File: mpchilim.pro

package info (click to toggle)
mpfit 1.85+2017.01.03-4
  • links: PTS, VCS
  • area: main
  • in suites: bullseye, sid
  • size: 788 kB
  • sloc: python: 126; makefile: 16; sh: 13
file content (598 lines) | stat: -rw-r--r-- 16,072 bytes parent folder | download | duplicates (3)
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
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
;+
; NAME:
;   MPCHILIM
;
; AUTHOR:
;   Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
;   craigm@lheamail.gsfc.nasa.gov
;   UPDATED VERSIONs can be found on my WEB PAGE: 
;      http://cow.physics.wisc.edu/~craigm/idl/idl.html
;
; PURPOSE:
;   Compute confidence limits for chi-square statistic
;
; MAJOR TOPICS:
;   Curve and Surface Fitting, Statistics
;
; CALLING SEQUENCE:
;   DELCHI = MPCHILIM(PROB, DOF, [/SIGMA, /CLEVEL, /SLEVEL ])
;
; DESCRIPTION:
;
;  The function MPCHILIM() computes confidence limits of the
;  chi-square statistic for a desired probability level.  The returned
;  values, DELCHI, are the limiting chi-squared values: a chi-squared
;  value of greater than DELCHI will occur by chance with probability
;  PROB:
;
;    P_CHI(CHI > DELCHI; DOF) = PROB
;
;  In specifying the probability level the user has three choices:
;
;    * give the confidence level (default);
;
;    * give the significance level (i.e., 1 - confidence level) and
;      pass the /SLEVEL keyword; OR
;
;    * give the "sigma" of the probability (i.e., compute the
;      probability based on the normal distribution) and pass the
;      /SIGMA keyword.
;
;  Note that /SLEVEL, /CLEVEL and /SIGMA are mutually exclusive.
;
; INPUTS:
;
;   PROB - scalar or vector number, giving the desired probability
;          level as described above.
;
;   DOF - scalar or vector number, giving the number of degrees of
;         freedom in the chi-square distribution.
;
; RETURNS:
;
;  Returns a scalar or vector of chi-square confidence limits.
;
; KEYWORD PARAMETERS:
;
;   SLEVEL - if set, then PROB describes the significance level.
;
;   CLEVEL - if set, then PROB describes the confidence level
;            (default).
;
;   SIGMA - if set, then PROB is the number of "sigma" away from the
;           mean in the normal distribution.
;
; EXAMPLES:
;
;   print, mpchilim(0.99d, 2d, /clevel)
;
;   Print the 99% confidence limit for a chi-squared of 2 degrees of
;   freedom.
;
;   print, mpchilim(5d, 2d, /sigma)
;
;   Print the "5 sigma" confidence limit for a chi-squared of 2
;   degrees of freedom.  Here "5 sigma" indicates the gaussian
;   probability of a 5 sigma event or greater. 
;       P_GAUSS(5D) = 1D - 5.7330314e-07
;
; REFERENCES:
;
;   Algorithms taken from CEPHES special function library, by Stephen
;   Moshier. (http://www.netlib.org/cephes/)
;
; MODIFICATION HISTORY:
;   Completed, 1999, CM
;   Documented, 16 Nov 2001, CM
;   Reduced obtrusiveness of common block and math error handling, 18
;     Nov 2001, CM
;   Convert to IDL 5 array syntax (!), 16 Jul 2006, CM
;   Move STRICTARR compile option inside each function/procedure, 9
;     Oct 2006
;   Add usage message, 24 Nov 2006, CM
;   Usage message with /CONTINUE, 23 Sep 2009, CM
;
;  $Id: mpchilim.pro,v 1.8 2009/09/23 20:12:46 craigm Exp $
;-
; Copyright (C) 1997-2001, 2006, 2009, Craig Markwardt
; This software is provided as is without any warranty whatsoever.
; Permission to use, copy, modify, and distribute modified or
; unmodified copies is granted, provided this copyright and disclaimer
; are included unchanged.
;-

forward_function cephes_ndtri, cephes_igam, cephes_igamc, cephes_igami

;; Set machine constants, once for this session.  Double precision
;; only.
pro cephes_setmachar
  COMPILE_OPT strictarr
  common cephes_machar, cephes_machar_vals
  if n_elements(cephes_machar_vals) GT 0 then return

  if (!version.release) LT 5 then dummy = check_math(1, 1)

  mch = machar(/double)
  machep = mch.eps
  maxnum = mch.xmax
  minnum = mch.xmin
  maxlog = alog(mch.xmax)
  minlog = alog(mch.xmin)
  maxgam = 171.624376956302725D

  cephes_machar_vals = {machep: machep, maxnum: maxnum, minnum: minnum, $
                        maxlog: maxlog, minlog: minlog, maxgam: maxgam}

  if (!version.release) LT 5 then dummy = check_math(0, 0)
  return
end

function cephes_polevl, x, coef
  COMPILE_OPT strictarr
  ans = coef[0]
  nc  = n_elements(coef)
  for i = 1L, nc-1 do ans = ans * x + coef[i]
  return, ans
end

function cephes_ndtri, y0
;   
;   	Inverse of Normal distribution function
;   
;   
;   
;    SYNOPSIS:
;   
;    double x, y, ndtri();
;   
;    x = ndtri( y );
;   
;   
;   
;    DESCRIPTION:
;   
;    Returns the argument, x, for which the area under the
;    Gaussian probability density function (integrated from
;    minus infinity to x) is equal to y.
;   
;   
;    For small arguments 0 < y < exp(-2), the program computes
;    z = sqrt( -2.0 * log(y) );  then the approximation is
;    x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z).
;    There are two rational functions P/Q, one for 0 < y < exp(-32)
;    and the other for y up to exp(-2).  For larger arguments,
;    w = y - 0.5, and  x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
;   
;   
;    ACCURACY:
;   
;                         Relative error:
;    arithmetic   domain        # trials      peak         rms
;       DEC      0.125, 1         5500       9.5e-17     2.1e-17
;       DEC      6e-39, 0.135     3500       5.7e-17     1.3e-17
;       IEEE     0.125, 1        20000       7.2e-16     1.3e-16
;       IEEE     3e-308, 0.135   50000       4.6e-16     9.8e-17
;   
;   
;    ERROR MESSAGES:
;   
;      message         condition    value returned
;    ndtri domain       x <= 0        -MAXNUM
;    ndtri domain       x >= 1         MAXNUM
  COMPILE_OPT strictarr
  common cephes_ndtri_data, s2pi, p0, q0, p1, q1, p2, q2

  if n_elements(s2pi) EQ 0 then begin
      s2pi = sqrt(2.D*!dpi)
      p0 = [ -5.99633501014107895267D1,  9.80010754185999661536D1, $
             -5.66762857469070293439D1,  1.39312609387279679503D1, $
             -1.23916583867381258016D0 ]
      q0 = [ 1.D, $
             1.95448858338141759834D0,   4.67627912898881538453D0, $
             8.63602421390890590575D1,  -2.25462687854119370527D2, $
             2.00260212380060660359D2,  -8.20372256168333339912D1, $
             1.59056225126211695515D1,  -1.18331621121330003142D0  ]
      p1 = [ 4.05544892305962419923D0,   3.15251094599893866154D1, $
             5.71628192246421288162D1,   4.40805073893200834700D1, $
             1.46849561928858024014D1,   2.18663306850790267539D0, $
             -1.40256079171354495875D-1,-3.50424626827848203418D-2,$
             -8.57456785154685413611D-4  ]
      q1 = [ 1.D, $
             1.57799883256466749731D1,   4.53907635128879210584D1, $
             4.13172038254672030440D1,   1.50425385692907503408D1, $
             2.50464946208309415979D0,  -1.42182922854787788574D-1,$
             -3.80806407691578277194D-2,-9.33259480895457427372D-4 ]
      p2 = [  3.23774891776946035970D0,  6.91522889068984211695D0, $
              3.93881025292474443415D0,  1.33303460815807542389D0, $
              2.01485389549179081538D-1, 1.23716634817820021358D-2,$
              3.01581553508235416007D-4, 2.65806974686737550832D-6,$
              6.23974539184983293730D-9 ]
      q2 = [  1.D, $
              6.02427039364742014255D0,  3.67983563856160859403D0, $
              1.37702099489081330271D0,  2.16236993594496635890D-1,$
              1.34204006088543189037D-2, 3.28014464682127739104D-4,$
              2.89247864745380683936D-6, 6.79019408009981274425D-9]
  endif

  common cephes_machar, machvals
  MAXNUM = machvals.maxnum

  if y0 LE 0 then begin
      message, 'ERROR: domain', /info
      return, -MAXNUM
  endif
  if y0 GE 1 then begin
      message, 'ERROR: domain', /info
      return, MAXNUM
  endif

  code = 1
  y = y0
  exp2 = exp(-2.D)
  if y GT (1.D - exp2) then begin
      y = 1.D - y
      code = 0
  endif
  if y GT exp2 then begin
      y = y - 0.5
      y2 = y * y
      x = y + y * y2 * cephes_polevl(y2, p0) / cephes_polevl(y2, q0)
      x = x * s2pi
      return, x
  endif
  
  x = sqrt( -2.D * alog(y))
  x0 = x - alog(x)/x
  z = 1.D/x
  if x LT 8. then $
    x1 = z * cephes_polevl(z, p1) / cephes_polevl(z, q1) $
  else $
    x1 = z * cephes_polevl(z, p2) / cephes_polevl(z, q2)

  x = x0 - x1
  if code NE 0 then x = -x
  return, x
end

function cephes_igam, a, x
;   
;   	Incomplete gamma integral
;   
;   
;   
;    SYNOPSIS:
;   
;    double a, x, y, igam();
;   
;    y = igam( a, x );
;   
;    DESCRIPTION:
;   
;    The function is defined by
;   
;                              x
;                               -
;                      1       | |  -t  a-1
;     igam(a,x)  =   -----     |   e   t   dt.
;                     -      | |
;                    | (a)    -
;                              0
;   
;   
;    In this implementation both arguments must be positive.
;    The integral is evaluated by either a power series or
;    continued fraction expansion, depending on the relative
;    values of a and x.
;   
;    ACCURACY:
;   
;                         Relative error:
;    arithmetic   domain     # trials      peak         rms
;       IEEE      0,30       200000       3.6e-14     2.9e-15
;       IEEE      0,100      300000       9.9e-14     1.5e-14
  COMPILE_OPT strictarr
  common cephes_machar, machvals
  MAXLOG = machvals.maxlog
  MACHEP = machvals.machep

  if x LE 0 OR a LE 0 then return, 0.D
  if x GT 1. AND x GT a then return, 1.D - cephes_igamc(a, x)
  
  ax = a * alog(x) - x - lngamma(a)
  if ax LT -MAXLOG then begin
;      message, 'WARNING: underflow', /info
      return, 0.D
  endif
  ax = exp(ax)
  r = a
  c = 1.D
  ans = 1.D
  
  repeat begin
      r = r + 1
      c = c * x/r
      ans = ans + c
  endrep until (c/ans LE MACHEP)

  return, ans*ax/a
end

function cephes_igamc, a, x
;   
;   	Complemented incomplete gamma integral
;   
;   
;   
;    SYNOPSIS:
;   
;    double a, x, y, igamc();
;   
;    y = igamc( a, x );
;   
;    DESCRIPTION:
;   
;    The function is defined by
;   
;   
;     igamc(a,x)   =   1 - igam(a,x)
;   
;                               inf.
;                                 -
;                        1       | |  -t  a-1
;                  =   -----     |   e   t   dt.
;                       -      | |
;                      | (a)    -
;                                x
;   
;   
;    In this implementation both arguments must be positive.
;    The integral is evaluated by either a power series or
;    continued fraction expansion, depending on the relative
;    values of a and x.
;   
;    ACCURACY:
;   
;    Tested at random a, x.
;                   a         x                      Relative error:
;    arithmetic   domain   domain     # trials      peak         rms
;       IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
;       IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15

  COMPILE_OPT strictarr
  common cephes_machar, machvals
  MAXLOG = machvals.maxlog
  MACHEP = machvals.machep

  big = 4.503599627370496D15
  biginv = 2.22044604925031308085D-16

  if x LE 0 OR a LE 0 then return, 1.D
  if x LT 1. OR x LT a then return, 1.D - cephes_igam(a, x)
  ax = a * alog(x) - x - lngamma(a)

  if ax LT -MAXLOG then begin
;      message, 'ERROR: underflow', /info
      return, 0.D
  endif
  
  ax = exp(ax)
  y = 1.D - a
  z = x + y + 1.D
  c = 0.D
  pkm2 = 1.D
  qkm2 = x
  pkm1 = x + 1.D
  qkm1 = z * x
  ans = pkm1 / qkm1

  repeat begin
      c = c + 1.D
      y = y + 1.D
      z = z + 2.D
      yc = y * c
      pk = pkm1 * z - pkm2 * yc
      qk = qkm1 * z - qkm2 * yc
      if qk NE 0 then begin
          r = pk/qk
          t = abs( (ans-r)/r )
          ans = r
      endif else begin
          t = 1.D
      endelse
      pkm2 = pkm1
      pkm1 = pk
      qkm2 = qkm1
      qkm1 = qk
      if abs(pk) GT big then begin
          pkm2 = pkm2 * biginv
          pkm1 = pkm1 * biginv
          qkm2 = qkm2 * biginv
          qkm1 = qkm1 * biginv
      endif
  endrep until t LE MACHEP

  return, ans * ax
end
  
function cephes_igami, a, y0
;  
;        Inverse of complemented imcomplete gamma integral
;  
;  
;  
;   SYNOPSIS:
;  
;   double a, x, p, igami();
;  
;   x = igami( a, p );
;  
;   DESCRIPTION:
;  
;   Given p, the function finds x such that
;  
;    igamc( a, x ) = p.
;  
;   Starting with the approximate value
;  
;           3
;    x = a t
;  
;    where
;  
;    t = 1 - d - ndtri(p) sqrt(d)
;   
;   and
;  
;    d = 1/9a,
;  
;   the routine performs up to 10 Newton iterations to find the
;   root of igamc(a,x) - p = 0.
;  
;   ACCURACY:
;  
;   Tested at random a, p in the intervals indicated.
;  
;                  a        p                      Relative error:
;   arithmetic   domain   domain     # trials      peak         rms
;      IEEE     0.5,100   0,0.5       100000       1.0e-14     1.7e-15
;      IEEE     0.01,0.5  0,0.5       100000       9.0e-14     3.4e-15
;      IEEE    0.5,10000  0,0.5        20000       2.3e-13     3.8e-14

  COMPILE_OPT strictarr
  common cephes_machar, machvals
  MAXNUM = machvals.maxnum
  MAXLOG = machvals.maxlog
  MACHEP = machvals.machep

  x0 = MAXNUM
  yl = 0.D
  x1 = 0.D
  yh = 1.D
  dithresh = 5.D * MACHEP

  d = 1.D/(9.D*a)
  y = (1.D - d - cephes_ndtri(y0) * sqrt(d))
  x = a * y * y * y
 
  lgm = lngamma(a)

  for i=0, 9 do begin
      if x GT x0 OR x LT x1 then goto, ihalve
      y = cephes_igamc(a, x)
      if y LT yl OR y GT yh then goto, ihalve
      if y LT y0 then begin
          x0 = x
          yl = y
      endif else begin
          x1 = x
          yh = y
      endelse
      
      d = (a-1.D) * alog(x) - x - lgm
      if d LT -MAXLOG then goto, ihalve
      d = -exp(d)
      d = (y - y0)/d
      if abs(d/x) LT MACHEP then goto, done
      x = x - d
  endfor

; Resort to interval halving if Newton iteration did not converge 
IHALVE:
  d = 0.0625D
  if x0 EQ MAXNUM then begin
      if x LE 0 then x = 1.D
      while x0 EQ MAXNUM do begin
          x = (1.D + d) * x
          y = cephes_igamc(a, x)
          if y LT y0 then begin
              x0 = x
              yl = y
              goto, DONELOOP1
          endif
          d = d + d
      endwhile
      DONELOOP1:
  endif

  d = 0.5
  dir = 0L
  for i=0, 399 do begin
      x = x1 + d * (x0-x1)
      y = cephes_igamc(a, x)
      lgm = (x0 - x1)/(x1 + x0)
      if abs(lgm) LT dithresh then goto, DONELOOP2
      lgm = (y - y0)/y0
      if abs(lgm) LT dithresh then goto, DONELOOP2
      if x LT 0 then goto, DONELOOP2
      if y GE y0 then begin
          x1 = x
          yh = y
          if dir LT 0 then begin
              dir = 0
              d = 0.5D
          endif else if dir GT 1 then begin
              d = 0.5 * d + 0.5
          endif else begin
              d = (y0 - yl)/(yh - yl)
          endelse
          dir = dir + 1
      endif else begin
          x0 = x
          yl = y
          if dir GT 0 then begin
              dir = 0
              d = 0.5
          endif else if dir LT -1 then begin
              d = 0.5 * d
          endif else begin
              d = (y0 - yl)/(yh - yl)
          endelse
          dir = dir - 1
      endelse
  endfor
  DONELOOP2:
  
  if x EQ 0 then begin
;      message, 'WARNING: underflow', /info
  endif

  DONE:
  return, x
end


function mpchilim, p, dof, sigma=sigma, clevel=clevel, slevel=slevel

  COMPILE_OPT strictarr

  if n_params() EQ 0 then begin
      message, 'USAGE: DELCHI = MPCHILIM(PROB, DOF, [/SIGMA, /CLEVEL, /SLEVEL ])', /cont
      return, !values.d_nan
  endif

  cephes_setmachar   ;; Set machine constants
  if n_elements(dof) EQ 0 then dof = 1.

  ;; Confidence level is the default
  if n_elements(clevel) EQ 0 then clevel = 1
  if keyword_set(sigma) then begin  ;; Significance in terms of SIGMA
      slev = 1D - errorf(p/sqrt(2.))
  endif else if keyword_set(slevel) then begin ;; in terms of SIGNIFICANCE LEVEL
      slev = p
  endif else if keyword_set(clevel) then begin ;; in terms of CONFIDENCE LEVEL
      slev = 1.D - double(p)        
  endif else begin
      message, 'ERROR: must specify one of SIGMA, CLEVEL, SLEVEL'
  endelse

  ;; Output will have same type as input
  y = p*0

  ;; Loop through, computing the inverse, incomplete gamma function
  ;; slev is the significance level
  for i = 0L, n_elements(p)-1 do begin
      y[i] = 2.D * cephes_igami(0.5D*double(dof), slev[i])
  end

  return, y
end