GRASS GIS 8 Programmer's Manual 8.2.0(2022)-exported
as177.c
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1
2/*-Algorithm AS 177
3 * Expected Normal Order Statistics (Exact and Approximate),
4 * by J.P. Royston, 1982.
5 * Applied Statistics, 31(2):161-165.
6 *
7 * Translation to C by James Darrell McCauley, mccauley@ecn.purdue.edu.
8 *
9 * The functions Cdhc_nscor1() and Cdhc_nscor2() calculate the expected values of
10 * normal order statistics in exact or approximate form, respectively.
11 *
12 */
13
14#define NSTEP 721
15#define H 0.025
16
17#include <math.h>
18#include <stdio.h>
19#include "local_proto.h"
20
21
22/* Local function prototypes */
23static double Cdhc_alnfac(int j);
24static double Cdhc_correc(int i, int n);
25
26
27/* exact calculation of normal scores */
28void Cdhc_nscor1(double s[], int n, int n2, double work[], int *ifault)
29{
30 double ani, c, c1, d, scor;
31 int i, j;
32
33 *ifault = 3;
34 if (n2 != n / 2)
35 return;
36
37 *ifault = 1;
38 if (n <= 1)
39 return;
40
41 *ifault = 0;
42 if (n > 2000)
43 *ifault = 2;
44
45 /* calculate the natural log of factorial(n) */
46 c1 = Cdhc_alnfac(n);
47 d = c1 - log((double)n);
48
49 /* accumulate ordinates for calculation of integral for rankits */
50 for (i = 0; i < n2; ++i) {
51 ani = (double)n - i - 1;
52 c = c1 - d;
53 for (scor = 0.0, j = 0; j < NSTEP; ++j)
54 scor += work[0 * NSTEP + j] *
55 exp(work[1 * NSTEP + j] + work[2 * NSTEP + j] * i
56 + work[3 * NSTEP + j] * ani + c);
57 s[i] = scor * H;
58 d += log((double)(i + 1.0) / ani);
59 }
60
61 return;
62}
63
64
65void init(double work[])
66{
67 double xstart = -9.0, pi2 = -0.918938533, xx;
68 int i;
69
70 xx = xstart;
71
72 /* set up arrays for calculation of integral */
73 for (i = 0; i < NSTEP; ++i) {
74 work[0 * NSTEP + i] = xx;
75 work[1 * NSTEP + i] = pi2 - xx * xx * 0.5;
76 work[2 * NSTEP + i] = log(Cdhc_alnorm(xx, 1));
77 work[3 * NSTEP + i] = log(Cdhc_alnorm(xx, 0));
78 xx = xstart + H * (i + 1.0);
79 }
80
81 return;
82}
83
84
85/*-Algorithm AS 177.2 Appl. Statist. (1982) Vol.31, No.2
86 * Natural logarithm of factorial for non-negative argument
87 */
88static double Cdhc_alnfac(int j)
89{
90 static double r[7] = { 0.0, 0.0, 0.69314718056, 1.79175946923,
91 3.17805383035, 4.78749174278, 6.57925121101
92 };
93 double w, z;
94
95 if (j == 1)
96 return (double)1.0;
97 else if (j <= 7)
98 return r[j];
99
100 w = (double)j + 1;
101 z = 1.0 / (w * w);
102
103 return (w - 0.5) * log(w) - w + 0.918938522305 +
104 (((4.0 - 3.0 * z) * z - 14.0) * z + 420.0) / (5040.0 * w);
105}
106
107
108/*-Algorithm AS 177.3 Appl. Statist. (1982) Vol.31, No.2
109 * Approximation for Rankits
110 */
111void Cdhc_nscor2(double s[], int n, int n2, int *ifault)
112{
113 static double eps[4] = { 0.419885, 0.450536, 0.456936, 0.468488 };
114 static double dl1[4] = { 0.112063, 0.121770, 0.239299, 0.215159 };
115 static double dl2[4] = { 0.080122, 0.111348, -0.211867, -0.115049 };
116 static double gam[4] = { 0.474798, 0.469051, 0.208597, 0.259784 };
117 static double lam[4] = { 0.282765, 0.304856, 0.407708, 0.414093 };
118 static double bb = -0.283833, d = -0.106136, b1 = 0.5641896;
119 double e1, e2, l1;
120 int i, k;
121
122 *ifault = 3;
123 if (n2 != n / 2)
124 return;
125
126 *ifault = 1;
127 if (n <= 1)
128 return;
129
130 *ifault = 0;
131 if (n > 2000)
132 *ifault = 2;
133
134 s[0] = b1;
135 if (n == 2)
136 return;
137
138 /* calculate normal areas for 3 largest rankits */
139 k = (n2 < 3) ? n2 : 3;
140 for (i = 0; i < k; ++i) {
141 e1 = (1.0 + i - eps[i]) / (n + gam[i]);
142 e2 = pow(e1, lam[i]);
143 s[i] = e1 + e2 * (dl1[i] + e2 * dl2[i]) / n - Cdhc_correc(1 + i, n);
144 }
145
146 if (n2 != k) {
147 /* calculate normal areas for remaining rankits */
148 for (i = 3; i < n2; ++i) {
149 l1 = lam[3] + bb / (1.0 + i + d);
150 e1 = (1.0 + i - eps[3]) / (n + gam[3]);
151 e2 = pow(e1, l1);
152 s[i] = e1 + e2 * (dl1[3] + e2 * dl2[3]) / n - Cdhc_correc(1 + i, n);
153 }
154 }
155
156 /* convert normal tail areas to normal deviates */
157 for (i = 0; i < n2; ++i)
158 s[i] = -ppnd16(s[i]);
159
160 return;
161}
162
163
164/*-Algorithm AS 177.4 Appl. Statist. (1982) Vol.31, No.2
165 * Calculates Cdhc_correction for tail area of noraml distribution
166 * corresponding to ith largest rankit in sample size n.
167 */
168static double Cdhc_correc(int i, int n)
169{
170 static double c1[7] = { 9.5, 28.7, 1.9, 0.0, -7.0, -6.2, -1.6 };
171 static double c2[7] = { -6.195e3, -9.569e3, -6.728e3, -17.614e3,
172 -8.278e3, -3.570e3, 1.075e3
173 };
174 static double c3[7] = { 9.338e4, 1.7516e5, 4.1040e5, 2.157e6,
175 2.376e6, 2.065e6, 2.065e6
176 };
177 static double mic = 1.0e-6, c14 = 1.9e-5;
178 double an;
179
180 if (i * n == 4)
181 return c14;
182
183 if (i < 1 || i > 7)
184 return 0.0;
185 else if (i != 4 && n > 20)
186 return 0.0;
187 else if (i == 4 && n > 40)
188 return 0.0;
189
190 /* else */
191 an = 1.0 / (double)(n * n);
192 return (c1[i - 1] + an * (c2[i - 1] + an * c3[i - 1])) * mic;
193}
#define NSTEP
Definition: as177.c:14
void init(double work[])
Definition: as177.c:65
void Cdhc_nscor1(double s[], int n, int n2, double work[], int *ifault)
Definition: as177.c:28
void Cdhc_nscor2(double s[], int n, int n2, int *ifault)
Definition: as177.c:111
#define H
Definition: as177.c:15
double ppnd16(double p)
Definition: as241.c:90
double Cdhc_alnorm(double x, int upper)
Definition: as66.c:35
double r