GeneralizedExtremeValueDistribution

Generalized extreme value probability distribution object

Description

A GeneralizedExtremeValueDistribution object consists of parameters, a model description, and sample data for a generalized extreme value probability 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.

Parameter	Description	Support
k	Shape parameter	$$-\infty \le k\le \infty$$
sigma	Scale parameter	$$\sigma \ge 0$$
mu	Location parameter	$$-\infty \le \mu \le \infty$$

Creation

There are several ways to create a GeneralizedExtremeValueDistribution probability distribution object.

• Create a distribution with specified parameter values using makedist.

• Fit a distribution to data using fitdist.

• Interactively fit a distribution to data using the Distribution Fitter app.

Properties

Distribution Parameters

k — Shape parameter

scalar value

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

Data Types: single | double

sigma — Scale parameter

nonnegative scalar value

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

Data Types: single | double

mu — Location parameter

scalar value

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

Data Types: single | double

Distribution Characteristics

IsTruncated — Logical flag for truncated distribution

0 | 1

This property is read-only.

Logical flag for truncated distribution, specified as a logical value. If IsTruncated equals 0, the distribution is not truncated. If IsTruncated equals 1, the distribution is truncated.

Data Types: logical

NumParameters — Number of parameters

positive integer value

This property is read-only.

Number of parameters for the probability distribution, specified as a positive integer value.

Data Types: double

ParameterCovariance — Covariance matrix of the parameter estimates

matrix of scalar values

This property is read-only.

Covariance matrix of the parameter estimates, specified 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.

Data Types: double

ParameterIsFixed — Logical flag for fixed parameters

array of logical values

This property is read-only.

Logical flag for fixed parameters, specified 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.

Data Types: logical

ParameterValues — Distribution parameter values

vector of scalar values

This property is read-only.

Distribution parameter values, specified as a vector of scalar values.

Data Types: single | double

Truncation — Truncation interval

vector of scalar values

This property is read-only.

Truncation interval for the probability distribution, specified as a vector of scalar values containing the lower and upper truncation boundaries.

Data Types: single | double

Other Object Properties

DistributionName — Probability distribution name

character vector

This property is read-only.

Probability distribution name, specified as a character vector.

Data Types: char

InputData — Data used for distribution fitting

structure

This property is read-only.

Data used for distribution fitting, specified 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.

Data Types: struct

ParameterDescription — Distribution parameter descriptions

cell array of character vectors

This property is read-only.

Distribution parameter descriptions, specified as a cell array of character vectors. Each cell contains a short description of one distribution parameter.

Data Types: char

ParameterNames — Distribution parameter names

cell array of character vectors

This property is read-only.

Distribution parameter names, specified as a cell array of character vectors.

Data Types: char

Object Functions

cdf	Cumulative distribution function
gather	Gather properties of Statistics and Machine Learning Toolbox object from GPU
icdf	Inverse cumulative distribution function
iqr	Interquartile range of probability distribution
mean	Mean of probability distribution
median	Median of probability distribution
negloglik	Negative loglikelihood of probability distribution
paramci	Confidence intervals for probability distribution parameters
pdf	Probability density function
plot	Plot probability distribution object
proflik	Profile likelihood function for probability distribution
random	Random numbers
std	Standard deviation of probability distribution
truncate	Truncate probability distribution object
var	Variance of probability distribution

Examples

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

Extended Capabilities

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• GeneralizedExtremeValueDistribution can be a probability distribution object fitted by using fitdist with GPU array input arguments.

• The object functions of GeneralizedExtremeValueDistribution fully support GPU arrays.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced in R2013a