Mercurial > octave
view scripts/statistics/base/kurtosis.m @ 24511:4f0e6ee6c9b8 stable
Make documentation Sec 26.1 more consistent and Sec 25.4 clearer (bug #52685)
* corr.m: Add space in LaTeX formula. For the example, place variables in a
@var qualifier.
* cov.m: Use @var in LaTeX for x and y when referring to function input vector.
Correct Octave-help formula by placing parentheses around N-1 so that -1 is
in the denominator. Define N after the formula in which it is used.
* gls.m: Define what GLS stands for. Use @var instead of @math for function
input and output variables. Move the description of matrix O and scalar s
to a third paragraph, ensuring s is lower case. Give a little more context
to the description of X and Y in the second paragraph. Add an expansive
paragraph three for details about the error variables E including the
description of O and s along with their dimensions. Add "matrix" before
B and "scalar" before s for clarity. Place @var around variables r, y, x
and beta to make those upper case in Octave-help.
* histc.m: Use LaTeX math rather than @code for the @tex scenario.
* kendall.m: Treat tau differently for LaTeX and Octave-help scenarios. Add
space in LaTeX formulas. Treat tau as @var in Octave-help case. Use lower
case 'i' for index variable and upper case 'N' for vector length.
* kurtosis.m: For mean value of x, use script rather than non-script. Define
N after the formula in which it is used for Octave-help case.
* mean.m: Indicate N is number of elements. Use @var on input vector x for
Octave-help case.
* meansq.m: Indicate N is number of elements, but drop the reference to mean
value because there is none. Use @var on input vector x for Octave-help
case. Use "If x is a matrix" consistent with all others.
* median.m: Indicate N is number of elements for LaTeX case. For Octave-help
place some vertical lines to represent case curly-bracket. Place @math
around N. Define an intermediate vector S representing sorted X and use
that in the math formula.
* moment.m: Define x-bar as mean and N as number of elements. Use @var on
x and p in the Octave-help formulas.
* ols.m: Define meaning of OLS. Add @var to LaTeX variables to make them
non-script vectors. Use @var instead of @math for function input and output
variables. Use hyphens for matrix dimensions in Octave-help formula. Move
the description of matrix S to a third paragraph. Give a little more context
to the description of X and Y in the second paragraph. Add an expansive
paragraph three for details about the error variables E including the
description of matrix S along with its dimensions, ensuring S is upper case.
Add "matrix" before B for clarity. Make the definition of SIGMA one line for
appearance in Octave-help.
* prctile.m: Change a mistaken 'y' to 'q' to work in LaTeX as well.
* quantile.m: Use @var{method} rather than METHOD. Break up all the method
formulas for p(k) into LaTeX and Octave-help versions for better control.
Use upper case N for the length of P.
* skewness.m: Remove @var from x when referring to vector elements in LaTeX.
Indicate N is number of elements.
* spearman.m: Break into separate LaTeX and Octave-help cases rather than
use @code for LaTeX. Use Greek symbol rho in LaTeX.
* std.m: Add @var to x variable to indicate LaTeX or Octave-help vector. Add
clarification about N being number elements of x to both LaTeX and
Octave-help formulas.
* var.m: Indicate N is number of elements. Apply @var to x to show it is a
vector. Change == to "is equal to" for normal text.
author | Daniel J Sebald <daniel.sebald@ieee.org> |
---|---|
date | Wed, 27 Dec 2017 23:38:25 -0600 |
parents | 3ac9f9ecfae5 |
children | 3fc1c8ebe5c3 |
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## Copyright (C) 2013-2017 Julien Bect ## Copyright (C) 1996-2016 John W. Eaton ## ## This file is part of Octave. ## ## Octave is free software; you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation; either version 3 of the License, or (at ## your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU ## General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## <http://www.gnu.org/licenses/>. ## -*- texinfo -*- ## @deftypefn {} {} kurtosis (@var{x}) ## @deftypefnx {} {} kurtosis (@var{x}, @var{flag}) ## @deftypefnx {} {} kurtosis (@var{x}, @var{flag}, @var{dim}) ## Compute the sample kurtosis of the elements of @var{x}. ## ## The sample kurtosis is defined as ## @tex ## $$ ## \kappa_1 = {{{1\over N}\, ## \sum_{i=1}^N (x_i - \bar{x})^4} \over \sigma^4}, ## $$ ## where $N$ is the length of @var{x}, $\bar{x}$ its mean, and $\sigma$ ## its (uncorrected) standard deviation. ## @end tex ## @ifnottex ## ## @example ## @group ## mean ((@var{x} - mean (@var{x})).^4) ## k1 = ------------------------ ## std (@var{x}).^4 ## @end group ## @end example ## ## @end ifnottex ## ## @noindent ## The optional argument @var{flag} controls which normalization is used. ## If @var{flag} is equal to 1 (default value, used when @var{flag} is omitted ## or empty), return the sample kurtosis as defined above. If @var{flag} is ## equal to 0, return the @w{"bias-corrected"} kurtosis coefficient instead: ## @tex ## $$ ## \kappa_0 = 3 + {\scriptstyle N - 1 \over \scriptstyle (N - 2)(N - 3)} \, ## \left( (N + 1)\, \kappa_1 - 3 (N - 1) \right) ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## N - 1 ## k0 = 3 + -------------- * ((N + 1) * k1 - 3 * (N - 1)) ## (N - 2)(N - 3) ## @end group ## @end example ## ## where @math{N} is the length of the @var{x} vector. ## ## @end ifnottex ## The bias-corrected kurtosis coefficient is obtained by replacing the sample ## second and fourth central moments by their unbiased versions. It is an ## unbiased estimate of the population kurtosis for normal populations. ## ## If @var{x} is a matrix, or more generally a multi-dimensional array, return ## the kurtosis along the first non-singleton dimension. If the optional ## @var{dim} argument is given, operate along this dimension. ## ## @seealso{var, skewness, moment} ## @end deftypefn ## Author: KH <Kurt.Hornik@wu-wien.ac.at> ## Created: 29 July 1994 ## Adapted-By: jwe function y = kurtosis (x, flag, dim) if (nargin < 1) || (nargin > 3) print_usage (); endif if (! (isnumeric (x) || islogical (x))) error ("kurtosis: X must be a numeric vector or matrix"); endif if (nargin < 2 || isempty (flag)) flag = 1; # default: do not use the "bias corrected" version else if (! isscalar (flag) || (flag != 0 && flag != 1)) error ("kurtosis: FLAG must be 0 or 1"); endif endif nd = ndims (x); sz = size (x); if (nargin < 3) ## Find the first non-singleton dimension. (dim = find (sz > 1, 1)) || (dim = 1); else if (! (isscalar (dim) && dim == fix (dim) && dim > 0)) error ("kurtosis: DIM must be an integer and a valid dimension"); endif endif n = size (x, dim); sz(dim) = 1; x = center (x, dim); # center also promotes integer, logical to double v = var (x, 1, dim); # normalize with 1/N y = sum (x .^ 4, dim); idx = (v != 0); y(idx) = y(idx) ./ (n * v(idx) .^ 2); y(! idx) = NaN; ## Apply bias correction to the second and fourth central sample moment if (flag == 0) if (n > 3) C = (n - 1) / ((n - 2) * (n - 3)); y = 3 + C * ((n + 1) * y - 3 * (n - 1)); else y(:) = NaN; endif endif endfunction %!test %! x = [-1; 0; 0; 0; 1]; %! y = [x, 2*x]; %! assert (kurtosis (y), [2.5, 2.5], sqrt (eps)); %!assert (kurtosis ([-3, 0, 1]) == kurtosis ([-1, 0, 3])) %!assert (kurtosis (ones (3, 5)), NaN (1, 5)) %!assert (kurtosis (1, [], 3), NaN) %!assert (kurtosis ([1:5 10; 1:5 10], 0, 2), 5.4377317925288901 * [1; 1], 8 * eps) %!assert (kurtosis ([1:5 10; 1:5 10], 1, 2), 2.9786509002956195 * [1; 1], 8 * eps) %!assert (kurtosis ([1:5 10; 1:5 10], [], 2), 2.9786509002956195 * [1; 1], 8 * eps) ## Test behavior on single input %!assert (kurtosis (single ([1:5 10])), single (2.9786513), eps ("single")) %!assert (kurtosis (single ([1 2]), 0), single (NaN)) ## Verify no "divide-by-zero" warnings %!test %! warning ("on", "Octave:divide-by-zero", "local"); %! lastwarn (""); # clear last warning %! kurtosis (1); %! assert (lastwarn (), ""); ## Test input validation %!error kurtosis () %!error kurtosis (1, 2, 3) %!error <X must be a numeric vector or matrix> kurtosis (['A'; 'B']) %!error <FLAG must be 0 or 1> kurtosis (1, 2) %!error <FLAG must be 0 or 1> kurtosis (1, [1 0]) %!error <DIM must be an integer> kurtosis (1, [], ones (2,2)) %!error <DIM must be an integer> kurtosis (1, [], 1.5) %!error <DIM must be .* a valid dimension> kurtosis (1, [], 0)