Statistics: LognormalDistribution

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

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

 ## Set parameter 'mu'
 pd.mu = 1

 ## 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.

ans = 3.2683e-03
pd =
  LognormalDistribution

  Lognormal distribution
       mu =       1
    sigma = 0.99306

Example: 2


 ## Create a Lognormal distribution object by calling its constructor
 pd = LognormalDistribution (1.5, 0.5)

 ## Query parameter 'mu'
 pd.mu

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

pd =
  LognormalDistribution

  Lognormal distribution
       mu = 1.5
    sigma = 0.5

ans = 1.5000

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

 ## Set parameter 'sigma'
 pd.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.

ans = 0.9955
pd =
  LognormalDistribution

  Lognormal distribution
       mu = -0.0175268
    sigma =        0.5

Example: 2


 ## Create a Lognormal distribution object by calling its constructor
 pd = LognormalDistribution (0, 1.5)

 ## Query parameter 'sigma'
 pd.sigma

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

pd =
  LognormalDistribution

  Lognormal distribution
       mu =   0
    sigma = 1.5

ans = 1.5000

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.

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")

 ## 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)")

 ## 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)

 ## 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.

iqr_value = 1.4811

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)
 mean2 = mean (pd2)

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

mean1 = 1.1371
mean2 = 1.6404

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)
 median2 = median (pd2)

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

median1 = 1.0013
median2 = 0.9864

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)

 ## 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.

nlogL_lognlike = 119.01

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.

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

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

ci =

  -0.047292   0.938386
   0.074275   1.024439

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")

 ## 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.

`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 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")

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")

 ## 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")

 ## 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");

 ## 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")

 ## 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)

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

std_value = 2.1528

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")

 ## 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)

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

var_value = 4.8417

Examples

 pd_fixed = makedist ("Lognormal", "mu", 0, "sigma", 1)
 randn ("seed", 2);
 data = random (pd_fixed, 5000, 1);
 pd_fitted = fitdist (data, "Lognormal")
 plot (pd_fitted)
 msg = "Fitted Lognormal distribution with mu = %0.2f and sigma = %0.2f";
 title (sprintf (msg, pd_fitted.mu, pd_fitted.sigma))

pd_fixed =
  LognormalDistribution

  Lognormal distribution
       mu = 0
    sigma = 1

pd_fitted =
  LognormalDistribution

  Lognormal distribution
       mu = 0.00166669   [-0.0251795, 0.0285129]
    sigma =   0.968309   [0.949696, 0.987671]

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Class Definition: LognormalDistribution

statistics: LognormalDistribution

Properties

mu

Example: 1

Example: 2

sigma

Example: 1

Example: 2

DistributionName

NumParameters

ParameterNames

ParameterDescription

ParameterValues

ParameterCovariance

ParameterIsFixed

Truncation

IsTruncated

InputData

Methods

cdf

LognormalDistribution: p = cdf (pd, x)

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

Example: 1

icdf

LognormalDistribution: x = icdf (pd, p)

Example: 1

iqr

LognormalDistribution: r = iqr (pd)

Example: 1

mean

LognormalDistribution: m = mean (pd)

Example: 1

median

LognormalDistribution: m = median (pd)

Example: 1

negloglik

LognormalDistribution: nlogL = negloglik (pd)

Example: 1

paramci

Class Definition: `LognormalDistribution`

`mu`

`sigma`

`DistributionName`

`NumParameters`

`ParameterNames`

`ParameterDescription`

`ParameterValues`

`ParameterCovariance`

`ParameterIsFixed`

`Truncation`

`IsTruncated`

`InputData`

`cdf`

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

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

`icdf`

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

`iqr`

`LognormalDistribution: r = iqr (pd)`

`mean`

`LognormalDistribution: m = mean (pd)`

`median`

`LognormalDistribution: m = median (pd)`

`negloglik`

`LognormalDistribution: nlogL = negloglik (pd)`

`paramci`

`LognormalDistribution: ci = paramci (pd)`

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

`pdf`

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

`plot`

`LognormalDistribution: plot (pd)`

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

`LognormalDistribution: h = plot (…)`

`proflik`

`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)`

`random`

`LognormalDistribution: r = random (pd)`

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

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

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

`std`

`LognormalDistribution: s = std (pd)`

`truncate`

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

`var`

`LognormalDistribution: v = var (pd)`