octave-statistics 1.3.0-4 / usr / share / octave / packages / statistics-1.3.0 / doc-cache

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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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bbscdf


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 -- bbscdf (X, LOCATION, SCALE, SHAPE)
     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Birnbaum-Saunders distribution with parameters
     LOCATION, SCALE and SHAPE.


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For each element of X, compute the cumulative distribution function
(CDF) at X o



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bbsinv


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 -- bbsinv (X, LOCATION, SCALE, SHAPE)
     For each element of X, compute the quantile (the inverse of the
     CDF) at X of the Birnbaum-Saunders distribution with parameters
     LOCATION, SCALE, and SHAPE.


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For each element of X, compute the quantile (the inverse of the CDF) at
X of the



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bbspdf


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 -- bbspdf (X, LOCATION, SCALE, SHAPE)
     For each element of X, compute the probability density function
     (PDF) at X of the Birnbaum-Saunders distribution with parameters
     LOCATION, SCALE and SHAPE.


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For each element of X, compute the probability density function (PDF) at
X of th



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bbsrnd


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 -- bbsrnd (LOCATION, SCALE, SHAPE)
 -- bbsrnd (LOCATION, SCALE, SHAPE, R)
 -- bbsrnd (LOCATION, SCALE, SHAPE, R, C, ...)
 -- bbsrnd (LOCATION, SCALE, SHAPE, [SZ])
     Return a matrix of random samples from the generalized Pareto
     distribution with parameters LOCATION, SCALE and SHAPE.

     When called with a single size argument, return 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 of
     dimensions SZ.

     If no size arguments are given then the result matrix is the common
     size of LOCATION, SCALE and SHAPE.


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Return a matrix of random samples from the generalized Pareto
distribution with 



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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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binotest


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 -- Function File: [H, PVAL, CI] = binotest (POS,N,P0)
 -- Function File: [H, PVAL, CI] = binotest (POS,N,P0,NAME,VALUE)
     Test for probability P of a binomial sample

     Perform a test of the null hypothesis P == P0 for a sample of size
     N with POS positive results

     Name-Value pair arguments can be used to set various options.
     "alpha" can be used to specify the significance level of the test
     (the default value is 0.05).  The option "tail", can be used to
     select the desired alternative hypotheses.  If the value is "both"
     (default) the null is tested against the two-sided alternative 'P
     != P0'.  The value of PVAL is determined by adding the
     probabilities of all event less or equally likely than the observed
     number POS of positive events.  If the value of "tail" is "right"
     the one-sided alternative 'P > P0' is considered.  Similarly for
     "left", the one-sided alternative 'P < P0' is considered.

     If H is 0 the null hypothesis is accepted, if it is 1 the null
     hypothesis is rejected.  The p-value of the test is returned in
     PVAL.  A 100(1-alpha)% confidence interval is returned in CI.


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Test for probability P of a binomial sample



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boxplot


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 -- Function File: S = boxplot (DATA, NOTCHED, SYMBOL, VERTICAL,
          MAXWHISKER, ...)
 -- Function File: S = boxplot (DATA, GROUP)
 -- 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 handles to the plot elements, allowing
     customization of the visualization using set/get functions.

     Example

          title ("Grade 3 heights");
          axis ([0,3]);
          set(gca (), "xtick", [1 2], "xticklabel", {"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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burrcdf


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 -- burrcdf (X, C, K)
     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Burr distribution with scale parameter ALPHA and
     shape parameters C and K.


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For each element of X, compute the cumulative distribution function
(CDF) at X o



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burrinv


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 -- burrinv (X, ALPHA, C, K)
     For each element of X, compute the quantile (the inverse of the
     CDF) at X of the Burr distribution with scale parameter ALPHA and
     shape parameters C and K.


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For each element of X, compute the quantile (the inverse of the CDF) at
X of the



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burrpdf


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 -- burrpdf (X, ALPHA, C, K)
     For each element of X, compute the probability density function
     (PDF) at X of the Burr distribution with scale parameter ALPHA and
     shape parameters C and K.


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For each element of X, compute the probability density function (PDF) at
X of th



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burrrnd


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 -- burrrnd (ALPHA, C, K)
 -- burrrnd (ALPHA, C, K, R)
 -- burrrnd (ALPHA, C, K, R, C, ...)
 -- burrrnd (ALPHA, C, K, [SZ])
     Return a matrix of random samples from the generalized Pareto
     distribution with scale parameter ALPHA and shape parameters C and
     K.

     When called with a single size argument, return 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 of
     dimensions SZ.

     If no size arguments are given then the result matrix is the common
     size of ALPHA, C and K.


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Return a matrix of random samples from the generalized Pareto
distribution with 



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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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cdf


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 -- Function File: RETVAL = cdf (NAME, X, ...)
     Return cumulative density function of NAME function for value X.
     This is a wrapper around various NAMEcdf and NAME_cdf functions.
     See the individual functions help to learn the signification of the
     arguments after X.  Supported functions and corresponding number of
     additional arguments are:

       function               alternative                      args
     -------------------------------------------------------------------------
       "beta"                 "beta"                           2
       "bino"                 "binomial"                       2
       "cauchy"                                                2
       "chi2"                 "chisquare"                      1
       "discrete"                                              2
       "exp"                  "exponential"                    1
       "f"                                                     2
       "gam"                  "gamma"                          2
       "geo"                  "geometric"                      1
       "gev"                  "generalized extreme value"      3
       "hyge"                 "hypergeometric"                 3
       "kolmogorov_smirnov"                                    1
       "laplace"                                               0
       "logistic"                                              0
       "logn"                 "lognormal"                      2
       "norm"                 "normal"                         2
       "poiss"                "poisson"                        1
       "rayl"                 "rayleigh"                       1
       "t"                                                     1
       "unif"                 "uniform"                        2
       "wbl"                  "weibull"                        2

     See also: betacdf, binocdf, cauchy_cdf, chi2cdf, discrete_cdf,
     expcdf, fcdf, gamcdf, geocdf, gevcdf, hygecdf,
     kolmogorov_smirnov_cdf, laplace_cdf, logistic_cdf, logncdf,
     normcdf, poisscdf, raylcdf, tcdf, unifcdf, wblcdf.


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Return cumulative density function of NAME function for value X.



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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.

     Takes an N by N distance (or difference, similarity, or
     dissimilarity) matrix D.  Returns Y, a matrix of N points with
     coordinates in P dimensional space which approximate those
     distances (or differences, similarities, or dissimilarities).  Also
     returns the eigenvalues E of 'B = -1/2 * J * (D.^2) * J', where 'J
     = eye(N) - ones(N,N)/N'.  P, the number of columns of Y, is equal
     to the number of positive real eigenvalues of B.

     D can be a full or sparse matrix or a vector of length 'N*(N-1)/2'
     containing the upper triangular elements (like the output of the
     'pdist' function).  It must be symmetric with non-negative entries
     whose values are further restricted by the type of matrix being
     represented:

     * If D is either a distance, dissimilarity, or difference matrix,
     then it must have zero entries along the main diagonal.  In this
     case the points Y equal or approximate the distances given by D.

     * If D is a similarity matrix, the elements must all be less than
     or equal to one, with ones along the the main diagonal.  In this
     case the points Y equal or approximate the distances given by 'D =
     sqrt(ones(N,N)-D)'.

     D is a Euclidean matrix if and only if B is positive semi-definite.
     When this is the case, then Y is an exact representation of the
     distances given in D.  If D is non-Euclidean, Y only approximates
     the distance given in D.  The approximation used by 'cmdscale'
     minimizes the statistical loss function known as STRAIN.

     The returned Y is an N by P matrix showing possible coordinates of
     the points in P dimensional space ('P < N').  The columns are
     correspond to the positive eigenvalues of B in descending order.  A
     translation, rotation, or reflection of the coordinates given by Y
     will satisfy the same distance matrix up to the limits of machine
     precision.

     For any 'K <= P', if the largest K positive eigenvalues of B are
     significantly greater in absolute magnitude than its other
     eigenvalues, the first K columns of Y provide a K-dimensional
     reduction of Y which approximates the distances given by D.  The
     optional return E can be used to consider various values of K, or
     to evaluate the accuracy of specific dimension reductions (e.g., 'K
     = 2').

     Reference: Ingwer Borg and Patrick J.F. Groenen (2005), Modern
     Multidimensional Scaling, Second Edition, Springer, ISBN:
     978-0-387-25150-9 (Print) 978-0-387-28981-6 (Online)

     See also: pdist.


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Classical multidimensional scaling of a matrix.



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combnk


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 -- Function File: C = combnk (DATA, K)
     Return all combinations of K elements in DATA.


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Return all combinations of K elements in DATA.



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copulacdf


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 -- 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.


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Compute the cumulative distribution function of a copula family.



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copulapdf


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 -- 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.


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Compute the probability density function of a copula family.



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copularnd


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 -- 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.


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Generate random samples from a copula family.



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crossval


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 -- Function File: RESULTS = crossval (F, X, Y[, PARAMS])
     Perform cross validation on given data.

     F should be a function that takes 4 inputs XTRAIN, YTRAIN, XTEST,
     YTEST, fits a model based on XTRAIN, YTRAIN, applies the fitted
     model to XTEST, and returns a goodness of fit measure based on
     comparing the predicted and actual YTEST.  'crossval' returns an
     array containing the values returned by F for every
     cross-validation fold or resampling applied to the given data.

     X should be an N by M matrix of predictor values

     Y should be an N by 1 vector of predicand values

     PARAMS may include parameter-value pairs as follows:

     "KFold"
          Divide set into K equal-size subsets, using each one
          successively for validation.

     "HoldOut"
          Divide set into two subsets, training and validation.  If the
          value K is a fraction, that is the fraction of values put in
          the validation subset (by default K=0.1); if it is a positive
          integer, that is the number of values in the validation
          subset.

     "LeaveOut"
          Leave-one-out partition (each element is placed in its own
          subset).  The value is ignored.

     "Partition"
          The value should be a CVPARTITION object.

     "Given"
          The value should be an N by 1 vector specifying in which
          partition to put each element.

     "stratify"
          The value should be an N by 1 vector containing class
          designations for the elements, in which case the "KFold" and
          "HoldOut" partitionings attempt to ensure each partition
          represents the classes proportionately.

     "mcreps"
          The value should be a positive integer specifying the number
          of times to resample based on different partitionings.
          Currently only works with the partition type "HoldOut".

     Only one of "KFold", "HoldOut", "LeaveOut", "Given", "Partition"
     should be specified.  If none is specified, the default is "KFold"
     with K = 10.

     See also: cvpartition.


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Perform cross validation on given data.



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dcov


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 -- Function File: [DCOR, DCOV, DVARX, DVARY] = dcov (X, Y, INDEX=1)
     Distance covariance and correlation statistics.

     It returns distace correlation (DCOR), distance covariance (DCOV),
     diatance variace on x (DVARX) and distance variance on y (DVARY).

     Reference: https://en.wikipedia.org/wiki/Distance_correlation

     See also: cov.


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Distance covariance and correlation statistics.



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dendrogram


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 -- Function File: P = dendrogram (TREE)
 -- Function File: P, T = dendrogram (TREE)
 -- Function File: P, T, PERM = dendrogram (TREE)
     Plots a dendrogram 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.


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Plots a dendrogram using the output of function 'linkage'.



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expstat


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 -- 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.


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Compute mean and variance of the exponential distribution.



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ff2n


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 -- Function File: ff2n (N)
     Full-factor design with n binary terms.

     See also: fullfact.


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Full-factor design with n binary terms.



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fstat


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 -- 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.


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Compute mean and variance of the F distribution.



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fullfact


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 -- 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.


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Full factorial design.



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gamfit


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 -- 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.


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Calculate gamma distribution parameters.



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gamlike


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 -- 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.


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Calculates the negative log-likelihood function for the Gamma
distribution over 



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gamstat


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 -- 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.


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Compute mean and variance of the gamma distribution.



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geomean


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 -- 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.


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Compute the geometric mean.



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geostat


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 -- 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.


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Compute mean and variance of the geometric distribution.



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gevcdf


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 -- 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.


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Compute the cumulative distribution function of the generalized extreme
value (G



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gevfit


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 -- 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 K=0 and
          SIGMA, MU determined by matching the data mean and standard
          deviation to their expected 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 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.


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Find the maximum likelihood estimator (PARAMHAT) of the generalized
extreme valu



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gevfit_lmom


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 -- 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


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Find an estimator (PARAMHAT) of the generalized extreme value (GEV)
distribution



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gevinv


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 -- 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.


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Compute a desired quantile (inverse CDF) of the generalized extreme
value (GEV) 



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gevlike


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 -- 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.


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Compute the negative log-likelihood of data under the generalized
extreme value 



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gevpdf


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 -- 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.


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Compute the probability density function of the generalized extreme
value (GEV) 



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gevrnd


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 -- 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.


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Return a matrix of random samples from the generalized extreme value
(GEV) distr



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gevstat


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 -- 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.


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Compute the mean and variance of the generalized extreme value (GEV)
distributio



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gpcdf


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 -- gpcdf (X, LOCATION, SCALE, SHAPE)
     Compute the cumulative distribution function (CDF) at X of the
     generalized Pareto distribution with parameters LOCATION, SCALE,
     and SHAPE.


# name: <cell-element>
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Compute the cumulative distribution function (CDF) at X of the
generalized Paret



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gpinv


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 -- gpinv (X, LOCATION, SCALE, SHAPE)
     For each element of X, compute the quantile (the inverse of the
     CDF) at X of the generalized Pareto distribution with parameters
     LOCATION, SCALE, and SHAPE.


# name: <cell-element>
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For each element of X, compute the quantile (the inverse of the CDF) at
X of the



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gppdf


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 -- gppdf (X, LOCATION, SCALE, SHAPE)
     Compute the probability density function (PDF) at X of the
     generalized Pareto distribution with parameters LOCATION, SCALE,
     and SHAPE.


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Compute the probability density function (PDF) at X of the generalized
Pareto di



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gprnd


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 -- gprnd (LOCATION, SCALE, SHAPE)
 -- gprnd (LOCATION, SCALE, SHAPE, R)
 -- gprnd (LOCATION, SCALE, SHAPE, R, C, ...)
 -- gprnd (LOCATION, SCALE, SHAPE, [SZ])
     Return a matrix of random samples from the generalized Pareto
     distribution with parameters LOCATION, SCALE and SHAPE.

     When called with a single size argument, return 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 of
     dimensions SZ.

     If no size arguments are given then the result matrix is the common
     size of LOCATION, SCALE and SHAPE.


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Return a matrix of random samples from the generalized Pareto
distribution with 



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grp2idx


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 -- Function File: [G, GN, GL] = grp2idx (S)
     Get index for group variables.

     For variable S, returns the indices G, into the variable groups GN
     and GL.  The first has a string representation of the groups while
     the later has its actual values.

     NaNs and empty strings in S appear as NaN in G and are not present
     on either GN and GL.

     See also: cellstr, num2str, unique.


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Get index for group variables.



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harmmean


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 -- 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.


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Compute the harmonic mean.



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hist3


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 -- Function File: hist3 (X)
 -- Function File: hist3 (X, NBINS)
 -- Function File: hist3 (X, "Nbins", NBINS)
 -- Function File: hist3 (X, CENTERS)
 -- Function File: hist3 (X, "Ctrs", CENTERS)
 -- Function File: hist3 (X, "Edges", EDGES)
 -- Function File: [N, C] = hist3 (...)
 -- Function File: hist3 (..., PROP, VAL, ...)
 -- Function File: hist3 (HAX, ...)
     Produce bivariate (2D) histogram counts or plots.

     The elements to produce the histogram are taken from the Nx2 matrix
     X.  Any row with NaN values are ignored.  The actual bins can be
     configured in 3 different: number, centers, or edges of the bins:

     Number of bins (default)
          Produces equally spaced bins between the minimum and maximum
          values of X.  Defined as a 2 element vector, NBINS, one for
          each dimension.  Defaults to '[10 10]'.

     Center of bins
          Defined as a cell array of 2 monotonically increasing vectors,
          CENTERS.  The width of each bin is determined from the
          adjacent values in the vector with the initial and final bin,
          extending to Infinity.

     Edge of bins
          Defined as a cell array of 2 monotonically increasing vectors,
          EDGES.  'N(i,j)' contains the number of elements in X for
          which:

               EDGES{1}(i) <= X(:,1) < EDGES{1}(i+1)
               EDGES{2}(j) <= X(:,2) < EDGES{2}(j+1)

          The consequence of this definition is that values outside the
          initial and final edge values are ignored, and that the final
          bin only contains the number of elements exactly equal to the
          final edge.

     The return values, N and C, are the bin counts and centers
     respectively.  These are specially useful to produce intensity
     maps:

          [counts, centers] = hist3 (data);
          imagesc (centers{1}, centers{2}, data)

     If there is no output argument, or if the axes graphics handle HAX
     is defined, the function will plot a 3 dimensional bar graph.  Any
     extra property/value pairs are passed directly to the underlying
     surface object.

     See also: hist, histc, lookup, mesh.


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Produce bivariate (2D) histogram counts or plots.



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histfit


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 -- 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.


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Plot histogram with superimposed fitted normal density.



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hmmestimate


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 -- 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.


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Estimate the matrix of transition probabilities and the matrix of output
probabi



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hmmgenerate


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 -- 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.


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Generate an output sequence and hidden states of a hidden Markov model.



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hmmviterbi


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 -- 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>
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Use the Viterbi algorithm to find the Viterbi path of a hidden Markov
model give



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hygestat


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 -- 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.


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Compute mean and variance of the hypergeometric distribution.



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iwishpdf


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 -- 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.


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Compute the probability density function of the Wishart distribution



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iwishrnd


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 -- 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.


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Return a random matrix sampled from the inverse Wishart distribution
with given 



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jackknife


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 -- 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


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Compute jackknife estimates of a parameter taking one or more given
samples as p



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jsucdf


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 -- 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.


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For each element of X, compute the cumulative distribution function
(CDF) at X o



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jsupdf


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 -- 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.


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For each element of X, compute the probability density function (PDF) at
X of th



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kmeans


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 -- [IDX, CENTERS, SUMD, DIST] = kmeans (DATA, K, PARAM1, VALUE1, ...)
     Perform a K-means clustering of the NxD table DATA.  If parameter
     start is specified, then K may be empty in which case K is set to
     the number of rows of START.

     The outputs are:
     'IDX'
          An Nx1 vector whose Ith element is the class to which row I of
          DATA is assigned.

     'CENTERS'
          A KxD array whose Ith row is the centroid of cluster I.

     'SUMD'
          A Kx1 vector whose Ith entry is the sum of the distances from
          samples in cluster I to centroid I.

     'DIST'
          An NxK matrix whose IJth element is the distance from sample I
          to centroid J.

     The following parameters may be placed in any order.  Each
     parameter must be followed by its value.
     'START'
          The initialization method for the centroids.
          'plus'
               (Default) The k-means++ algorithm.
          'sample'
               A subset of K rows from DATA, sampled uniformly without
               replacement.
          'cluster'
               Perform a pilot clustering on 10% of the rows of DATA.
          'uniform'
               Each component of each centroid is drawn uniformly from
               the interval between the maximum and minimum values of
               that component within DATA.  This performs poorly and is
               implemented only for Matlab compatibility.
          'A'
               A KxDxR matrix, where R is the number of replicates.

     'REPLICATES'
          An positive integer specifying the number of independent
          clusterings to perform.  The output values are the values for
          the best clustering, i.e., the one with the smallest value of
          SUMD.  If START is numeric, then REPLICATES defaults to (and
          must equal) the size of the third dimension of START.
          Otherwise it defaults to 1.

     'MAXITER'
          The maximum number of iterations to perform for each
          replicate.  If the maximum change of any centroid is less than
          0.001, then the replicate terminates even if MAXITER
          iterations have no occurred.  The default is 100.

     'DISTANCE'
          The distance measure used for partitioning and calculating
          centroids.
          'sqeuclidean'
               The squared Euclidean distance, i.e., the sum of the
               squares of the differences between corresponding
               components.  In this case, the centroid is the arithmetic
               mean of all samples in its cluster.  This is the only
               distance for which this algorithm is truly "k-means".

          'cityblock'
               The sum metric, or L1 distance, i.e., the sum of the
               absolute differences between corresponding components.
               In this case, the centroid is the median of all samples
               in its cluster.  This gives the k-medians algorithm.

          'cosine'
               (Documentation incomplete.)

          'correlation'
               (Documentation incomplete.)

          'hamming'
               The number of components in which the sample and the
               centroid differ.  In this case, the centroid is the
               median of all samples in its cluster.  Unlike Matlab,
               Octave allows non-logical DATA.

     'EMPTYACTION'
          What to do when a centroid is not the closest to any data
          sample.
          'error'
               (Default) Throw an error.
          'singleton'
               Select the row of DATA that has the highest error and use
               that as the new centroid.
          'drop'
               Remove the centroid, and continue computation with one
               fewer centroid.  The dimensions of the outputs CENTROIDS
               and D are unchanged, with values for omitted centroids
               replaced by NA.

     Example:

     [~,c] = kmeans (rand(10, 3), 2, "emptyaction", "singleton");

     See also: linkage.


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Perform a K-means clustering of the NxD table DATA.



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linkage


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 -- 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.


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Produce a hierarchical clustering dendrogram



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lognstat


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 -- 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.


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Compute mean and variance of the lognormal distribution.



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mad


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 -- 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.


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Compute the mean/median absolute deviation of X.



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mahal


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 -- Function File: mahal (Y, X)
     Mahalanobis' D-square distance.

     Return the Mahalanobis' D-square distance of the points in Y from
     the distribution implied by points X.

     Specifically, it uses a Cholesky decomposition to set

           answer(i) = (Y(i,:) - mean (X)) * inv (A) * (Y(i,:)-mean (X))'

     where A is the covariance of X.

     The data X and Y must have the same number of components (columns),
     but may have a different number of observations (rows).


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Mahalanobis' D-square distance.



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mnpdf


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 -- 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.


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Compute the probability density function of the multinomial
distribution.



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mnrnd


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 -- 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.


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Generate random samples from the multinomial distribution.



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monotone_smooth


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 -- 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)


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Produce a smooth monotone increasing approximation to a sampled
functional depen



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mvncdf


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 -- 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.


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Compute the cumulative distribution function of the multivariate normal
distribu



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mvnpdf


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 -- 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.


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Compute multivariate normal pdf for X given mean MU and covariance
matrix SIGMA.



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mvnrnd


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 -- 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")'.


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Draw N random D-dimensional vectors from a multivariate Gaussian
distribution wi



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mvtcdf


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 -- 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.


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Compute the cumulative distribution function of the multivariate
Student's t dis



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mvtpdf


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 -- Function File: P = mvtpdf (X, SIGMA, NU)
     Compute the probability density function of the multivariate
     Student's t distribution.

     Arguments
     ---------

        * X are the points at which to find the probability, where each
          row corresponds to an observation.  (N by D matrix)

        * SIGMA is the scale matrix.  (D by D symmetric positive
          definite matrix)

        * NU is the degrees of freedom.  (scalar or N vector)

     The distribution is assumed to be centered (zero mean).

     Return values
     -------------

        * P is the probability density for each row of X.  (N by 1
          vector)

     Examples
     --------

          x = [1 2];
          sigma = [1.0 0.5; 0.5 1.0];
          nu = 4;
          p = mvtpdf (x, sigma, nu)

     References
     ----------

       1. Michael Roth, On the Multivariate t Distribution, Technical
          report from Automatic Control at Linkoepings universitet,
          <http://users.isy.liu.se/en/rt/roth/student.pdf>


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Compute the probability density function of the multivariate Student's t
distrib



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mvtrnd


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 -- 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.


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Generate random samples from the multivariate t-distribution.



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nakacdf


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 -- nakacdf (X, M, W)
     For each element of X, compute the cumulative distribution function
     (CDF) at X of the Nakagami distribution with shape parameter M and
     scale parameter W.


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For each element of X, compute the cumulative distribution function
(CDF) at X o



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nakainv


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 -- nakainv (X, M, W)
     For each element of X, compute the quantile (the inverse of the
     CDF) at X of the Nakagami distribution with shape parameter M and
     scale parameter W.


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For each element of X, compute the quantile (the inverse of the CDF) at
X of the



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nakapdf


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 -- nakapdf (X, M, W)
     For each element of X, compute the probability density function
     (PDF) at X of the Nakagami distribution with shape parameter M and
     scale parameter W.


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For each element of X, compute the probability density function (PDF) at
X of th



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nakarnd


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 -- nakarnd (M, W)
 -- nakarnd (M, W, R)
 -- nakarnd (M, W, R, C, ...)
 -- nakarnd (M, W, [SZ])
     Return a matrix of random samples from the Nakagami distribution
     with shape parameter M and scale W.

     When called with a single size argument, return 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 of
     dimensions SZ.

     If no size arguments are given then the result matrix is the common
     size of M and W.


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Return a matrix of random samples from the Nakagami distribution with
shape para



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nanmax


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 -- 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.


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Find the maximal element while ignoring NaN values.



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nanmean


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 -- 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.


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Compute the mean value while ignoring NaN values.



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nanmedian


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 -- 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.


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Compute the median of data while ignoring NaN values.



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nanmin


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 -- 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.


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Find the minimal element while ignoring NaN values.



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nanstd


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 -- 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.


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Compute the standard deviation while ignoring NaN values.



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nansum


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 -- Built-in Function: nansum (X)
 -- Built-in Function: nansum (X, DIM)
 -- Built-in Function: nansum (..., "native")
 -- Built-in Function: nansum (..., "double")
 -- Built-in Function: nansum (..., "extra")
     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 help text of 'sum' for details on the options.

     See also: sum, nanmin, nanmax, nanmean, nanmedian.


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Compute the sum while ignoring NaN values.



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nanvar


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 -- 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.


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Compute the variance while ignoring NaN values.



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nbinstat


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 -- 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.


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Compute mean and variance of the negative binomial distribution.



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normalise_distribution


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 -- 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)


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Transform a set of data so as to be N(0,1) distributed according to an
idea by v



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normplot


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 -- 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'.


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Produce normal probability plot for each column of X.



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normstat


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 -- 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.


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Compute mean and variance of the normal distribution.



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pcacov


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 -- 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


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   * pcacov performs principal component analysis on the nxn covariance
     mat



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pcares


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 -- 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


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   * X : N x P Matrix with N observations and P variables, the variables
     wi



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# length: 3
pdf


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 -- Function File: RETVAL = pdf (NAME, X, ...)
     Return probability density function of NAME function for value X.
     This is a wrapper around various NAMEpdf and NAME_pdf functions.
     See the individual functions help to learn the signification of the
     arguments after X.  Supported functions and corresponding number of
     additional arguments are:

       function               alternative                      args
     -------------------------------------------------------------------------
       "beta"                 "beta"                           2
       "bino"                 "binomial"                       2
       "cauchy"                                                2
       "chi2"                 "chisquare"                      1
       "discrete"                                              2
       "exp"                  "exponential"                    1
       "f"                                                     2
       "gam"                  "gamma"                          2
       "geo"                  "geometric"                      1
       "gev"                  "generalized extreme value"      3
       "hyge"                 "hypergeometric"                 3
       "kolmogorov_smirnov"                                    1
       "laplace"                                               2
       "logistic"                                              0
       "logn"                 "lognormal"                      2
       "norm"                 "normal"                         2
       "poiss"                "poisson"                        1
       "rayl"                 "rayleigh"                       1
       "t"                                                     1
       "unif"                 "uniform"                        2
       "wbl"                  "weibull"                        2

     See also: betapdf, binopdf, cauchy_pdf, chi2pdf, discrete_pdf,
     exppdf, fpdf, gampdf, geopdf, gevpdf, hygepdf, laplace_pdf,
     logistic_pdf, lognpdf, normpdf, poisspdf, raylpdf, tpdf, unifpdf,
     wblpdf.


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Return probability density function of NAME function for value X.



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pdist


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 -- 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, pdist2.


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Return the distance between any two rows in X.



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pdist2


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 -- Function File: pdist2 (X, Y)
 -- Function File: pdist2 (X, Y, METRIC)
     Compute pairwise distance between two sets of vectors.

     Let X be an MxP matrix representing m points in P-dimensional space
     and Y be an NxP matrix representing another set of points in the
     same space.  This function computes the M-by-N distance matrix D
     where 'D(i,j)' is the distance between 'X(i,:)' and 'Y(j,:)'.

     The optional argument METRIC can be used to select different
     distances:

     "euclidean" (default)

     "sqeuclidean"
          Compute the squared euclidean distance, i.e., the euclidean
          distance before computing square root.  This is ideal when the
          interest is on the order of the euclidean distances rather
          than the actual distance value because it performs
          significantly faster while preserving the order.

     "chisq'"
          The chi-squared distance between two vectors is defined as:
          'd(x, y) = sum ((xi-yi)^2 / (xi+yi)) / 2'.  The chi-squared
          distance is useful when comparing histograms.

     "cosine"
          Distance is defined as the cosine of the angle between two
          vectors.

     "emd"
          Earth Mover's Distance (EMD) between positive vectors
          (histograms).  Note for 1D, with all histograms having equal
          weight, there is a simple closed form for the calculation of
          the EMD. The EMD between histograms X and Y is given by 'sum
          (abs (cdf (x) - cdf (y)))', where 'cdf' is the cumulative
          distribution function (computed simply by 'cumsum').

     "L1"
          The L1 distance between two vectors is defined as: 'sum (abs
          (x-y))'

     See also: pdist.


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Compute pairwise distance between two sets of vectors.



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plsregress


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 521
 -- 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>
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plsregress(X, Y, NCOMP)
   * X: Matrix of observations
   * Y: Is a vector or ma



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poisstat


# name: <cell-element>
# type: sq_string
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# length: 862
 -- 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>
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Compute mean and variance of the Poisson distribution.



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princomp


# name: <cell-element>
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 -- 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>
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   * princomp performs principal component analysis on a NxP data matrix
     X




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qrandn


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 -- Function File: Z = qrandn (Q, R,C)
 -- Function File: Z = qrandn (Q, [R,C])
     Returns random deviates drawn from a q-Gaussian distribution.

     Parameter Q charcterizes the q-Gaussian distribution.  The result
     has the size indicated by S.

     Reference: W. Thistleton, J. A. Marsh, K. Nelson, C. Tsallis (2006)
     "Generalized Box-Muller method for generating q-Gaussian random
     deviates" arXiv:cond-mat/0605570
     http://arxiv.org/abs/cond-mat/0605570

     See also: rand, randn.


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Returns random deviates drawn from a q-Gaussian distribution.



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random


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 -- 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>
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Generates pseudo-random numbers from a given one-, two-, or
three-parameter dist



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randsample


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 -- Function File: Y = randsample (V, K, REPLACEMENT=false [, W])
     Elements sampled from a vector.

     Returns K random elements from a vector V with N elements, sampled
     without or with REPLACEMENT.

     If V is a scalar, samples from 1:V.

     If a weight vector W of the same size as V is specified, the
     probablility of each element being sampled is proportional to W.
     Unlike Matlab's function of the same name, this can be done for
     sampling with or without replacement.

     Randomization is performed using rand().

     See also: randperm.


# name: <cell-element>
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Elements sampled from a vector.



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raylcdf


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 -- 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.


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Compute the cumulative distribution function of the Rayleigh
distribution.



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raylinv


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 -- 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.


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Compute the quantile of the Rayleigh distribution.



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raylpdf


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 -- 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.


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Compute the probability density function of the Rayleigh distribution.



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raylrnd


# name: <cell-element>
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 -- 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>
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Generate a matrix of random samples from the Rayleigh distribution.



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raylstat


# name: <cell-element>
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 -- 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.


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Compute mean and variance of the Rayleigh distribution.



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regress


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 -- 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>
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Multiple Linear Regression using Least Squares Fit of Y on X with the
model 'y =



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regress_gp


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 -- Function File: [M, K] = regress_gp (X, Y, SP)
 -- Function File: [... YI DY] = regress_gp (..., 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.


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Linear scalar regression using gaussian processes.



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repanova


# name: <cell-element>
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 -- 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!


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Perform a repeated measures analysis of variance (Repeated ANOVA).



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runstest


# name: <cell-element>
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 -- 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>
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Runs test for detecting serial correlation in the vector X.



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signtest


# name: <cell-element>
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 -- Function File: [PVAL, H, STATS] = signtest (X)
 -- Function File: [PVAL, H, STATS] = signtest (X, M)
 -- Function File: [PVAL, H, STATS] = signtest (X, Y)
 -- Function File: [PVAL, H, STATS] = signtest (X, Y, NAME, VALUE)
     Test for median.

     Perform a signtest of the null hypothesis that X is from a
     distribution that has a zero median.

     If the second argument M is a scalar, the null hypothesis is that X
     has median m.

     If the second argument Y is a vector, the null hypothesis is that
     the distribution of 'X - Y' has zero median.

     The argument "alpha" can be used to specify the significance level
     of the test (the default value is 0.05).  The string argument
     "tail", can be used to select the desired alternative hypotheses.
     If "alt" is "both" (default) the null is tested against the
     two-sided alternative 'median (X) != M'.  If "alt" is "right" the
     one-sided alternative 'median (X) > M' is considered.  Similarly
     for "left", the one-sided alternative 'median (X) < M' is
     considered.  When "method" is "exact" the p-value is computed using
     an exact method (this is the default).  When "method" is
     "approximate" a normal approximation is used for the test
     statistic.

     The p-value of the test is returned in PVAL.  If H is 0 the null
     hypothesis is accepted, if it is 1 the null hypothesis is rejected.
     STATS is a structure containing the value of the test statistic
     (SIGN) and the value of the z statistic (ZVAL) (only computed when
     the 'method' is 'approximate'.


# name: <cell-element>
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Test for median.



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squareform


# name: <cell-element>
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 -- Function File: Z = squareform (Y)
 -- Function File: Y = squareform (Z)
 -- Function File: Z = squareform (Y, "tovector")
 -- Function File: Y = squareform (Z, "tomatrix")
     Interchange between distance matrix and distance vector formats.

     Converts between an hollow (diagonal filled with zeros), square,
     and symmetric matrix and a vector with of the lower triangular
     part.

     Its target application is the conversion of the vector returned by
     'pdist' into a distance matrix.  It performs the opposite operation
     if input is a matrix.

     If X is a vector, its number of elements must fit into the
     triangular part of a matrix (main diagonal excluded).  In other
     words, 'numel (X) = N * (N - 1) / 2' for some integer N.  The
     resulting matrix will be N by N.

     If X is a distance matrix, it must be square and the diagonal
     entries of X must all be zeros.  'squareform' will generate a
     warning if X is not symmetric.

     The second argument is used to specify the output type in case
     there is a single element.  It will defaults to "tomatrix"
     otherwise.

     See also: pdist.


# name: <cell-element>
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Interchange between distance matrix and distance vector formats.



# name: <cell-element>
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stepwisefit


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1254
 -- 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.


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Linear regression with stepwise variable selection.



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tabulate


# name: <cell-element>
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 -- 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
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Compute a frequency table.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
tblread


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 770
 -- 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: 757
 -- 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: 6
tricdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 177
 -- tricdf (X, A, B, C)
     Compute the cumulative distribution function (CDF) at X of the
     triangular distribution with parameters A, B, and C on the interval
     [A, B].


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Compute the cumulative distribution function (CDF) at X of the
triangular distri



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
triinv


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 195
 -- triinv (X, A, B, C)
     For each element of X, compute the quantile (the inverse of the
     CDF) at X of the triangular distribution with parameters A, B, and
     C on the interval [A, B].


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
For each element of X, compute the quantile (the inverse of the CDF) at
X of the



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
trimmean


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 386
 -- 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: 6
tripdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 173
 -- tripdf (X, A, B, C)
     Compute the probability density function (PDF) at X of the
     triangular distribution with parameters A, B, and C on the interval
     [A, B].


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Compute the probability density function (PDF) at X of the triangular
distributi



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
trirnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 701
 -- trirnd (A, B, C)
 -- trirnd (A, B, C, R)
 -- trirnd (A, B, C, R, C, ...)
 -- trirnd (A, B, C, [SZ])
     Return a matrix of random samples from the rectangular distribution
     with parameters A, B, and C on the interval [A, B].

     When called with a single size argument, return 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 of
     dimensions SZ.

     If no size arguments are given then the result matrix is the common
     size of A, B and C.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Return a matrix of random samples from the rectangular distribution with
paramet



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
tstat


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 840
 -- 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: 5
ttest


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1791
 -- Function File: [H, PVAL, CI, STATS] = ttest (X)
 -- Function File: [H, PVAL, CI, STATS] = ttest (X, M)
 -- Function File: [H, PVAL, CI, STATS] = ttest (X, Y)
 -- Function File: [H, PVAL, CI, STATS] = ttest (X, M, NAME, VALUE)
 -- Function File: [H, PVAL, CI, STATS] = ttest (X, Y, NAME, VALUE)
     Test for mean of a normal sample with known variance.

     Perform a T-test of the null hypothesis 'mean (X) == M' for a
     sample X from a normal distribution with unknown mean and unknown
     std deviation.  Under the null, the test statistic T has a
     Student's t distribution.  The default value of M is 0.

     If the second argument Y is a vector, a paired-t test of the
     hypothesis 'mean (X) = mean (Y)' is performed.

     Name-Value pair arguments can be used to set various options.
     "alpha" can be used to specify the significance level of the test
     (the default value is 0.05).  "tail", can be used to select the
     desired alternative hypotheses.  If the value is "both" (default)
     the null is tested against the two-sided alternative 'mean (X) !=
     M'.  If it is "right" the one-sided alternative 'mean (X) > M' is
     considered.  Similarly for "left", the one-sided alternative 'mean
     (X) < M' is considered.  When argument X is a matrix, "dim" can be
     used to selection the dimension over which to perform the test.
     (The default is the first non-singleton dimension).

     If H is 0 the null hypothesis is accepted, if it is 1 the null
     hypothesis is rejected.  The p-value of the test is returned in
     PVAL.  A 100(1-alpha)% confidence interval is returned in CI.
     STATS is a structure containing the value of the test statistic
     (TSTAT), the degrees of freedom (DF) and the sample standard
     deviation (SD).


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 53
Test for mean of a normal sample with known variance.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
ttest2


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1738
 -- Function File: [H, PVAL, CI, STATS] = ttest2 (X, Y)
 -- Function File: [H, PVAL, CI, STATS] = ttest2 (X, Y, NAME, VALUE)
     Test for mean of a normal sample with known variance.

     Perform a T-test of the null hypothesis 'mean (X) == M' for a
     sample X from a normal distribution with unknown mean and unknown
     std deviation.  Under the null, the test statistic T has a
     Student's t distribution.

     If the second argument Y is a vector, a paired-t test of the
     hypothesis 'mean (X) = mean (Y)' is performed.

     The argument "alpha" can be used to specify the significance level
     of the test (the default value is 0.05).  The string argument
     "tail", can be used to select the desired alternative hypotheses.
     If "alt" is "both" (default) the null is tested against the
     two-sided alternative 'mean (X) != M'.  If "alt" is "right" the
     one-sided alternative 'mean (X) > M' is considered.  Similarly for
     "left", the one-sided alternative 'mean (X) < M' is considered.
     When "vartype" is "equal" the variances are assumed to be equal
     (this is the default).  When "vartype" is "unequal" the variances
     are not assumed equal.  When argument X is a matrix the "dim"
     argument can be used to selection the dimension over which to
     perform the test.  (The default is the first non-singleton
     dimension.)

     If H is 0 the null hypothesis is accepted, if it is 1 the null
     hypothesis is rejected.  The p-value of the test is returned in
     PVAL.  A 100(1-alpha)% confidence interval is returned in CI.
     STATS is a structure containing the value of the test statistic
     (TSTAT), the degrees of freedom (DF) and the sample standard
     deviation (SD).


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 53
Test for mean of a normal sample with known variance.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
unidstat


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 882
 -- 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.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 63
Compute mean and variance of the discrete uniform distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
unifstat


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1093
 -- 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.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 65
Compute mean and variance of the continuous uniform distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
vartest


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1490
 -- Function File: [H, PVAL, CI, STATS] = vartest (X, Y)
 -- Function File: [H, PVAL, CI, STATS] = vartest (X, Y, NAME, VALUE)
     Perform a F-test for equal variances.

     If the second argument Y is a vector, a paired-t test of the
     hypothesis 'mean (X) = mean (Y)' is performed.

     The argument "alpha" can be used to specify the significance level
     of the test (the default value is 0.05).  The string argument
     "tail", can be used to select the desired alternative hypotheses.
     If "alt" is "both" (default) the null is tested against the
     two-sided alternative 'mean (X) != M'.  If "alt" is "right" the
     one-sided alternative 'mean (X) > M' is considered.  Similarly for
     "left", the one-sided alternative 'mean (X) < M' is considered.
     When "vartype" is "equal" the variances are assumed to be equal
     (this is the default).  When "vartype" is "unequal" the variances
     are not assumed equal.  When argument X is a matrix the "dim"
     argument can be used to selection the dimension over which to
     perform the test.  (The default is the first non-singleton
     dimension.)

     If H is 0 the null hypothesis is accepted, if it is 1 the null
     hypothesis is rejected.  The p-value of the test is returned in
     PVAL.  A 100(1-alpha)% confidence interval is returned in CI.
     STATS is a structure containing the value of the test statistic
     (TSTAT), the degrees of freedom (DF) and the sample standard
     deviation (SD).


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 37
Perform a F-test for equal variances.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 8
vartest2


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1492
 -- Function File: [H, PVAL, CI, STATS] = vartest2 (X, Y)
 -- Function File: [H, PVAL, CI, STATS] = vartest2 (X, Y, NAME, VALUE)
     Perform a F-test for equal variances.

     If the second argument Y is a vector, a paired-t test of the
     hypothesis 'mean (X) = mean (Y)' is performed.

     The argument "alpha" can be used to specify the significance level
     of the test (the default value is 0.05).  The string argument
     "tail", can be used to select the desired alternative hypotheses.
     If "alt" is "both" (default) the null is tested against the
     two-sided alternative 'mean (X) != M'.  If "alt" is "right" the
     one-sided alternative 'mean (X) > M' is considered.  Similarly for
     "left", the one-sided alternative 'mean (X) < M' is considered.
     When "vartype" is "equal" the variances are assumed to be equal
     (this is the default).  When "vartype" is "unequal" the variances
     are not assumed equal.  When argument X is a matrix the "dim"
     argument can be used to selection the dimension over which to
     perform the test.  (The default is the first non-singleton
     dimension.)

     If H is 0 the null hypothesis is accepted, if it is 1 the null
     hypothesis is rejected.  The p-value of the test is returned in
     PVAL.  A 100(1-alpha)% confidence interval is returned in CI.
     STATS is a structure containing the value of the test statistic
     (TSTAT), the degrees of freedom (DF) and the sample standard
     deviation (SD).


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 37
Perform a F-test for equal variances.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 6
violin


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 2627
 -- Function: H = violin (X)
 -- Function: H = violin (..., PROPERTY, VALUE, ...)
 -- Function: H = violin (HAX, ...)
 -- Function: H = violin (..., "horizontal")
     Produce a Violin plot of the data X.

     The input data X can be a N-by-m array containg N observations of m
     variables.  It can also be a cell with m elements, for the case in
     which the varibales are not uniformly sampled.

     The following PROPERTY can be set using PROPERTY/VALUE pairs
     (default values in parenthesis).  The value of the property can be
     a scalar indicating that it applies to all the variables in the
     data.  It can also be a cell/array, indicating the property for
     each variable.  In this case it should have m columns (as many as
     variables).

     Color
          ("y") Indicates the filling color of the violins.

     Nbins
          (50) Internally, the function calls 'hist' to compute the
          histogram of the data.  This property indicates how many bins
          to use.  See 'help hist' for more details.

     SmoothFactor
          (4) The fuction performs simple kernel density estimation and
          automatically finds the bandwith of the kernel function that
          best approximates the histogram using optimization ('sqp').
          The result is in general very noisy.  To smooth the result the
          bandwidth is multiplied by the value of this property.  The
          higher the value the smoother the violings, but values too
          high might remove features from the data distribution.

     Bandwidth
          (NA) If this property is given a value other than NA, it sets
          the bandwith of the kernel function.  No optimization is
          peformed and the property SmoothFactor is ignored.

     Width
          (0.5) Sets the maximum width of the violins.  Violins are
          centered at integer axis values.  The distance between two
          violin middle axis is 1.  Setting a value higher thna 1 in
          this property will cause the violins to overlap.

     If the string "Horizontal" is among the input arguments, the violin
     plot is rendered along the x axis with the variables in the y axis.

     The returned structure H has handles to the plot elements, allowing
     customization of the visualization using set/get functions.

     Example:

          title ("Grade 3 heights");
          axis ([0,3]);
          set (gca, "xtick", 1:2, "xticklabel", {"girls"; "boys"});
          h = violin ({randn(100,1)*5+140, randn(130,1)*8+135}, "Nbins", 10);
          set (h.violin, "linewidth", 2)

     See also: boxplot, hist.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 36
Produce a Violin plot of the data X.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
vmpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 318
 -- 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.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 53
Evaluates the Von Mises probability density function.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 5
vmrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 517
 -- 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.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 80
Draw random angles from a Von Mises distribution with mean MU and
concentration 



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
wblstat


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1096
 -- 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.


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 54
Compute mean and variance of the Weibull distribution.



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
wishpdf


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 644
 -- 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.


# name: <cell-element>
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Compute the probability density function of the Wishart distribution



# name: <cell-element>
# type: sq_string
# elements: 1
# length: 7
wishrnd


# name: <cell-element>
# type: sq_string
# elements: 1
# length: 1004
 -- 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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ztest


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 -- Function File: [H, PVAL, CI, Z, ZCRIT] = ztest (X, M, S)
 -- Function File: [H, PVAL, CI, Z, ZCRIT] = ztest (X, M, S, NAME,
          VALUE)
     Test for mean of a normal sample with known variance.

     Perform a Z-test of the null hypothesis 'mean (X) == M' for a
     sample X from a normal distribution with unknown mean and known std
     deviation S.  Under the null, the test statistic Z follows a
     standard normal distribution.

     Name-Value pair arguments can be used to set various options.
     "alpha" can be used to specify the significance level of the test
     (the default value is 0.05).  "tail", can be used to select the
     desired alternative hypotheses.  If the value is "both" (default)
     the null is tested against the two-sided alternative 'mean (X) !=
     M'.  If it is "right" the one-sided alternative 'mean (X) > M' is
     considered.  Similarly for "left", the one-sided alternative 'mean
     (X) < M' is considered.  When argument X is a matrix, "dim" can be
     used to selection the dimension over which to perform the test.
     (The default is the first non-singleton dimension.)

     If H is 0 the null hypothesis is accepted, if it is 1 the null
     hypothesis is rejected.  The p-value of the test is returned in
     PVAL.  A 100(1-alpha)% confidence interval is returned in CI.  The
     test statistic value is returned in Z and the z critical value in
     ZCRIT.


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Test for mean of a normal sample with known variance.