribution to a given set of observations.
(Inherited from UnivariateContinuousDistribution.) | |
FromBayesian |
Constructs a Gamma distribution using α and β parameterization.
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FromMean |
Constructs a Gamma distribution using k and μ parameterization.
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Generate |
Generates a random observation from the current distribution.
(Inherited from UnivariateContinuousDistribution.) | |
Generate(Random) |
Generates a random observation from the current distribution.
(Overrides UnivariateContinuousDistributionGenerate(Random).) | |
Generate(Int32) |
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.) | |
Generate(Int32, Double) |
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.) | |
Generate(Int32, Random) |
Generates a random vector of observations from the current distribution.
(Inherited from UnivariateContinuousDistribution.) | |
Generate(Int32, Double, Random) |
Generates a random vector of observations from the current distribution.
(Overrides UnivariateContinuousDistributionGenerate(Int32, Double, Random).) | |
GetHashCode | Serves as the default hash function. (Inherited from Object.) | |
GetRange |
Gets the distribution range within a given percentile.
(Inherited from UnivariateContinuousDistribution.) | |
GetType | Gets the Type of the current instance. (Inherited from Object.) | |
HazardFunction |
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.) | |
InnerComplementaryDistributionFunction |
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.) | |
InnerDistributionFunction |
Gets the cumulative distribution function (cdf) for
this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionInnerDistributionFunction(Double).) | |
InnerInverseDistributionFunction |
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.
(Overrides UnivariateContinuousDistributionInnerInverseDistributionFunction(Double).) | |
InnerLogProbabilityDensityFunction |
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionInnerLogProbabilityDensityFunction(Double).) | |
InnerProbabilityDensityFunction |
Gets the probability density function (pdf) for
this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionInnerProbabilityDensityFunction(Double).) | |
InverseDistributionFunction |
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.) | |
LogCumulativeHazardFunction |
Gets the log of the cumulative hazard function for this
distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.) | |
LogProbabilityDensityFunction |
Gets the log-probability density function (pdf) for
this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.) | |
Marsaglia(Double, Double) |
Random Gamma-distribution number generation
based on Marsaglia's Simple Method (2000).
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Marsaglia(Double, Double, Random) |
Random Gamma-distribution number generation
based on Marsaglia's Simple Method (2000).
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MemberwiseClone | Creates a shallow copy of the current Object. (Inherited from Object.) | |
ProbabilityDensityFunction |
Gets the probability density function (pdf) for
this distribution evaluated at point x.
(Inherited from UnivariateContinuousDistribution.) | |
QuantileDensityFunction |
Gets the first derivative of the
inverse distribution function (icdf) for this distribution evaluated
at probability p.
(Inherited from UnivariateContinuousDistribution.) | |
Random(Double, Double) |
Generates a random observation from the
Gamma distribution with the given parameters.
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Random(Double, Double, Int32) |
Generates a random vector of observations from the
Gamma distribution with the given parameters.
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Random(Double, Double, Random) |
Generates a random observation from the
Gamma distribution with the given parameters.
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Random(Double, Double, Int32, Double) |
Generates a random vector of observations from the
Gamma distribution with the given parameters.
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Random(Double, Double, Int32, Random) |
Generates a random vector of observations from the
Gamma distribution with the given parameters.
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Random(Double, Double, Int32, Double, Random) |
Generates a random vector of observations from the
Gamma distribution with the given parameters.
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ToString |
Returns a String that represents this instance.
(Inherited from DistributionBase.) | |
ToString(IFormatProvider) |
Returns a String that represents this instance.
(Inherited from DistributionBase.) | |
ToString(String) |
Returns a String that represents this instance.
(Inherited from DistributionBase.) | |
ToString(String, IFormatProvider) |
Returns a String that represents this instance.
(Overrides DistributionBaseToString(String, IFormatProvider).) |
Name | Description | |
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HasMethod |
Checks whether an object implements a method with the given name.
(Defined by ExtensionMethods.) | |
IsEqual |
Compares two objects for equality, performing an elementwise
comparison if the elements are vectors or matrices.
(Defined by Matrix.) | |
To(Type) | Overloaded.
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.) | |
ToT | Overloaded.
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.) |
The gamma distribution is a two-parameter family of continuous probability distributions. There are three different parameterizations in common use:
In each of these three forms, both parameters are positive real numbers. The parameterization with k and θ appears to be more common in econometrics and certain other applied fields, where e.g. the gamma distribution is frequently used to model waiting times. For instance, in life testing, the waiting time until death is a random variable that is frequently modeled with a gamma distribution. This is the default construction method for this class.
The parameterization with α and β is more common in Bayesian statistics, where the gamma distribution is used as a conjugate prior distribution for various types of inverse scale (aka rate) parameters, such as the λ of an exponential distribution or a Poisson distribution – or for that matter, the β of the gamma distribution itself. (The closely related inverse gamma distribution is used as a conjugate prior for scale parameters, such as the variance of a normal distribution.) In order to create a Gamma distribution using the Bayesian parameterization, you can use FromBayesian(Double, Double).
If k is an integer, then the distribution represents an Erlang distribution; i.e., the sum of k independent exponentially distributed random variables, each of which has a mean of θ (which is equivalent to a rate parameter of 1/θ).
The gamma distribution is the maximum entropy probability distribution for a random variable X for which E[X] = kθ = α/β is fixed and greater than zero, and E[ln(X)] = ψ(k) + ln(θ) = ψ(α) − ln(β) is fixed (ψ is the digamma function).
References:
The following example shows how to create, test and compute the main functions of a Gamma distribution given parameters θ = 4 and k = 2:
// Create a Γ-distribution with k = 2 and θ = 4 var gamma = new GammaDistribution(theta: 4, k: 2); // Common measures double mean = gamma.Mean; // 8.0 double median = gamma.Median; // 6.7133878418421506 double var = gamma.Variance; // 32.0 double mode = gamma.Mode; // 4.0 // Cumulative distribution functions double cdf = gamma.DistributionFunction(x: 0.27); // 0.002178158242390601 double ccdf = gamma.ComplementaryDistributionFunction(x: 0.27); // 0.99782184175760935 double icdf = gamma.InverseDistributionFunction(p: cdf); // 0.26999998689819171 // Probability density functions double pdf = gamma.ProbabilityDensityFunction(x: 0.27); // 0.015773530285395465 double lpdf = gamma.LogProbabilityDensityFunction(x: 0.27); // -4.1494220422235433 // Hazard (failure rate) functions double hf = gamma.HazardFunction(x: 0.27); // 0.015807962529274005 double chf = gamma.CumulativeHazardFunction(x: 0.27); // 0.0021805338793574793 // String representation string str = gamma.ToString(CultureInfo.InvariantCulture); // "Γ(x; k = 2, θ = 4)"