@c Copyright (C) 1996-2018 John W. Eaton
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@c This file is part of Octave.
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@c (at your option) any later version.
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@c MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
@c GNU General Public License for more details.
@c
@c You should have received a copy of the GNU General Public License
@c along with Octave; see the file COPYING.  If not, see
@c <https://www.gnu.org/licenses/>.

@node Statistics
@chapter Statistics

Octave has support for various statistical methods.  The emphasis is on basic
descriptive statistics, but the Octave Forge statistics package includes
probability distributions, statistical tests, random number generation, and
much more.

The functions that analyze data all assume that multi-dimensional data is
arranged in a matrix where each row is an observation, and each column is a
variable.  Thus, the matrix defined by

@example
@group
a = [ 0.9, 0.7;
      0.1, 0.1;
      0.5, 0.4 ];
@end group
@end example

@noindent
contains three observations from a two-dimensional distribution.  While this is
the default data arrangement, most functions support different arrangements.

It should be noted that the statistics functions don't test for data containing
NaN, NA, or Inf.  These values need to be detected and dealt with explicitly.
See @ref{XREFisnan,,isnan}, @ref{XREFisna,,isna}, @ref{XREFisinf,,isinf},
@ref{XREFisfinite,,isfinite}.

@menu
* Descriptive Statistics::
* Basic Statistical Functions::
* Correlation and Regression Analysis::
* Distributions::
* Random Number Generation::
@end menu

@node Descriptive Statistics
@section Descriptive Statistics

One principal goal of descriptive statistics is to represent the essence of a
large data set concisely.  Octave provides the mean, median, and mode functions
which all summarize a data set with just a single number corresponding to
the central tendency of the data.

@DOCSTRING(mean)

@DOCSTRING(median)

@DOCSTRING(mode)

Using just one number, such as the mean, to represent an entire data set may
not give an accurate picture of the data.  One way to characterize the fit is
to measure the dispersion of the data.  Octave provides several functions for
measuring dispersion.

@DOCSTRING(bounds)

@DOCSTRING(range)

@DOCSTRING(iqr)

@DOCSTRING(mad)

@DOCSTRING(meansq)

@DOCSTRING(std)

In addition to knowing the size of a dispersion it is useful to know the shape
of the data set.  For example, are data points massed to the left or right
of the mean?  Octave provides several common measures to describe the shape
of the data set.  Octave can also calculate moments allowing arbitrary shape
measures to be developed.

@DOCSTRING(var)

@DOCSTRING(skewness)

@DOCSTRING(kurtosis)

@DOCSTRING(moment)

@DOCSTRING(quantile)

@DOCSTRING(prctile)

A summary view of a data set can be generated quickly with the
@code{statistics} function.

@DOCSTRING(statistics)

@node Basic Statistical Functions
@section Basic Statistical Functions

Octave supports various helpful statistical functions.  Many are useful as
initial steps to prepare a data set for further analysis.  Others provide
different measures from those of the basic descriptive statistics.

@DOCSTRING(center)

@DOCSTRING(zscore)

@DOCSTRING(histc)

@noindent
@code{unique} function documented at @ref{XREFunique,,unique} is often
useful for statistics.

@DOCSTRING(nchoosek)

@DOCSTRING(perms)

@DOCSTRING(ranks)

@DOCSTRING(run_count)

@DOCSTRING(runlength)

@node Correlation and Regression Analysis
@section Correlation and Regression Analysis

@c FIXME: Need Intro Here

@DOCSTRING(cov)

@DOCSTRING(corr)

@DOCSTRING(corrcoef)

@DOCSTRING(spearman)

@DOCSTRING(kendall)

@node Distributions
@section Distributions

Octave has functions for computing the Probability Density Function (PDF), the
Cumulative Distribution function (CDF), and the quantile (the inverse of the
CDF) for arbitrary user-defined distributions (discrete) and for experimental
data (empirical).

The following table summarizes the supported distributions (in alphabetical
order).

@tex
\vskip 6pt
{\hbox to \hsize {\hfill\vbox{\offinterlineskip \tabskip=0pt
\halign{
\vrule height2.0ex depth1.ex width 0.6pt #\tabskip=0.3em &
# \hfil & \vrule # & # \hfil & \vrule # & # \hfil & \vrule # & # \hfil &
# \vrule width 0.6pt \tabskip=0pt\cr
\noalign{\hrule height 0.6pt}
& {\bf Distribution} && {\bf PDF}      && {\bf CDF}     && {\bf Quantile}&\cr
\noalign{\hrule}
&Univariate Discrete       && discrete\_pdf  && discrete\_cdf && discrete\_inv&\cr
&Empirical    && empirical\_pdf  && empirical\_cdf && empirical\_inv&\cr
\noalign{\hrule height 0.6pt}
}}\hfill}}
@end tex
@ifnottex
@multitable @columnfractions .31 .23 .23 .23
@headitem Distribution
  @tab PDF
  @tab CDF
  @tab Quantile
@item Univariate Discrete Distribution
  @tab @code{discrete_pdf}
  @tab @code{discrete_cdf}
  @tab @code{discrete_inv}
@item Empirical Distribution
  @tab @code{empirical_pdf}
  @tab @code{empirical_cdf}
  @tab @code{empirical_inv}
@end multitable
@end ifnottex

@DOCSTRING(discrete_pdf)

@DOCSTRING(discrete_cdf)

@DOCSTRING(discrete_inv)

@DOCSTRING(empirical_pdf)

@DOCSTRING(empirical_cdf)

@DOCSTRING(empirical_inv)

@node Random Number Generation
@section Random Number Generation

Octave can generate random numbers from a large number of distributions.  The
random number generators are based on the random number generators described in
@ref{Special Utility Matrices}.

The following table summarizes the available random number generators (in
alphabetical order).

@tex
\vskip 6pt
{\hbox to \hsize {\hfill\vbox{\offinterlineskip \tabskip=0pt
\halign{
\vrule height2.0ex depth1.ex width 0.6pt #\tabskip=0.3em &
# \hfil & \vrule # & # \hfil & # \vrule width 0.6pt \tabskip=0pt\cr
\noalign{\hrule height 0.6pt}
& {\bf Distribution}                && {\bf Function} &\cr
\noalign{\hrule}
& Univariate Discrete Distribution  && discrete\_rnd &\cr
& Empirical Distribution            && empirical\_rnd &\cr
& Exponential Distribution          && rande &\cr
& Gamma Distribution                && randg &\cr
& Poisson Distribution              && randp &\cr
& Standard Normal Distribution      && randn &\cr
& Uniform Distribution              && rand &\cr
& Uniform Distribution (integers)   && randi &\cr
\noalign{\hrule height 0.6pt}
}}\hfill}}
@end tex
@ifnottex
@multitable @columnfractions .4 .3
@headitem Distribution                  @tab Function
@item Univariate Discrete Distribution  @tab @code{discrete_rnd}
@item Empirical Distribution            @tab @code{empirical_rnd}
@item Exponential Distribution          @tab @code{rande}
@item Gamma Distribution                @tab @code{randg}
@item Poisson Distribution              @tab @code{randp}
@item Standard Normal Distribution      @tab @code{randn}
@item Uniform Distribution              @tab @code{rand}
@item Uniform Distribution (integers)   @tab @code{randi}
@end multitable
@end ifnottex

@DOCSTRING(discrete_rnd)

@DOCSTRING(empirical_rnd)
