fitdist

Fit probability distribution object to data

Syntax

pd = fitdist(x,distname)

pd = fitdist(x,distname,Name,Value)

[pdca,gn,gl] = fitdist(x,distname,'By',groupvar)

[pdca,gn,gl] = fitdist(x,distname,'By',groupvar,Name,Value)

Description

pd = fitdist(x,distname) creates a probability distribution object by fitting the distribution specified by distname to the data in column vector x.

pd = fitdist(x,distname,Name,Value) creates the probability distribution object with additional options specified by one or more name-value pair arguments. For example, you can indicate censored data or specify control parameters for the iterative fitting algorithm.

[pdca,gn,gl] = fitdist(x,distname,'By',groupvar) creates probability distribution objects by fitting the distribution specified by distname to the data in x based on the grouping variable groupvar. It returns a cell array of fitted probability distribution objects, pdca, a cell array of group labels, gn, and a cell array of grouping variable levels, gl.

[pdca,gn,gl] = fitdist(x,distname,'By',groupvar,Name,Value) returns the above output arguments using additional options specified by one or more name-value pair arguments. For example, you can indicate censored data or specify control parameters for the iterative fitting algorithm.

Examples

Fit a Normal Distribution to Data

Load the sample data. Create a vector containing the patients' weight data.

load hospital
x = hospital.Weight;

Create a normal distribution object by fitting it to the data.

pd = fitdist(x,'Normal')

pd = 
  NormalDistribution

  Normal distribution
       mu =     154   [148.728, 159.272]
    sigma = 26.5714   [23.3299, 30.8674]

The intervals next to the parameter estimates are the 95% confidence intervals for the distribution parameters.

Plot the pdf of the distribution.

x_values = 50:1:250;
y = pdf(pd,x_values);
plot(x_values,y,'LineWidth',2)

Fit a Kernel Distribution to Data

Load the sample data. Create a vector containing the patients' weight data.

load hospital
x = hospital.Weight;

Create a kernel distribution object by fitting it to the data. Use the Epanechnikov kernel function.

pd = fitdist(x,'Kernel','Kernel','epanechnikov')

pd = 
  KernelDistribution

    Kernel = epanechnikov
    Bandwidth = 14.3792
    Support = unbounded

Plot the pdf of the distribution.

x_values = 50:1:250;
y = pdf(pd,x_values);
plot(x_values,y)

Fit Normal Distributions to Grouped Data

Load the sample data. Create a vector containing the patients' weight data.

load hospital
x = hospital.Weight;

Create normal distribution objects by fitting them to the data, grouped by patient gender.

gender = hospital.Sex;
[pdca,gn,gl] = fitdist(x,'Normal','By',gender)

pdca=1×2 cell array
    {1x1 prob.NormalDistribution}    {1x1 prob.NormalDistribution}

gn = 2x1 cell
    {'Female'}
    {'Male'  }

gl = 2x1 cell
    {'Female'}
    {'Male'  }

The cell array pdca contains two probability distribution objects, one for each gender group. The cell array gn contains two group labels. The cell array gl contains two group levels.

View each distribution in the cell array pdca to compare the mean, mu, and the standard deviation, sigma, grouped by patient gender.

female = pdca{1}  % Distribution for females

female = 
  NormalDistribution

  Normal distribution
       mu = 130.472   [128.183, 132.76]
    sigma = 8.30339   [6.96947, 10.2736]

male = pdca{2}  % Distribution for males

male = 
  NormalDistribution

  Normal distribution
       mu = 180.532   [177.833, 183.231]
    sigma = 9.19322   [7.63933, 11.5466]

Compute the pdf of each distribution.

x_values = 50:1:250;
femalepdf = pdf(female,x_values);
malepdf = pdf(male,x_values);

Plot the pdfs for a visual comparison of weight distribution by gender.

figure
plot(x_values,femalepdf,'LineWidth',2)
hold on
plot(x_values,malepdf,'Color','r','LineStyle',':','LineWidth',2)
legend(gn,'Location','NorthEast')
hold off

Fit Kernel Distributions to Grouped Data

Load the sample data. Create a vector containing the patients' weight data.

load hospital
x = hospital.Weight;

Create kernel distribution objects by fitting them to the data, grouped by patient gender. Use a triangular kernel function.

gender = hospital.Sex;
[pdca,gn,gl] = fitdist(x,'Kernel','By',gender,'Kernel','triangle');

View each distribution in the cell array pdca to see the kernel distributions for each gender.

female = pdca{1}  % Distribution for females

female = 
  KernelDistribution

    Kernel = triangle
    Bandwidth = 4.25894
    Support = unbounded

male = pdca{2}  % Distribution for males

male = 
  KernelDistribution

    Kernel = triangle
    Bandwidth = 5.08961
    Support = unbounded

Compute the pdf of each distribution.

x_values = 50:1:250;
femalepdf = pdf(female,x_values);
malepdf = pdf(male,x_values);

Plot the pdfs for a visual comparison of weight distribution by gender.

figure
plot(x_values,femalepdf,'LineWidth',2)
hold on
plot(x_values,malepdf,'Color','r','LineStyle',':','LineWidth',2)
legend(gn,'Location','NorthEast')
hold off

Input Arguments

x — Input data

column vector

Input data, specified as a column vector. fitdist ignores NaN values in x. Additionally, any NaN values in the censoring vector or frequency vector cause fitdist to ignore the corresponding values in x.

Data Types: double

distname — Distribution name

character vector | string scalar

Distribution name, specified as one of the following character vectors or string scalars. The distribution specified by distname determines the type of the returned probability distribution object.

Distribution Name	Description	Distribution Object
'Beta'	Beta distribution	BetaDistribution
'Binomial'	Binomial distribution	BinomialDistribution
'BirnbaumSaunders'	Birnbaum-Saunders distribution	BirnbaumSaundersDistribution
'Burr'	Burr distribution	BurrDistribution
'Exponential'	Exponential distribution	ExponentialDistribution
'ExtremeValue'	Extreme Value distribution	ExtremeValueDistribution
'Gamma'	Gamma distribution	GammaDistribution
'GeneralizedExtremeValue'	Generalized Extreme Value distribution	GeneralizedExtremeValueDistribution
'GeneralizedPareto'	Generalized Pareto distribution	GeneralizedParetoDistribution
'HalfNormal'	Half-normal distribution	HalfNormalDistribution
'InverseGaussian'	Inverse Gaussian distribution	InverseGaussianDistribution
'Kernel'	Kernel distribution	KernelDistribution
'Logistic'	Logistic distribution	LogisticDistribution
'Loglogistic'	Loglogistic distribution	LoglogisticDistribution
'Lognormal'	Lognormal distribution	LognormalDistribution
'Nakagami'	Nakagami distribution	NakagamiDistribution
'NegativeBinomial'	Negative Binomial distribution	NegativeBinomialDistribution
'Normal'	Normal distribution	NormalDistribution
'Poisson'	Poisson distribution	PoissonDistribution
'Rayleigh'	Rayleigh distribution	RayleighDistribution
'Rician'	Rician distribution	RicianDistribution
'Stable'	Stable distribution	StableDistribution
'tLocationScale'	t Location-Scale distribution	tLocationScaleDistribution
'Weibull'	Weibull distribution	WeibullDistribution

groupvar — Grouping variable

categorical array | logical or numeric vector | character array | string array | cell array of character vectors

Grouping variable, specified as a categorical array, logical or numeric vector, character array, string array, or cell array of character vectors. Each unique value in a grouping variable defines a group.

For example, if Gender is a cell array of character vectors with values 'Male' and 'Female', you can use Gender as a grouping variable to fit a distribution to your data by gender.

More than one grouping variable can be used by specifying a cell array of grouping variables. Observations are placed in the same group if they have common values of all specified grouping variables.

For example, if Smoker is a logical vector with values 0 for nonsmokers and 1 for smokers, then specifying the cell array {Gender,Smoker} divides observations into four groups: Male Smoker, Male Nonsmoker, Female Smoker, and Female Nonsmoker.

Example: {Gender,Smoker}

Data Types: categorical | logical | single | double | char | string | cell

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: fitdist(x,'Kernel','Kernel','triangle') fits a kernel distribution object to the data in x using a triangular kernel function.

'Censoring' — Logical flag for censored data

0 (default) | vector of logical values

Logical flag for censored data, specified as the comma-separated pair consisting of 'Censoring' and a vector of logical values that is the same size as input vector x. The value is 1 when the corresponding element in x is a right-censored observation and 0 when the corresponding element is an exact observation. The default is a vector of 0s, indicating that all observations are exact.

fitdist ignores any NaN values in this censoring vector. Additionally, any NaN values in x or the frequency vector cause fitdist to ignore the corresponding values in the censoring vector.

This argument is valid only if distname is 'BirnbaumSaunders', 'Burr', 'Exponential', 'ExtremeValue', 'Gamma', 'InverseGaussian', 'Kernel', 'Logistic', 'Loglogistic', 'Lognormal', 'Nakagami', 'Normal', 'Rician', 'tLocationScale', or 'Weibull'.

Data Types: logical

'Frequency' — Observation frequency

1 (default) | vector of nonnegative integer values

Observation frequency, specified as the comma-separated pair consisting of 'Frequency' and a vector of nonnegative integer values that is the same size as input vector x. Each element of the frequency vector specifies the frequencies for the corresponding elements in x. The default is a vector of 1s, indicating that each value in x only appears once.

fitdist ignores any NaN values in this frequency vector. Additionally, any NaN values in x or the censoring vector cause fitdist to ignore the corresponding values in the frequency vector.

Data Types: single | double

'Options' — Control parameters

structure

Control parameters for the iterative fitting algorithm, specified as the comma-separated pair consisting of 'Options' and a structure you create using statset.

Data Types: struct

'NTrials' — Number of trials

positive integer value

Number of trials for the binomial distribution, specified as the comma-separated pair consisting of 'NTrials' and a positive integer value. You must specify distname as 'Binomial' to use this option.

Data Types: single | double

'Theta' — Threshold parameter

0 (default) | scalar value

Threshold parameter for the generalized Pareto distribution, specified as the comma-separated pair consisting of 'Theta' and a scalar value. You must specify distname as 'GeneralizedPareto' to use this option.

Data Types: single | double

'mu' — Location parameter

0 (default) | scalar value

Location parameter for the half-normal distribution, specified as the comma-separated pair consisting of 'mu' and a scalar value. You must specify distname as 'HalfNormal' to use this option.

Data Types: single | double

'Kernel' — Kernel smoother type

'normal' (default) | 'box' | 'triangle' | 'epanechnikov'

Kernel smoother type, specified as the comma-separated pair consisting of 'Kernel' and one of the following:

• 'normal'

• 'box'

• 'triangle'

• 'epanechnikov'

You must specify distname as 'Kernel' to use this option.

'Support' — Kernel density support

'unbounded' (default) | 'positive' | two-element vector

Kernel density support, specified as the comma-separated pair consisting of 'Support' and 'unbounded', 'positive', or a two-element vector.

'unbounded'	Density can extend over the whole real line.
'positive'	Density is restricted to positive values.

Alternatively, you can specify a two-element vector giving finite lower and upper limits for the support of the density.

You must specify distname as 'Kernel' to use this option.

Data Types: single | double | char | string

'Width' — Bandwidth of kernel smoothing window

scalar value

Bandwidth of the kernel smoothing window, specified as the comma-separated pair consisting of 'Width' and a scalar value. The default value used by fitdist is optimal for estimating normal densities, but you might want to choose a smaller value to reveal features such as multiple modes. You must specify distname as 'Kernel' to use this option.

Data Types: single | double

Output Arguments

pd — Probability distribution

probability distribution object

Probability distribution, returned as a probability distribution object. The distribution specified by distname determines the class type of the returned probability distribution object.

pdca — Probability distribution objects

cell array

Probability distribution objects of the type specified by distname, returned as a cell array.

gn — Group labels

cell array of character vectors

Group labels, returned as a cell array of character vectors.

gl — Grouping variable levels

cell array of character vectors

Grouping variable levels, returned as a cell array of character vectors containing one column for each grouping variable.

Algorithms

The fitdist function fits most distributions using maximum likelihood estimation. Two exceptions are the normal and lognormal distributions with uncensored data.

• For the uncensored normal distribution, the estimated value of the sigma parameter is the square root of the unbiased estimate of the variance.

• For the uncensored lognormal distribution, the estimated value of the sigma parameter is the square root of the unbiased estimate of the variance of the log of the data.

Alternative Functionality

App

The Distribution Fitter app opens a graphical user interface for you to import data from the workspace and interactively fit a probability distribution to that data. You can then save the distribution to the workspace as a probability distribution object. Open the Distribution Fitter app using distributionFitter, or click Distribution Fitter on the Apps tab.

References

[1] Johnson, N. L., S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions. Vol. 1, Hoboken, NJ: Wiley-Interscience, 1993.

[2] Johnson, N. L., S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions. Vol. 2, Hoboken, NJ: Wiley-Interscience, 1994.

[3] Bowman, A. W., and A. Azzalini. Applied Smoothing Techniques for Data Analysis. New York: Oxford University Press, 1997.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Supported syntaxes are:

  pd = fitdist(x,distname)
  pd = fitdist(x,distname,Name,Value)

  Code generation does not support the syntaxes that include the grouping variable 'By',groupvar and the related output arguments pdca, gn, and gl.

• fitdist supports code generation for beta, exponential, extreme value, lognormal, normal, and Weibull distributions.

  • The value of distname can be 'Beta', 'Exponential', 'ExtremeValue', 'Lognormal', 'Normal' or 'Weibull'.

  • The value of distname must be a compile-time constant.

• The values of x, 'Censoring', and 'Frequency' must not contain NaN values.

• Code generation ignores the 'Frequency' value for the beta distribution. Instead of specifying the 'Frequency' value, manually add duplicated values to x so that the values in x have the frequency you want.

• Code generation does not support these input arguments: groupvar, NTrials, Theta, mu, Kernel, Support, and Width.

• Names in name-value pair arguments must be compile-time constants.

• These object functions of pd support code generation: cdf, icdf, iqr, mean, median, pdf, std, truncate, and var.

For more information on code generation, see Introduction to Code Generation and Code Generation for Probability Distribution Objects.