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Class Definition: LognormalDistribution

statistics: LognormalDistribution

Lognormal probability distribution object.

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

The lognormal distribution is a continuous probability distribution whose logarithm is normally distributed. It is defined by mean parameter mu and standard deviation parameter sigma of the logarithmic values.

There are several ways to create a LognormalDistribution object.

  • Fit a distribution to data using the fitdist function.
  • Create a distribution with fixed parameter values using the makedist function.
  • Use the constructor LognormalDistribution (mu, sigma) to create a lognormal distribution with fixed parameter values mu and sigma.
  • Use the static method LognormalDistribution.fit (x, censor, freq, options) to fit a distribution to the data in x using the same input arguments as the lognfit 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 lognormal distribution can be found at https://en.wikipedia.org/wiki/Log-normal_distribution

See also: fitdist, makedist, logncdf, logninv, lognpdf, lognrnd, lognfit, lognlike, lognstat

Source Code: LognormalDistribution

The LognormalDistribution class contains the following properties:

A scalar value characterizing the mean of the logarithmic values of the lognormal distribution. You can access the mu property using dot name assignment.

Example: 1

Create a Lognormal distribution with default parameters

 data = lognrnd (0, 1, 10000, 1);
 pd = fitdist (data, "Lognormal");

Query parameter 'mu' (mean of logarithmic values)

 pd.mu
ans = 3.2683e-03

Set parameter 'mu'

 pd.mu = 1
pd =
  LognormalDistribution

  Lognormal distribution
       mu =       1
    sigma = 0.99306

Use this to initialize or modify the mean parameter of the logarithmic values in a Lognormal distribution. The mu parameter must be a real scalar, often representing the log-mean of multiplicative processes, like growth rates.

Example: 2

Create a Lognormal distribution object by calling its constructor

 pd = LognormalDistribution (1.5, 0.5)
pd =
  LognormalDistribution

  Lognormal distribution
       mu = 1.5
    sigma = 0.5

Query parameter 'mu'

 pd.mu
ans = 1.5000

This demonstrates direct construction with a specific mu parameter, useful for modeling skewed positive data shifted in log-scale, such as incomes or sizes.

A positive scalar value characterizing the standard deviation of the logarithmic values of the lognormal distribution. You can access the sigma property using dot name assignment.

Example: 1

Create a Lognormal distribution with default parameters

 data = lognrnd (0, 1, 10000, 1);
 pd = fitdist (data, "Lognormal");

Query parameter 'sigma' (standard deviation of logarithmic values)

 pd.sigma
ans = 0.9955

Set parameter 'sigma'

 pd.sigma = 0.5
pd =
  LognormalDistribution

  Lognormal distribution
       mu = -0.0175268
    sigma =        0.5

Use this to initialize or modify the sigma parameter in a Lognormal distribution. The sigma parameter must be a positive real scalar, controlling the skewness and spread of the distribution.

Example: 2

Create a Lognormal distribution object by calling its constructor

 pd = LognormalDistribution (0, 1.5)
pd =
  LognormalDistribution

  Lognormal distribution
       mu =   0
    sigma = 1.5

Query parameter 'sigma'

 pd.sigma
ans = 1.5000

This shows how to set the sigma parameter directly via the constructor, ideal for modeling variability in positive skewed data, such as stock returns.

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 LognormalDistribution class offers the following public methods:

LognormalDistribution: p = cdf (pd, x)
LognormalDistribution: 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 Lognormal distribution

 x = 0:0.01:10;
 data1 = lognrnd (0, 0.5, 10000, 1);
 data2 = lognrnd (0, 1.0, 10000, 1);
 data3 = lognrnd (0, 1.5, 10000, 1);
 pd1 = fitdist (data1, "Lognormal");
 pd2 = fitdist (data2, "Lognormal");
 pd3 = fitdist (data3, "Lognormal");
 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 = 1.5"}, ...
         "location", "southeast")
 title ("Lognormal CDF")
 xlabel ("values in x (x > 0)")
 ylabel ("Cumulative probability")
plotted figure

Use this to compute and visualize the cumulative distribution function for different Lognormal distributions, showing how probability accumulates for positive skewed data, useful in finance or biology modeling.

LognormalDistribution: 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 Lognormal distribution

 p = 0.001:0.001:0.999;
 data1 = lognrnd (0, 0.5, 10000, 1);
 data2 = lognrnd (0, 1.0, 10000, 1);
 data3 = lognrnd (0, 1.5, 10000, 1);
 pd1 = fitdist (data1, "Lognormal");
 pd2 = fitdist (data2, "Lognormal");
 pd3 = fitdist (data3, "Lognormal");
 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 = 1.5"}, ...
         "location", "northwest")
 title ("Lognormal iCDF")
 xlabel ("Probability")
 ylabel ("values in x (x > 0)")
plotted figure

This demonstrates the inverse CDF (quantiles) for Lognormal distributions, useful for finding values corresponding to given probabilities, such as risk thresholds in finance.

LognormalDistribution: r = iqr (pd)

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

Example: 1

Compute the interquartile range for a Lognormal distribution

 data = lognrnd (0, 1, 10000, 1);
 pd = fitdist (data, "Lognormal");
 iqr_value = iqr (pd)
iqr_value = 1.4811

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

LognormalDistribution: m = mean (pd)

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

Example: 1

Compute the mean for different Lognormal distributions

 data1 = lognrnd (0, 0.5, 10000, 1);
 data2 = lognrnd (0, 1.0, 10000, 1);
 pd1 = fitdist (data1, "Lognormal");
 pd2 = fitdist (data2, "Lognormal");
 mean1 = mean (pd1)
mean1 = 1.1371
 mean2 = mean (pd2)
mean2 = 1.6404

This shows how to compute the expected value for Lognormal distributions with different sigma parameters, representing average outcomes in multiplicative processes.

LognormalDistribution: m = median (pd)

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

Example: 1

Compute the median for different Lognormal distributions

 data1 = lognrnd (0, 0.5, 10000, 1);
 data2 = lognrnd (0, 1.0, 10000, 1);
 pd1 = fitdist (data1, "Lognormal");
 pd2 = fitdist (data2, "Lognormal");
 median1 = median (pd1)
median1 = 1.0013
 median2 = median (pd2)
median2 = 0.9864

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

LognormalDistribution: 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 Lognormal distribution

 rand ("seed", 21);
 data = lognrnd (0, 1, 100, 1);
 pd_fitted = fitdist (data, "Lognormal");
 params = [pd_fitted.mu, pd_fitted.sigma];
 nlogL_lognlike = lognlike (params, data)
nlogL_lognlike = 119.01

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

LognormalDistribution: ci = paramci (pd)
LognormalDistribution: 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.

NameValue
'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 Lognormal distribution

 rand ("seed", 21);
 data = lognrnd (0, 1, 1000, 1);
 pd_fitted = fitdist (data, "Lognormal");
 ci = paramci (pd_fitted, "Alpha", 0.05)
ci =

  -0.047292   0.938386
   0.074275   1.024439

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

LognormalDistribution: 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 Lognormal distribution

 x = 0:0.01:10;
 data1 = lognrnd (0, 0.5, 10000, 1);
 data2 = lognrnd (0, 1.0, 10000, 1);
 data3 = lognrnd (0, 1.5, 10000, 1);
 pd1 = fitdist (data1, "Lognormal");
 pd2 = fitdist (data2, "Lognormal");
 pd3 = fitdist (data3, "Lognormal");
 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 = 1.5"}, ...
         "location", "northeast")
 title ("Lognormal PDF")
 xlabel ("values in x (x > 0)")
 ylabel ("Probability density")
plotted figure

This visualizes the probability density function for Lognormal distributions, showing the likelihood for positive skewed values.

LognormalDistribution: plot (pd)
LognormalDistribution: plot (pd, Name, Value)
LognormalDistribution: 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.

NameValue
'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 Lognormal distribution with fixed parameters mu = 0 and sigma = 1 and plot its PDF.

 data = lognrnd (0, 1, 10000, 1);
 pd = fitdist (data, "Lognormal");
 plot (pd)
 title ("Fixed Lognormal distribution with mu = 0 and sigma = 1")
plotted figure

Example: 2

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

 rand ("seed", 21);
 data = lognrnd (0, 1, 100, 1);
 pd_fitted = fitdist (data, "Lognormal");
 plot (pd_fitted, "PlotType", "cdf")
 txt = "Fitted Lognormal 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")
plotted figure

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 Lognormal distribution with parameters mu = 0 and sigma = 1. Display a probability plot for the Lognormal distribution fit to the data.

 rand ("seed", 21);
 data = lognrnd (0, 1, 200, 1);
 pd_fitted = fitdist (data, "Lognormal");
 plot (pd_fitted, "PlotType", "probability")
 txt = strcat ("Probability plot of fitted Lognormal", ...
               " 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")
plotted figure

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

LognormalDistribution: [nlogL, param] = proflik (pd, pnum)
LognormalDistribution: [nlogL, param] = proflik (pd, pnum, 'Display', display)
LognormalDistribution: [nlogL, param] = proflik (pd, pnum, setparam)
LognormalDistribution: [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 Lognormal 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 sigma parameter of a fitted Lognormal distribution

 rand ("seed", 21);
 data = lognrnd (0, 1, 1000, 1);
 pd_fitted = fitdist (data, "Lognormal");
 [nlogL, param] = proflik (pd_fitted, 2, "Display", "on");
plotted figure

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

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

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

When called with a single size argument, lognrnd 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 Lognormal distribution

 rand ("seed", 21);
 samples = lognrnd (0, 1, 500, 1);
 hist (samples, 50)
 title ("Histogram of 500 random samples from Lognormal(mu=0, sigma=1)")
 xlabel ("values in x (x > 0)")
 ylabel ("Frequency")
plotted figure

This generates random samples from a Lognormal distribution, useful for simulating skewed positive data like asset prices or biological measurements.

LognormalDistribution: s = std (pd)

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

Example: 1

Compute the standard deviation for a Lognormal distribution

 data = lognrnd (0, 1, 10000, 1);
 pd = fitdist (data, "Lognormal");
 std_value = std (pd)
std_value = 2.1528

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

LognormalDistribution: 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 Lognormal distribution, with parameters mu = 0 and sigma = 1, truncated at [0.5, 5] intervals. Generate 10000 random samples from this truncated distribution and superimpose a histogram scaled accordingly

 rand ("seed", 21);
 data_all = lognrnd (0, 1, 20000, 1);
 data = data_all(data_all >= 0.5 & data_all <= 5);
 data = data(1:10000);
 pd = fitdist (data, "Lognormal");
 t = truncate (pd, 0.5, 5);
 [counts, centers] = hist (data, 50);
 bin_width = centers(2) - centers(1);
 bar (centers, counts / (sum (counts) * bin_width), 1);
 hold on;
 x = linspace (0.5, 5, 500);
 y = pdf (t, x);
 plot (x, y, "r", "linewidth", 2);
 title ("Lognormal distribution (mu = 0, sigma = 1) truncated at [0.5, 5]")
 legend ("Truncated PDF", "Histogram")
plotted figure

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

LognormalDistribution: v = var (pd)

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

Example: 1

Compute the variance for a Lognormal distribution

 data = lognrnd (0, 1, 10000, 1);
 pd = fitdist (data, "Lognormal");
 var_value = var (pd)
var_value = 4.8417

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

Examples 

 pd_fixed = makedist ('Lognormal', 'mu', 0, 'sigma', 1)
pd_fixed =
  LognormalDistribution

  Lognormal distribution
       mu = 0
    sigma = 1
 randn ('seed', 2);
 data = random (pd_fixed, 5000, 1);
 pd_fitted = fitdist (data, 'Lognormal')
pd_fitted =
  LognormalDistribution

  Lognormal distribution
       mu = 0.00166669   [-0.0251795, 0.0285129]
    sigma =   0.968309   [0.949696, 0.987671]
 plot (pd_fitted)
 msg = 'Fitted Lognormal distribution with mu = %0.2f and sigma = %0.2f';
 title (sprintf (msg, pd_fitted.mu, pd_fitted.sigma))
plotted figure