Statistics: WeibullDistribution

Class Definition: `WeibullDistribution`

statistics: WeibullDistribution

Weibull probability distribution object.

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

The Weibull distribution is a continuous probability distribution that models the time to failure of materials or the lifetime of mechanical systems. It is defined by scale parameter lambda and shape parameter k.

There are several ways to create a WeibullDistribution object.

Fit a distribution to data using the fitdist function.
Create a distribution with fixed parameter values using the makedist function.
Use the constructor WeibullDistribution (lambda, k) to create a Weibull distribution with fixed parameter values lambda and k.
Use the static method WeibullDistribution.fit (x, alpha, censor, freq) to fit a distribution to the data in x using the same input arguments as the wblfit 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 Weibull distribution can be found at https://en.wikipedia.org/wiki/Weibull_distribution

See also: fitdist, makedist, wblcdf, wblinv, wblpdf, wblrnd, wblfit, wbllike, wblstat

Source Code: WeibullDistribution

The WeibullDistribution class contains the following properties:

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

Example: 1

Create a Weibull distribution with default parameters

 data = wblrnd (1, 1, 10000, 1);
 pd = fitdist (data, "Weibull");

Query parameter 'lambda' (scale parameter)

 pd.lambda

ans = 1.0008

Set parameter 'lambda'

 pd.lambda = 1.5

pd =
  WeibullDistribution

  Weibull distribution
   lambda =     1.5
        k = 1.00376

Use this to initialize or modify the scale parameter of a Weibull distribution. The scale parameter must be a positive real scalar, often representing the characteristic life in reliability analysis or the spread in lifetime modeling.

Example: 2

Create a Weibull distribution object by calling its constructor

 pd = WeibullDistribution (2, 1.5)

pd =
  WeibullDistribution

  Weibull distribution
   lambda =   2
        k = 1.5

Query parameter 'lambda'

 pd.lambda

ans = 2

This demonstrates direct construction with a specific scale parameter, useful for modeling failure times where lambda shifts the distribution, such as in survival or wind speed data.

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

Example: 1

Create a Weibull distribution with default parameters

 data = wblrnd (1, 1, 10000, 1);
 pd = fitdist (data, "Weibull");

Query parameter 'k' (shape parameter)

 pd.k

ans = 1.0141

Set parameter 'k'

 pd.k = 2

pd =
  WeibullDistribution

  Weibull distribution
   lambda = 1.00089
        k =       2

Use this to initialize or modify the shape parameter in a Weibull distribution. The shape parameter must be a positive real scalar, controlling the failure rate behavior (k<1 decreasing, k=1 constant, k>1 increasing).

Example: 2

Create a Weibull distribution object by calling its constructor

 pd = WeibullDistribution (1, 3)

pd =
  WeibullDistribution

  Weibull distribution
   lambda = 1
        k = 3

Query parameter 'k'

 pd.k

ans = 3

This shows how to set the shape parameter directly via the constructor, ideal for modeling different hazard functions in reliability engineering.

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 lambda and k 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 WeibullDistribution class offers the following public methods:

WeibullDistribution: p = cdf (pd, x)
WeibullDistribution: 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 Weibull distribution

 x = 0:0.01:5;
 data1 = wblrnd (1, 0.5, 10000, 1);
 data2 = wblrnd (1, 1, 10000, 1);
 data3 = wblrnd (1, 2, 10000, 1);
 pd1 = fitdist (data1, "Weibull");
 pd2 = fitdist (data2, "Weibull");
 pd3 = fitdist (data3, "Weibull");
 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 ({"lambda = 1, k = 0.5", "lambda = 1, k = 1", "lambda = 1, k = 2"}, ...
         "location", "southeast")
 title ("Weibull CDF")
 xlabel ("values in x (x >= 0)")
 ylabel ("Cumulative probability")

Use this to compute and visualize the cumulative distribution function for different Weibull distributions, showing how probability accumulates for positive values, useful in survival analysis or time-to-event modeling.

WeibullDistribution: 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 Weibull distribution

 p = 0.001:0.001:0.999;
 data1 = wblrnd (1, 0.5, 10000, 1);
 data2 = wblrnd (1, 1, 10000, 1);
 data3 = wblrnd (1, 2, 10000, 1);
 pd1 = fitdist (data1, "Weibull");
 pd2 = fitdist (data2, "Weibull");
 pd3 = fitdist (data3, "Weibull");
 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 ({"lambda = 1, k = 0.5", "lambda = 1, k = 1", "lambda = 1, k = 2"}, ...
         "location", "northwest")
 title ("Weibull iCDF")
 xlabel ("Probability")
 ylabel ("values in x (x >= 0)")

This demonstrates the inverse CDF (quantiles) for Weibull distributions, useful for finding values corresponding to given probabilities, such as predicting failure times in engineering applications.

WeibullDistribution: r = iqr (pd)

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

Example: 1

Compute the interquartile range for a Weibull distribution

 data = wblrnd (1, 1, 10000, 1);
 pd = fitdist (data, "Weibull");
 iqr_value = iqr (pd)

iqr_value = 1.0869

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

WeibullDistribution: m = mean (pd)

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

Example: 1

Compute the mean for different Weibull distributions

 data1 = wblrnd (1, 0.5, 10000, 1);
 data2 = wblrnd (1, 2, 10000, 1);
 pd1 = fitdist (data1, "Weibull");
 pd2 = fitdist (data2, "Weibull");
 mean1 = mean (pd1)

mean1 = 2.0656

 mean2 = mean (pd2)

mean2 = 0.8876

This shows how to compute the expected value for Weibull distributions with different shape parameters, representing average lifetime or duration.

WeibullDistribution: m = median (pd)

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

Example: 1

Compute the median for different Weibull distributions

 data1 = wblrnd (1, 0.5, 10000, 1);
 data2 = wblrnd (1, 2, 10000, 1);
 pd1 = fitdist (data1, "Weibull");
 pd2 = fitdist (data2, "Weibull");
 median1 = median (pd1)

median1 = 0.4913

 median2 = median (pd2)

median2 = 0.8400

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

WeibullDistribution: 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 Weibull distribution

 rand ("seed", 21);
 data = wblrnd (1, 1, 100, 1);
 pd_fitted = fitdist (data, "Weibull");
 params = [pd_fitted.lambda, pd_fitted.k];
 nlogL_wbllike = wbllike (params, data)

nlogL_wbllike = 101.83

This is useful for assessing the fit of a Weibull distribution to data, with lower values indicating a better fit, often used in model selection for reliability data.

WeibullDistribution: ci = paramci (pd)
WeibullDistribution: 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 Weibull distribution

 rand ("seed", 21);
 data = wblrnd (1, 1, 1000, 1);
 pd_fitted = fitdist (data, "Weibull");
 ci = paramci (pd_fitted, "Alpha", 0.05)

ci =

   0.9109   0.9536
   1.0378   1.0509

Use this to obtain confidence intervals for the estimated parameters (lambda and k), providing a range of plausible values given the data, essential in uncertainty quantification for failure models.

WeibullDistribution: 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 Weibull distribution

 x = 0:0.01:5;
 data1 = wblrnd (1, 0.5, 10000, 1);
 data2 = wblrnd (1, 1, 10000, 1);
 data3 = wblrnd (1, 2, 10000, 1);
 pd1 = fitdist (data1, "Weibull");
 pd2 = fitdist (data2, "Weibull");
 pd3 = fitdist (data3, "Weibull");
 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 ({"lambda = 1, k = 0.5", "lambda = 1, k = 1", "lambda = 1, k = 2"}, ...
         "location", "northeast")
 title ("Weibull PDF")
 xlabel ("values in x (x >= 0)")
 ylabel ("Probability density")

This visualizes the probability density function for Weibull distributions, showing the likelihood for positive values, common in wind speed or failure rate modeling.

WeibullDistribution: plot (pd)
WeibullDistribution: plot (pd, Name, Value)
WeibullDistribution: 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 Weibull distribution with fixed parameters lambda = 1 and k = 1 and plot its PDF.

 data = wblrnd (1, 1, 10000, 1);
 pd = fitdist (data, "Weibull");
 plot (pd)
 title ("Fixed Weibull distribution with lambda = 1 and k = 1")

Example: 2

Generate a data set of 100 random samples from a Weibull distribution with parameters lambda = 1 and k = 1. Fit a Weibull distribution to this data and plot its CDF superimposed over an empirical CDF.

 rand ("seed", 21);
 data = wblrnd (1, 1, 100, 1);
 pd_fitted = fitdist (data, "Weibull");
 plot (pd_fitted, "PlotType", "cdf")
 txt = "Fitted Weibull distribution with lambda = %0.2f and k = %0.2f";
 title (sprintf (txt, pd_fitted.lambda, pd_fitted.k))
 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 in time-to-failure scenarios.

Example: 3

Generate a data set of 200 random samples from a Weibull distribution with parameters lambda = 1 and k = 1. Display a probability plot for the Weibull distribution fit to the data.

 rand ("seed", 21);
 data = wblrnd (1, 1, 200, 1);
 pd_fitted = fitdist (data, "Weibull");
 plot (pd_fitted, "PlotType", "probability")
 txt = strcat ("Probability plot of fitted Weibull distribution", ...
               " with lambda = %0.2f and k = %0.2f");
 title (sprintf (txt, pd_fitted.lambda, pd_fitted.k))
 legend ({"empirical CDF", "fitted CDF"}, "location", "southeast")

This creates a probability plot to compare the fitted distribution to the data, useful for validating the Weibull assumption in reliability studies.

WeibullDistribution: [nlogL, param] = proflik (pd, pnum)
WeibullDistribution: [nlogL, param] = proflik (pd, pnum, 'Display', display)
WeibullDistribution: [nlogL, param] = proflik (pd, pnum, setparam)
WeibullDistribution: [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 Weibull distribution, pnum = 1 selects the parameter lambda and pnum = 2 selects the parameter k.

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 Weibull distribution

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

Use this to analyze the profile likelihood of the shape parameter (k), helping to understand the uncertainty in parameter estimates for failure rate models.

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

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

When called with a single size argument, wblrnd 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 Weibull distribution

 rand ("seed", 21);
 samples = wblrnd (1, 1, 500, 1);
 hist (samples, 50)
 title ("Histogram of 500 random samples from Weibull(lambda=1, k=1)")
 xlabel ("values in x (x >= 0)")
 ylabel ("Frequency")

This generates random samples from a Weibull distribution, useful for simulating lifetime data or positive skewed distributions.

WeibullDistribution: s = std (pd)

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

Example: 1

Compute the standard deviation for a Weibull distribution

 data = wblrnd (1, 1, 10000, 1);
 pd = fitdist (data, "Weibull");
 std_value = std (pd)

std_value = 0.9948

Use this to calculate the standard deviation, which measures the variability in lifetime or positive data modeled by Weibull.

WeibullDistribution: 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 Weibull distribution, with parameters lambda = 1 and k = 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 = wblrnd (1, 1, 20000, 1);
 data = data_all(data_all >= 0.5 & data_all <= 2);
 data = data(1:9000);
 pd = fitdist (data, "Weibull");
 t = truncate (pd, 0.5, 2);
 [counts, centers] = hist (data, 50);
 bin_width = centers(2) - centers(1);
 bar (centers, counts / (sum (counts) * bin_width), 1);
 hold on;

Plot histogram and truncated PDF

 x = linspace (0.5, 5, 500);
 y = pdf (t, x);
 plot (x, y, "r", "linewidth", 2);
 title ("Weibull distribution (lambda = 1, k = 1) truncated at [0.5, 2]")
 legend ("Truncated PDF", "Histogram")

This demonstrates truncating a Weibull distribution to a specific range and visualizing the resulting distribution with random samples, useful for bounded lifetime analysis.

WeibullDistribution: v = var (pd)

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

Example: 1

Compute the variance for a Weibull distribution

 data = wblrnd (1, 1, 10000, 1);
 pd = fitdist (data, "Weibull");
 var_value = var (pd)

var_value = 0.9512

Use this to calculate the variance, which quantifies the spread of the positive values in lifetime or reliability data.

Examples

 pd_fixed = makedist ('Weibull', 'lambda', 1, 'k', 2)

pd_fixed =
  WeibullDistribution

  Weibull distribution
   lambda = 1
        k = 2

 rand ('seed', 2);
 data = random (pd_fixed, 5000, 1);
 pd_fitted = fitdist (data, 'Weibull')

pd_fitted =
  WeibullDistribution

  Weibull distribution
   lambda = 1.00578   [0.991097, 1.02068]
        k = 1.98409   [1.94163, 2.02748]

 plot (pd_fitted)
 msg = 'Fitted Weibull distribution with lambda = %0.2f and k = %0.2f';
 title (sprintf (msg, pd_fitted.lambda, pd_fitted.k))

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Class Definition: WeibullDistribution

statistics: WeibullDistribution

lambda

Example: 1

Example: 2

k

Example: 1

Example: 2

DistributionName

NumParameters

ParameterNames

ParameterDescription

ParameterValues

ParameterCovariance

ParameterIsFixed

Truncation

IsTruncated

InputData

cdf

WeibullDistribution: p = cdf (pd, x)

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

Example: 1

icdf

WeibullDistribution: x = icdf (pd, p)

Example: 1

iqr

WeibullDistribution: r = iqr (pd)

Example: 1

mean

WeibullDistribution: m = mean (pd)

Example: 1

median

WeibullDistribution: m = median (pd)

Example: 1

negloglik

WeibullDistribution: nlogL = negloglik (pd)

Example: 1

paramci

Class Definition: `WeibullDistribution`

`lambda`

`k`

`DistributionName`

`NumParameters`

`ParameterNames`

`ParameterDescription`

`ParameterValues`

`ParameterCovariance`

`ParameterIsFixed`

`Truncation`

`IsTruncated`

`InputData`

`cdf`

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

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

`icdf`

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

`iqr`

`WeibullDistribution: r = iqr (pd)`

`mean`

`WeibullDistribution: m = mean (pd)`

`median`

`WeibullDistribution: m = median (pd)`

`negloglik`

`WeibullDistribution: nlogL = negloglik (pd)`

`paramci`

`WeibullDistribution: ci = paramci (pd)`

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

`pdf`

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

`plot`

`WeibullDistribution: plot (pd)`

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

`WeibullDistribution: h = plot (…)`

`proflik`

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

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

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

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

`random`

`WeibullDistribution: r = random (pd)`

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

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

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

`std`

`WeibullDistribution: s = std (pd)`

`truncate`

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

`var`

`WeibullDistribution: v = var (pd)`