Statistics: GammaDistribution

Class Definition: `GammaDistribution`

statistics: GammaDistribution

Gamma probability distribution object.

A GammaDistribution object consists of parameters, a model description, and sample data for a gamma probability distribution.

The gamma distribution is a continuous probability distribution that models the time to failure of a process. It is defined by shape parameter a and scale parameter b.

There are several ways to create a GammaDistribution object.

Fit a distribution to data using the fitdist function.
Create a distribution with fixed parameter values using the makedist function.
Use the constructor GammaDistribution (a, b) to create a gamma distribution with fixed parameter values a and b.
Use the static method GammaDistribution.fit (x, alpha, censor, freq, options) to fit a distribution to the data in x using the same input arguments as the gamfit 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 gamma distribution can be found at https://en.wikipedia.org/wiki/Gamma_distribution

See also: fitdist, makedist, gamcdf, gaminv, gampdf, gamrnd, gamfit, gamlike, gamstat

Source Code: GammaDistribution

Properties

A positive scalar value characterizing the shape of the gamma distribution. You can access the a property using dot name assignment.

Example: 1


 ## Create a Gamma distribution with default parameters
 pd = makedist ("Gamma")

 ## Query parameter 'a' (shape parameter)
 pd.a

 ## Set parameter 'a'
 pd.a = 2

 ## Use this to initialize or modify the shape parameter of a Gamma
 ## distribution. The shape parameter must be a positive real scalar.

pd =
  GammaDistribution

  gamma distribution
        a = 1
        b = 1

ans = 1
pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

Example: 2


 ## Create a Gamma distribution object by calling its constructor
 pd = GammaDistribution (2, 1)

 ## Query parameter 'a'
 pd.a

 ## This demonstrates direct construction with a specific shape parameter,
 ## useful for modeling waiting times with known variability.

pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

ans = 2

A positive scalar value characterizing the scale of the gamma distribution. You can access the b property using dot name assignment.

Example: 1


 ## Create a Gamma distribution with default parameters
 pd = makedist ("Gamma")

 ## Query parameter 'b' (scale parameter)
 pd.b

 ## Set parameter 'b'
 pd.b = 2

 ## Use this to initialize or modify the scale parameter in a Gamma
 ## distribution. The scale parameter must be a positive real scalar.

pd =
  GammaDistribution

  gamma distribution
        a = 1
        b = 1

ans = 1
pd =
  GammaDistribution

  gamma distribution
        a = 1
        b = 2

Example: 2


 ## Create a Gamma distribution object by calling its constructor
 pd = GammaDistribution (2, 1)

 ## Query parameter 'b'
 pd.b

 ## This shows how to set the scale parameter directly via the constructor,
 ## ideal for scaling the distribution in lifetime models.

pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

ans = 1

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 a and b 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.

Methods

GammaDistribution: p = cdf (pd, x)
GammaDistribution: 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 Gamma distribution
 x = 0:0.01:10;
 pd1 = makedist ("Gamma", "a", 1, "b", 1);
 pd2 = makedist ("Gamma", "a", 2, "b", 1);
 pd3 = makedist ("Gamma", "a", 5, "b", 1);
 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 ({"a = 1, b = 1", "a = 2, b = 1", "a = 5, b = 1"}, ...
         "location", "southeast")
 title ("Gamma CDF")
 xlabel ("Values")
 ylabel ("Cumulative probability")

 ## Use this to compute and visualize the cumulative distribution function
 ## for different Gamma distributions, showing how probability accumulates
 ## over values.

GammaDistribution: 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 Gamma distribution
 p = 0.001:0.001:0.999;
 pd1 = makedist ("Gamma", "a", 1, "b", 1);
 pd2 = makedist ("Gamma", "a", 2, "b", 1);
 pd3 = makedist ("Gamma", "a", 5, "b", 1);
 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 ({"a = 1, b = 1", "a = 2, b = 1", "a = 5, b = 1"}, ...
         "location", "northwest")
 title ("Gamma iCDF")
 xlabel ("Probability")
 ylabel ("Values")

 ## This demonstrates the inverse CDF (quantiles) for Gamma
 ## distributions, useful for finding values corresponding to
 ## given probabilities.

GammaDistribution: r = iqr (pd)

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

Example: 1


 ## Compute the interquartile range for a Gamma distribution
 pd = makedist ("Gamma", "a", 2, "b", 1)
 iqr_value = iqr (pd)

 ## Use this to calculate the interquartile range, which measures the spread
 ## of the middle 50% of the distribution, useful for understanding variability
 ## in waiting times.

pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

iqr_value = 1.7314

GammaDistribution: m = mean (pd)

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

Example: 1


 ## Compute the mean for different Gamma distributions
 pd1 = makedist ("Gamma", "a", 1, "b", 1);
 pd2 = makedist ("Gamma", "a", 2, "b", 1);
 mean1 = mean (pd1)
 mean2 = mean (pd2)

 ## This shows how to compute the expected value for Gamma
 ## distributions with different shape parameters.

mean1 = 1
mean2 = 2

GammaDistribution: m = median (pd)

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

Example: 1


 ## Compute the median for different Gamma distributions
 pd1 = makedist ("Gamma", "a", 1, "b", 1);
 pd2 = makedist ("Gamma", "a", 2, "b", 1);
 median1 = median (pd1)
 median2 = median (pd2)

 ## Use this to find the median value, which splits the distribution
 ## into two equal probability halves.

median1 = 0.6931
median2 = 1.6783

GammaDistribution: 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 Gamma distribution
 pd = makedist ("Gamma", "a", 2, "b", 1)
 rand ("seed", 21);
 data = gamrnd (2, 1, 100, 1);
 pd_fitted = fitdist (data, "Gamma")
 nlogL = negloglik (pd_fitted)

 ## This is useful for assessing the fit of a Gamma distribution to
 ## data, lower values indicate a better fit.

pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

pd_fitted =
  GammaDistribution

  gamma distribution
        a = 1.50419   [1.16867, 1.93604]
        b =  1.2765   [0.946859, 1.72091]

nlogL = -160.73

GammaDistribution: ci = paramci (pd)
GammaDistribution: 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 Gamma
 ## distribution
 pd = makedist ("Gamma", "a", 2, "b", 1)
 rand ("seed", 21);
 data = gamrnd (2, 1, 1000, 1);
 pd_fitted = fitdist (data, "Gamma")
 ci = paramci (pd_fitted, "Alpha", 0.05)

 ## Use this to obtain confidence intervals for the estimated parameters (a
 ## and b), providing a range of plausible values given the data.

pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

pd_fitted =
  GammaDistribution

  gamma distribution
        a =  2.07915   [1.91623, 2.25592]
        b = 0.997778   [0.909871, 1.09418]

ci =

   1.9162   0.9099
   2.2559   1.0942

GammaDistribution: 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 Gamma distribution
 x = 0:0.01:10;
 pd1 = makedist ("Gamma", "a", 1, "b", 1);
 pd2 = makedist ("Gamma", "a", 2, "b", 1);
 pd3 = makedist ("Gamma", "a", 5, "b", 1);
 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 ({"a = 1, b = 1", "a = 2, b = 1", "a = 5, b = 1"}, ...
         "location", "northeast")
 title ("Gamma PDF")
 xlabel ("Values")
 ylabel ("Probability density")

 ## This visualizes the probability density function for Gamma
 ## distributions, showing the likelihood of different values.

GammaDistribution: plot (pd)
GammaDistribution: plot (pd, Name, Value)
GammaDistribution: 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 Gamma distribution with fixed parameters a = 2 and
 ## b = 1 and plot its PDF.

 pd = makedist ("Gamma", "a", 2, "b", 1)
 plot (pd)
 title ("Fixed Gamma distribution with a = 2 and b = 1")

pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

Example: 2


 ## Generate a data set of 100 random samples from a Gamma
 ## distribution with parameters a = 2 and b = 1. Fit a Gamma
 ## distribution to this data and plot its CDF superimposed over an empirical
 ## CDF.

 pd_fixed = makedist ("Gamma", "a", 2, "b", 1)
 rand ("seed", 21);
 data = gamrnd (2, 1, 100, 1);
 pd_fitted = fitdist (data, "Gamma")
 plot (pd_fitted, "PlotType", "cdf")
 txt = "Fitted Gamma distribution with a = %0.2f and b = %0.2f";
 title (sprintf (txt, pd_fitted.a, pd_fitted.b))
 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.

pd_fixed =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

pd_fitted =
  GammaDistribution

  gamma distribution
        a =  2.64912   [2.03932, 3.44127]
        b = 0.806472   [0.604664, 1.07564]

Example: 3


 ## Generate a data set of 200 random samples from a Gamma
 ## distribution with parameters a = 2 and b = 1. Display a probability
 ## plot for the Gamma distribution fit to the data.

 pd_fixed = makedist ("Gamma", "a", 2, "b", 1)
 rand ("seed", 21);
 data = gamrnd (2, 1, 200, 1);
 pd_fitted = fitdist (data, "Gamma")
 plot (pd_fitted, "PlotType", "probability")
 txt = strcat ("Probability plot of fitted Gamma", ...
               " distribution with a = %0.2f and b = %0.2f");
 title (sprintf (txt, pd_fitted.a, pd_fitted.b))
 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 Gamma model is appropriate.

pd_fixed =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

pd_fitted =
  GammaDistribution

  gamma distribution
        a = 1.90063   [1.58529, 2.27871]
        b = 1.06516   [0.865639, 1.31067]

GammaDistribution: [nlogL, param] = proflik (pd, pnum)
GammaDistribution: [nlogL, param] = proflik (pd, pnum, "Display", display)
GammaDistribution: [nlogL, param] = proflik (pd, pnum, setparam)
GammaDistribution: [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 gamma distribution, pnum = 1 selects the parameter a and pnum = 2 selects the parameter b.

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 shape parameter of a fitted
 ## Gamma distribution
 pd = makedist ("Gamma", "a", 2, "b", 1)
 rand ("seed", 21);
 data = gamrnd (2, 1, 1000, 1);
 pd_fitted = fitdist (data, "Gamma")
 [nlogL, param] = proflik (pd_fitted, 1, "Display", "on");

 ## Use this to analyze the profile likelihood of the shape parameter (a),
 ## helping to understand the uncertainty in parameter estimates.

pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

pd_fitted =
  GammaDistribution

  gamma distribution
        a = 2.02691   [1.86833, 2.19896]
        b = 1.01569   [0.926067, 1.11398]

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

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

When called with a single size argument, gamrnd 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 Gamma distribution
 pd = makedist ("Gamma", "a", 2, "b", 1)
 rand ("seed", 21);
 samples = random (pd, 500, 1);
 hist (samples, 50)
 title ("Histogram of 500 random samples from Gamma(a=2, b=1)")
 xlabel ("Values")
 ylabel ("Frequency")

 ## This generates random samples from a Gamma distribution, useful
 ## for simulating waiting times or other positive skewed data.

pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

GammaDistribution: s = std (pd)

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

Example: 1


 ## Compute the standard deviation for a Gamma distribution
 pd = makedist ("Gamma", "a", 2, "b", 1)
 std_value = std (pd)

 ## Use this to calculate the standard deviation, which measures the variability
 ## in the distribution.

pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

std_value = 1.4142

GammaDistribution: 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 Gamma distribution, with parameters a = 2
 ## and b = 1, truncated at [1, 5] intervals. Generate 10000 random
 ## samples from this truncated distribution and superimpose a histogram scaled
 ## accordingly

 pd = makedist ("Gamma", "a", 2, "b", 1)
 t = truncate (pd, -1, 2)
 rand ("seed", 21);
 data = random (t, 10000, 1);
 ## Plot histogram and fitted PDF
 plot (t)
 hold on
 hist (data, 100, 50)
 hold off
 title ("Gamma distribution (a = 2, b = 1) truncated at [-1, 2]")
 legend ("Truncated PDF", "Histogram")

 ## This demonstrates truncating a Gamma distribution to a specific
 ## range and visualizing the resulting distribution with random samples.

pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

t =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1
  Truncated to the interval [-1, 2]

GammaDistribution: v = var (pd)

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

Example: 1


 ## Compute the variance for a Gamma distribution
 pd = makedist ("Gamma", "a", 2, "b", 1)
 var_value = var (pd)

 ## Use this to calculate the variance, which quantifies the spread of the
 ## distribution.

pd =
  GammaDistribution

  gamma distribution
        a = 2
        b = 1

var_value = 2

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Class Definition: GammaDistribution

statistics: GammaDistribution

Properties

a

Example: 1

Example: 2

b

Example: 1

Example: 2

DistributionName

NumParameters

ParameterNames

ParameterDescription

ParameterValues

ParameterCovariance

ParameterIsFixed

Truncation

IsTruncated

InputData

Methods

cdf

GammaDistribution: p = cdf (pd, x)

GammaDistribution: p = cdf (pd, x, "upper")

Example: 1

icdf

GammaDistribution: x = icdf (pd, p)

Example: 1

iqr

GammaDistribution: r = iqr (pd)

Example: 1

mean

GammaDistribution: m = mean (pd)

Example: 1

median

GammaDistribution: m = median (pd)

Example: 1

negloglik

GammaDistribution: nlogL = negloglik (pd)

Example: 1

paramci

Class Definition: `GammaDistribution`

`a`

`b`

`DistributionName`

`NumParameters`

`ParameterNames`

`ParameterDescription`

`ParameterValues`

`ParameterCovariance`

`ParameterIsFixed`

`Truncation`

`IsTruncated`

`InputData`

`cdf`

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

`GammaDistribution: p = cdf (pd, x, "upper")`

`icdf`

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

`iqr`

`GammaDistribution: r = iqr (pd)`

`mean`

`GammaDistribution: m = mean (pd)`

`median`

`GammaDistribution: m = median (pd)`

`negloglik`

`GammaDistribution: nlogL = negloglik (pd)`

`paramci`

`GammaDistribution: ci = paramci (pd)`

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

`pdf`

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

`plot`

`GammaDistribution: plot (pd)`

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

`GammaDistribution: h = plot (…)`

`proflik`

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

`GammaDistribution: [nlogL, param] = proflik (pd, pnum, "Display", display)`

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

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

`random`

`GammaDistribution: r = random (pd)`

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

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

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

`std`

`GammaDistribution: s = std (pd)`

`truncate`

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

`var`

`GammaDistribution: v = var (pd)`