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prob.LoglogisticDistribution class

Package: prob
Superclasses: prob.ToolboxFittableParametricDistribution

Loglogistic probability distribution object

Description

prob.LoglogisticDistribution is an object consisting of parameters, a model description, and sample data for a loglogistic 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('Loglogistic') creates a loglogistic probability distribution object using the default parameter values.

pd = makedist('Loglogistic','mu',mu,'sigma',sigma) creates a loglogistic probability distribution object using the specified parameter values .

Input Arguments

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Log mean for the loglogistic distribution, specified as a positive scalar value.

Data Types: single | double

Log scale parameter for the loglogistic distribution, specified as a positive scalar value.

Data Types: single | double

Properties

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Log mean for the loglogistic distribution, stored as a positive scalar value.

Data Types: single | double

Log scale parameter for the loglogistic distribution, stored as a positive scalar value.

Data Types: single | double

Probability distribution name, stored as a character vector. This property is read-only.

Data Types: char

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

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

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

Data Types: single | double

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

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

Data Types: char

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

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

Data Types: char

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

Data Types: single | double

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 log likelihood 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

Loglogistic Distribution

The loglogistic distribution is closely related to the logistic distribution. If x is distributed loglogistically with parameters μ and σ, then log(x) is distributed logistically with mean and standard deviation. This distribution is often used in survival analysis to model events that experience an initial rate increase, followed by a rate decrease.

The loglogistic distribution uses the following parameters.

ParameterDescriptionSupport
muLog meanμ>0
sigmaLog scale parameterσ>0

The probability density function (pdf) is

f(x|μ,σ)=1σ1xez(1+ez)2;x0,

where z=log(x)μσ.

Examples

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Create a loglogistic distribution object using the default parameter values.

pd = makedist('Loglogistic')
pd = 

  LoglogisticDistribution

  Log-Logistic distribution
       mu = 0
    sigma = 1

Create a loglogistic distribution object by specifying the parameter values.

pd = makedist('Loglogistic','mu',5,'sigma',2)
pd = 

  LoglogisticDistribution

  Log-Logistic distribution
       mu = 5
    sigma = 2

Generate random numbers from the loglogistic distribution and compute their log values.

rng(19) % for reproducibility
x = random(pd,10000,1);
logx = log(x);

Compute the mean of the log values.

m = mean(logx)
m =

    4.9828

The mean of the log of x is equal to the mu parameter of x, since x has a loglogistic distribution.

Plot logx.

histogram(logx,50)

The plot shows that the log values of x have a logistic distribution.

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