/usr/share/octave/packages/statistics-1.2.3/doc-cache is in octave-statistics 1.2.3-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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anderson_darling_cdf
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-- Function File: P = anderson_darling_cdf (A, N)
Return the CDF for the given Anderson-Darling coefficient A
computed from N values sampled from a distribution. For a vector
of random variables X of length N, compute the CDF of the values
from the distribution from which they are drawn. You can uses
these values to compute A as follows:
A = -N - sum( (2*i-1) .* (log(X) + log(1 - X(N:-1:1,:))) )/N;
From the value A, 'anderson_darling_cdf' returns the probability
that A could be returned from a set of samples.
The algorithm given in [1] claims to be an approximation for the
Anderson-Darling CDF accurate to 6 decimal points.
Demonstrate using:
n = 300; reps = 10000;
z = randn(n, reps);
x = sort ((1 + erf (z/sqrt (2)))/2);
i = [1:n]' * ones (1, size (x, 2));
A = -n - sum ((2*i-1) .* (log (x) + log (1 - x (n:-1:1, :))))/n;
p = anderson_darling_cdf (A, n);
hist (100 * p, [1:100] - 0.5);
You will see that the histogram is basically flat, which is to say
that the probabilities returned by the Anderson-Darling CDF are
distributed uniformly.
You can easily determine the extreme values of P:
[junk, idx] = sort (p);
The histograms of various P aren't very informative:
histfit (z (:, idx (1)), linspace (-3, 3, 15));
histfit (z (:, idx (end/2)), linspace (-3, 3, 15));
histfit (z (:, idx (end)), linspace (-3, 3, 15));
More telling is the qqplot:
qqplot (z (:, idx (1))); hold on; plot ([-3, 3], [-3, 3], ';;'); hold off;
qqplot (z (:, idx (end/2))); hold on; plot ([-3, 3], [-3, 3], ';;'); hold off;
qqplot (z (:, idx (end))); hold on; plot ([-3, 3], [-3, 3], ';;'); hold off;
Try a similarly analysis for Z uniform:
z = rand (n, reps); x = sort(z);
and for Z exponential:
z = rande (n, reps); x = sort (1 - exp (-z));
[1] Marsaglia, G; Marsaglia JCW; (2004) "Evaluating the Anderson
Darling distribution", Journal of Statistical Software, 9(2).
See also: anderson_darling_test
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Return the CDF for the given Anderson-Darling coefficient A computed
from N valu
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anderson_darling_test
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-- Function File: [Q, ASQ, INFO] = = anderson_darling_test (X,
DISTRIBUTION)
Test the hypothesis that X is selected from the given distribution
using the Anderson-Darling test. If the returned Q is small,
reject the hypothesis at the Q*100% level.
The Anderson-Darling A^2 statistic is calculated as follows:
n
A^2_n = -n - SUM (2i-1)/n log(z_i (1 - z_{n-i+1}))
i=1
where z_i is the ordered position of the X's in the CDF of the
distribution. Unlike the Kolmogorov-Smirnov statistic, the
Anderson-Darling statistic is sensitive to the tails of the
distribution.
The DISTRIBUTION argument must be a either "uniform", "normal", or
"exponential".
For "normal"' and "exponential" distributions, estimate the
distribution parameters from the data, convert the values to CDF
values, and compare the result to tabluated critical values. This
includes an correction for small N which works well enough for N >=
8, but less so from smaller N. The returned 'info.Asq_corrected'
contains the adjusted statistic.
For "uniform", assume the values are uniformly distributed in
(0,1), compute A^2 and return the corresponding p-value from
'1-anderson_darling_cdf(A^2,n)'.
If you are selecting from a known distribution, convert your values
into CDF values for the distribution and use "uniform". Do not use
"uniform" if the distribution parameters are estimated from the
data itself, as this sharply biases the A^2 statistic toward
smaller values.
[1] Stephens, MA; (1986), "Tests based on EDF statistics", in
D'Agostino, RB; Stephens, MA; (eds.) Goodness-of-fit Techinques.
New York: Dekker.
See also: anderson_darling_cdf
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Test the hypothesis that X is selected from the given distribution using
the And
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anovan
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-- Function File: [PVAL, F, DF_B, DF_E] = anovan (DATA, GRPS)
-- Function File: [PVAL, F, DF_B, DF_E] = anovan (DATA, GRPS, 'param1',
VALUE1)
Perform a multi-way analysis of variance (ANOVA). The goal is to
test whether the population means of data taken from K different
groups are all equal.
Data is a single vector DATA with groups specified by a
corresponding matrix of group labels GRPS, where GRPS has the same
number of rows as DATA. For example, if DATA = [1.1;1.2]; GRPS=
[1,2,1; 1,5,2]; then data point 1.1 was measured under conditions
1,2,1 and data point 1.2 was measured under conditions 1,5,2. Note
that groups do not need to be sequentially numbered.
By default, a 'linear' model is used, computing the N main effects
with no interactions. this may be modified by param 'model'
p= anovan(data,groups, 'model', modeltype) - modeltype = 'linear':
compute N main effects - modeltype = 'interaction': compute N
effects and N*(N-1) two-factor interactions - modeltype = 'full':
compute interactions at all levels
Under the null of constant means, the statistic F follows an F
distribution with DF_B and DF_E degrees of freedom.
The p-value (1 minus the CDF of this distribution at F) is returned
in PVAL.
If no output argument is given, the standard one-way ANOVA table is
printed.
BUG: DFE is incorrect for modeltypes != full
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Perform a multi-way analysis of variance (ANOVA).
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betastat
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-- Function File: [M, V] = betastat (A, B)
Compute mean and variance of the beta distribution.
Arguments
--------------
* A is the first parameter of the beta distribution. A must be
positive
* B is the second parameter of the beta distribution. B must be
positive
A and B must be of common size or one of them must be scalar
Return values
------------------
* M is the mean of the beta distribution
* V is the variance of the beta distribution
Examples
-------------
a = 1:6;
b = 1:0.2:2;
[m, v] = betastat (a, b)
[m, v] = betastat (a, 1.5)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
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Compute mean and variance of the beta distribution.
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binostat
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-- Function File: [M, V] = binostat (N, P)
Compute mean and variance of the binomial distribution.
Arguments
--------------
* N is the first parameter of the binomial distribution. The
elements of N must be natural numbers
* P is the second parameter of the binomial distribution. The
elements of P must be probabilities
N and P must be of common size or one of them must be scalar
Return values
------------------
* M is the mean of the binomial distribution
* V is the variance of the binomial distribution
Examples
-------------
n = 1:6;
p = 0:0.2:1;
[m, v] = binostat (n, p)
[m, v] = binostat (n, 0.5)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
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Compute mean and variance of the binomial distribution.
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boxplot
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-- Function File: S = boxplot (DATA, NOTCHED, SYMBOL, VERTICAL,
MAXWHISKER, ...)
-- Function File: [... H]= boxplot (...)
Produce a box plot.
The box plot is a graphical display that simultaneously describes
several important features of a data set, such as center, spread,
departure from symmetry, and identification of observations that
lie unusually far from the bulk of the data.
DATA is a matrix with one column for each data set, or data is a
cell vector with one cell for each data set.
NOTCHED = 1 produces a notched-box plot. Notches represent a
robust estimate of the uncertainty about the median.
NOTCHED = 0 (default) produces a rectangular box plot.
NOTCHED in (0,1) produces a notch of the specified depth. notched
values outside (0,1) are amusing if not exactly practical.
SYMBOL sets the symbol for the outlier values, default symbol for
points that lie outside 3 times the interquartile range is 'o',
default symbol for points between 1.5 and 3 times the interquartile
range is '+'.
SYMBOL = '.' points between 1.5 and 3 times the IQR is marked with
'.' and points outside 3 times IQR with 'o'.
SYMBOL = ['x','*'] points between 1.5 and 3 times the IQR is marked
with 'x' and points outside 3 times IQR with '*'.
VERTICAL = 0 makes the boxes horizontal, by default VERTICAL = 1.
MAXWHISKER defines the length of the whiskers as a function of the
IQR (default = 1.5). If MAXWHISKER = 0 then 'boxplot' displays all
data values outside the box using the plotting symbol for points
that lie outside 3 times the IQR.
Supplemental arguments are concatenated and passed to plot.
The returned matrix S has one column for each data set as follows:
1 Minimum
2 1st quartile
3 2nd quartile (median)
4 3rd quartile
5 Maximum
6 Lower confidence limit for median
7 Upper confidence limit for median
The returned structure H has hanldes to the plot elements, allowing
customization of the visualization using set/get functions.
Example
title ("Grade 3 heights");
axis ([0,3]);
tics ("x", 1:2, {"girls"; "boys"});
boxplot ({randn(10,1)*5+140, randn(13,1)*8+135});
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Produce a box plot.
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caseread
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-- Function File: NAMES = caseread (FILENAME)
Read case names from an ascii file.
Essentially, this reads all lines from a file as text and returns
them in a string matrix.
See also: casewrite, tblread, tblwrite, csv2cell, cell2csv, fopen
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Read case names from an ascii file.
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casewrite
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-- Function File: casewrite (STRMAT, FILENAME)
Write case names to an ascii file.
Essentially, this writes all lines from STRMAT to FILENAME (after
deblanking them).
See also: caseread, tblread, tblwrite, csv2cell, cell2csv, fopen
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Write case names to an ascii file.
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chi2stat
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-- Function File: [M, V] = chi2stat (N)
Compute mean and variance of the chi-square distribution.
Arguments
--------------
* N is the parameter of the chi-square distribution. The
elements of N must be positive
Return values
------------------
* M is the mean of the chi-square distribution
* V is the variance of the chi-square distribution
Example
------------
n = 1:6;
[m, v] = chi2stat (n)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
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Compute mean and variance of the chi-square distribution.
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cl_multinom
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-- Function File: CL = cl_multinom (X, N, B, CALCULATION_TYPE ) -
Confidence level of multinomial portions
Returns confidence level of multinomial parameters estimated p = x
/ sum(x) with predefined confidence interval B. Finite population
is also considered.
This function calculates the level of confidence at which the
samples represent the true distribution given that there is a
predefined tolerance (confidence interval). This is the upside
down case of the typical excercises at which we want to get the
confidence interval given the confidence level (and the estimated
parameters of the underlying distribution). But once we accept
(lets say at elections) that we have a standard predefined maximal
acceptable error rate (e.g. B=0.02 ) in the estimation and we just
want to know that how sure we can be that the measured proportions
are the same as in the entire population (ie. the expected value
and mean of the samples are roghly the same) we need to use this
function.
Arguments
--------------
* X : int vector : sample frequencies bins
* N : int : Population size that was sampled by x. If N<sum(x),
infinite number assumed
* B : real, vector : confidence interval if vector, it should be
the size of x containing confence interval for each cells if
scalar, each cell will have the same value of b unless it is
zero or -1 if value is 0, b=.02 is assumed which is standard
choice at elections otherwise it is calculated in a way that
one sample in a cell alteration defines the confidence
interval
* CALCULATION_TYPE : string : (Optional), described below
"bromaghin" (default) - do not change it unless you have a
good reason to do so "cochran" "agresti_cull" this is not
exactly the solution at reference given below but an
adjustment of the solutions above
Returns
------------
Confidence level.
Example
------------
CL = cl_multinom( [27;43;19;11], 10000, 0.05 ) returns 0.69
confidence level.
References
---------------
"bromaghin" calculation type (default) is based on is based on the
article Jeffrey F. Bromaghin, "Sample Size Determination for
Interval Estimation of Multinomial Probabilities", The American
Statistician vol 47, 1993, pp 203-206.
"cochran" calculation type is based on article Robert T. Tortora,
"A Note on Sample Size Estimation for Multinomial Populations", The
American Statistician, , Vol 32. 1978, pp 100-102.
"agresti_cull" calculation type is based on article in which
Quesenberry Hurst and Goodman result is combined A. Agresti and
B.A. Coull, "Approximate is better than \"exact\" for interval
estimation of binomial portions", The American Statistician, Vol.
52, 1998, pp 119-126
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Returns confidence level of multinomial parameters estimated p = x /
sum(x) with
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cmdscale
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-- Function File: Y = cmdscale (D)
-- Function File: [ Y , E] = cmdscale (D)
Classical multidimensional scaling of a matrix. Also known as
principal coordinates analysis.
Given an N by N Euclidean distance matrix D, find N points in P
dimensional space which have this distance matrix. The coordinates
of the points Y are returned.
D should be a full distance matrix (hollow, symmetric, entries
obeying the triangle inequality), or can be a vector of length
'n(n-1)/2' containing the upper triangular elements of the distance
matrix (such as that returned by the pdist function). If D is not
a valid distance matrix, points Y will be returned whose distance
matrix approximates D.
The returned Y is an N by P matrix showing possible coordinates of
the points in P dimensional space ('p < n'). Of course, any
translation, rotation, or reflection of these would also have the
same distance matrix.
Can also return the eigenvalues E of '(D(1, :) .^ 2 + D(:, 1) .^ 2
- D .^ 2) / 2', where the number of positive eigenvalues determines
P.
Reference: Rudolf Mathar (1985), The best Euclidian fit to a given
distance matrix in prescribed dimensions, Linear Algebra and its
Applications, 67: 1-6, doi: 10.1016/0024-3795(85)90181-8
See also: pdist, squareform
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Classical multidimensional scaling of a matrix.
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combnk
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# elements: 1
# length: 93
-- Function File: C = combnk (DATA, K)
Return all combinations of K elements in DATA.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 46
Return all combinations of K elements in DATA.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
copulacdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2185
-- Function File: P = copulacdf (FAMILY, X, THETA)
-- Function File: copulacdf ('t', X, THETA, NU)
Compute the cumulative distribution function of a copula family.
Arguments
--------------
* FAMILY is the copula family name. Currently, FAMILY can be
''Gaussian'' for the Gaussian family, ''t'' for the Student's
t family, ''Clayton'' for the Clayton family, ''Gumbel'' for
the Gumbel-Hougaard family, ''Frank'' for the Frank family,
''AMH'' for the Ali-Mikhail-Haq family, or ''FGM'' for the
Farlie-Gumbel-Morgenstern family.
* X is the support where each row corresponds to an observation.
* THETA is the parameter of the copula. For the Gaussian and
Student's t copula, THETA must be a correlation matrix. For
bivariate copulas THETA can also be a correlation coefficient.
For the Clayton family, the Gumbel-Hougaard family, the Frank
family, and the Ali-Mikhail-Haq family, THETA must be a vector
with the same number of elements as observations in X or be
scalar. For the Farlie-Gumbel-Morgenstern family, THETA must
be a matrix of coefficients for the Farlie-Gumbel-Morgenstern
polynomial where each row corresponds to one set of
coefficients for an observation in X. A single row is
expanded. The coefficients are in binary order.
* NU is the degrees of freedom for the Student's t family. NU
must be a vector with the same number of elements as
observations in X or be scalar.
Return values
------------------
* P is the cumulative distribution of the copula at each row of
X and corresponding parameter THETA.
Examples
-------------
x = [0.2:0.2:0.6; 0.2:0.2:0.6];
theta = [1; 2];
p = copulacdf ("Clayton", x, theta)
x = [0.2:0.2:0.6; 0.2:0.1:0.4];
theta = [0.2, 0.1, 0.1, 0.05];
p = copulacdf ("FGM", x, theta)
References
---------------
1. Roger B. Nelsen. 'An Introduction to Copulas'. Springer, New
York, second edition, 2006.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 64
Compute the cumulative distribution function of a copula family.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
copulapdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1496
-- Function File: P = copulapdf (FAMILY, X, THETA)
Compute the probability density function of a copula family.
Arguments
--------------
* FAMILY is the copula family name. Currently, FAMILY can be
''Clayton'' for the Clayton family, ''Gumbel'' for the
Gumbel-Hougaard family, ''Frank'' for the Frank family, or
''AMH'' for the Ali-Mikhail-Haq family.
* X is the support where each row corresponds to an observation.
* THETA is the parameter of the copula. The elements of THETA
must be greater than or equal to '-1' for the Clayton family,
greater than or equal to '1' for the Gumbel-Hougaard family,
arbitrary for the Frank family, and greater than or equal to
'-1' and lower than '1' for the Ali-Mikhail-Haq family.
Moreover, THETA must be non-negative for dimensions greater
than '2'. THETA must be a column vector with the same number
of rows as X or be scalar.
Return values
------------------
* P is the probability density of the copula at each row of X
and corresponding parameter THETA.
Examples
-------------
x = [0.2:0.2:0.6; 0.2:0.2:0.6];
theta = [1; 2];
p = copulapdf ("Clayton", x, theta)
p = copulapdf ("Gumbel", x, 2)
References
---------------
1. Roger B. Nelsen. 'An Introduction to Copulas'. Springer, New
York, second edition, 2006.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 60
Compute the probability density function of a copula family.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
copularnd
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1888
-- Function File: X = copularnd (FAMILY, THETA, N)
-- Function File: copularnd (FAMILY, THETA, N, D)
-- Function File: copularnd ('t', THETA, NU, N)
Generate random samples from a copula family.
Arguments
--------------
* FAMILY is the copula family name. Currently, FAMILY can be
''Gaussian'' for the Gaussian family, ''t'' for the Student's
t family, or ''Clayton'' for the Clayton family.
* THETA is the parameter of the copula. For the Gaussian and
Student's t copula, THETA must be a correlation matrix. For
bivariate copulas THETA can also be a correlation coefficient.
For the Clayton family, THETA must be a vector with the same
number of elements as samples to be generated or be scalar.
* NU is the degrees of freedom for the Student's t family. NU
must be a vector with the same number of elements as samples
to be generated or be scalar.
* N is the number of rows of the matrix to be generated. N must
be a non-negative integer and corresponds to the number of
samples to be generated.
* D is the number of columns of the matrix to be generated. D
must be a positive integer and corresponds to the dimension of
the copula.
Return values
------------------
* X is a matrix of random samples from the copula with N samples
of distribution dimension D.
Examples
-------------
theta = 0.5;
x = copularnd ("Gaussian", theta);
theta = 0.5;
nu = 2;
x = copularnd ("t", theta, nu);
theta = 0.5;
n = 2;
x = copularnd ("Clayton", theta, n);
References
---------------
1. Roger B. Nelsen. 'An Introduction to Copulas'. Springer, New
York, second edition, 2006.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 45
Generate random samples from a copula family.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
dendogram
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 578
-- Function File: P = dendogram (TREE)
-- Function File: P, T = dendogram (TREE)
-- Function File: P, T, PERM = dendogram (TREE)
Plots a dendogram using the output of function 'linkage'.
t is a vector containing the leaf node number for each object in
the original dataset. For now, all objects are leaf nodes.
perm is the permutation of the input objects used to display the
dendrogram, in left-to-right order.
TODO: Return handle to lines to set properties TODO: Rescale the
plot automatically based on data.
See also: linkage
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 57
Plots a dendogram using the output of function 'linkage'.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
expstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 846
-- Function File: [M, V] = expstat (L)
Compute mean and variance of the exponential distribution.
Arguments
--------------
* L is the parameter of the exponential distribution. The
elements of L must be positive
Return values
------------------
* M is the mean of the exponential distribution
* V is the variance of the exponential distribution
Example
------------
l = 1:6;
[m, v] = expstat (l)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 58
Compute mean and variance of the exponential distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4
ff2n
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 100
-- Function File: ff2n ( N)
Full-factor design with n binary terms.
See also: fullfact
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 39
Full-factor design with n binary terms.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
fstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1167
-- Function File: [MN, V] = fstat (M, N)
Compute mean and variance of the F distribution.
Arguments
--------------
* M is the first parameter of the F distribution. The elements
of M must be positive
* N is the second parameter of the F distribution. The elements
of N must be positive
M and N must be of common size or one of them must be scalar
Return values
------------------
* MN is the mean of the F distribution. The mean is undefined
for N not greater than 2
* V is the variance of the F distribution. The variance is
undefined for N not greater than 4
Examples
-------------
m = 1:6;
n = 5:10;
[mn, v] = fstat (m, n)
[mn, v] = fstat (m, 5)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Compute mean and variance of the F distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
fullfact
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 264
-- Function File: fullfact ( N)
Full factorial design.
If N is a scalar, return the full factorial design with N binary
choices, 0 and 1.
If N is a vector, return the full factorial design with choices 1
through N_I for each factor I.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 22
Full factorial design.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gamfit
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 296
-- Function File: MLE = gamfit (DATA)
Calculate gamma distribution parameters.
Find the maximum likelihood estimators (MLEs) of the Gamma
distribution of DATA. MLE is a two element vector with shape
parameter A and scale B.
See also: gampdf, gaminv, gamrnd, gamlike
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 40
Calculate gamma distribution parameters.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
gamlike
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 225
-- Function File: X = gamlike ([A B], R)
Calculates the negative log-likelihood function for the Gamma
distribution over vector R, with the given parameters A and B.
See also: gampdf, gaminv, gamrnd, gamfit
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Calculates the negative log-likelihood function for the Gamma
distribution over
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
gamstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1040
-- Function File: [M, V] = gamstat (A, B)
Compute mean and variance of the gamma distribution.
Arguments
--------------
* A is the first parameter of the gamma distribution. A must be
positive
* B is the second parameter of the gamma distribution. B must
be positive
A and B must be of common size or one of them must be scalar
Return values
------------------
* M is the mean of the gamma distribution
* V is the variance of the gamma distribution
Examples
-------------
a = 1:6;
b = 1:0.2:2;
[m, v] = gamstat (a, b)
[m, v] = gamstat (a, 1.5)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 52
Compute mean and variance of the gamma distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
geomean
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 178
-- Function File: geomean ( X)
-- Function File: geomean ( X, DIM)
Compute the geometric mean.
This function does the same as 'mean (x, "g")'.
See also: mean
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 27
Compute the geometric mean.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
geostat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 855
-- Function File: [M, V] = geostat (P)
Compute mean and variance of the geometric distribution.
Arguments
--------------
* P is the rate parameter of the geometric distribution. The
elements of P must be probabilities
Return values
------------------
* M is the mean of the geometric distribution
* V is the variance of the geometric distribution
Example
------------
p = 1 ./ (1:6);
[m, v] = geostat (p)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 56
Compute mean and variance of the geometric distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevcdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1261
-- Function File: P = gevcdf (X, K, SIGMA, MU)
Compute the cumulative distribution function of the generalized
extreme value (GEV) distribution.
Arguments
--------------
* X is the support.
* K is the shape parameter of the GEV distribution. (Also
denoted gamma or xi.)
* SIGMA is the scale parameter of the GEV distribution. The
elements of SIGMA must be positive.
* MU is the location parameter of the GEV distribution.
The inputs must be of common size, or some of them must be scalar.
Return values
------------------
* P is the cumulative distribution of the GEV distribution at
each element of X and corresponding parameter values.
Examples
-------------
x = 0:0.5:2.5;
sigma = 1:6;
k = 1;
mu = 0;
y = gevcdf (x, k, sigma, mu)
y = gevcdf (x, k, 0.5, mu)
References
---------------
1. Rolf-Dieter Reiss and Michael Thomas. 'Statistical Analysis
of Extreme Values with Applications to Insurance, Finance,
Hydrology and Other Fields'. Chapter 1, pages 16-17,
Springer, 2007.
See also: gevfit, gevinv, gevlike, gevpdf, gevrnd, gevstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Compute the cumulative distribution function of the generalized extreme
value (G
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevfit
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1159
-- Function File: PARAMHAT, PARAMCI = gevfit (DATA, PARMGUESS)
Find the maximum likelihood estimator (PARAMHAT) of the generalized
extreme value (GEV) distribution to fit DATA.
Arguments
--------------
* DATA is the vector of given values.
* PARMGUESS is an initial guess for the maximum likelihood
parameter vector. If not given, this defaults to [0; 1; 0].
Return values
------------------
* PARMHAT is the 3-parameter maximum-likelihood parameter vector
[K; SIGMA; MU], where K is the shape parameter of the GEV
distribution, SIGMA is the scale parameter of the GEV
distribution, and MU is the location parameter of the GEV
distribution.
* PARAMCI has the approximate 95% confidence intervals of the
parameter values based on the Fisher information matrix at the
maximum-likelihood position.
Examples
-------------
data = 1:50;
[pfit, pci] = gevfit (data);
p1 = gevcdf(data,pfit(1),pfit(2),pfit(3));
plot(data, p1)
See also: gevcdf, gevinv, gevlike, gevpdf, gevrnd, gevstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Find the maximum likelihood estimator (PARAMHAT) of the generalized
extreme valu
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
gevfit_lmom
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1194
-- Function File: PARAMHAT, PARAMCI = gevfit_lmom (DATA)
Find an estimator (PARAMHAT) of the generalized extreme value (GEV)
distribution fitting DATA using the method of L-moments.
Arguments
--------------
* DATA is the vector of given values.
Return values
------------------
* PARMHAT is the 3-parameter maximum-likelihood parameter vector
[K; SIGMA; MU], where K is the shape parameter of the GEV
distribution, SIGMA is the scale parameter of the GEV
distribution, and MU is the location parameter of the GEV
distribution.
* PARAMCI has the approximate 95% confidence intervals of the
parameter values (currently not implemented).
Examples
-------------
data = gevrnd (0.1, 1, 0, 100, 1);
[pfit, pci] = gevfit_lmom (data);
p1 = gevcdf (data,pfit(1),pfit(2),pfit(3));
[f, x] = ecdf (data);
plot(data, p1, 's', x, f)
See also: gevfit
References
---------------
1. Ailliot, P.; Thompson, C. & Thomson, P. Mixed methods for
fitting the GEV distribution, Water Resources Research, 2011,
47, W05551
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Find an estimator (PARAMHAT) of the generalized extreme value (GEV)
distribution
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevinv
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1215
-- Function File: X = gevinv (P, K, SIGMA, MU)
Compute a desired quantile (inverse CDF) of the generalized extreme
value (GEV) distribution.
Arguments
--------------
* P is the desired quantile of the GEV distribution. (Between 0
and 1.)
* K is the shape parameter of the GEV distribution. (Also
denoted gamma or xi.)
* SIGMA is the scale parameter of the GEV distribution. The
elements of SIGMA must be positive.
* MU is the location parameter of the GEV distribution.
The inputs must be of common size, or some of them must be scalar.
Return values
------------------
* X is the value corresponding to each quantile of the GEV
distribution
References
---------------
1. Rolf-Dieter Reiss and Michael Thomas. 'Statistical Analysis
of Extreme Values with Applications to Insurance, Finance,
Hydrology and Other Fields'. Chapter 1, pages 16-17,
Springer, 2007.
2. J. R. M. Hosking (2012). 'L-moments'. R package, version
1.6. URL: http://CRAN.R-project.org/package=lmom.
See also: gevcdf, gevfit, gevlike, gevpdf, gevrnd, gevstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Compute a desired quantile (inverse CDF) of the generalized extreme
value (GEV)
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
gevlike
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1420
-- Function File: NLOGL, GRAD, ACOV = gevlike (PARAMS, DATA)
Compute the negative log-likelihood of data under the generalized
extreme value (GEV) distribution with given parameter values.
Arguments
--------------
* PARAMS is the 3-parameter vector [K, SIGMA, MU], where K is
the shape parameter of the GEV distribution, SIGMA is the
scale parameter of the GEV distribution, and MU is the
location parameter of the GEV distribution.
* DATA is the vector of given values.
Return values
------------------
* NLOGL is the negative log-likelihood.
* GRAD is the 3 by 1 gradient vector (first derivative of the
negative log likelihood with respect to the parameter values)
* ACOV is the 3 by 3 Fisher information matrix (second
derivative of the negative log likelihood with respect to the
parameter values)
Examples
-------------
x = -5:-1;
k = -0.2;
sigma = 0.3;
mu = 0.5;
[L, ~, C] = gevlike ([k sigma mu], x);
References
---------------
1. Rolf-Dieter Reiss and Michael Thomas. 'Statistical Analysis
of Extreme Values with Applications to Insurance, Finance,
Hydrology and Other Fields'. Chapter 1, pages 16-17,
Springer, 2007.
See also: gevcdf, gevfit, gevinv, gevpdf, gevrnd, gevstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Compute the negative log-likelihood of data under the generalized
extreme value
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevpdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1253
-- Function File: Y = gevpdf (X, K, SIGMA, MU)
Compute the probability density function of the generalized extreme
value (GEV) distribution.
Arguments
--------------
* X is the support.
* K is the shape parameter of the GEV distribution. (Also
denoted gamma or xi.)
* SIGMA is the scale parameter of the GEV distribution. The
elements of SIGMA must be positive.
* MU is the location parameter of the GEV distribution.
The inputs must be of common size, or some of them must be scalar.
Return values
------------------
* Y is the probability density of the GEV distribution at each
element of X and corresponding parameter values.
Examples
-------------
x = 0:0.5:2.5;
sigma = 1:6;
k = 1;
mu = 0;
y = gevpdf (x, k, sigma, mu)
y = gevpdf (x, k, 0.5, mu)
References
---------------
1. Rolf-Dieter Reiss and Michael Thomas. 'Statistical Analysis
of Extreme Values with Applications to Insurance, Finance,
Hydrology and Other Fields'. Chapter 1, pages 16-17,
Springer, 2007.
See also: gevcdf, gevfit, gevinv, gevlike, gevrnd, gevstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Compute the probability density function of the generalized extreme
value (GEV)
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
gevrnd
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 858
-- Function File: gevrnd (K, SIGMA, MU)
-- Function File: gevrnd (K, SIGMA, MU, R)
-- Function File: gevrnd (K, SIGMA, MU, R, C, ...)
-- Function File: gevrnd (K, SIGMA, MU, [SZ])
Return a matrix of random samples from the generalized extreme
value (GEV) distribution with parameters K, SIGMA, MU.
When called with a single size argument, 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 vector SZ
of dimensions.
If no size arguments are given, then the result matrix is the
common size of the input parameters.
See also: gevcdf, gevfit, gevinv, gevlike, gevpdf, gevstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Return a matrix of random samples from the generalized extreme value
(GEV) distr
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
gevstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 732
-- Function File: [M, V] = gevstat (K, SIGMA, MU)
Compute the mean and variance of the generalized extreme value
(GEV) distribution.
Arguments
--------------
* K is the shape parameter of the GEV distribution. (Also
denoted gamma or xi.)
* SIGMA is the scale parameter of the GEV distribution. The
elements of SIGMA must be positive.
* MU is the location parameter of the GEV distribution.
The inputs must be of common size, or some of them must be scalar.
Return values
------------------
* M is the mean of the GEV distribution
* V is the variance of the GEV distribution
See also: gevcdf, gevfit, gevinv, gevlike, gevpdf, gevrnd
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Compute the mean and variance of the generalized extreme value (GEV)
distributio
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
harmmean
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 179
-- Function File: harmmean ( X)
-- Function File: harmmean ( X, DIM)
Compute the harmonic mean.
This function does the same as 'mean (x, "h")'.
See also: mean
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 26
Compute the harmonic mean.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
hist3
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1340
-- Function File: hist3( X )
-- Function File: hist3( X , NBINS)
-- Function File: hist3( X , 'Nbins', NBINS)
-- Function File: hist3( X , CENTERS)
-- Function File: hist3( X , 'Centers', CENTERS)
-- Function File: hist3( X , 'Edges', EDGES)
-- Function File: N = hist3( X, ...)
-- Function File: [N, C] = hist3( X, ...)
Plots a 2D histogram of the N x 2 matrix X with 10 equally spaced
bins in both the x and y direction using the 'mesh' function
The number of equally spaced bins to compute histogram can be
specified with NBINS. If NBINS is a 2 element vector, use the two
values as the number of bins in the x and y axis, respectively,
otherwise, use the same value for each.
The centers of the histogram bins can be specified with CENTERS.
CENTERS should be a cell array containing two arrays of the bin
centers on the x and y axis, respectively.
The edges of the histogram bins can be specified with EDGES. EDGES
should be a cell array containing two arrays of the bin edges on
the x and y axis, respectively.
N returns the 2D array of bin counts, and does not plot the
histogram
N and C returns the 2D array of bin counts in N and the bin centers
in the 2 element cell array C, and does not plot the histogram
See also: hist, mesh
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Plots a 2D histogram of the N x 2 matrix X with 10 equally spaced bins
in both t
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
histfit
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 413
-- Function File: histfit ( DATA, NBINS)
Plot histogram with superimposed fitted normal density.
'histfit (DATA, NBINS)' plots a histogram of the values in the
vector DATA using NBINS bars in the histogram. With one input
argument, NBINS is set to the square root of the number of elements
in data.
Example
histfit (randn (100, 1))
See also: bar, hist, pareto
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 55
Plot histogram with superimposed fitted normal density.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
hmmestimate
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 4214
-- Function File: [TRANSPROBEST, OUTPROBEST] = hmmestimate (SEQUENCE,
STATES)
-- Function File: hmmestimate (..., 'statenames', STATENAMES)
-- Function File: hmmestimate (..., 'symbols', SYMBOLS)
-- Function File: hmmestimate (..., 'pseudotransitions',
PSEUDOTRANSITIONS)
-- Function File: hmmestimate (..., 'pseudoemissions', PSEUDOEMISSIONS)
Estimate the matrix of transition probabilities and the matrix of
output probabilities of a given sequence of outputs and states
generated by a hidden Markov model. The model assumes that the
generation starts in state '1' at step '0' but does not include
step '0' in the generated states and sequence.
Arguments
--------------
* SEQUENCE is a vector of a sequence of given outputs. The
outputs must be integers ranging from '1' to the number of
outputs of the hidden Markov model.
* STATES is a vector of the same length as SEQUENCE of given
states. The states must be integers ranging from '1' to the
number of states of the hidden Markov model.
Return values
------------------
* TRANSPROBEST is the matrix of the estimated transition
probabilities of the states. 'transprobest(i, j)' is the
estimated probability of a transition to state 'j' given state
'i'.
* OUTPROBEST is the matrix of the estimated output
probabilities. 'outprobest(i, j)' is the estimated
probability of generating output 'j' given state 'i'.
If ''symbols'' is specified, then SEQUENCE is expected to be a
sequence of the elements of SYMBOLS instead of integers. SYMBOLS
can be a cell array.
If ''statenames'' is specified, then STATES is expected to be a
sequence of the elements of STATENAMES instead of integers.
STATENAMES can be a cell array.
If ''pseudotransitions'' is specified then the integer matrix
PSEUDOTRANSITIONS is used as an initial number of counted
transitions. 'pseudotransitions(i, j)' is the initial number of
counted transitions from state 'i' to state 'j'. TRANSPROBEST will
have the same size as PSEUDOTRANSITIONS. Use this if you have
transitions that are very unlikely to occur.
If ''pseudoemissions'' is specified then the integer matrix
PSEUDOEMISSIONS is used as an initial number of counted outputs.
'pseudoemissions(i, j)' is the initial number of counted outputs
'j' given state 'i'. If ''pseudoemissions'' is also specified then
the number of rows of PSEUDOEMISSIONS must be the same as the
number of rows of PSEUDOTRANSITIONS. OUTPROBEST will have the same
size as PSEUDOEMISSIONS. Use this if you have outputs or states
that are very unlikely to occur.
Examples
-------------
transprob = [0.8, 0.2; 0.4, 0.6];
outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1];
[sequence, states] = hmmgenerate (25, transprob, outprob);
[transprobest, outprobest] = hmmestimate (sequence, states)
symbols = {'A', 'B', 'C'};
statenames = {'One', 'Two'};
[sequence, states] = hmmgenerate (25, transprob, outprob,
'symbols', symbols, 'statenames', statenames);
[transprobest, outprobest] = hmmestimate (sequence, states,
'symbols', symbols,
'statenames', statenames)
pseudotransitions = [8, 2; 4, 6];
pseudoemissions = [2, 4, 4; 7, 2, 1];
[sequence, states] = hmmgenerate (25, transprob, outprob);
[transprobest, outprobest] = hmmestimate (sequence, states, 'pseudotransitions', pseudotransitions, 'pseudoemissions', pseudoemissions)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Lawrence R. Rabiner. A Tutorial on Hidden Markov Models and
Selected Applications in Speech Recognition. 'Proceedings of
the IEEE', 77(2), pages 257-286, February 1989.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Estimate the matrix of transition probabilities and the matrix of output
probabi
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
hmmgenerate
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2471
-- Function File: [SEQUENCE, STATES] = hmmgenerate (LEN, TRANSPROB,
OUTPROB)
-- Function File: hmmgenerate (..., 'symbols', SYMBOLS)
-- Function File: hmmgenerate (..., 'statenames', STATENAMES)
Generate an output sequence and hidden states of a hidden Markov
model. The model starts in state '1' at step '0' but will not
include step '0' in the generated states and sequence.
Arguments
--------------
* LEN is the number of steps to generate. SEQUENCE and STATES
will have LEN entries each.
* TRANSPROB is the matrix of transition probabilities of the
states. 'transprob(i, j)' is the probability of a transition
to state 'j' given state 'i'.
* OUTPROB is the matrix of output probabilities. 'outprob(i,
j)' is the probability of generating output 'j' given state
'i'.
Return values
------------------
* SEQUENCE is a vector of length LEN of the generated outputs.
The outputs are integers ranging from '1' to 'columns
(outprob)'.
* STATES is a vector of length LEN of the generated hidden
states. The states are integers ranging from '1' to 'columns
(transprob)'.
If ''symbols'' is specified, then the elements of SYMBOLS are used
for the output sequence instead of integers ranging from '1' to
'columns (outprob)'. SYMBOLS can be a cell array.
If ''statenames'' is specified, then the elements of STATENAMES are
used for the states instead of integers ranging from '1' to
'columns (transprob)'. STATENAMES can be a cell array.
Examples
-------------
transprob = [0.8, 0.2; 0.4, 0.6];
outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1];
[sequence, states] = hmmgenerate (25, transprob, outprob)
symbols = {'A', 'B', 'C'};
statenames = {'One', 'Two'};
[sequence, states] = hmmgenerate (25, transprob, outprob,
'symbols', symbols, 'statenames', statenames)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Lawrence R. Rabiner. A Tutorial on Hidden Markov Models and
Selected Applications in Speech Recognition. 'Proceedings of
the IEEE', 77(2), pages 257-286, February 1989.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 71
Generate an output sequence and hidden states of a hidden Markov model.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
hmmviterbi
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2624
-- Function File: VPATH = hmmviterbi (SEQUENCE, TRANSPROB, OUTPROB)
-- Function File: hmmviterbi (..., 'symbols', SYMBOLS)
-- Function File: hmmviterbi (..., 'statenames', STATENAMES)
Use the Viterbi algorithm to find the Viterbi path of a hidden
Markov model given a sequence of outputs. The model assumes that
the generation starts in state '1' at step '0' but does not include
step '0' in the generated states and sequence.
Arguments
--------------
* SEQUENCE is the vector of length LEN of given outputs. The
outputs must be integers ranging from '1' to 'columns
(outprob)'.
* TRANSPROB is the matrix of transition probabilities of the
states. 'transprob(i, j)' is the probability of a transition
to state 'j' given state 'i'.
* OUTPROB is the matrix of output probabilities. 'outprob(i,
j)' is the probability of generating output 'j' given state
'i'.
Return values
------------------
* VPATH is the vector of the same length as SEQUENCE of the
estimated hidden states. The states are integers ranging from
'1' to 'columns (transprob)'.
If ''symbols'' is specified, then SEQUENCE is expected to be a
sequence of the elements of SYMBOLS instead of integers ranging
from '1' to 'columns (outprob)'. SYMBOLS can be a cell array.
If ''statenames'' is specified, then the elements of STATENAMES are
used for the states in VPATH instead of integers ranging from '1'
to 'columns (transprob)'. STATENAMES can be a cell array.
Examples
-------------
transprob = [0.8, 0.2; 0.4, 0.6];
outprob = [0.2, 0.4, 0.4; 0.7, 0.2, 0.1];
[sequence, states] = hmmgenerate (25, transprob, outprob)
vpath = hmmviterbi (sequence, transprob, outprob)
symbols = {'A', 'B', 'C'};
statenames = {'One', 'Two'};
[sequence, states] = hmmgenerate (25, transprob, outprob,
'symbols', symbols, 'statenames', statenames)
vpath = hmmviterbi (sequence, transprob, outprob,
'symbols', symbols, 'statenames', statenames)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Lawrence R. Rabiner. A Tutorial on Hidden Markov Models and
Selected Applications in Speech Recognition. 'Proceedings of
the IEEE', 77(2), pages 257-286, February 1989.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Use the Viterbi algorithm to find the Viterbi path of a hidden Markov
model give
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
hygestat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1335
-- Function File: [MN, V] = hygestat (T, M, N)
Compute mean and variance of the hypergeometric distribution.
Arguments
--------------
* T is the total size of the population of the hypergeometric
distribution. The elements of T must be positive natural
numbers
* M is the number of marked items of the hypergeometric
distribution. The elements of M must be natural numbers
* N is the size of the drawn sample of the hypergeometric
distribution. The elements of N must be positive natural
numbers
T, M, and N must be of common size or scalar
Return values
------------------
* MN is the mean of the hypergeometric distribution
* V is the variance of the hypergeometric distribution
Examples
-------------
t = 4:9;
m = 0:5;
n = 1:6;
[mn, v] = hygestat (t, m, n)
[mn, v] = hygestat (t, m, 2)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 61
Compute mean and variance of the hypergeometric distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
iwishpdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 659
-- Function File: Y = iwishpdf (W, TAU, DF, LOG_Y=false)
Compute the probability density function of the Wishart
distribution
Inputs: A P x P matrix W where to find the PDF and the P x P
positive definite scale matrix TAU and scalar degrees of freedom
parameter DF characterizing the inverse Wishart distribution. (For
the density to be finite, need DF > (P - 1).) If the flag LOG_Y is
set, return the log probability density - this helps avoid
underflow when the numerical value of the density is very small
Output: Y is the probability density of Wishart(SIGMA, DF) at W.
See also: iwishrnd, wishpdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 69
Compute the probability density function of the Wishart distribution
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
iwishrnd
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1043
-- Function File: [ W[, DI]] = iwishrnd (PSI, DF[, DI][, N=1])
Return a random matrix sampled from the inverse Wishart
distribution with given parameters
Inputs: the P x P positive definite matrix TAU and scalar degrees
of freedom parameter DF (and optionally the transposed Cholesky
factor DI of SIGMA = 'inv(Tau)'). DF can be non-integer as long as
DF > D
Output: a random P x P matrix W from the inverse Wishart(TAU, DF)
distribution. ('inv(W)' is from the Wishart('inv(Tau)', DF)
distribution.) If N > 1, then W is P x P x N and holds N such
random matrices. (Optionally, the transposed Cholesky factor DI of
SIGMA is also returned.)
Averaged across many samples, the mean of W should approach TAU /
(DF - P - 1).
Reference: Yu-Cheng Ku and Peter Bloomfield (2010), Generating
Random Wishart Matrices with Fractional Degrees of Freedom in OX,
http://www.gwu.edu/~forcpgm/YuChengKu-030510final-WishartYu-ChengKu.pdf
See also: wishrnd, iwishpdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Return a random matrix sampled from the inverse Wishart distribution
with given
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
jackknife
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2010
-- Function File: JACKSTAT = jackknife (E, X, ...)
Compute jackknife estimates of a parameter taking one or more given
samples as parameters. In particular, E is the estimator to be
jackknifed as a function name, handle, or inline function, and X is
the sample for which the estimate is to be taken. The I-th entry
of JACKSTAT will contain the value of the estimator on the sample X
with its I-th row omitted.
jackstat(I) = E(X(1 : I - 1, I + 1 : length(X)))
Depending on the number of samples to be used, the estimator must
have the appropriate form: If only one sample is used, then the
estimator need not be concerned with cell arrays, for example
jackknifing the standard deviation of a sample can be performed
with 'JACKSTAT = jackknife (@std, rand (100, 1))'. If, however,
more than one sample is to be used, the samples must all be of
equal size, and the estimator must address them as elements of a
cell-array, in which they are aggregated in their order of
appearance:
JACKSTAT = jackknife(@(x) std(x{1})/var(x{2}), rand (100, 1), randn (100, 1)
If all goes well, a theoretical value P for the parameter is
already known, N is the sample size, 'T = N * E(X) - (N - 1) *
mean(JACKSTAT)', and 'V = sumsq(N * E(X) - (N - 1) * JACKSTAT - T)
/ (N * (N - 1))', then '(T-P)/sqrt(V)' should follow a
t-distribution with N-1 degrees of freedom.
Jackknifing is a well known method to reduce bias; further details
can be found in:
* Rupert G. Miller: The jackknife-a review; Biometrika (1974)
61(1): 1-15; doi:10.1093/biomet/61.1.1
* Rupert G. Miller: Jackknifing Variances; Ann. Math. Statist.
Volume 39, Number 2 (1968), 567-582;
doi:10.1214/aoms/1177698418
* M. H. Quenouille: Notes on Bias in Estimation; Biometrika Vol.
43, No. 3/4 (Dec., 1956), pp. 353-360;
doi:10.1093/biomet/43.3-4.353
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Compute jackknife estimates of a parameter taking one or more given
samples as p
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
jsucdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 262
-- Function File: jsucdf (X, ALPHA1, ALPHA2)
For each element of X, compute the cumulative distribution function
(CDF) at X of the Johnson SU distribution with shape parameters
ALPHA1 and ALPHA2.
Default values are ALPHA1 = 1, ALPHA2 = 1.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For each element of X, compute the cumulative distribution function
(CDF) at X o
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
jsupdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 258
-- Function File: jsupdf (X, ALPHA1, ALPHA2)
For each element of X, compute the probability density function
(PDF) at X of the Johnson SU distribution with shape parameters
ALPHA1 and ALPHA2.
Default values are ALPHA1 = 1, ALPHA2 = 1.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For each element of X, compute the probability density function (PDF) at
X of th
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
kmeans
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 134
-- Function File: [IDX, CENTERS] = kmeans (DATA, K, PARAM1, VALUE1,
...)
K-means clustering.
See also: linkage
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 19
K-means clustering.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
linkage
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2935
-- Function File: Y = linkage (D)
-- Function File: Y = linkage (D, METHOD)
-- Function File: Y = linkage (X, METHOD, METRIC)
-- Function File: Y = linkage (X, METHOD, ARGLIST)
Produce a hierarchical clustering dendrogram
D is the dissimilarity matrix relative to n observations, formatted
as a (n-1)*n/2x1 vector as produced by 'pdist'. Alternatively, X
contains data formatted for input to 'pdist', METRIC is a metric
for 'pdist' and ARGLIST is a cell array containing arguments that
are passed to 'pdist'.
'linkage' starts by putting each observation into a singleton
cluster and numbering those from 1 to n. Then it merges two
clusters, chosen according to METHOD, to create a new cluster
numbered n+1, and so on until all observations are grouped into a
single cluster numbered 2(n-1). Row k of the (m-1)x3 output matrix
relates to cluster n+k: the first two columns are the numbers of
the two component clusters and column 3 contains their distance.
METHOD defines the way the distance between two clusters is
computed and how they are recomputed when two clusters are merged:
'"single" (default)'
Distance between two clusters is the minimum distance between
two elements belonging each to one cluster. Produces a
cluster tree known as minimum spanning tree.
'"complete"'
Furthest distance between two elements belonging each to one
cluster.
'"average"'
Unweighted pair group method with averaging (UPGMA). The mean
distance between all pair of elements each belonging to one
cluster.
'"weighted"'
Weighted pair group method with averaging (WPGMA). When two
clusters A and B are joined together, the new distance to a
cluster C is the mean between distances A-C and B-C.
'"centroid"'
Unweighted Pair-Group Method using Centroids (UPGMC). Assumes
Euclidean metric. The distance between cluster centroids,
each centroid being the center of mass of a cluster.
'"median"'
Weighted pair-group method using centroids (WPGMC). Assumes
Euclidean metric. Distance between cluster centroids. When
two clusters are joined together, the new centroid is the
midpoint between the joined centroids.
'"ward"'
Ward's sum of squared deviations about the group mean (ESS).
Also known as minimum variance or inner squared distance.
Assumes Euclidean metric. How much the moment of inertia of
the merged cluster exceeds the sum of those of the individual
clusters.
*Reference* Ward, J. H. Hierarchical Grouping to Optimize an
Objective Function J. Am. Statist. Assoc. 1963, 58, 236-244,
<http://iv.slis.indiana.edu/sw/data/ward.pdf>.
See also: pdist, squareform
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 45
Produce a hierarchical clustering dendrogram
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
lognstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1078
-- Function File: [M, V] = lognstat (MU, SIGMA)
Compute mean and variance of the lognormal distribution.
Arguments
--------------
* MU is the first parameter of the lognormal distribution
* SIGMA is the second parameter of the lognormal distribution.
SIGMA must be positive or zero
MU and SIGMA must be of common size or one of them must be scalar
Return values
------------------
* M is the mean of the lognormal distribution
* V is the variance of the lognormal distribution
Examples
-------------
mu = 0:0.2:1;
sigma = 0.2:0.2:1.2;
[m, v] = lognstat (mu, sigma)
[m, v] = lognstat (0, sigma)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 56
Compute mean and variance of the lognormal distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 3
mad
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 771
-- Function File: mad ( X)
-- Function File: mad ( X, FLAG)
-- Function File: mad ( X, FLAG, DIM)
Compute the mean/median absolute deviation of X.
The mean absolute deviation is computed as
mean (abs (X - mean (X)))
and the median absolute deviation is computed as
median (abs (X - median (X)))
Elements of X containing NaN or NA values are ignored during
computations.
If FLAG is 0, the absolute mean deviation is computed, and if FLAG
is 1, the absolute median deviation is computed. By default FLAG
is 0.
This is done along the dimension DIM of X. If this variable is not
given, the mean/median absolute deviation s computed along the
smallest dimension of X.
See also: std
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 48
Compute the mean/median absolute deviation of X.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
mnpdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1688
-- Function File: Y = mnpdf (X, P)
Compute the probability density function of the multinomial
distribution.
Arguments
--------------
* X is vector with a single sample of a multinomial distribution
with parameter P or a matrix of random samples from
multinomial distributions. In the latter case, each row of X
is a sample from a multinomial distribution with the
corresponding row of P being its parameter.
* P is a vector with the probabilities of the categories or a
matrix with each row containing the probabilities of a
multinomial sample.
Return values
------------------
* Y is a vector of probabilites of the random samples X from the
multinomial distribution with corresponding parameter P. The
parameter N of the multinomial distribution is the sum of the
elements of each row of X. The length of Y is the number of
columns of X. If a row of P does not sum to '1', then the
corresponding element of Y will be 'NaN'.
Examples
-------------
x = [1, 4, 2];
p = [0.2, 0.5, 0.3];
y = mnpdf (x, p);
x = [1, 4, 2; 1, 0, 9];
p = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8];
y = mnpdf (x, p);
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Merran Evans, Nicholas Hastings and Brian Peacock.
'Statistical Distributions'. pages 134-136, Wiley, New York,
third edition, 2000.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 73
Compute the probability density function of the multinomial
distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
mnrnd
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2224
-- Function File: X = mnrnd (N, P)
-- Function File: X = mnrnd (N, P, S)
Generate random samples from the multinomial distribution.
Arguments
--------------
* N is the first parameter of the multinomial distribution. N
can be scalar or a vector containing the number of trials of
each multinomial sample. The elements of N must be
non-negative integers.
* P is the second parameter of the multinomial distribution. P
can be a vector with the probabilities of the categories or a
matrix with each row containing the probabilities of a
multinomial sample. If P has more than one row and N is
non-scalar, then the number of rows of P must match the number
of elements of N.
* S is the number of multinomial samples to be generated. S
must be a non-negative integer. If S is specified, then N
must be scalar and P must be a vector.
Return values
------------------
* X is a matrix of random samples from the multinomial
distribution with corresponding parameters N and P. Each row
corresponds to one multinomial sample. The number of columns,
therefore, corresponds to the number of columns of P. If S is
not specified, then the number of rows of X is the maximum of
the number of elements of N and the number of rows of P. If a
row of P does not sum to '1', then the corresponding row of X
will contain only 'NaN' values.
Examples
-------------
n = 10;
p = [0.2, 0.5, 0.3];
x = mnrnd (n, p);
n = 10 * ones (3, 1);
p = [0.2, 0.5, 0.3];
x = mnrnd (n, p);
n = (1:2)';
p = [0.2, 0.5, 0.3; 0.1, 0.1, 0.8];
x = mnrnd (n, p);
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Merran Evans, Nicholas Hastings and Brian Peacock.
'Statistical Distributions'. pages 134-136, Wiley, New York,
third edition, 2000.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 58
Generate random samples from the multinomial distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 15
monotone_smooth
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1785
-- Function File: YY = monotone_smooth (X, Y, H)
Produce a smooth monotone increasing approximation to a sampled
functional dependence y(x) using a kernel method (an Epanechnikov
smoothing kernel is applied to y(x); this is integrated to yield
the monotone increasing form. See Reference 1 for details.)
Arguments
--------------
* X is a vector of values of the independent variable.
* Y is a vector of values of the dependent variable, of the same
size as X. For best performance, it is recommended that the Y
already be fairly smooth, e.g. by applying a kernel smoothing
to the original values if they are noisy.
* H is the kernel bandwidth to use. If H is not given, a
"reasonable" value is computed.
Return values
------------------
* YY is the vector of smooth monotone increasing function values
at X.
Examples
-------------
x = 0:0.1:10;
y = (x .^ 2) + 3 * randn(size(x)); %typically non-monotonic from the added noise
ys = ([y(1) y(1:(end-1))] + y + [y(2:end) y(end)])/3; %crudely smoothed via
moving average, but still typically non-monotonic
yy = monotone_smooth(x, ys); %yy is monotone increasing in x
plot(x, y, '+', x, ys, x, yy)
References
---------------
1. Holger Dette, Natalie Neumeyer and Kay F. Pilz (2006), A
simple nonparametric estimator of a strictly monotone
regression function, 'Bernoulli', 12:469-490
2. Regine Scheder (2007), R Package 'monoProc', Version 1.0-6,
<http://cran.r-project.org/web/packages/monoProc/monoProc.pdf>
(The implementation here is based on the monoProc function
mono.1d)
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Produce a smooth monotone increasing approximation to a sampled
functional depen
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
mvncdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1240
-- Function File: P = mvncdf (X, MU, SIGMA)
-- Function File: mvncdf (A, X, MU, SIGMA)
-- Function File: [P, ERR] = mvncdf (...)
Compute the cumulative distribution function of the multivariate
normal distribution.
Arguments
--------------
* X is the upper limit for integration where each row
corresponds to an observation.
* MU is the mean.
* SIGMA is the correlation matrix.
* A is the lower limit for integration where each row
corresponds to an observation. A must have the same size as
X.
Return values
------------------
* P is the cumulative distribution at each row of X and A.
* ERR is the estimated error.
Examples
-------------
x = [1 2];
mu = [0.5 1.5];
sigma = [1.0 0.5; 0.5 1.0];
p = mvncdf (x, mu, sigma)
a = [-inf 0];
p = mvncdf (a, x, mu, sigma)
References
---------------
1. Alan Genz and Frank Bretz. Numerical Computation of
Multivariate t-Probabilities with Application to Power
Calculation of Multiple Constrasts. 'Journal of Statistical
Computation and Simulation', 63, pages 361-378, 1999.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Compute the cumulative distribution function of the multivariate normal
distribu
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
mvnpdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1600
-- Function File: Y = mvnpdf (X)
-- Function File: Y = mvnpdf (X, MU)
-- Function File: Y = mvnpdf (X, MU, SIGMA)
Compute multivariate normal pdf for X given mean MU and covariance
matrix SIGMA. The dimension of X is D x P, MU is 1 x P and SIGMA
is P x P. The normal pdf is defined as
1/Y^2 = (2 pi)^P |SIGMA| exp { (X-MU)' inv(SIGMA) (X-MU) }
*References*
NIST Engineering Statistics Handbook 6.5.4.2
http://www.itl.nist.gov/div898/handbook/pmc/section5/pmc542.htm
*Algorithm*
Using Cholesky factorization on the positive definite covariance
matrix:
R = chol (SIGMA);
where R'*R = SIGMA. Being upper triangular, the determinant of R
is trivially the product of the diagonal, and the determinant of
SIGMA is the square of this:
DET = prod (diag (R))^2;
The formula asks for the square root of the determinant, so no need
to square it.
The exponential argument A = X' * inv (SIGMA) * X
A = X' * inv (SIGMA) * X
= X' * inv (R' * R) * X
= X' * inv (R) * inv(R') * X
Given that inv (R') == inv(R)', at least in theory if not
numerically,
A = (X' / R) * (X'/R)' = sumsq (X'/R)
The interface takes the parameters to the multivariate normal in
columns rather than rows, so we are actually dealing with the
transpose:
A = sumsq (X/r)
and the final result is:
R = chol (SIGMA)
Y = (2*pi)^(-P/2) * exp (-sumsq ((X-MU)/R, 2)/2) / prod (diag (R))
See also: mvncdf, mvnrnd
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Compute multivariate normal pdf for X given mean MU and covariance
matrix SIGMA.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
mvnrnd
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 503
-- Function File: S = mvnrnd (MU, SIGMA)
-- Function File: S = mvnrnd (MU, SIGMA, N)
-- Function File: S = mvnrnd (..., TOL)
Draw N random D-dimensional vectors from a multivariate Gaussian
distribution with mean MU(NxD) and covariance matrix SIGMA(DxD).
MU must be N-by-D (or 1-by-D if N is given) or a scalar.
If the argument TOL is given the eigenvalues of SIGMA are checked
for positivity against -100*tol. The default value of tol is
'eps*norm (Sigma, "fro")'.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Draw N random D-dimensional vectors from a multivariate Gaussian
distribution wi
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
mvtcdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1251
-- Function File: P = mvtcdf (X, SIGMA, NU)
-- Function File: mvtcdf (A, X, SIGMA, NU)
-- Function File: [P, ERR] = mvtcdf (...)
Compute the cumulative distribution function of the multivariate
Student's t distribution.
Arguments
--------------
* X is the upper limit for integration where each row
corresponds to an observation.
* SIGMA is the correlation matrix.
* NU is the degrees of freedom.
* A is the lower limit for integration where each row
corresponds to an observation. A must have the same size as
X.
Return values
------------------
* P is the cumulative distribution at each row of X and A.
* ERR is the estimated error.
Examples
-------------
x = [1 2];
sigma = [1.0 0.5; 0.5 1.0];
nu = 4;
p = mvtcdf (x, sigma, nu)
a = [-inf 0];
p = mvtcdf (a, x, sigma, nu)
References
---------------
1. Alan Genz and Frank Bretz. Numerical Computation of
Multivariate t-Probabilities with Application to Power
Calculation of Multiple Constrasts. 'Journal of Statistical
Computation and Simulation', 63, pages 361-378, 1999.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Compute the cumulative distribution function of the multivariate
Student's t dis
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
mvtrnd
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2023
-- Function File: X = mvtrnd (SIGMA, NU)
-- Function File: X = mvtrnd (SIGMA, NU, N)
Generate random samples from the multivariate t-distribution.
Arguments
--------------
* SIGMA is the matrix of correlation coefficients. If there are
any non-unit diagonal elements then SIGMA will be normalized,
so that the resulting covariance of the obtained samples X
follows: 'cov (x) = nu/(nu-2) * sigma ./ (sqrt (diag (sigma) *
diag (sigma)))'. In order to obtain samples distributed
according to a standard multivariate t-distribution, SIGMA
must be equal to the identity matrix. To generate
multivariate t-distribution samples X with arbitrary
covariance matrix SIGMA, the following scaling might be used:
'x = mvtrnd (sigma, nu, n) * diag (sqrt (diag (sigma)))'.
* NU is the degrees of freedom for the multivariate
t-distribution. NU must be a vector with the same number of
elements as samples to be generated or be scalar.
* N is the number of rows of the matrix to be generated. N must
be a non-negative integer and corresponds to the number of
samples to be generated.
Return values
------------------
* X is a matrix of random samples from the multivariate
t-distribution with N row samples.
Examples
-------------
sigma = [1, 0.5; 0.5, 1];
nu = 3;
n = 10;
x = mvtrnd (sigma, nu, n);
sigma = [1, 0.5; 0.5, 1];
nu = [2; 3];
n = 2;
x = mvtrnd (sigma, nu, 2);
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Samuel Kotz and Saralees Nadarajah. 'Multivariate t
Distributions and Their Applications'. Cambridge University
Press, Cambridge, 2004.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 61
Generate random samples from the multivariate t-distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nanmax
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 378
-- Function File: [V, IDX] = nanmax (X)
-- Function File: [V, IDX] = nanmax (X, Y)
Find the maximal element while ignoring NaN values.
'nanmax' is identical to the 'max' function except that NaN values
are ignored. If all values in a column are NaN, the maximum is
returned as NaN rather than [].
See also: max, nansum, nanmin, nanmean, nanmedian
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 51
Find the maximal element while ignoring NaN values.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
nanmean
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 338
-- Function File: V = nanmean (X)
-- Function File: V = nanmean (X, DIM)
Compute the mean value while ignoring NaN values.
'nanmean' is identical to the 'mean' function except that NaN
values are ignored. If all values are NaN, the mean is returned as
NaN.
See also: mean, nanmin, nanmax, nansum, nanmedian
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 49
Compute the mean value while ignoring NaN values.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 9
nanmedian
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 354
-- Function File: V = nanmedian (X)
-- Function File: V = nanmedian (X, DIM)
Compute the median of data while ignoring NaN values.
This function is identical to the 'median' function except that NaN
values are ignored. If all values are NaN, the median is returned
as NaN.
See also: median, nanmin, nanmax, nansum, nanmean
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 53
Compute the median of data while ignoring NaN values.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nanmin
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 378
-- Function File: [V, IDX] = nanmin (X)
-- Function File: [V, IDX] = nanmin (X, Y)
Find the minimal element while ignoring NaN values.
'nanmin' is identical to the 'min' function except that NaN values
are ignored. If all values in a column are NaN, the minimum is
returned as NaN rather than [].
See also: min, nansum, nanmax, nanmean, nanmedian
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 51
Find the minimal element while ignoring NaN values.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nanstd
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 927
-- Function File: V = nanstd (X)
-- Function File: V = nanstd (X, OPT)
-- Function File: V = nanstd (X, OPT, DIM)
Compute the standard deviation while ignoring NaN values.
'nanstd' is identical to the 'std' function except that NaN values
are ignored. If all values are NaN, the standard deviation is
returned as NaN. If there is only a single non-NaN value, the
deviation is returned as 0.
The argument OPT determines the type of normalization to use.
Valid values are
0:
normalizes with N-1, provides the square root of best unbiased
estimator of the variance [default]
1:
normalizes with N, this provides the square root of the second
moment around the mean
The third argument DIM determines the dimension along which the
standard deviation is calculated.
See also: std, nanmin, nanmax, nansum, nanmedian, nanmean
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 57
Compute the standard deviation while ignoring NaN values.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nansum
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 344
-- Function File: V = nansum (X)
-- Function File: V = nansum (X, DIM)
Compute the sum while ignoring NaN values.
'nansum' is identical to the 'sum' function except that NaN values
are treated as 0 and so ignored. If all values are NaN, the sum is
returned as 0.
See also: sum, nanmin, nanmax, nanmean, nanmedian
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 42
Compute the sum while ignoring NaN values.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
nanvar
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 782
-- Function File: nanvar (X)
-- Function File: V = nanvar (X, OPT)
-- Function File: V = nanvar (X, OPT, DIM)
Compute the variance while ignoring NaN values.
For vector arguments, return the (real) variance of the values.
For matrix arguments, return a row vector containing the variance
for each column.
The argument OPT determines the type of normalization to use.
Valid values are
0:
Normalizes with N-1, provides the best unbiased estimator of
the variance [default].
1:
Normalizes with N, this provides the second moment around the
mean.
The third argument DIM determines the dimension along which the
variance is calculated.
See also: var, nanmean, nanstd, nanmax, nanmin
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 47
Compute the variance while ignoring NaN values.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
nbinstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1151
-- Function File: [M, V] = nbinstat (N, P)
Compute mean and variance of the negative binomial distribution.
Arguments
--------------
* N is the first parameter of the negative binomial
distribution. The elements of N must be natural numbers
* P is the second parameter of the negative binomial
distribution. The elements of P must be probabilities
N and P must be of common size or one of them must be scalar
Return values
------------------
* M is the mean of the negative binomial distribution
* V is the variance of the negative binomial distribution
Examples
-------------
n = 1:4;
p = 0.2:0.2:0.8;
[m, v] = nbinstat (n, p)
[m, v] = nbinstat (n, 0.5)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 64
Compute mean and variance of the negative binomial distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 22
normalise_distribution
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2094
-- Function File: NORMALISED = normalise_distribution (DATA)
-- Function File: NORMALISED = normalise_distribution (DATA,
DISTRIBUTION)
-- Function File: NORMALISED = normalise_distribution (DATA,
DISTRIBUTION, DIMENSION)
Transform a set of data so as to be N(0,1) distributed according to
an idea by van Albada and Robinson. This is achieved by first
passing it through its own cumulative distribution function (CDF)
in order to get a uniform distribution, and then mapping the
uniform to a normal distribution. The data must be passed as a
vector or matrix in DATA. If the CDF is unknown, then [] can be
passed in DISTRIBUTION, and in this case the empirical CDF will be
used. Otherwise, if the CDFs for all data are known, they can be
passed in DISTRIBUTION, either in the form of a single function
name as a string, or a single function handle, or a cell array
consisting of either all function names as strings, or all function
handles. In the latter case, the number of CDFs passed must match
the number of rows, or columns respectively, to normalise. If the
data are passed as a matrix, then the transformation will operate
either along the first non-singleton dimension, or along DIMENSION
if present.
Notes: The empirical CDF will map any two sets of data having the
same size and their ties in the same places after sorting to some
permutation of the same normalised data:
normalise_distribution([1 2 2 3 4])
=> -1.28 0.00 0.00 0.52 1.28
normalise_distribution([1 10 100 10 1000])
=> -1.28 0.00 0.52 0.00 1.28
Original source: S.J. van Albada, P.A. Robinson "Transformation of
arbitrary distributions to the normal distribution with application
to EEG test-retest reliability" Journal of Neuroscience Methods,
Volume 161, Issue 2, 15 April 2007, Pages 205-211 ISSN 0165-0270,
10.1016/j.jneumeth.2006.11.004.
(http://www.sciencedirect.com/science/article/pii/S0165027006005668)
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Transform a set of data so as to be N(0,1) distributed according to an
idea by v
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
normplot
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 444
-- Function File: normplot ( X)
Produce normal probability plot for each column of X.
The line joing the 1st and 3rd quantile is drawn on the graph. If
the underlying distribution is normal, the points will cluster
around this line.
Note that this function sets the title, xlabel, ylabel, axis, grid,
tics and hold properties of the graph. These need to be cleared
before subsequent graphs using 'clf'.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 53
Produce normal probability plot for each column of X.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
normstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1010
-- Function File: [MN, V] = normstat (M, S)
Compute mean and variance of the normal distribution.
Arguments
--------------
* M is the mean of the normal distribution
* S is the standard deviation of the normal distribution. S
must be positive
M and S must be of common size or one of them must be scalar
Return values
------------------
* MN is the mean of the normal distribution
* V is the variance of the normal distribution
Examples
-------------
m = 1:6;
s = 0:0.2:1;
[mn, v] = normstat (m, s)
[mn, v] = normstat (0, s)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 53
Compute mean and variance of the normal distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
pcacov
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 669
-- Function File: [COEFF] = pcacov(X)
-- Function File: [COEFF,LATENT] = pcacov(X)
-- Function File: [COEFF,LATENT,EXPLAINED] = pcacov(X)
* pcacov performs principal component analysis on the nxn
covariance matrix X
* COEFF : a nxn matrix with columns containing the principal
component coefficients
* LATENT : a vector containing the principal component variances
* EXPLAINED : a vector containing the percentage of the total
variance explained by each principal component
References
---------------
1. Jolliffe, I. T., Principal Component Analysis, 2nd Edition,
Springer, 2002
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
* pcacov performs principal component analysis on the nxn covariance
mat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
pcares
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 414
-- Function File: [RESIDUALS,RECONSTRUCTED] =pcares( X, NDIM)
* X : N x P Matrix with N observations and P variables, the
variables will be mean centered
* NDIM : Is a scalar indicating the number of principal
components to use and should be <= P
References
---------------
1. Jolliffe, I. T., Principal Component Analysis, 2nd Edition,
Springer, 2002
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
* X : N x P Matrix with N observations and P variables, the variables
wi
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
pdist
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2491
-- Function File: Y = pdist (X)
-- Function File: Y = pdist (X, METRIC)
-- Function File: Y = pdist (X, METRIC, METRICARG, ...)
Return the distance between any two rows in X.
X is the NxD matrix representing Q row vectors of size D.
The output is a dissimilarity matrix formatted as a row vector Y,
(n-1)*n/2 long, where the distances are in the order [(1, 2) (1, 3)
... (2, 3) ... (n-1, n)]. You can use the 'squareform' function to
display the distances between the vectors arranged into an NxN
matrix.
'metric' is an optional argument specifying how the distance is
computed. It can be any of the following ones, defaulting to
"euclidean", or a user defined function that takes two arguments X
and Y plus any number of optional arguments, where X is a row
vector and and Y is a matrix having the same number of columns as
X. 'metric' returns a column vector where row I is the distance
between X and row I of Y. Any additional arguments after the
'metric' are passed as metric (X, Y, METRICARG1, METRICARG2 ...).
Predefined distance functions are:
'"euclidean"'
Euclidean distance (default).
'"seuclidean"'
Standardized Euclidean distance. Each coordinate in the sum
of squares is inverse weighted by the sample variance of that
coordinate.
'"mahalanobis"'
Mahalanobis distance: see the function mahalanobis.
'"cityblock"'
City Block metric, aka Manhattan distance.
'"minkowski"'
Minkowski metric. Accepts a numeric parameter P: for P=1 this
is the same as the cityblock metric, with P=2 (default) it is
equal to the euclidean metric.
'"cosine"'
One minus the cosine of the included angle between rows, seen
as vectors.
'"correlation"'
One minus the sample correlation between points (treated as
sequences of values).
'"spearman"'
One minus the sample Spearman's rank correlation between
observations, treated as sequences of values.
'"hamming"'
Hamming distance: the quote of the number of coordinates that
differ.
'"jaccard"'
One minus the Jaccard coefficient, the quote of nonzero
coordinates that differ.
'"chebychev"'
Chebychev distance: the maximum coordinate difference.
See also: linkage, mahalanobis, squareform
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 46
Return the distance between any two rows in X.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
plsregress
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 523
-- Function File:
[XLOADINGS,YLOADINGS,XSCORES,YSCORES,COEFFICIENTS,FITTED] =
...
plsregress(X, Y, NCOMP)
* X: Matrix of observations
* Y: Is a vector or matrix of responses
* NCOMP: number of components used for modelling
* X and Y will be mean centered to improve accuracy
References
---------------
1. SIMPLS: An alternative approach to partial least squares
regression. Chemometrics and Intelligent Laboratory Systems
(1993)
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
plsregress(X, Y, NCOMP)
* X: Matrix of observations
* Y: Is a vector or ma
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
poisstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 864
-- Function File: [M, V] = poisstat (LAMBDA)
Compute mean and variance of the Poisson distribution.
Arguments
--------------
* LAMBDA is the parameter of the Poisson distribution. The
elements of LAMBDA must be positive
Return values
------------------
* M is the mean of the Poisson distribution
* V is the variance of the Poisson distribution
Example
------------
lambda = 1 ./ (1:6);
[m, v] = poisstat (lambda)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 54
Compute mean and variance of the Poisson distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
princomp
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1263
-- Function File: [COEFF] = princomp(X)
-- Function File: [COEFF,SCORE] = princomp(X)
-- Function File: [COEFF,SCORE,LATENT] = princomp(X)
-- Function File: [COEFF,SCORE,LATENT,TSQUARE] = princomp(X)
-- Function File: [...] = princomp(X,'econ')
* princomp performs principal component analysis on a NxP data
matrix X
* COEFF : returns the principal component coefficients
* SCORE : returns the principal component scores, the
representation of X in the principal component space
* LATENT : returns the principal component variances, i.e., the
eigenvalues of the covariance matrix X.
* TSQUARE : returns Hotelling's T-squared Statistic for each
observation in X
* [...] = princomp(X,'econ') returns only the elements of
latent that are not necessarily zero, and the corresponding
columns of COEFF and SCORE, that is, when n <= p, only the
first n-1. This can be significantly faster when p is much
larger than n. In this case the svd will be applied on the
transpose of the data matrix X
References
---------------
1. Jolliffe, I. T., Principal Component Analysis, 2nd Edition,
Springer, 2002
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
* princomp performs principal component analysis on a NxP data matrix
X
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
random
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 3138
-- Function File: R = random(NAME, ARG1)
-- Function File: R = random(NAME, ARG1, ARG2)
-- Function File: R = random(NAME, ARG1, ARG2, ARG3)
-- Function File: R = random(NAME, ..., S1, ...)
Generates pseudo-random numbers from a given one-, two-, or
three-parameter distribution.
The variable NAME must be a string that names the distribution from
which to sample. If this distribution is a one-parameter
distribution ARG1 should be supplied, if it is a two-paramter
distribution ARG2 must also be supplied, and if it is a
three-parameter distribution ARG3 must also be present. Any
arguments following the distribution paramters will determine the
size of the result.
As an example, the following code generates a 10 by 20 matrix
containing random numbers from a normal distribution with mean 5
and standard deviation 2.
R = random("normal", 5, 2, [10, 20]);
The variable NAME can be one of the following strings
"beta"
"beta distribution"
Samples are drawn from the Beta distribution.
"bino"
"binomial"
"binomial distribution"
Samples are drawn from the Binomial distribution.
"chi2"
"chi-square"
"chi-square distribution"
Samples are drawn from the Chi-Square distribution.
"exp"
"exponential"
"exponential distribution"
Samples are drawn from the Exponential distribution.
"f"
"f distribution"
Samples are drawn from the F distribution.
"gam"
"gamma"
"gamma distribution"
Samples are drawn from the Gamma distribution.
"geo"
"geometric"
"geometric distribution"
Samples are drawn from the Geometric distribution.
"hyge"
"hypergeometric"
"hypergeometric distribution"
Samples are drawn from the Hypergeometric distribution.
"logn"
"lognormal"
"lognormal distribution"
Samples are drawn from the Log-Normal distribution.
"nbin"
"negative binomial"
"negative binomial distribution"
Samples are drawn from the Negative Binomial distribution.
"norm"
"normal"
"normal distribution"
Samples are drawn from the Normal distribution.
"poiss"
"poisson"
"poisson distribution"
Samples are drawn from the Poisson distribution.
"rayl"
"rayleigh"
"rayleigh distribution"
Samples are drawn from the Rayleigh distribution.
"t"
"t distribution"
Samples are drawn from the T distribution.
"unif"
"uniform"
"uniform distribution"
Samples are drawn from the Uniform distribution.
"unid"
"discrete uniform"
"discrete uniform distribution"
Samples are drawn from the Uniform Discrete distribution.
"wbl"
"weibull"
"weibull distribution"
Samples are drawn from the Weibull distribution.
See also: rand, betarnd, binornd, chi2rnd, exprnd, frnd, gamrnd,
geornd, hygernd, lognrnd, nbinrnd, normrnd, poissrnd, raylrnd,
trnd, unifrnd, unidrnd, wblrnd
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Generates pseudo-random numbers from a given one-, two-, or
three-parameter dist
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylcdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1121
-- Function File: P = raylcdf (X, SIGMA)
Compute the cumulative distribution function of the Rayleigh
distribution.
Arguments
--------------
* X is the support. The elements of X must be non-negative.
* SIGMA is the parameter of the Rayleigh distribution. The
elements of SIGMA must be positive.
X and SIGMA must be of common size or one of them must be scalar.
Return values
------------------
* P is the cumulative distribution of the Rayleigh distribution
at each element of X and corresponding parameter SIGMA.
Examples
-------------
x = 0:0.5:2.5;
sigma = 1:6;
p = raylcdf (x, sigma)
p = raylcdf (x, 0.5)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. pages 104 and 148, McGraw-Hill, New
York, second edition, 1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 74
Compute the cumulative distribution function of the Rayleigh
distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylinv
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1179
-- Function File: X = raylinv (P, SIGMA)
Compute the quantile of the Rayleigh distribution. The quantile is
the inverse of the cumulative distribution function.
Arguments
--------------
* P is the cumulative distribution. The elements of P must be
probabilities.
* SIGMA is the parameter of the Rayleigh distribution. The
elements of SIGMA must be positive.
P and SIGMA must be of common size or one of them must be scalar.
Return values
------------------
* X is the quantile of the Rayleigh distribution at each element
of P and corresponding parameter SIGMA.
Examples
-------------
p = 0:0.1:0.5;
sigma = 1:6;
x = raylinv (p, sigma)
x = raylinv (p, 0.5)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. pages 104 and 148, McGraw-Hill, New
York, second edition, 1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 50
Compute the quantile of the Rayleigh distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylpdf
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1113
-- Function File: Y = raylpdf (X, SIGMA)
Compute the probability density function of the Rayleigh
distribution.
Arguments
--------------
* X is the support. The elements of X must be non-negative.
* SIGMA is the parameter of the Rayleigh distribution. The
elements of SIGMA must be positive.
X and SIGMA must be of common size or one of them must be scalar.
Return values
------------------
* Y is the probability density of the Rayleigh distribution at
each element of X and corresponding parameter SIGMA.
Examples
-------------
x = 0:0.5:2.5;
sigma = 1:6;
y = raylpdf (x, sigma)
y = raylpdf (x, 0.5)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. pages 104 and 148, McGraw-Hill, New
York, second edition, 1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 70
Compute the probability density function of the Rayleigh distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
raylrnd
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1527
-- Function File: X = raylrnd (SIGMA)
-- Function File: X = raylrnd (SIGMA, SZ)
-- Function File: X = raylrnd (SIGMA, R, C)
Generate a matrix of random samples from the Rayleigh distribution.
Arguments
--------------
* SIGMA is the parameter of the Rayleigh distribution. The
elements of SIGMA must be positive.
* SZ is the size of the matrix to be generated. SZ must be a
vector of non-negative integers.
* R is the number of rows of the matrix to be generated. R must
be a non-negative integer.
* C is the number of columns of the matrix to be generated. C
must be a non-negative integer.
Return values
------------------
* X is a matrix of random samples from the Rayleigh distribution
with corresponding parameter SIGMA. If neither SZ nor R and C
are specified, then X is of the same size as SIGMA.
Examples
-------------
sigma = 1:6;
x = raylrnd (sigma)
sz = [2, 3];
x = raylrnd (0.5, sz)
r = 2;
c = 3;
x = raylrnd (0.5, r, c)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. pages 104 and 148, McGraw-Hill, New
York, second edition, 1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 67
Generate a matrix of random samples from the Rayleigh distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
raylstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 859
-- Function File: [M, V] = raylstat (SIGMA)
Compute mean and variance of the Rayleigh distribution.
Arguments
--------------
* SIGMA is the parameter of the Rayleigh distribution. The
elements of SIGMA must be positive.
Return values
------------------
* M is the mean of the Rayleigh distribution.
* V is the variance of the Rayleigh distribution.
Example
------------
sigma = 1:6;
[m, v] = raylstat (sigma)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 55
Compute mean and variance of the Rayleigh distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
regress
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1321
-- Function File: [B, BINT, R, RINT, STATS] = regress (Y, X, [ALPHA])
Multiple Linear Regression using Least Squares Fit of Y on X with
the model 'y = X * beta + e'.
Here,
* 'y' is a column vector of observed values
* 'X' is a matrix of regressors, with the first column filled
with the constant value 1
* 'beta' is a column vector of regression parameters
* 'e' is a column vector of random errors
Arguments are
* Y is the 'y' in the model
* X is the 'X' in the model
* ALPHA is the significance level used to calculate the
confidence intervals BINT and RINT (see 'Return values'
below). If not specified, ALPHA defaults to 0.05
Return values are
* B is the 'beta' in the model
* BINT is the confidence interval for B
* R is a column vector of residuals
* RINT is the confidence interval for R
* STATS is a row vector containing:
* The R^2 statistic
* The F statistic
* The p value for the full model
* The estimated error variance
R and RINT can be passed to 'rcoplot' to visualize the residual
intervals and identify outliers.
NaN values in Y and X are removed before calculation begins.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Multiple Linear Regression using Least Squares Fit of Y on X with the
model 'y =
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
regress_gp
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1119
-- Function File: [M, K] = regress_gp (X, Y, SP)
-- Function File: [... YI DY] = sqp (..., XI)
Linear scalar regression using gaussian processes.
It estimates the model Y = X'*m for X R^D and Y in R. The
information about errors of the predictions
(interpolation/extrapolation) is given by the covarianve matrix K.
If D==1 the inputs must be column vectors, if D>1 then X is n-by-D,
with n the number of data points. SP defines the prior covariance
of M, it should be a (D+1)-by-(D+1) positive definite matrix, if it
is empty, the default is 'Sp = 100*eye(size(x,2)+1)'.
If XI inputs are provided, the model is evaluated and returned in
YI. The estimation of the variation of YI are given in DY.
Run 'demo regress_gp' to see an examples.
The function is a direc implementation of the formulae in pages
11-12 of Gaussian Processes for Machine Learning. Carl Edward
Rasmussen and Christopher K. I. Williams. The MIT Press, 2006.
ISBN 0-262-18253-X. available online at
<http://gaussianprocess.org/gpml/>.
See also: regress
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 50
Linear scalar regression using gaussian processes.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
repanova
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 696
-- Function File: [PVAL, TABLE, ST] = repanova (X, COND)
-- Function File: [PVAL, TABLE, ST] = repanova (X, COND, ['string' |
'cell'])
Perform a repeated measures analysis of variance (Repeated ANOVA).
X is formated such that each row is a subject and each column is a
condition.
condition is typically a point in time, say t=1 then t=2, etc
condition can also be thought of as groups.
The optional flag can be either 'cell' or 'string' and reflects the
format of the table returned. Cell is the default.
NaNs are ignored using nanmean and nanstd.
This fuction does not currently support multiple columns of the
same condition!
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 66
Perform a repeated measures analysis of variance (Repeated ANOVA).
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
runstest
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1123
-- Function File: H, P, STATS = runstest (X, V)
Runs test for detecting serial correlation in the vector X.
Arguments
--------------
* X is the vector of given values.
* V is the value to subtract from X to get runs (defaults to
'median(x)')
Return values
------------------
* H is true if serial correlation is detected at the 95%
confidence level (two-tailed), false otherwise.
* P is the probablity of obtaining a test statistic of the
magnitude found under the null hypothesis of no serial
correlation.
* STATS is the structure containing as fields the number of runs
NRUNS; the numbers of positive and negative values of 'x - v',
N1 and N0; and the test statistic Z.
Note: the large-sample normal approximation is used to find H and
P. This is accurate if N1, N0 are both greater than 10.
Reference: NIST Engineering Statistics Handbook, 1.3.5.13. Runs
Test for Detecting Non-randomness,
http://www.itl.nist.gov/div898/handbook/eda/section3/eda35d.htm
See also:
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 59
Runs test for detecting serial correlation in the vector X.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 10
squareform
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 383
-- Function File: Y = squareform (X)
-- Function File: Y = squareform (X, "tovector")
-- Function File: Y = squareform (X, "tomatrix")
Convert a vector from the pdist function into a square matrix or
from a square matrix back to the vector form.
The second argument is used to specify the output type in case
there is a single element.
See also: pdist
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Convert a vector from the pdist function into a square matrix or from a
square m
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 11
stepwisefit
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1255
-- Function File: X_USE, B, BINT, R, RINT, STATS = stepwisefit (Y, X,
PENTER = 0.05, PREMOVE = 0.1)
Linear regression with stepwise variable selection.
Arguments
--------------
* Y is an N by 1 vector of data to fit.
* X is an N by K matrix containing the values of K potential
predictors. No constant term should be included (one will
always be added to the regression automatically).
* PENTER is the maximum p-value to enter a new variable into the
regression (default: 0.05).
* PREMOVE is the minimum p-value to remove a variable from the
regression (default: 0.1).
Return values
------------------
* X_USE contains the indices of the predictors included in the
final regression model. The predictors are listed in the
order they were added, so typically the first ones listed are
the most significant.
* B, BINT, R, RINT, STATS are the results of '[b, bint, r, rint,
stats] = regress(y, [ones(size(y)) X(:, X_use)], penter);'
References
---------------
1. N. R. Draper and H. Smith (1966). 'Applied Regression
Analysis'. Wiley. Chapter 6.
See also: regress
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 51
Linear regression with stepwise variable selection.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
tabulate
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1886
-- Function File: TABLE = tabulate (DATA, EDGES)
Compute a frequency table.
For vector data, the function counts the number of values in data
that fall between the elements in the edges vector (which must
contain monotonically non-decreasing values). TABLE is a matrix.
The first column of TABLE is the number of bin, the second is the
number of instances in each class (absolute frequency). The third
column contains the percentage of each value (relative frequency)
and the fourth column contains the cumulative frequency.
If EDGES is missed the width of each class is unitary, if EDGES is
a scalar then represent the number of classes, or you can define
the width of each bin. TABLE(K, 2) will count the value DATA (I)
if EDGES (K) <= DATA (I) < EDGES (K+1). The last bin will count
the value of DATA (I) if EDGES(K) <= DATA (I) <= EDGES (K+1).
Values outside the values in EDGES are not counted. Use -inf and
inf in EDGES to include all values. Tabulate with no output
arguments returns a formatted table in the command window.
Example
sphere_radius = [1:0.05:2.5];
tabulate (sphere_radius)
Tabulate returns 2 bins, the first contains the sphere with radius
between 1 and 2 mm excluded, and the second one contains the sphere
with radius between 2 and 3 mm.
tabulate (sphere_radius, 10)
Tabulate returns ten bins.
tabulate (sphere_radius, [1, 1.5, 2, 2.5])
Tabulate returns three bins, the first contains the sphere with
radius between 1 and 1.5 mm excluded, the second one contains the
sphere with radius between 1.5 and 2 mm excluded, and the third
contains the sphere with radius between 2 and 2.5 mm.
bar (table (:, 1), table (:, 2))
draw histogram.
See also: bar, pareto
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 26
Compute a frequency table.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
tblread
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 771
-- Function File: [DATA, VARNAMES, CASENAMES] = tblread (FILENAME)
-- Function File: [DATA, VARNAMES, CASENAMES] = tblread (FILENAME,
DELIMETER)
Read tabular data from an ascii file.
DATA is read from an ascii data file named FILENAME with an
optional DELIMETER. The delimeter may be any single character or
* "space" " " (default)
* "tab" "\t"
* "comma" ","
* "semi" ";"
* "bar" "|"
The DATA is read starting at cell (2,2) where the VARNAMES form a
char matrix from the first row (starting at (1,2)) vertically
concatenated, and the CASENAMES form a char matrix read from the
first column (starting at (2,1)) vertically concatenated.
See also: tblwrite, csv2cell, cell2csv
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 37
Read tabular data from an ascii file.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
tblwrite
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 758
-- Function File: tblwrite (DATA, VARNAMES, CASENAMES, FILENAME)
-- Function File: tblwrite (DATA, VARNAMES, CASENAMES, FILENAME,
DELIMETER)
Write tabular data to an ascii file.
DATA is written to an ascii data file named FILENAME with an
optional DELIMETER. The delimeter may be any single character or
* "space" " " (default)
* "tab" "\t"
* "comma" ","
* "semi" ";"
* "bar" "|"
The DATA is written starting at cell (2,2) where the VARNAMES are a
char matrix or cell vector written to the first row (starting at
(1,2)), and the CASENAMES are a char matrix (or cell vector)
written to the first column (starting at (2,1)).
See also: tblread, csv2cell, cell2csv
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 36
Write tabular data to an ascii file.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
trimmean
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 387
-- Function File: A = trimmean (X, P)
Compute the trimmed mean.
The trimmed mean of X is defined as the mean of X excluding the
highest and lowest P percent of the data.
For example
mean ([-inf, 1:9, inf])
is NaN, while
trimmean ([-inf, 1:9, inf], 10)
excludes the infinite values, which make the result 5.
See also: mean
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 25
Compute the trimmed mean.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
tstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 842
-- Function File: [M, V] = tstat (N)
Compute mean and variance of the t (Student) distribution.
Arguments
--------------
* N is the parameter of the t (Student) distribution. The
elements of N must be positive
Return values
------------------
* M is the mean of the t (Student) distribution
* V is the variance of the t (Student) distribution
Example
------------
n = 3:8;
[m, v] = tstat (n)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 58
Compute mean and variance of the t (Student) distribution.
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
unidstat
# name: <cell-element>
# type: sq_string
# elements: 1
# length: 884
-- Function File: [M, V] = unidstat (N)
Compute mean and variance of the discrete uniform distribution.
Arguments
--------------
* N is the parameter of the discrete uniform distribution. The
elements of N must be positive natural numbers
Return values
------------------
* M is the mean of the discrete uniform distribution
* V is the variance of the discrete uniform distribution
Example
------------
n = 1:6;
[m, v] = unidstat (n)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
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Compute mean and variance of the discrete uniform distribution.
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unifstat
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-- Function File: [M, V] = unifstat (A, B)
Compute mean and variance of the continuous uniform distribution.
Arguments
--------------
* A is the first parameter of the continuous uniform
distribution
* B is the second parameter of the continuous uniform
distribution
A and B must be of common size or one of them must be scalar and A
must be less than B
Return values
------------------
* M is the mean of the continuous uniform distribution
* V is the variance of the continuous uniform distribution
Examples
-------------
a = 1:6;
b = 2:2:12;
[m, v] = unifstat (a, b)
[m, v] = unifstat (a, 10)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
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Compute mean and variance of the continuous uniform distribution.
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vmpdf
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-- Function File: THETA = vmpdf (X, MU, K)
Evaluates the Von Mises probability density function.
The Von Mises distribution has probability density function
f (X) = exp (K * cos (X - MU)) / Z ,
where Z is a normalisation constant. By default, MU is 0 and K is
1.
See also: vmrnd
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Evaluates the Von Mises probability density function.
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vmrnd
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-- Function File: THETA = vmrnd (MU, K)
-- Function File: THETA = vmrnd (MU, K, SZ)
Draw random angles from a Von Mises distribution with mean MU and
concentration K.
The Von Mises distribution has probability density function
f (X) = exp (K * cos (X - MU)) / Z ,
where Z is a normalisation constant.
The output, THETA, is a matrix of size SZ containing random angles
drawn from the given Von Mises distribution. By default, MU is 0
and K is 1.
See also: vmpdf
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Draw random angles from a Von Mises distribution with mean MU and
concentration
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wblstat
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-- Function File: [M, V] = wblstat (SCALE, SHAPE)
Compute mean and variance of the Weibull distribution.
Arguments
--------------
* SCALE is the scale parameter of the Weibull distribution.
SCALE must be positive
* SHAPE is the shape parameter of the Weibull distribution.
SHAPE must be positive
SCALE and SHAPE must be of common size or one of them must be
scalar
Return values
------------------
* M is the mean of the Weibull distribution
* V is the variance of the Weibull distribution
Examples
-------------
scale = 3:8;
shape = 1:6;
[m, v] = wblstat (scale, shape)
[m, v] = wblstat (6, shape)
References
---------------
1. Wendy L. Martinez and Angel R. Martinez. 'Computational
Statistics Handbook with MATLAB'. Appendix E, pages 547-557,
Chapman & Hall/CRC, 2001.
2. Athanasios Papoulis. 'Probability, Random Variables, and
Stochastic Processes'. McGraw-Hill, New York, second edition,
1984.
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Compute mean and variance of the Weibull distribution.
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wishpdf
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-- Function File: Y = wishpdf (W, SIGMA, DF, LOG_Y=false)
Compute the probability density function of the Wishart
distribution
Inputs: A P x P matrix W where to find the PDF. The P x P positive
definite matrix SIGMA and scalar degrees of freedom parameter DF
characterizing the Wishart distribution. (For the density to be
finite, need DF > (P - 1).) If the flag LOG_Y is set, return the
log probability density - this helps avoid underflow when the
numerical value of the density is very small
Output: Y is the probability density of Wishart(SIGMA, DF) at W.
See also: wishrnd, iwishpdf
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Compute the probability density function of the Wishart distribution
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wishrnd
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-- Function File: [ W[, D]] = wishrnd (SIGMA, DF[, D][, N=1])
Return a random matrix sampled from the Wishart distribution with
given parameters
Inputs: the P x P positive definite matrix SIGMA and scalar degrees
of freedom parameter DF (and optionally the Cholesky factor D of
SIGMA). DF can be non-integer as long as DF > P
Output: a random P x P matrix W from the Wishart(SIGMA, DF)
distribution. If N > 1, then W is P x P x N and holds N such
random matrices. (Optionally, the Cholesky factor D of SIGMA is
also returned.)
Averaged across many samples, the mean of W should approach
DF*SIGMA, and the variance of each element W_ij should approach
DF*(SIGMA_ij^2 + SIGMA_ii*SIGMA_jj)
Reference: Yu-Cheng Ku and Peter Bloomfield (2010), Generating
Random Wishart Matrices with Fractional Degrees of Freedom in OX,
http://www.gwu.edu/~forcpgm/YuChengKu-030510final-WishartYu-ChengKu.pdf
See also: iwishrnd, wishpdf
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Return a random matrix sampled from the Wishart distribution with given
paramete
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