Statistics: RicianDistribution

Class Definition: `RicianDistribution`

statistics: RicianDistribution

Rician probability distribution object.

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

The Rician distribution is a continuous probability distribution that models the magnitude of a signal in the presence of Gaussian noise. It is defined by noncentrality parameter s and scale parameter sigma.

There are several ways to create a RicianDistribution object.

Fit a distribution to data using the fitdist function.
Create a distribution with fixed parameter values using the makedist function.
Use the constructor RicianDistribution (s, sigma) to create a Rician distribution with fixed parameter values s and sigma.
Use the static method RicianDistribution.fit (x, censor, freq, options) to fit a distribution to data x.

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 Rician distribution can be found at https://en.wikipedia.org/wiki/Rice_distribution

See also: fitdist, makedist, ricecdf, riceinv, ricepdf, ricernd, ricefit, ricelike, ricestat

Source Code: RicianDistribution

Properties

A non-negative scalar value characterizing the noncentrality of the Rician distribution. You can access the s property using dot name assignment.

Example: 1


 ## Create a Rician distribution by fitting to data
 data = ricernd (1, 1, [10000, 1]);  % Generate data with s=1, sigma=1
 pd = fitdist (data, "Rician");

 ## Query parameter 's' (noncentrality parameter)
 pd.s

 ## Set parameter 's'
 pd.s = 1.5

 ## Use this to initialize or modify the noncentrality parameter of a Rician
 ## distribution. The noncentrality parameter 's' must be a non-negative real
 ## scalar, representing the magnitude of the signal in the presence of noise.

ans = 0.9969
pd =
  RicianDistribution

  Rician distribution
        s =      1.5
    sigma = 0.991154

Example: 2


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

 ## Query parameter 's'
 pd.s

 ## This demonstrates direct construction with a specific noncentrality
 ## parameter, useful for modeling data with a known signal strength.

pd =
  RicianDistribution

  Rician distribution
        s = 2
    sigma = 1

ans = 2

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

Example: 1


 ## Create a Rician distribution with fitted parameters
 data = ricernd (1, 1, [10000, 1]);
 pd = fitdist (data, "Rician");

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

 ## Set parameter 'sigma'
 pd.sigma = 1.2

 ## Use this to initialize or modify the scale parameter in a Rician
 ## distribution. The scale parameter 'sigma' must be a positive real scalar,
 ## controlling the spread due to Gaussian noise.

ans = 0.9913
pd =
  RicianDistribution

  Rician distribution
        s = 1.00301
    sigma =     1.2

Example: 2


 ## Create a Rician distribution object by calling its constructor
 pd = RicianDistribution (1, 1.5)

 ## Query parameter 'sigma'
 pd.sigma

 ## This shows how to set the scale parameter directly via the constructor,
 ## ideal for modeling variability in signal magnitude data.

pd =
  RicianDistribution

  Rician distribution
        s =   1
    sigma = 1.5

ans = 1.5000

Example: 3


 ## Create a Rician distribution with specific parameters
 pd = RicianDistribution (2, 1)

 ## Display the distribution parameters
 pd.s
 pd.sigma

 ## Use the constructor to create a Rician distribution with fixed parameters
 ## 's' and 'sigma', suitable for scenarios where signal and noise parameters
 ## are known, such as in communication systems or image processing.

pd =
  RicianDistribution

  Rician distribution
        s = 2
    sigma = 1

ans = 2
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 s 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

RicianDistribution: p = cdf (pd, x)
RicianDistribution: 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 Rician distribution
 x = 0:0.01:5;
 data1 = ricernd (1, 0.5, [10000, 1]);
 data2 = ricernd (1, 1, [10000, 1]);
 data3 = ricernd (1, 2, [10000, 1]);
 pd1 = fitdist (data1, "Rician");
 pd2 = fitdist (data2, "Rician");
 pd3 = fitdist (data3, "Rician");
 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 ({"s = 1, sigma = 0.5", "s = 1, sigma = 1", "s = 1, sigma = 2"}, ...
         "location", "southeast")
 title ("Rician CDF")
 xlabel ("values in x (x >= 0)")
 ylabel ("Cumulative probability")

 ## Use this to compute and visualize the cumulative distribution function
 ## for different Rician distributions, showing how probability accumulates
 ## for non-negative signal magnitudes, useful in signal processing.

warning: ricefit: maximum number of iterations are exceeded.
warning: called from
    ricefit at line 171 column 7
    fit at line 752 column 8
    fitdist at line 652 column 9
    build_DEMOS at line 94 column 11
    classdef_texi2html at line 310 column 7
    package_texi2html at line 298 column 9

RicianDistribution: 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 Rician distribution
 p = 0.001:0.001:0.999;
 data1 = ricernd (1, 0.5, [10000, 1]);
 data2 = ricernd (1, 1, [10000, 1]);
 data3 = ricernd (1, 2, [10000, 1]);
 pd1 = fitdist (data1, "Rician");
 pd2 = fitdist (data2, "Rician");
 pd3 = fitdist (data3, "Rician");
 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 ({"s = 1, sigma = 0.5", "s = 1, sigma = 1", "s = 1, sigma = 2"}, ...
         "location", "northwest")
 title ("Rician iCDF")
 xlabel ("Probability")
 ylabel ("values in x (x >= 0)")

 ## This demonstrates the inverse CDF (quantiles) for Rician distributions,
 ## useful for finding signal magnitude thresholds in applications like radar.

RicianDistribution: r = iqr (pd)

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

Example: 1


 ## Compute the interquartile range for a Rician distribution
 data = ricernd (1, 1, [10000, 1]);
 pd = fitdist (data, "Rician");
 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 variability
 ## in signal magnitudes.

iqr_value = 1.0924

RicianDistribution: m = mean (pd)

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

Example: 1


 ## Compute the mean for different Rician distributions
 data1 = ricernd (1, 0.5, [10000, 1]);
 data2 = ricernd (1, 1, [10000, 1]);
 pd1 = fitdist (data1, "Rician");
 pd2 = fitdist (data2, "Rician");
 mean1 = mean (pd1)
 mean2 = mean (pd2)

 ## This shows how to compute the expected value for Rician distributions
 ## with different scale parameters, representing the average signal magnitude.

mean1 = 1.1363
mean2 = 1.5349

RicianDistribution: m = median (pd)

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

Example: 1


 ## Compute the median for different Rician distributions
 data1 = ricernd (1, 0.5, [10000, 1]);
 data2 = ricernd (1, 1, [10000, 1]);
 pd1 = fitdist (data1, "Rician");
 pd2 = fitdist (data2, "Rician");
 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 signal data.

median1 = 1.1198
median2 = 1.4845

RicianDistribution: 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 Rician distribution
 rand ("seed", 21);
 data = ricernd (1, 1, [100, 1]);
 pd_fitted = fitdist (data, "Rician");
 nlogL = negloglik (pd_fitted)

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

nlogL = -113.17

RicianDistribution: ci = paramci (pd)
RicianDistribution: 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 Rician distribution
 rand ("seed", 21);
 data = ricernd (1, 1, [1000, 1]);
 pd_fitted = fitdist (data, "Rician");
 ci = paramci (pd_fitted, "Alpha", 0.05)

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

ci =

   0.7184   0.9400
   0.9412   1.2697

RicianDistribution: 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 Rician distribution
 x = 0:0.01:5;
 data1 = ricernd (1, 0.5, [10000, 1]);
 data2 = ricernd (1, 1, [10000, 1]);
 data3 = ricernd (1, 2, [10000, 1]);
 pd1 = fitdist (data1, "Rician");
 pd2 = fitdist (data2, "Rician");
 pd3 = fitdist (data3, "Rician");
 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 ({"s = 1, sigma = 0.5", "s = 1, sigma = 1", "s = 1, sigma = 2"}, ...
         "location", "northeast")
 title ("Rician PDF")
 xlabel ("values in x (x >= 0)")
 ylabel ("Probability density")

 ## This visualizes the probability density function for Rician distributions,
 ## showing the likelihood for non-negative signal magnitudes.

RicianDistribution: plot (pd)
RicianDistribution: plot (pd, Name, Value)
RicianDistribution: 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 Rician distribution with fixed parameters s = 1 and sigma = 1
 ## and plot its PDF.
 pd = RicianDistribution (1, 1);
 plot (pd)
 title ("Rician distribution with s = 1 and sigma = 1")

 ## Use this to visualize the PDF of a Rician distribution with fixed parameters,
 ## useful for understanding the shape of the distribution.

Example: 2


 ## Generate a data set of 100 random samples from a Rician distribution
 ## with parameters s = 1 and sigma = 1. Fit a Rician distribution to this
 ## data and plot its CDF superimposed over an empirical CDF.
 rand ("seed", 21);
 data = ricernd (1, 1, [100, 1]);
 pd_fitted = fitdist (data, "Rician");
 plot (pd_fitted, "PlotType", "cdf")
 txt = "Fitted Rician distribution with s = %0.2f and sigma = %0.2f";
 title (sprintf (txt, pd_fitted.s, 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 Rician distribution
 ## with parameters s = 1 and sigma = 1. Display a probability plot for the
 ## Rician distribution fit to the data.
 rand ("seed", 21);
 data = ricernd (1, 1, [200, 1]);
 pd_fitted = fitdist (data, "Rician");
 plot (pd_fitted, "PlotType", "probability")
 txt = strcat ("Probability plot of fitted Rician distribution", ...
               " with s = %0.2f and sigma = %0.2f");
 title (sprintf (txt, pd_fitted.s, 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 Rician model is appropriate.

RicianDistribution: [nlogL, param] = proflik (pd, pnum)
RicianDistribution: [nlogL, param] = proflik (pd, pnum, "Display", display)
RicianDistribution: [nlogL, param] = proflik (pd, pnum, setparam)
RicianDistribution: [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 Rician distribution, pnum = 1 selects the parameter s 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 scale parameter of a fitted
 ## Rician distribution
 rand ("seed", 21);
 data = ricernd (1, 1, [1000, 1]);
 pd_fitted = fitdist (data, "Rician");
 [nlogL, param] = proflik (pd_fitted, 2, "Display", "on");

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

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

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

When called with a single size argument, ricernd 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 Rician distribution
 rand ("seed", 21);
 samples = ricernd (1, 1, [500, 1]);
 hist (samples, 50)
 title ("Histogram of 500 random samples from Rician(s=1, sigma=1)")
 xlabel ("values in x (x >= 0)")
 ylabel ("Frequency")

 ## This generates random samples from a Rician distribution, useful for
 ## simulating signal magnitudes in applications like wireless communications.

RicianDistribution: s = std (pd)

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

Example: 1


 ## Compute the standard deviation for a Rician distribution
 data = ricernd (1, 1, [10000, 1]);
 pd = fitdist (data, "Rician");
 std_value = std (pd)

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

std_value = 0.7810

RicianDistribution: 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 Rician distribution, with parameters s = 1 and sigma = 1,
 ## truncated at [0.5, 3] intervals. Generate 10000 random samples from this
 ## truncated distribution and superimpose a histogram scaled accordingly
 rand ("seed", 21);
 data_all = ricernd (1, 1, [20000, 1]);
 data = data_all(data_all >= 0.5 & data_all <= 3);
 data = data(1:10000);

 pd = fitdist (data, "Rician");
 t = truncate (pd, 0.5, 3);

 [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 ("Rician distribution (s = 1, sigma = 1) truncated at [0.5, 3]")
 legend ("Truncated PDF", "Histogram")

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

RicianDistribution: v = var (pd)

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

Example: 1


 ## Compute the variance for a Rician distribution
 data = ricernd (1, 1, [10000, 1]);
 pd = fitdist (data, "Rician");
 var_value = var (pd)

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

var_value = 0.6012

Examples

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

pd_fixed =
  RicianDistribution

  Rician distribution
        s = 2
    sigma = 1

pd_fitted =
  RicianDistribution

  Rician distribution
        s = 2.01215   [1.982, 2.04275]
    sigma = 1.01103   [0.986101, 1.0366]

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Class Definition: RicianDistribution

statistics: RicianDistribution

Properties

s

Example: 1

Example: 2

sigma

Example: 1

Example: 2

Example: 3

DistributionName

NumParameters

ParameterNames

ParameterDescription

ParameterValues

ParameterCovariance

ParameterIsFixed

Truncation

IsTruncated

InputData

Methods

cdf

RicianDistribution: p = cdf (pd, x)

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

Example: 1

icdf

RicianDistribution: x = icdf (pd, p)

Example: 1

iqr

RicianDistribution: r = iqr (pd)

Example: 1

mean

RicianDistribution: m = mean (pd)

Example: 1

median

RicianDistribution: m = median (pd)

Example: 1

negloglik

RicianDistribution: nlogL = negloglik (pd)

Example: 1

Class Definition: `RicianDistribution`

`s`

`sigma`

`DistributionName`

`NumParameters`

`ParameterNames`

`ParameterDescription`

`ParameterValues`

`ParameterCovariance`

`ParameterIsFixed`

`Truncation`

`IsTruncated`

`InputData`

`cdf`

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

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

`icdf`

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

`iqr`

`RicianDistribution: r = iqr (pd)`

`mean`

`RicianDistribution: m = mean (pd)`

`median`

`RicianDistribution: m = median (pd)`

`negloglik`

`RicianDistribution: nlogL = negloglik (pd)`

`paramci`

`RicianDistribution: ci = paramci (pd)`

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

`pdf`

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

`plot`

`RicianDistribution: plot (pd)`

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

`RicianDistribution: h = plot (…)`

`proflik`

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

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

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

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

`random`

`RicianDistribution: r = random (pd)`

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

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

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

`std`

`RicianDistribution: s = std (pd)`

`truncate`

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

`var`

`RicianDistribution: v = var (pd)`