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MixtureT Class

Mixture of univariate probability distributions.
Inheritance Hierarchy
SystemObject
  Accord.Statistics.DistributionsDistributionBase
    Accord.Statistics.Distributions.UnivariateUnivariateContinuousDistribution
      Accord.Statistics.Distributions.UnivariateMixtureT

Namespace:  Accord.Statistics.Distributions.Univariate
Assembly:  Accord.Statistics (in Accord.Statistics.dll) Version: 3.5.0
Syntax
[SerializableAttribute]
public class Mixture<T> : UnivariateContinuousDistribution, IMixture<T>, 
	IDistribution, ICloneable, IFittableDistribution<double, MixtureOptions>, 
	IFittable<double, MixtureOptions>, IFittable<double>, 
	IFittableDistribution<double>, IDistribution<double>, ISampleableDistribution<double>, 
	IRandomNumberGenerator<double>
where T : Object, IUnivariateDistribution<double>
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Type Parameters

T
The type of the univariate component distributions.

The MixtureT type exposes the following members.

Constructors
Properties
  NameDescription
Public propertyCoefficients
Gets the weight coefficients.
Public propertyComponents
Gets the mixture components.
Public propertyEntropy
This method is not supported.
(Overrides UnivariateContinuousDistributionEntropy.)
Public propertyMean
Gets the mean for this distribution.
(Overrides UnivariateContinuousDistributionMean.)
Public propertyMedian
Gets the median for this distribution.
(Inherited from UnivariateContinuousDistribution.)
Public propertyMode
This method is not supported.
(Overrides UnivariateContinuousDistributionMode.)
Public propertyQuartiles
Gets the Quartiles for this distribution.
(Inherited from UnivariateContinuousDistribution.)
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
Gets the variance for this distribution.
(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.
(Overrides UnivariateContinuousDistributionDistributionFunction(Double).)
Public methodDistributionFunction(Int32, Double)
Gets the cumulative distribution function (cdf) for one component of this distribution evaluated at point x.
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.)
Public methodStatic memberEstimate(Double, T)
Estimates a new mixture model from a given set of observations.
Public methodStatic memberEstimate(Double, Double, T)
Estimates a new mixture model from a given set of observations.
Public methodStatic memberEstimate(Double, Double, Double, T)
Estimates a new mixture model from a given set of observations.
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 methodFit(Double, Double, IFittingOptions)
Fits the underlying distribution to a given set of observations.
(Overrides UnivariateContinuousDistributionFit(Double, Double, IFittingOptions).)
Public methodFit(Double, Double, MixtureOptions)
Fits the underlying distribution to a given set of observations.
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.
(Overrides UnivariateContinuousDistributionGenerate.)
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.
(Overrides UnivariateContinuousDistributionGenerate(Int32, Double).)
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.)
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 methodLogLikelihood(Double)
Computes the log-likelihood of the distribution for a given set of observations.
Public methodLogLikelihood(Double, Double)
Computes the log-likelihood of the distribution for a given set of observations.
Public methodLogProbabilityDensityFunction(Double)
Gets the log-probability density function (pdf) for this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionLogProbabilityDensityFunction(Double).)
Public methodLogProbabilityDensityFunction(Int32, Double)
Gets the log-probability density function (pdf) for one of the component distributions evaluated at point x.
Protected methodMemberwiseClone
Creates a shallow copy of the current Object.
(Inherited from Object.)
Public methodProbabilityDensityFunction(Double)
Gets the probability density function (pdf) for this distribution evaluated at point x.
(Overrides UnivariateContinuousDistributionProbabilityDensityFunction(Double).)
Public methodProbabilityDensityFunction(Int32, Double)
Gets the probability density function (pdf) for one of the component distributions evaluated at point x.
Public methodQuantileDensityFunction
Gets the first derivative of the inverse distribution function (icdf) for this distribution evaluated at probability p.
(Inherited from UnivariateContinuousDistribution.)
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 MethodToTOverloaded.
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.)
Public Extension MethodToTOverloaded.
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 Matrix.)
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Remarks

A mixture density is a probability density function which is expressed as a convex combination (i.e. a weighted sum, with non-negative weights that sum to 1) of other probability density functions. The individual density functions that are combined to make the mixture density are called the mixture components, and the weights associated with each component are called the mixture weights.

References:

Examples
// Create a new mixture containing two Normal distributions
Mixture<NormalDistribution> mix = new Mixture<NormalDistribution>(
    new NormalDistribution(2, 1), new NormalDistribution(5, 1));

// Common measures
double mean   = mix.Mean;     // 3.5
double median = mix.Median;   // 3.4999998506015895
double var    = mix.Variance; // 3.25

// Cumulative distribution functions
double cdf = mix.DistributionFunction(x: 4.2);               // 0.59897597553494908
double ccdf = mix.ComplementaryDistributionFunction(x: 4.2); // 0.40102402446505092

// Probability mass functions
double pmf1 = mix.ProbabilityDensityFunction(x: 1.2); // 0.14499174984363708
double pmf2 = mix.ProbabilityDensityFunction(x: 2.3); // 0.19590437513747333
double pmf3 = mix.ProbabilityDensityFunction(x: 3.7); // 0.13270883471234715
double lpmf = mix.LogProbabilityDensityFunction(x: 4.2); // -1.8165661905848629

// Quantile function
double icdf1 = mix.InverseDistributionFunction(p: 0.17); // 1.5866611690305095
double icdf2 = mix.InverseDistributionFunction(p: 0.46); // 3.1968506765456883
double icdf3 = mix.InverseDistributionFunction(p: 0.87); // 5.6437596300843076

// Hazard (failure rate) functions
double hf = mix.HazardFunction(x: 4.2);            // 0.40541978256972522
double chf = mix.CumulativeHazardFunction(x: 4.2); // 0.91373394208601633

// String representation:
// Mixture(x; 0.5 * N(x; μ = 5, σ² = 1) + 0.5 * N(x; μ = 5, σ² = 1))
string str = mix.ToString(CultureInfo.InvariantCulture);

The following example shows how to estimate (fit) a Mixture of Normal distributions from weighted data:

// Randomly initialize some mixture components
NormalDistribution[] components = new NormalDistribution[2];
components[0] = new NormalDistribution(2, 1);
components[1] = new NormalDistribution(5, 1);

// Create an initial mixture
var mixture = new Mixture<NormalDistribution>(components);

// Now, suppose we have a weighted data
// set. Those will be the input points:

double[] points = { 0, 3, 1, 7, 3, 5, 1, 2, -1, 2, 7, 6, 8, 6 }; // (14 points)

// And those are their respective unnormalized weights:
double[] weights = { 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 3, 1, 1 }; // (14 weights)

// Let's normalize the weights so they sum up to one:
weights = weights.Divide(weights.Sum());

// Now we can fit our model to the data:
mixture.Fit(points, weights);   // done!

// Our model will be:
double mean1 = mixture.Components[0].Mean; // 1.41126
double mean2 = mixture.Components[1].Mean; // 6.53301

// With mixture weights
double pi1 = mixture.Coefficients[0]; // 0.51408
double pi2 = mixture.Coefficients[0]; // 0.48591

// If we need the GaussianMixtureModel functionality, we can
// use the estimated mixture to initialize a new model:
GaussianMixtureModel gmm = new GaussianMixtureModel(mixture);

mean1 = gmm.Gaussians[0].Mean[0]; // 1.41126 (same)
mean2 = gmm.Gaussians[1].Mean[0]; // 6.53301 (same)

p1 = gmm.Gaussians[0].Proportion; // 0.51408 (same)
p2 = gmm.Gaussians[1].Proportion; // 0.48591 (same)
See Also