Statistics: HalfNormalDistribution

Class Definition: `HalfNormalDistribution`

statistics: HalfNormalDistribution

Half-normal probability distribution object.

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

The half-normal distribution is a continuous probability distribution that models the time to failure of materials subjected to cyclic loading. It is defined by location parameter mu and scale parameter sigma.

There are several ways to create a HalfNormalDistribution object.

Fit a distribution to data using the fitdist function.
Create a distribution with fixed parameter values using the makedist function.
Use the constructor HalfNormalDistribution (mu, sigma) to create a half-normal distribution with fixed parameter values mu and sigma.
Use the static method HalfNormalDistribution.fit (x, mu, freq) to fit a distribution to the data in x using the same input arguments as the hnfit function.

It is highly recommended to use fitdist and makedist functions to create probability distribution objects, instead of the class constructor or the aforementioned static method.

Further information about the half-normal distribution can be found at https://en.wikipedia.org/wiki/Half-normal_distribution

See also: fitdist, makedist, hncdf, hninv, hnpdf, hnrnd, hnfit, hnlike, hnstat

Source Code: HalfNormalDistribution

The HalfNormalDistribution class contains the following properties:

A scalar value characterizing the location of the half-normal distribution. You can access the mu property using dot name assignment.

Example: 1

Create a Half-normal distribution with default parameters

 data = abs (randn (10000, 1));
 pd = fitdist (data, "HalfNormal");

Query parameter 'mu' (location parameter)

 pd.mu

ans = 0

Set parameter 'mu'

 pd.mu = 1

pd =
  HalfNormalDistribution

  Half-normal distribution
       mu =       1
    sigma = 1.00214

Use this to initialize or modify the location parameter of a Half-normal distribution. The location parameter must be a real scalar, often set to 0 for modeling positive deviations from zero, such as measurement errors.

Example: 2

Create a Half-normal distribution object by calling its constructor

 pd = HalfNormalDistribution (1.5, 2)

pd =
  HalfNormalDistribution

  Half-normal distribution
       mu = 1.5
    sigma =   2

Query parameter 'mu'

 pd.mu

ans = 1.5000

This demonstrates direct construction with a specific location parameter, useful for modeling data shifted from zero, like distances or folded normals.

A positive scalar value characterizing the scale of the half-normal distribution. You can access the sigma property using dot name assignment.

Example: 1

Create a Half-normal distribution with default parameters

 data = abs (randn (10000, 1));
 pd = fitdist (data, "HalfNormal");

Query parameter 'sigma' (scale parameter)

 pd.sigma

ans = 1.0006

Set parameter 'sigma'

 pd.sigma = 2

pd =
  HalfNormalDistribution

  Half-normal distribution
       mu = 0
    sigma = 2

Use this to initialize or modify the scale parameter in a Half-normal distribution. The scale parameter must be a positive real scalar, controlling the spread of the distribution.

Example: 2

Create a Half-normal distribution object by calling its constructor

 pd = HalfNormalDistribution (0, 1.5)

pd =
  HalfNormalDistribution

  Half-normal distribution
       mu =   0
    sigma = 1.5

Query parameter 'sigma'

 pd.sigma

ans = 1.5000

This shows how to set the scale parameter directly via the constructor, ideal for modeling variability in positive data, such as absolute residuals.

A character vector specifying the name of the probability distribution object. This property is read-only.

A scalar integer value specifying the number of parameters characterizing the probability distribution. This property is read-only.

A $2×1$ cell array of character vectors with each element containing the name of a distribution parameter. This property is read-only.

A $2×1$ cell array of character vectors with each element containing a short description of a distribution parameter. This property is read-only.

A $2×1$ numeric vector containing the values of the distribution parameters. This property is read-only. You can change the distribution parameters by assigning new values to the mu and sigma properties.

A $2×2$ numeric matrix containing the variance-covariance of the parameter estimates. Diagonal elements contain the variance of each estimated parameter, and non-diagonal elements contain the covariance between the parameter estimates. The covariance matrix is only meaningful when the distribution was fitted to data. If the distribution object was created with fixed parameters, or a parameter of a fitted distribution is modified, then all elements of the variance-covariance are zero. This property is read-only.

A $1×2$ logical vector specifying which parameters are fixed and which are estimated. true values correspond to fixed parameters, false values correspond to parameter estimates. This property is read-only.

A $1×2$ numeric vector specifying the truncation interval for the probability distribution. First element contains the lower boundary, second element contains the upper boundary. This property is read-only. You can only truncate a probability distribution with the truncate method.

A logical scalar value specifying whether a probability distribution is truncated or not. This property is read-only.

A scalar structure containing the following fields:

data: a numeric vector containing the data used for distribution fitting.
cens: a numeric vector of logical values indicating censoring information corresponding to the elements of the data used for distribution fitting. If no censoring vector was used for distribution fitting, then this field defaults to an empty array.
freq: a numeric vector of non-negative integer values containing the frequency information corresponding to the elements of the data used for distribution fitting. If no frequency vector was used for distribution fitting, then this field defaults to an empty array.

The HalfNormalDistribution class offers the following public methods:

HalfNormalDistribution: p = cdf (pd, x)
HalfNormalDistribution: p = cdf (pd, x, 'upper')

p = cdf (pd, x) computes the CDF of the probability distribution object, pd, evaluated at the values in x.

p = cdf (…, 'upper') returns the complement of the CDF of the probability distribution object, pd, evaluated at the values in x.

Example: 1

Plot various CDFs from the Half-normal distribution

 x = -1:0.01:5;
 data1 = 0.5 * abs (randn (10000, 1));
 data2 = 1.0 * abs (randn (10000, 1));
 data3 = 2.0 * abs (randn (10000, 1));
 pd1 = fitdist (data1, "HalfNormal");
 pd2 = fitdist (data2, "HalfNormal");
 pd3 = fitdist (data3, "HalfNormal");
 p1 = cdf (pd1, x);
 p2 = cdf (pd2, x);
 p3 = cdf (pd3, x);
 plot (x, p1, "-b", x, p2, "-g", x, p3, "-r")
 grid on
 legend ({"mu = 0, sigma = 0.5", "mu = 0, sigma = 1", "mu = 0, sigma = 2"}, ...
         "location", "southeast")
 title ("Half-normal CDF")
 xlabel ("values in x (x >= mu)")
 ylabel ("Cumulative probability")

Use this to compute and visualize the cumulative distribution function for different Half-normal distributions, showing how probability accumulates for positive values, useful in reliability or error modeling.

HalfNormalDistribution: x = icdf (pd, p)

x = icdf (pd, p) computes the quantile (the inverse of the CDF) of the probability distribution object, pd, evaluated at the values in p.

Example: 1

Plot various iCDFs from the Half-normal distribution

 p = 0.001:0.001:0.999;
 data1 = 0.5 * abs (randn (10000, 1));
 data2 = 1.0 * abs (randn (10000, 1));
 data3 = 2.0 * abs (randn (10000, 1));
 pd1 = fitdist (data1, "HalfNormal");
 pd2 = fitdist (data2, "HalfNormal");
 pd3 = fitdist (data3, "HalfNormal");
 x1 = icdf (pd1, p);
 x2 = icdf (pd2, p);
 x3 = icdf (pd3, p);
 plot (p, x1, "-b", p, x2, "-g", p, x3, "-r")
 grid on
 legend ({"mu = 0, sigma = 0.5", "mu = 0, sigma = 1", "mu = 0, sigma = 2"}, ...
         "location", "northwest")
 title ("Half-normal iCDF")
 xlabel ("Probability")
 ylabel ("values in x (x >= mu)")

This demonstrates the inverse CDF (quantiles) for Half-normal distributions, useful for finding values corresponding to given probabilities, such as thresholds in quality control.

HalfNormalDistribution: r = iqr (pd)

r = iqr (pd) computes the interquartile range of the probability distribution object, pd.

Example: 1

Compute the interquartile range for a Half-normal distribution

 data = abs (randn (10000, 1));
 pd = fitdist (data, "HalfNormal");
 iqr_value = iqr (pd)

iqr_value = 0.8336

Use this to calculate the interquartile range, which measures the spread of the middle 50% of the distribution, helpful for understanding central variability in positive data.

HalfNormalDistribution: m = mean (pd)

m = mean (pd) computes the mean of the probability distribution object, pd.

Example: 1

Compute the mean for different Half-normal distributions

 data1 = 0.5 * abs (randn (10000, 1));
 data2 = 1.0 * abs (randn (10000, 1));
 pd1 = fitdist (data1, "HalfNormal");
 pd2 = fitdist (data2, "HalfNormal");
 mean1 = mean (pd1)

mean1 = 0.3956

 mean2 = mean (pd2)

mean2 = 0.8034

This shows how to compute the expected value for Half-normal distributions with different scale parameters, representing average deviation or magnitude.

HalfNormalDistribution: m = median (pd)

m = median (pd) computes the median of the probability distribution object, pd.

Example: 1

Compute the median for different Half-normal distributions

 data1 = 0.5 * abs (randn (10000, 1));
 data2 = 1.0 * abs (randn (10000, 1));
 pd1 = fitdist (data1, "HalfNormal");
 pd2 = fitdist (data2, "HalfNormal");
 median1 = median (pd1)

median1 = 0.3343

 median2 = median (pd2)

median2 = 0.6733

Use this to find the median value, which splits the distribution into two equal probability halves, robust to skewness in positive data.

HalfNormalDistribution: nlogL = negloglik (pd)

nlogL = negloglik (pd) computes the negative loglikelihood of the probability distribution object, pd.

Example: 1

Compute the negative loglikelihood for a fitted Half-normal distribution

 rand ("seed", 21);
 data = abs (randn (100, 1));
 pd_fitted = fitdist (data, "HalfNormal");
 params = [pd_fitted.mu, pd_fitted.sigma];
 nlogL_hnlike = hnlike (params, data)

nlogL_hnlike = 72.265

This is useful for assessing the fit of a Half-normal distribution to data, with lower values indicating a better fit, often used in model comparison.

HalfNormalDistribution: ci = paramci (pd)
HalfNormalDistribution: ci = paramci (pd, Name, Value)

ci = paramci (pd) computes the lower and upper boundaries of the 95% confidence interval for each parameter of the probability distribution object, pd.

ci = paramci (pd, Name, Value) computes the confidence intervals with additional options specified by Name-Value pair arguments listed below.

`Name`		`Value`
`'Alpha'`		A scalar value in the range $(0,1)$ specifying the significance level for the confidence interval. The default value 0.05 corresponds to a 95% confidence interval.
`'Parameter'`		A character vector or a cell array of character vectors specifying the parameter names for which to compute confidence intervals. By default, `paramci` computes confidence intervals for all distribution parameters.

paramci is meaningful only when pd is fitted to data, otherwise an empty array, [], is returned.

Example: 1

Compute confidence intervals for parameters of a fitted Half-normal distribution

 rand ("seed", 21);
 data = abs (randn (1000, 1));
 pd_fitted = fitdist (data, "HalfNormal");
 ci = paramci (pd_fitted, "Alpha", 0.05)

Use this to obtain confidence intervals for the estimated parameters (mu fixed, sigma estimated), providing a range of plausible values given the data.

HalfNormalDistribution: y = pdf (pd, x)

y = pdf (pd, x) computes the PDF of the probability distribution object, pd, evaluated at the values in x.

Example: 1

Plot various PDFs from the Half-normal distribution

 x = -1:0.01:5;
 data1 = 0.5 * abs (randn (10000, 1));
 data2 = 1.0 * abs (randn (10000, 1));
 data3 = 2.0 * abs (randn (10000, 1));
 pd1 = fitdist (data1, "HalfNormal");
 pd2 = fitdist (data2, "HalfNormal");
 pd3 = fitdist (data3, "HalfNormal");
 y1 = pdf (pd1, x);
 y2 = pdf (pd2, x);
 y3 = pdf (pd3, x);
 plot (x, y1, "-b", x, y2, "-g", x, y3, "-r")
 grid on
 legend ({"mu = 0, sigma = 0.5", "mu = 0, sigma = 1", "mu = 0, sigma = 2"}, ...
         "location", "northeast")
 title ("Half-normal PDF")
 xlabel ("values in x (x >= mu)")
 ylabel ("Probability density")

This visualizes the probability density function for Half-normal distributions, showing the likelihood for positive values.

HalfNormalDistribution: plot (pd)
HalfNormalDistribution: plot (pd, Name, Value)
HalfNormalDistribution: h = plot (…)

plot (pd) plots a probability density function (PDF) of the probability distribution object pd. If pd contains data, which have been fitted by fitdist, the PDF is superimposed over a histogram of the data.

plot (pd, Name, Value) specifies additional options with the Name-Value pair arguments listed below.

`Name`		`Value`
`'PlotType'`		A character vector specifying the plot type. `'pdf'` plots the probability density function (PDF). When `pd` is fit to data, the PDF is superimposed on a histogram of the data. `'cdf'` plots the cumulative density function (CDF). When `pd` is fit to data, the CDF is superimposed over an empirical CDF. `'probability'` plots a probability plot using a CDF of the data and a CDF of the fitted probability distribution. This option is available only when `pd` is fitted to data.
`'Discrete'`		A logical scalar to specify whether to plot the PDF or CDF of a discrete distribution object as a line plot or a stem plot, by specifying `false` or `true`, respectively. By default, it is `true` for discrete distributions and `false` for continuous distributions. When `pd` is a continuous distribution object, option is ignored.
`'Parent'`		An axes graphics object for plot. If not specified, the `plot` function plots into the current axes or creates a new axes object if one does not exist.

h = plot (…) returns a graphics handle to the plotted objects.

Example: 1

Create a Half-normal distribution with fixed parameters mu = 0 and sigma = 1 and plot its PDF.

 data = abs (randn (10000, 1));
 pd = fitdist (data, "HalfNormal");
 plot (pd)
 title ("Fixed Half-normal distribution with mu = 0 and sigma = 1")

Example: 2

Generate a data set of 100 random samples from a Half-normal distribution with parameters mu = 0 and sigma = 1. Fit a Half-normal distribution to this data and plot its CDF superimposed over an empirical CDF.

 rand ("seed", 21);
 data = abs (randn (100, 1));
 pd_fitted = fitdist (data, "HalfNormal");
 plot (pd_fitted, "PlotType", "cdf")
 txt = "Fitted Half-normal distribution with mu = %0.2f and sigma = %0.2f";
 title (sprintf (txt, pd_fitted.mu, pd_fitted.sigma))
 legend ({"empirical CDF", "fitted CDF"}, "location", "southeast")

Use this to visualize the fitted CDF compared to the empirical CDF of the data, useful for assessing model fit.

Example: 3

Generate a data set of 200 random samples from a Half-normal distribution with parameters mu = 0 and sigma = 1. Display a probability plot for the Half-normal distribution fit to the data.

 rand ("seed", 21);
 data = abs (randn (200, 1));
 pd_fitted = fitdist (data, "HalfNormal");
 plot (pd_fitted, "PlotType", "probability")
 txt = strcat ("Probability plot of fitted Half-normal", ...
               " distribution with mu = %0.2f and sigma = %0.2f");
 title (sprintf (txt, pd_fitted.mu, pd_fitted.sigma))
 legend ({"empirical CDF", "fitted CDF"}, "location", "southeast")

This creates a probability plot to compare the fitted distribution to the data, useful for checking if the Half-normal model is appropriate.

HalfNormalDistribution: [nlogL, param] = proflik (pd, pnum)
HalfNormalDistribution: [nlogL, param] = proflik (pd, pnum, 'Display', display)
HalfNormalDistribution: [nlogL, param] = proflik (pd, pnum, setparam)
HalfNormalDistribution: [nlogL, param] = proflik (pd, pnum, setparam, 'Display', display)

[nlogL, param] = proflik (pd, pnum) returns a vector nlogL of negative loglikelihood values and a vector param of corresponding parameter values for the parameter in the position indicated by pnum. By default, proflik uses the lower and upper bounds of the 95% confidence interval and computes 100 equispaced values for the selected parameter. pd must be fitted to data.

[nlogL, param] = proflik (pd, pnum, 'Display', 'on') also plots the profile likelihood against the default range of the selected parameter.

[nlogL, param] = proflik (pd, pnum, setparam) defines a user-defined range of the selected parameter.

[nlogL, param] = proflik (pd, pnum, setparam, 'Display', 'on') also plots the profile likelihood against the user-defined range of the selected parameter.

For the Half-normal distribution, pnum = 1 selects the parameter mu and pnum = 2 selects the parameter sigma.

When opted to display the profile likelihood plot, proflik also plots the baseline loglikelihood computed at the lower bound of the 95% confidence interval and estimated maximum likelihood. The latter might not be observable if it is outside of the used-defined range of parameter values.

Example: 1

Compute and plot the profile likelihood for the scale parameter of a fitted Half-normal distribution

 rand ("seed", 21);
 data = abs (randn (1000, 1));
 pd_fitted = fitdist (data, "HalfNormal");
 [nlogL, param] = proflik (pd_fitted, 2, "Display", "on");

Use this to analyze the profile likelihood of the scale parameter (sigma), helping to understand the uncertainty in parameter estimates.

HalfNormalDistribution: r = random (pd)
HalfNormalDistribution: r = random (pd, rows)
HalfNormalDistribution: r = random (pd, rows, cols, …)
HalfNormalDistribution: r = random (pd, [sz])

r = random (pd) returns a random number from the distribution object pd.

When called with a single size argument, hnrnd returns a square matrix with the dimension specified. When called with more than one scalar argument, the first two arguments are taken as the number of rows and columns and any further arguments specify additional matrix dimensions. The size may also be specified with a row vector of dimensions, sz.

Example: 1

Generate random samples from a Half-normal distribution

 rand ("seed", 21);
 samples = abs (randn (500, 1));
 hist (samples, 50)
 title ("Histogram of 500 random samples from Half-normal(mu=0, sigma=1)")
 xlabel ("values in x (x >= mu)")
 ylabel ("Frequency")

This generates random samples from a Half-normal distribution, useful for simulating positive data like absolute errors or magnitudes.

HalfNormalDistribution: s = std (pd)

s = std (pd) computes the standard deviation of the probability distribution object, pd.

Example: 1

Compute the standard deviation for a Half-normal distribution

 data = abs (randn (10000, 1));
 pd = fitdist (data, "HalfNormal");
 std_value = std (pd)

std_value = 0.6062

Use this to calculate the standard deviation, which measures the variability in the positive values of the distribution.

HalfNormalDistribution: t = truncate (pd, lower, upper)

t = truncate (pd, lower, upper) returns a probability distribution t, which is the probability distribution pd truncated to the specified interval with lower limit, lower, and upper limit, upper. If pd is fitted to data with fitdist, the returned probability distribution t is not fitted, does not contain any data or estimated values, and it is as it has been created with the makedist function, but it includes the truncation interval.

Example: 1

Plot the PDF of a Half-normal distribution, with parameters mu = 0 and sigma = 1, truncated at [0.5, 2] intervals. Generate 10000 random samples from this truncated distribution and superimpose a histogram scaled accordingly

 rand ("seed", 21);
 data_all = abs (randn (20000, 1));
 data = data_all(data_all >= 0.5 & data_all <= 2);
 data = data(1:10000);
 pd = fitdist (data, "HalfNormal");
 t = truncate (pd, 0.5, 2);

Plot histogram and truncated PDF

 plot (t)
 hold on
 hist (data, 50)
 hold off
 title ("Half-normal distribution (mu = 0, sigma = 1) truncated at [0.5, 2]")
 legend ("Truncated PDF", "Histogram")

This demonstrates truncating a Half-normal distribution to a specific range and visualizing the resulting distribution with random samples.

HalfNormalDistribution: v = var (pd)

v = var (pd) computes the variance of the probability distribution object, pd.

Example: 1

Compute the variance for a Half-normal distribution

 data = abs (randn (10000, 1));
 pd = fitdist (data, "HalfNormal");
 var_value = var (pd)

var_value = 0.3652

Use this to calculate the variance, which quantifies the spread of the positive values in the distribution.

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Class Definition: HalfNormalDistribution

statistics: HalfNormalDistribution

mu

Example: 1

Example: 2

sigma

Example: 1

Example: 2

DistributionName

NumParameters

ParameterNames

ParameterDescription

ParameterValues

ParameterCovariance

ParameterIsFixed

Truncation

IsTruncated

InputData

cdf

HalfNormalDistribution: p = cdf (pd, x)

HalfNormalDistribution: p = cdf (pd, x, 'upper')

Example: 1

icdf

HalfNormalDistribution: x = icdf (pd, p)

Example: 1

iqr

HalfNormalDistribution: r = iqr (pd)

Example: 1

mean

HalfNormalDistribution: m = mean (pd)

Example: 1

median

HalfNormalDistribution: m = median (pd)

Example: 1

negloglik

HalfNormalDistribution: nlogL = negloglik (pd)

Example: 1

paramci

Class Definition: `HalfNormalDistribution`

`mu`

`sigma`

`DistributionName`

`NumParameters`

`ParameterNames`

`ParameterDescription`

`ParameterValues`

`ParameterCovariance`

`ParameterIsFixed`

`Truncation`

`IsTruncated`

`InputData`

`cdf`

`HalfNormalDistribution: p = cdf (pd, x)`

`HalfNormalDistribution: p = cdf (pd, x, 'upper')`

`icdf`

`HalfNormalDistribution: x = icdf (pd, p)`

`iqr`

`HalfNormalDistribution: r = iqr (pd)`

`mean`

`HalfNormalDistribution: m = mean (pd)`

`median`

`HalfNormalDistribution: m = median (pd)`

`negloglik`

`HalfNormalDistribution: nlogL = negloglik (pd)`

`paramci`

`HalfNormalDistribution: ci = paramci (pd)`

`HalfNormalDistribution: ci = paramci (pd, Name, Value)`

`pdf`

`HalfNormalDistribution: y = pdf (pd, x)`

`plot`

`HalfNormalDistribution: plot (pd)`

`HalfNormalDistribution: plot (pd, Name, Value)`

`HalfNormalDistribution: h = plot (…)`

`proflik`

`HalfNormalDistribution: [nlogL, param] = proflik (pd, pnum)`

`HalfNormalDistribution: [nlogL, param] = proflik (pd, pnum, 'Display', display)`

`HalfNormalDistribution: [nlogL, param] = proflik (pd, pnum, setparam)`

`HalfNormalDistribution: [nlogL, param] = proflik (pd, pnum, setparam, 'Display', display)`

`random`

`HalfNormalDistribution: r = random (pd)`

`HalfNormalDistribution: r = random (pd, rows)`

`HalfNormalDistribution: r = random (pd, rows, cols, …)`

`HalfNormalDistribution: r = random (pd, [sz])`

`std`

`HalfNormalDistribution: s = std (pd)`

`truncate`

`HalfNormalDistribution: t = truncate (pd, lower, upper)`

`var`

`HalfNormalDistribution: v = var (pd)`