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.
fitdist
function.
makedist
function.
RicianDistribution (s, sigma)
to create a Rician distribution with fixed parameter values s and
sigma.
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
A non-negative scalar value characterizing the noncentrality of the
Rician distribution. You can access the s
property using dot
name assignment.
## 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 |
## 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.
## 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 |
## 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 |
## 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 cell array of character vectors with each element containing the name of a distribution parameter. This property is read-only.
A cell array of character vectors with each element containing a short description of a distribution parameter. This property is read-only.
A 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 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 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 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.
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 (…,
returns the complement of
the CDF of the probability distribution object, pd, evaluated at
the values in x.
"upper"
)
## 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.
## 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.
## 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.
## 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.
## 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.
## 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 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.
## 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.
## 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.
## 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. |
## 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. |
## 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,
also plots the profile likelihood
against the default range of the selected parameter.
"Display"
, "on"
)
[nlogL, param] = proflik (pd, pnum,
setparam)
defines a user-defined range of the selected parameter.
[nlogL, param] = proflik (pd, pnum,
setparam,
also plots the profile
likelihood against the user-defined range of the selected parameter.
"Display"
, "on"
)
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.
## 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.
## 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.
## 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.
## 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.
## 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 |
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] |