LoglogisticDistribution

Loglogistic probability distribution object

Description

A LoglogisticDistribution object consists of parameters, a model description, and sample data for a loglogistic probability 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.

Parameter	Description	Support
mu	Mean of logarithmic values	$$\mu >0$$
sigma	Scale parameter of logarithmic values	$$\sigma >0$$

Creation

There are several ways to create a LoglogisticDistribution 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

mu — Mean of logarithmic values

positive scalar value

Mean of logarithmic values for the loglogistic distribution, specified as a positive scalar value.

Data Types: single | double

sigma — Scale parameter of logarithmic values

positive scalar value

Scale parameter of logarithmic values for the loglogistic distribution, specified as a positive 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.

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 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
icdf	Inverse cumulative distribution function
iqr	Interquartile range
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
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 Loglogistic Distribution Object Using Default Parameters

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 Using Specified Parameters

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.