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
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
|
/* statistics/Sn_source.c
*
* Copyright (C) 2018 Patrick Alken
* Copyright (C) 2005, 2006, 2007 Martin Maechler, ETH Zurich
*
* This program 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.
*
* This program 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 this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
/* This is a merge of the C version of original files qn.f and sn.f,
* translated by f2c (version 20010821). ==== ====
* and then by f2c-clean,v 1.9 2000/01/13 13:46:53
* and further clean-edited manually by Martin Maechler.
*
* Further added interface functions to be called via .C() from R or S-plus
* Note that Peter Rousseeuw has explicitely given permission to
* use his code under the GPL for the R project.
*/
/* Original comments by the authors of the Fortran original code,
* (merged for Qn & Sn in one file by M.M.):
This file contains fortran functions for two new robust estimators
of scale denoted as Qn and Sn, decribed in Rousseeuw and Croux (1993).
These estimators have a high breakdown point and a bounded influence
function. The implementation given here is very fast (running in
O(n logn) time) and needs little storage space.
Rousseeuw, P.J. and Croux, C. (1993)
Alternatives to the Median Absolute Deviation",
Journal of the American Statistical Association, Vol. 88, 1273-1283.
For both estimators, implementations in the pascal language can be
obtained from the original authors.
This software may be used and copied freely for scientific
and/or non-commercial purposes, provided reference is made
to the abovementioned paper.
Note by MM: We have explicit permission from P.Rousseeuw to
licence it under the GNU Public Licence.
*/
/*
gsl_stats_Sn0_from_sorted_data()
Efficient algorithm for the scale estimator:
S_n0 = LOMED_{i} HIMED_{i} |x_i - x_j|
which can equivalently be written as
S_n0 = LOMED_{i} LOMED_{j != i} |x_i - x_j|
Inputs: sorted_data - sorted array containing the observations
stride - stride
n - length of 'sorted_data'
work - workspace of length n
work[i] := LOMED_{j != i} | x_i - x_j |
Return: S_n statistic (without scale/correction factor)
*/
BASE
FUNCTION(gsl_stats,Sn0_from_sorted_data) (const BASE sorted_data[],
const size_t stride,
const size_t n,
BASE work[])
{
/* Local variables */
double medA, medB;
int i, diff, half, Amin, Amax, even, length;
int leftA, leftB, nA, nB, tryA, tryB, rightA, rightB;
int np1_2 = (n + 1) / 2;
work[0] = sorted_data[n / 2 * stride] - sorted_data[0];
/* first half for() loop : */
for (i = 2; i <= np1_2; ++i)
{
nA = i - 1;
nB = n - i;
diff = nB - nA;
leftA = leftB = 1;
rightA = rightB = nB;
Amin = diff / 2 + 1;
Amax = diff / 2 + nA;
while (leftA < rightA)
{
length = rightA - leftA + 1;
even = 1 - length % 2;
half = (length - 1) / 2;
tryA = leftA + half;
tryB = leftB + half;
if (tryA < Amin)
{
rightB = tryB;
leftA = tryA + even;
}
else
{
if (tryA > Amax)
{
rightA = tryA;
leftB = tryB + even;
}
else
{
medA = sorted_data[(i - 1) * stride] - sorted_data[(i - tryA + Amin - 2) * stride];
medB = sorted_data[(tryB + i - 1) * stride] - sorted_data[(i - 1) * stride];
if (medA >= medB)
{
rightA = tryA;
leftB = tryB + even;
}
else
{
rightB = tryB;
leftA = tryA + even;
}
}
}
} /* while */
if (leftA > Amax)
{
work[i - 1] = sorted_data[(leftB + i - 1) * stride] - sorted_data[(i - 1) * stride];
}
else
{
medA = sorted_data[(i - 1) * stride] - sorted_data[(i - leftA + Amin - 2) * stride];
medB = sorted_data[(leftB + i - 1) * stride] - sorted_data[(i - 1) * stride];
work[i - 1] = GSL_MIN(medA, medB);
}
}
/* second half for() loop : */
for (i = np1_2 + 1; i <= (int) n - 1; ++i)
{
nA = n - i;
nB = i - 1;
diff = nB - nA;
leftA = leftB = 1;
rightA = rightB = nB;
Amin = diff / 2 + 1;
Amax = diff / 2 + nA;
while (leftA < rightA)
{
length = rightA - leftA + 1;
even = 1 - length % 2;
half = (length - 1) / 2;
tryA = leftA + half;
tryB = leftB + half;
if (tryA < Amin)
{
rightB = tryB;
leftA = tryA + even;
}
else
{
if (tryA > Amax)
{
rightA = tryA;
leftB = tryB + even;
}
else
{
medA = sorted_data[(i + tryA - Amin) * stride] - sorted_data[(i - 1) * stride];
medB = sorted_data[(i - 1) * stride] - sorted_data[(i - tryB - 1) * stride];
if (medA >= medB)
{
rightA = tryA;
leftB = tryB + even;
}
else
{
rightB = tryB;
leftA = tryA + even;
}
}
}
} /* while */
if (leftA > Amax)
{
work[i - 1] = sorted_data[(i - 1) * stride] - sorted_data[(i - leftB - 1) * stride];
}
else
{
medA = sorted_data[(i + leftA - Amin) * stride] - sorted_data[(i - 1) * stride];
medB = sorted_data[(i - 1) * stride] - sorted_data[(i - leftB - 1) * stride];
work[i - 1] = GSL_MIN(medA, medB);
}
}
work[n - 1] = sorted_data[(n - 1) * stride] - sorted_data[(np1_2 - 1) * stride];
/* sort work array */
TYPE (gsl_sort) (work, 1, n);
return work[np1_2 - 1];
}
/*
gsl_stats_Sn_from_sorted_data()
Efficient algorithm for the scale estimator:
S_n = 1.1926 * c_n LOMED_{i} HIMED_{i} |x_i - x_j|
which can equivalently be written as
S_n = 1.1926 * c_n * LOMED_{i} LOMED_{j != i} |x_i - x_j|
and c_n is a correction factor for small sample bias
Inputs: sorted_data - sorted array containing the observations
stride - stride
n - length of 'sorted_data'
work - workspace of length n
work[i] := LOMED_{j != i} | x_i - x_j |
Return: S_n statistic
*/
double
FUNCTION(gsl_stats,Sn_from_sorted_data) (const BASE sorted_data[],
const size_t stride,
const size_t n,
BASE work[])
{
const double scale = 1.1926; /* asymptotic consistency for sigma^2 */
double Sn0 = (double) FUNCTION(gsl_stats,Sn0_from_sorted_data)(sorted_data, stride, n, work);
double cn = 1.0;
double Sn;
/* determine correction factor for finite sample bias */
if (n <= 9)
{
if (n == 2) cn = 0.743;
else if (n == 3) cn = 1.851;
else if (n == 4) cn = 0.954;
else if (n == 5) cn = 1.351;
else if (n == 6) cn = 0.993;
else if (n == 7) cn = 1.198;
else if (n == 8) cn = 1.005;
else if (n == 9) cn = 1.131;
}
else if (n % 2 == 1) /* n odd, >= 11 */
{
cn = (double) n / (n - 0.9);
}
Sn = scale * cn * Sn0;
return Sn;
}
|
