1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
|
########################################################################
##
## Copyright (C) 1995-2024 The Octave Project Developers
##
## See the file COPYRIGHT.md in the top-level directory of this
## distribution or <https://octave.org/copyright/>.
##
## 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
## <https://www.gnu.org/licenses/>.
##
########################################################################
## -*- texinfo -*-
## @deftypefn {} {@var{m} =} moment (@var{x}, @var{p})
## @deftypefnx {} {@var{m} =} moment (@var{x}, @var{p}, @var{type})
## @deftypefnx {} {@var{m} =} moment (@var{x}, @var{p}, @var{dim})
## @deftypefnx {} {@var{m} =} moment (@var{x}, @var{p}, @var{type}, @var{dim})
## @deftypefnx {} {@var{m} =} moment (@var{x}, @var{p}, @var{dim}, @var{type})
## Compute the @var{p}-th central moment of the vector @var{x}.
##
## The @var{p}-th central moment of @var{x} is defined as:
##
## @tex
## $$
## {\sum_{i=1}^N (x_i - \bar{x})^p \over N}
## $$
## where $\bar{x}$ is the mean value of @var{x} and $N$ is the number of elements of @var{x}.
##
##
## @end tex
## @ifnottex
##
## @example
## @group
## 1/N SUM_i (@var{x}(i) - mean(@var{x}))^@var{p}
## @end group
## @end example
##
## @noindent
## where @math{N} is the length of the @var{x} vector.
##
## @end ifnottex
##
## If @var{x} is a matrix, return the row vector containing the @var{p}-th
## central moment of each column.
##
## If the optional argument @var{dim} is given, operate along this dimension.
##
## The optional string @var{type} specifies the type of moment to be computed.
## Valid options are:
##
## @table @asis
## @item @qcode{"c"}
## Central Moment (default).
##
## @item @qcode{"a"}
## @itemx @qcode{"ac"}
## Absolute Central Moment. The moment about the mean ignoring sign
## defined as
## @tex
## $$
## {\sum_{i=1}^N {\left| x_i - \bar{x} \right|}^p \over N}
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## 1/N SUM_i (abs (@var{x}(i) - mean(@var{x})))^@var{p}
## @end group
## @end example
##
## @end ifnottex
##
## @item @qcode{"r"}
## Raw Moment. The moment about zero defined as
##
## @tex
## $$
## {\rm moment} (x) = { \sum_{i=1}^N {x_i}^p \over N }
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## moment (@var{x}) = 1/N SUM_i @var{x}(i)^@var{p}
## @end group
## @end example
##
## @end ifnottex
##
## @item @nospell{@qcode{"ar"}}
## Absolute Raw Moment. The moment about zero ignoring sign defined as
## @tex
## $$
## {\sum_{i=1}^N {\left| x_i \right|}^p \over N}
## $$
## @end tex
## @ifnottex
##
## @example
## @group
## 1/N SUM_i ( abs (@var{x}(i)) )^@var{p}
## @end group
## @end example
##
## @end ifnottex
## @end table
##
## If both @var{type} and @var{dim} are given they may appear in any order.
## @seealso{var, skewness, kurtosis}
## @end deftypefn
## Can easily be made to work for continuous distributions (using quad)
## as well, but how does the general case work?
function m = moment (x, p, opt1, opt2)
if (nargin < 2)
print_usage ();
endif
if (! (isnumeric (x) || islogical (x)) || isempty (x))
error ("moment: X must be a non-empty numeric matrix or vector");
endif
if (! (isnumeric (p) && isscalar (p)))
error ("moment: P must be a numeric scalar");
endif
need_dim = false;
if (nargin == 2)
type = "";
need_dim = true;
elseif (nargin == 3)
if (ischar (opt1))
type = opt1;
need_dim = true;
else
dim = opt1;
type = "";
endif
elseif (nargin == 4)
if (ischar (opt1))
type = opt1;
dim = opt2;
elseif (ischar (opt2))
type = opt2;
dim = opt1;
else
error ("moment: TYPE must be a string");
endif
endif
nd = ndims (x);
sz = size (x);
if (need_dim)
## Find the first non-singleton dimension.
(dim = find (sz > 1, 1)) || (dim = 1);
else
if (! (isscalar (dim) && dim == fix (dim) && dim > 0))
error ("moment: DIM must be an integer and a valid dimension");
endif
endif
n = size (x, dim);
if (! any (type == "r"))
x = center (x, dim);
endif
if (any (type == "a"))
x = abs (x);
endif
m = sum (x .^ p, dim) / n;
endfunction
%!shared x
%! x = rand (10);
%!assert (moment (x,1), mean (center (x)), eps)
%!assert (moment (x,2), meansq (center (x)), eps)
%!assert (moment (x,1,2), mean (center (x, 2), 2), eps)
%!assert (moment (x,1,"a"), mean (abs (center (x))), eps)
%!assert (moment (x,1,"r"), mean (x), eps)
%!assert (moment (x,1,"ar"), mean (abs (x)), eps)
%!assert (moment (single ([1 2 3]), 1, "r"), single (2))
%!assert (moment (1, 2, 4), 0)
## Test input validation
%!error <Invalid call> moment ()
%!error <Invalid call> moment (1)
%!error <X must be a non-empty numeric matrix> moment (['A'; 'B'], 2)
%!error <X must be a non-empty numeric matrix> moment (ones (2,0,3), 2)
%!error <P must be a numeric scalar> moment (1, true)
%!error <P must be a numeric scalar> moment (1, ones (2,2))
%!error <TYPE must be a string> moment (1, 2, 3, 4)
%!error <DIM must be an integer and a valid dimension> moment (1, 2, ones (2,2))
%!error <DIM must be an integer and a valid dimension> moment (1, 2, 1.5)
|
