BetaDistribution
Beta probability distribution object.
A BetaDistribution
object consists of parameters, a model
description, and sample data for a beta probability distribution.
The beta distribution is a family of continuous probability distributions
defined on the interval in terms of two positive parameters,
denoted by alpha (a)
and beta (b)
, that appear
as exponents of the variable and its complement to 1, respectively, and
control the shape of the distribution.
There are several ways to create a BetaDistribution
object.
fitdist
function.
makedist
function.
BetaDistribution (a, b)
to create a beta distribution with specified parameter values.
BetaDistribution.fit (x,
censor, freq, options)
to a distribution to data x.
It is highly recommended to use fitdist
and makedist
functions to create probability distribution objects, instead of the
constructor and the aforementioned static method.
Further information about the Beta distribution can be found at https://en.wikipedia.org/wiki/Beta_distribution
See also: fitdist, makedist, betacdf, betainv, betapdf, betarnd, lognfit, betalike, betastat
Source Code: BetaDistribution
First shape parameter
A positive scalar value characterizing the shape of the beta
distribution. You can access the a
property using dot name
assignment.
## Create a beta distribution with default parameters pd = makedist ("beta") ## Query parameter 'alpha' pd.a ## Set parameter 'alpha' pd.a = 2 pd = BetaDistribution beta distribution a = 1 b = 1 ans = 1 pd = BetaDistribution beta distribution a = 2 b = 1 |
## Create a beta distribution object by calling its constructor pd = BetaDistribution (2, 3) ## Query parameter 'alpha' pd.a pd = BetaDistribution beta distribution a = 2 b = 3 ans = 2 |
Second shape parameter
A positive scalar value characterizing the shape of the beta
distribution. You can access the a
property using dot name
assignment.
## Create a beta distribution with default parameters pd = makedist ("beta") ## Query parameter 'beta' pd.b ## Set parameter 'beta' pd.b = 2 pd = BetaDistribution beta distribution a = 1 b = 1 ans = 1 pd = BetaDistribution beta distribution a = 1 b = 2 |
## Create a beta distribution object by calling its constructor pd = BetaDistribution (2, 3) ## Query parameter 'beta' pd.b pd = BetaDistribution beta distribution a = 2 b = 3 ans = 3 |
Probability distribution name
A character vector specifying the name of the probability distribution object. This property is read-only.
Number of parameters
A scalar integer value specifying the number of parameters characterizing the probability distribution. This property is read-only.
Names of parameters
A cell array of character vectors with each element containing the name of a distribution parameter. This property is read-only.
Description of parameters
A cell array of character vectors with each element containing a short description of a distribution parameter. This property is read-only.
Distribution parameter values
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 a
and b
properties.
Covariance matrix of the parameter estimates
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.
## Generate a data set of 5000 random samples from a Beta distribution with ## parameters a = 2 and b = 4. Fit a Beta distribution to this data and plot ## a PDF of the fitted distribution superimposed on a histogram of the data pd = makedist ("Beta", "a", 2, "b", 4) randg ("seed", 21); data = random (pd, 5000, 1); pd = fitdist (data, "Beta") plot (pd) title (sprintf ("Fitted Beta distribution with a = %0.2f and b = %0.2f", ... pd.a, pd.b)) pd.ParameterCovariance pd = BetaDistribution beta distribution a = 2 b = 4 pd = BetaDistribution beta distribution a = 1.97369 [1.90215, 2.04792] b = 3.91487 [3.76164, 4.07435] ans = 1.3822e-03 2.4731e-03 2.4731e-03 6.3606e-03 |
Flag for fixed parameters
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.
Truncation interval
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.
Flag for truncated probability distribution
A logical scalar value specifying whether a probability distribution is truncated or not. This property is read-only.
Data used for fitting a probability distribution
A scalar structure containing the following fields:
data
: a numeric vector containing the data used for
distribution fitting.
cens
: an empty array, since BetaDistribution
does
not allow censoring.
frequency
: 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.
"upper"
)Compute the cumulative distribution function (CDF).
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"
)
Compute the inverse cumulative distribution function (iCDF).
p = icdf (pd, x)
computes the quantile (the
inverse of the CDF) of the probability distribution object, pd,
evaluated at the values in x.
Compute the interquartile range of a probability distribution.
r = iqr (pd)
computes the interquartile range of the
probability distribution object, pd.
Compute the mean of a probability distribution.
m = mean (pd)
computes the mean of the probability
distribution object, pd.
Compute the median of a probability distribution.
m = median (pd)
computes the median of the probability
distribution object, pd.
Compute the negative loglikelihood of a probability distribution.
m = negloglik (pd)
computes the negative loglikelihood
of the probability distribution object, pd.
Compute the confidence intervals for probability distribution parameters.
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 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 the probability distribution function (PDF).
y = pdf (pd, x)
computes the PDF of the
probability distribution object, pd, evaluated at the values in
x.
Plot a probability distribution object.
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.
## Generate a data set of 100 random samples from a Beta distribution with ## parameters a = 2 and b = 4. Fit a Beta distribution to this data and plot ## its CDF superimposed over an empirical CDF of the data pd = makedist ("Beta", "a", 2, "b", 4) randg ("seed", 21); data = random (pd, 100, 1); pd = fitdist (data, "Beta") plot (pd, "plottype", "cdf") title (sprintf ("Fitted Beta distribution with a = %0.2f and b = %0.2f", ... pd.a, pd.b)) legend ({"empirical CDF", "fitted CDF"}, "location", "east") pd = BetaDistribution beta distribution a = 2 b = 4 pd = BetaDistribution beta distribution a = 1.58844 [1.2184, 2.07086] b = 3.3298 [2.43615, 4.55126] |
## Generate a data set of 200 random samples from a Beta distribution with ## parameters a = 2 and b = 4. Display a probability plot for the Beta ## distribution fit to the data. pd = makedist ("Beta", "a", 2, "b", 4) randg ("seed", 21); data = random (pd, 200, 1); pd = fitdist (data, "Beta") plot (pd, "plottype", "probability") title (sprintf ("Probability plot of a fitted Beta distribution with a = %0.2f and b = %0.2f", ... pd.a, pd.b)) legend ({"empirical CDF", "fitted CDF"}, "location", "southeast") pd = BetaDistribution beta distribution a = 2 b = 4 pd = BetaDistribution beta distribution a = 1.68235 [1.42401, 1.98756] b = 3.7591 [3.1139, 4.53798] |
"Display"
, display)"Display"
, display)Profile likelihood function for a probability distribution object.
[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 beta distribution, pnum = 1
selects the parameter
a
and pnum = 2
selects the parameter b.
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.
Generate random arrays from the probability distribution object.
r = random (pd)
returns a random number from the
distribution object pd.
When called with a single size argument, betarnd
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.
Compute the standard deviation of a probability distribution.
s = std (pd)
computes the standard deviation of the
probability distribution object, pd.
Truncate a probability distribution.
t = truncate (pd)
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 Beta distribution, with parameters a = 2 and b = 4, ## truncated at [0.1, 0.8] intervals. Generate 10000 random samples from ## this truncated distribution and superimpose a histogram with 100 bins ## scaled accordingly pd = makedist ("Beta", "a", 2, "b", 4) t = truncate (pd, 0.1, 0.8) randg ("seed", 21); data = random (t, 10000, 1); plot (t) title ("Beta distribution (a = 2, b = 4) truncated at [0.1, 0.8]") hold on hist (data, 100, 140) hold off pd = BetaDistribution beta distribution a = 2 b = 4 t = BetaDistribution beta distribution a = 2 b = 4 Truncated to the interval [0.1, 0.8] |
Compute the variance of a probability distribution.
v = var (pd)
computes the standard deviation of the
probability distribution object, pd.