HalfNormalDistribution

Half-normal probability distribution object

Description

A HalfNormalDistribution object consists of parameters, a model description, and sample data for a half-normal probability distribution.

The half-normal distribution is a special case of the folded normal and truncated normal distribution. Applications of the half-normal distribution include modeling measurement data and lifetime data.

The half-normal distribution uses the following parameters:

Parameter	Description	Support
mu	Location	$$-\infty <\mu <\infty$$
sigma	Scale	$$\sigma \ge 0$$

For more information about the half-normal distribution, see Half-Normal Distribution.

Creation

There are several ways to create a HalfNormalDistribution 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 — Location parameter

scalar value

Location parameter of the half-normal distribution, specified as a scalar value. The mu parameter is also the lower limit of the half-normal distribution.

The Statistics and Machine Learning Toolbox™ implementation of the half-normal distribution assumes a fixed value for the location parameter μ. You can specify a value for the μ parameter when creating a HalfNormalDistribution object.

Data Types: single | double

sigma — Scale parameter

nonnegative scalar value

Scale parameter of the half-normal distribution, specified as a nonnegative 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 Half-Normal Distribution Object Using Default Parameters

pd = makedist('HalfNormal')

pd = 
  HalfNormalDistribution

  Half Normal distribution
       mu = 0
    sigma = 1

Create a Half-Normal Distribution Object Using Specified Parameters

Create a half-normal distribution object. Specify mu equal to 0 and sigma equal to 1.5.

pd = makedist('HalfNormal','mu',0,'sigma',1.5)

pd = 
  HalfNormalDistribution

  Half Normal distribution
       mu =   0
    sigma = 1.5

Compute the mean and standard deviation of the distribution.

m = mean(pd)

m = 1.1968

s = std(pd)

s = 0.9042

Fit a Half-Normal Distribution Object

Generate 100 random numbers from a standard normal distribution and compute their absolute value.

rng default  % For reproducibility
x = abs(random(makedist('Normal'),100,1));

Fit a half-normal distribution object to the sample data.

pd = fitdist(x,'HalfNormal')

pd = 
  HalfNormalDistribution

  Half Normal distribution
       mu =      0
    sigma = 1.1631   [1.02184, 1.35006]

Calculate the mean of the fitted half-normal distribution using the probability distribution object.

m = mean(pd)

m = 0.9280

Calculate the mean of the half-normal distribution by substituting the fitted mu and sigma parameter values into the formula

$$mean=\mu +\sigma \sqrt{\frac{2}{\pi }}.$$

mcalc = pd.mu + pd.sigma*(sqrt(2/pi))

mcalc = 0.9280

References

[1] Cooray, K. and M.M.A. Ananda. “A Generalization of the Half-Normal Distribution with Applications to Lifetime Data.” Communications in Statistics – Theory and Methods. Vol. 37, Number 9, 2008, pp. 1323–1337.

[2] Pewsey, A. Large-Sample Inference for the General Half-Normal Distribution. Communications in Statistics – Theory and Methods. Vol. 31, Number 7, 2002, pp. 1045–1054.