Half-Normal Distribution

Overview

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

Parameters

The half-normal distribution uses the following parameters:

Parameter	Description
$- \infty < μ < \infty$	Location parameter
$σ \geq 0$	Scale parameter

The support for the half-normal distribution isx≥μ.

Usemakedistwith specified parameter values to create a half-normal probability distribution objectHalfNormalDistribution. Usefitdistto fit a half-normal probability distribution object to sample data. Usemleto estimate the half-normal distribution parameter values from sample data without creating a probability distribution object. For more information about working with probability distributions, seeWorking with Probability Distributions.

The Statistics and Machine Learning Toolbox™ implementation of the half-normal distribution assumes a fixed value for the location parameterμ. Therefore, neitherfitdistnormleestimates the value of the parameterμwhen fitting a half-normal distribution to sample data. You can specify a value for theμparameter by using the name-value pair argument'mu'. The default value for the'mu'argument is 0 in bothfitdistandmle.

Probability Density Function

The probability density function (pdf) of the half-normal distribution is

$y = f (x | μ, σ) = \sqrt{\frac{2}{π}} \frac{1}{σ} e^{- \frac{1}{2} {(\frac{x - μ}{σ})}^{2}}; x \geq μ,$

whereμis the location parameter andσis the scale parameter. Ifx≤μ, then the pdf is undefined.

To compute the pdf of the half-normal distribution, create aHalfNormalDistributionprobability distribution object usingfitdistormakedist, then use thepdfmethod to work with the object.

PDF of Half-Normal Probability Distribution

Open Live Script

This example shows how changing the values of themuandsigmaparameters alters the shape of the pdf.

Create four probability distribution objects with different parameters.

pd1 = makedist('HalfNormal'); pd2 = makedist('HalfNormal','mu',0,'sigma',2); pd3 = makedist('HalfNormal','mu',0,'sigma',3); pd4 = makedist('HalfNormal','mu',0,'sigma',5);

Compute the probability density functions (pdfs) of each distribution.

x = 0:0.1:10; pdf1 = pdf(pd1,x); pdf2 = pdf(pd2,x); pdf3 = pdf(pd3,x); pdf4 = pdf(pd4,x);

Plot the pdfs on the same figure.

figure; plot(x,pdf1,'r','LineWidth',2) holdon; plot(x,pdf2,'k:','LineWidth',2); plot(x,pdf3,'b-.','LineWidth',2); plot(x,pdf4,'g--','LineWidth',2); legend({'mu = 0, sigma = 1','mu = 0, sigma = 2',...'mu = 0, sigma = 3','mu = 0, sigma = 5'},'Location','NE'); holdoff;

Assigmaincreases, the curve flattens and the peak value becomes smaller.

Cumulative Distribution Function

The cumulative distribution function (cdf) of the half-normal distribution is

$y = F (x) = e r f (\frac{x - μ}{\sqrt{2} σ}) = 2 Φ (\frac{x - μ}{σ}) - 1; x \geq μ,$

whereμis the location parameter,σis the scale parameter,erf(•)is the error function, andΦ(•)is the cdf of the standard normal distribution. Ifx≤μ, then the cdf is undefined.

To compute the cdf of the half-normal distribution, create aHalfNormalDistributionprobability distribution object usingfitdistormakedist, then use thecdfmethod to work with the object.

CDF of Half-Normal Probability Distribution

Open Live Script

This example shows how changing the values of themuandsigmaparameters alters the shape of the cdf.

Create four probability distribution objects with different parameters.

pd1 = makedist('HalfNormal'); pd2 = makedist('HalfNormal','mu',0,'sigma',2); pd3 = makedist('HalfNormal','mu',0,'sigma',3); pd4 = makedist('HalfNormal','mu',0,'sigma',5);

计算(cdf的累积分布函数s) for each probability distribution.

x = 0:0.1:10; cdf1 = cdf(pd1,x); cdf2 = cdf(pd2,x); cdf3 = cdf(pd3,x); cdf4 = cdf(pd4,x);

Plot all four cdfs on the same figure.

figure; plot(x,cdf1,'r','LineWidth',2) holdon; plot(x,cdf2,'k:','LineWidth',2); plot(x,cdf3,'b-.','LineWidth',2); plot(x,cdf4,'g--','LineWidth',2); legend({'mu = 0, sigma = 1','mu = 0, sigma = 2',...'mu = 0, sigma = 3','mu = 0, sigma = 5'},'Location','SE'); holdoff;

Assigmaincreases, the curve of the cdf flattens.

Descriptive Statistics

The mean of the half-normal distribution is

$mean = μ + σ \sqrt{\frac{2}{π},}$

whereμis the location parameter andσis the scale parameter.

The variance of the half-normal distribution is

$var = σ^{2} (1 - \frac{2}{π}),$

whereσis the scale parameter.

Relationship to Other Distributions

If a random variableZhas a standard normal distribution with a meanμequal to zero and standard deviationσequal to one, then $X = μ + σ | Z |$ has a half-normal distribution with parametersμandσ.

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.