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prob.GeneralizedExtremeValueDistribution class

Package: prob
Superclasses: prob.ToolboxFittableParametricDistribution

Generalized extreme value probability distribution object

Description

prob.GeneralizedExtremeValueDistribution is an object consisting of parameters, a model description, and sample data for a generalized extreme value probability distribution.

Create a probability distribution object with specified parameter values using makedist. Alternatively, fit a distribution to data using fitdist or the Distribution Fitting app.

Construction

pd = makedist('GeneralizedExtremeValue') creates a generalized extreme value probability distribution object using the default parameter values.

pd = makedist('GeneralizedExtremeValue','k',k,'sigma',sigma,'mu',mu) creates a generalized extreme value probability distribution object using the specified parameter values.

Input Arguments

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k — Shape parameter0 (default) | scalar value

Shape parameter for the generalized extreme value distribution, specified as a scalar value.

Data Types: single | double

sigma — Scale parameter1 (default) | nonnegative scalar value

Scale parameter for the generalized extreme value distribution, specified as a nonnegative scalar value.

Data Types: single | double

mu — Location parameter0 (default) | scalar value

Location parameter for the generalized extreme value distribution, specified as a scalar value.

Data Types: single | double

Properties

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k — Shape parameterscalar value

Shape parameter of the generalized extreme value distribution, stored as a scalar value.

Data Types: single | double

sigma — Scale parameternonnegative scalar value

Scale parameter of the generalized extreme value distribution, stored as a nonnegative scalar value.

Data Types: single | double

mu — Location parameterscalar value

Location parameter of the generalized extreme value distribution, stored as a scalar value.

Data Types: single | double

DistributionName — Probability distribution nameprobability distribution name string

Probability distribution name, stored as a valid probability distribution name string. This property is read-only.

Data Types: char

InputData — Data used for distribution fittingstructure

Data used for distribution fitting, stored as a structure containing the following:

  • data: Data vector used for distribution fitting.

  • cens: Censoring vector, or empty if none.

  • freq: Frequency vector, or empty if none.

This property is read-only.

Data Types: struct

IsTruncated — Logical flag for truncated distribution0 | 1

Logical flag for truncated distribution, stored as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated. This property is read-only.

Data Types: logical

NumParameters — Number of parameterspositive integer value

Number of parameters for the probability distribution, stored as a positive integer value. This property is read-only.

Data Types: single | double

ParameterCovariance — Covariance matrix of the parameter estimatesmatrix of scalar values

Covariance matrix of the parameter estimates, stored as a p-by-p matrix, where p is the number of parameters in the distribution. The (i,j) element is the covariance between the estimates of the ith parameter and the jth parameter. The (i,i) element is the estimated variance of the ith parameter. If parameter i is fixed rather than estimated by fitting the distribution to data, then the (i,i) elements of the covariance matrix are 0. This property is read-only.

Data Types: single | double

ParameterDescription — Distribution parameter descriptionscell array of strings

Distribution parameter descriptions, stored as a cell array of strings. Each cell contains a short description of one distribution parameter. This property is read-only.

Data Types: char

ParameterIsFixed — Logical flag for fixed parametersarray of logical values

Logical flag for fixed parameters, stored as an array of logical values. If 0, the corresponding parameter in the ParameterNames array is not fixed. If 1, the corresponding parameter in the ParameterNames array is fixed. This property is read-only.

Data Types: logical

ParameterNames — Distribution parameter namescell array of strings

Distribution parameter names, stored as a cell array of strings. This property is read-only.

Data Types: char

ParameterValues — Distribution parameter valuesvector of scalar values

Distribution parameter values, stored as a vector. This property is read-only.

Data Types: single | double

Truncation — Truncation intervalvector of scalar values

Truncation interval for the probability distribution, stored as a vector containing the lower and upper truncation boundaries. This property is read-only.

Data Types: single | double

Methods

Inherited Methods

cdf Cumulative distribution function of probability distribution object
icdfInverse cumulative distribution function of probability distribution object
iqrInterquartile range of probability distribution object
median Median of probability distribution object
pdfProbability density function of probability distribution object
randomGenerate random numbers from probability distribution object
truncateTruncate probability distribution object
meanMean of probability distribution object
negloglikNegative loglikelihood of probability distribution object
paramciConfidence intervals for probability distribution parameters
proflikProfile likelihood function for probability distribution object
std Standard deviation of probability distribution object
varVariance of probability distribution object

Definitions

Generalized Extreme Value Distribution

The generalized extreme value distribution is often used to model the smallest or largest value among a large set of independent, identically distributed random values representing measurements or observations. It combines three simpler distributions into a single form, allowing a continuous range of possible shapes that include all three of the simpler distributions.

The three distribution types correspond to the limiting distribution of block maxima from different classes of underlying distributions:

  • Type 1 — Distributions whose tails decrease exponentially, such as the normal distribution

  • Type 2 — Distributions whose tails decrease as a polynomial, such as Student's t distribution

  • Type 3 — Distributions whose tails are finite, such as the beta distribution

The generalized extreme value distribution uses the following parameters.

ParameterDescriptionSupport
kShape parameter
sigmaScale parameter
muLocation parameter

The probability density function (pdf) for a Type 1 distribution, where shape parameter , is

When , the pdf is

for

For the Type 2 case, and . For the Type 3 case, and .

Examples

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Create a Generalized Extreme Value Distribution Object Using Default Parameters

Create a generalized extreme value distribution object using the default parameter values.

pd = makedist('GeneralizedExtremeValue')
pd = 

  GeneralizedExtremeValueDistribution

  Generalized Extreme Value distribution
        k = 0
    sigma = 1
       mu = 0

Create a Generalized Extreme Value Distribution Object Using Specified Parameters

Create a generalized extreme value distribution object by specifying values for the parameters.

pd = makedist('GeneralizedExtremeValue','k',0,'sigma',2,'mu',1)
pd = 

  GeneralizedExtremeValueDistribution

  Generalized Extreme Value distribution
        k = 0
    sigma = 2
       mu = 1

Compute the mean of the distribution.

m = mean(pd)
m =

    2.1544

See Also

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