forked from dsimcha/dstats
-
Notifications
You must be signed in to change notification settings - Fork 9
/
distrib.d
1788 lines (1532 loc) · 61 KB
/
distrib.d
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
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
/**Probability distribution CDFs, PDFs/PMFs, and a few inverse CDFs.
*
* Authors: David Simcha, Don Clugston*/
/*
* Acknowledgements: Some of this module was borrowed the mathstat module
* of Don Clugston's MathExtra library. This was done to create a
* coherent, complete library without massive dependencies, and without
* reinventing the wheel. These functions have been renamed to
* fit the naming conventions of this library, and are noted below.
* The code from Don Clugston's MathExtra library was based on the Cephes
* library by Stephen Moshier.
*
* Conventions:
* Cumulative distribution functions are named <distribution>CDF. For
* discrete distributions, are the P(X <= x) where X is the random variable,
* NOT P(X < x).
*
* All CDFs have a complement, named <distribution>CDFR, which stands for
* "Cumulative Distribution Function Right". For discrete distributions,
* this is P(X >= x), NOT P(X > x) and is therefore NOT equal to
* 1 - <distribution>CDF. Also, even for continuous distributions, the
* numerical accuracy is higher for small p-values if the CDFR is used than
* if 1 - CDF is used.
*
* If a PDF/PMF function is included for a distribution, it is named
* <distribution>PMF or <distribution>PDF (PMF for discrete, PDF for
* continuous distributions).
*
* If an inverse CDF is included, it is named inv<Distribution>CDF.
*
* For all distributions, the test statistic is the first function parameter
* and the distribution parameters are further down the function parameter
* list. This is important for certain generic code, such as tests and
* the parametrize template.
*
* The following functions are identical or functionally equivalent to
* functions found in MathExtra/Tango.Math.probability. This information
* might be useful if someone is trying to integrate this library into other code:
*
* normalCDF <=> normalDistribution
*
* normalCDFR <=> normalDistributionCompl
*
* invNormalCDF <=> normalDistributionComplInv
*
* studentsTCDF <=> studentsTDistribution (Note reversal in argument order)
*
* invStudentsTCDF <=> studentsTDistributionInv (Again, arg order reversed)
*
* binomialCDF <=> binomialDistribution
*
* negBinomCDF <=> negativeBinomialDistribution
*
* poissonCDF <=> poissonDistribution
*
* chiSqrCDF <=> chiSqrDistribution (Note reversed arg order)
*
* chiSqrCDFR <=> chiSqrDistributionCompl (Note reversed arg order)
*
* invChiSqCDFR <=> chiSqrDistributionComplInv
*
* fisherCDF <=> fDistribution (Note reversed arg order)
*
* fisherCDFR <=> fDistributionCompl (Note reversed arg order)
*
* invFisherCDFR <=> fDistributionComplInv
*
* gammaCDF <=> gammaDistribution (Note arg reversal)
*
* gammaCDFR <=> gammaDistributionCompl (Note arg reversal)
*
* Note that CDFRs/Compls of continuous distributions are not equivalent,
* because in Tango/MathExtra they represent P(X > x) while in dstats they
* represent P(X >= x).
*
*
* Copyright (c) 2008-2009, David Simcha and Don Clugston
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* * Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
*
* * Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* * Neither the name of the authors nor the
* names of its contributors may be used to endorse or promote products
* derived from this software without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED ''AS IS'' AND ANY
* EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
* WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDERS BE LIABLE FOR ANY
* DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
* (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
* ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
* SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.*/
module dstats.distrib;
import std.algorithm, std.conv, std.exception, std.math, std.traits,
std.mathspecial, std.range;
alias std.mathspecial.erfc erfc;
alias std.mathspecial.erf erf;
import dstats.base;
// CTFE doesn't work yet for sqrt() in GDC. This value is sqrt(2 * PI).
enum SQ2PI = 2.50662827463100050241576528481104525300698674060993831662992;
version(unittest) {
import std.stdio, std.random;
alias std.math.approxEqual ae;
}
/**Takes a distribution function (CDF or PDF/PMF) as a template argument, and
* parameters as function arguments in the order that they appear in the
* function declaration and returns a delegate that binds the supplied
* parameters to the distribution function. Assumes the non-parameter
* argument is the first argument to the distribution function.
*
* Examples:
* ---
* auto stdNormal = parametrize!(normalCDF)(0.0L, 1.0L);
* ---
*
* stdNormal is now a delegate for the normal(0, 1) distribution.*/
double delegate(ParameterTypeTuple!(distrib)[0])
parametrize(alias distrib)(ParameterTypeTuple!(distrib)[1..$]
parameters) {
double calculate(ParameterTypeTuple!(distrib)[0] arg) {
return distrib(arg, parameters);
}
return &calculate;
}
unittest {
// Just basically see if this compiles.
auto stdNormal = parametrize!normalCDF(0, 1);
assert(approxEqual(stdNormal(2.5), normalCDF(2.5, 0, 1)));
}
///
struct ParamFunctor(alias distrib) {
ParameterTypeTuple!(distrib)[1..$] parameters;
double opCall(ParameterTypeTuple!(distrib)[0] arg) {
return distrib(arg, parameters);
}
}
/**Takes a distribution function (CDF or PDF/PMF) as a template argument, and
* parameters as function arguments in the order that they appear in the
* function declaration and returns a functor that binds the supplied
* parameters to the distribution function. Assumes the non-parameter
* argument is the first argument to the distribution function.
*
* Examples:
* ---
* auto stdNormal = paramFunctor!(normalCDF)(0.0L, 1.0L);
* ---
*
* stdNormal is now a functor for the normal(0, 1) distribution.*/
ParamFunctor!(distrib) paramFunctor(alias distrib)
(ParameterTypeTuple!(distrib)[1..$] parameters) {
ParamFunctor!(distrib) ret;
foreach(ti, elem; parameters) {
ret.tupleof[ti] = elem;
}
return ret;
}
unittest {
// Just basically see if this compiles.
auto stdNormal = paramFunctor!normalCDF(0, 1);
assert(approxEqual(stdNormal(2.5), normalCDF(2.5, 0, 1)));
}
///
double uniformPDF(double X, double lower, double upper) {
dstatsEnforce(X >= lower, "Can't have X < lower bound in uniform distribution.");
dstatsEnforce(X <= upper, "Can't have X > upper bound in uniform distribution.");
return 1.0L / (upper - lower);
}
///
double uniformCDF(double X, double lower, double upper) {
dstatsEnforce(X >= lower, "Can't have X < lower bound in uniform distribution.");
dstatsEnforce(X <= upper, "Can't have X > upper bound in uniform distribution.");
return (X - lower) / (upper - lower);
}
///
double uniformCDFR(double X, double lower, double upper) {
dstatsEnforce(X >= lower, "Can't have X < lower bound in uniform distribution.");
dstatsEnforce(X <= upper, "Can't have X > upper bound in uniform distribution.");
return (upper - X) / (upper - lower);
}
///
double poissonPMF(ulong k, double lambda) {
dstatsEnforce(lambda > 0, "Cannot have a Poisson with lambda <= 0 or nan.");
return exp(cast(double) k * log(lambda) -
(lambda + logFactorial(k))); //Grouped for best precision.
}
unittest {
assert(approxEqual(poissonPMF(1, .1), .0904837));
}
enum POISSON_NORMAL = 1UL << 12; // Where to switch to normal approx.
// The gamma incomplete function is too unstable and the distribution
// is for all practical purposes normal anyhow.
private double normApproxPoisCDF(ulong k, double lambda)
in {
assert(lambda > 0);
} body {
double sd = sqrt(lambda);
// mean == lambda.
return normalCDF(k + 0.5L, lambda, sd);
}
/**P(K <= k) where K is r.v.*/
double poissonCDF(ulong k, double lambda) {
dstatsEnforce(lambda > 0, "Cannot have a poisson with lambda <= 0 or nan.");
return (max(k, lambda) >= POISSON_NORMAL) ?
normApproxPoisCDF(k, lambda) :
gammaIncompleteCompl(k + 1, lambda);
}
unittest {
// Make sure this jives with adding up PMF elements, since this is a
// discrete distribution.
static double pmfSum(uint k, double lambda) {
double ret = 0;
foreach(i; 0..k + 1) {
ret += poissonPMF(i, lambda);
}
return ret;
}
assert(approxEqual(poissonCDF(1, 0.5), pmfSum(1, 0.5)));
assert(approxEqual(poissonCDF(3, 0.7), pmfSum(3, 0.7)));
// Absurdly huge values: Test normal approximation.
// Values from R.
double ans = poissonCDF( (1UL << 50) - 10_000_000, 1UL << 50);
assert(approxEqual(ans, 0.3828427));
// Make sure cutoff is reasonable, i.e. make sure gamma incomplete branch
// and normal branch get roughly the same answer near the cutoff.
for(double lambda = POISSON_NORMAL / 2; lambda <= POISSON_NORMAL * 2; lambda += 100) {
for(ulong k = POISSON_NORMAL / 2; k <= POISSON_NORMAL * 2; k += 100) {
double normAns = normApproxPoisCDF(k, lambda);
double gammaAns = gammaIncompleteCompl(k + 1, lambda);
assert(abs(normAns - gammaAns) < 0.01, text(normAns, '\t', gammaAns));
}
}
}
// The gamma incomplete function is too unstable and the distribution
// is for all practical purposes normal anyhow.
private double normApproxPoisCDFR(ulong k, double lambda)
in {
assert(lambda > 0);
} body {
double sd = sqrt(lambda);
// mean == lambda.
return normalCDFR(k - 0.5L, lambda, sd);
}
/**P(K >= k) where K is r.v.*/
double poissonCDFR(ulong k, double lambda) {
dstatsEnforce(lambda > 0, "Can't have a poisson with lambda <= 0 or nan.");
return (max(k, lambda) >= POISSON_NORMAL) ?
normApproxPoisCDFR(k, lambda) :
gammaIncomplete(k, lambda);
}
unittest {
// Make sure this jives with adding up PMF elements, since this is a
// discrete distribution.
static double pmfSum(uint k, double lambda) {
double ret = 0;
foreach(i; 0..k + 1) {
ret += poissonPMF(i, lambda);
}
return ret;
}
assert(approxEqual(poissonCDFR(1, 0.5), 1 - pmfSum(0, 0.5)));
assert(approxEqual(poissonCDFR(3, 0.7), 1 - pmfSum(2, 0.7)));
// Absurdly huge value to test normal approximation.
// Values from R.
double ans = poissonCDFR( (1UL << 50) - 10_000_000, 1UL << 50);
assert(approxEqual(ans, 0.6171573));
// Make sure cutoff is reasonable, i.e. make sure gamma incomplete branch
// and normal branch get roughly the same answer near the cutoff.
for(double lambda = POISSON_NORMAL / 2; lambda <= POISSON_NORMAL * 2; lambda += 100) {
for(ulong k = POISSON_NORMAL / 2; k <= POISSON_NORMAL * 2; k += 100) {
double normAns = normApproxPoisCDFR(k, lambda);
double gammaAns = gammaIncomplete(k, lambda);
assert(abs(normAns - gammaAns) < 0.01, text(normAns, '\t', gammaAns));
}
}
}
/**Returns the value of k for the given p-value and lambda. If p-val
* doesn't exactly map to a value of k, the k for which poissonCDF(k, lambda)
* is closest to pVal is used.*/
uint invPoissonCDF(double pVal, double lambda) {
dstatsEnforce(lambda > 0, "Cannot have a poisson with lambda <= 0 or nan.");
dstatsEnforce(pVal >= 0 && pVal <= 1, "P-values must be between 0, 1.");
// Use normal approximation to get approx answer, then brute force search.
// This works better than you think because for small n, there's not much
// search space and for large n, the normal approx. is doublely good.
uint guess = cast(uint) max(round(
invNormalCDF(pVal, lambda, sqrt(lambda)) + 0.5), 0.0L);
double guessP = poissonCDF(guess, lambda);
if(guessP == pVal) {
return guess;
} else if(guessP < pVal) {
for(uint k = guess + 1; ; k++) {
double newP = guessP + poissonPMF(k, lambda);
if(newP >= 1)
return k;
if(abs(newP - pVal) > abs(guessP - pVal)) {
return k - 1;
} else {
guessP = newP;
}
}
} else {
for(uint k = guess - 1; k != uint.max; k--) {
double newP = guessP - poissonPMF(k + 1, lambda);
if(abs(newP - pVal) > abs(guessP - pVal)) {
return k + 1;
} else {
guessP = newP;
}
}
return 0;
}
}
unittest {
foreach(i; 0..1_000) {
// Restricted variable ranges are because, in the tails, more than one
// value of k can map to the same p-value at machine precision.
// Obviously, this is one of those corner cases that nothing can be
// done about.
double lambda = uniform(.05L, 8.0L);
uint k = uniform(0U, cast(uint) ceil(3.0L * lambda));
double pVal = poissonCDF(k, lambda);
assert(invPoissonCDF(pVal, lambda) == k);
}
}
///
double binomialPMF(ulong k, ulong n, double p) {
dstatsEnforce(k <= n, "k cannot be > n in binomial distribution.");
dstatsEnforce(p >= 0 && p <= 1, "p must be between 0, 1 in binomial distribution.");
return exp(logNcomb(n, k) + k * log(p) + (n - k) * log(1 - p));
}
unittest {
assert(approxEqual(binomialPMF(0, 10, .5), cast(double) 1/1024));
assert(approxEqual(binomialPMF(100, 1000, .11), .024856));
}
// Determines what value of n we switch to normal approximation at b/c
// betaIncomplete becomes unstable.
private enum BINOM_APPROX = 1UL << 24;
// Cutoff value of n * p for deciding whether to go w/ normal or poisson approx
// when betaIncomplete becomes unstable.
private enum BINOM_POISSON = 1_024;
// betaIncomplete is numerically unstable for huge values of n.
// Luckily this is exactly when the normal approximation becomes
// for all practical purposes exact.
private double normApproxBinomCDF(double k, double n, double p)
in {
assert(k <= n);
assert(p >= 0 && p <= 1);
} body {
double mu = p * n;
double sd = sqrt( to!double(n) ) * sqrt(p) * sqrt(1 - p);
double xCC = k + 0.5L;
return normalCDF(xCC, mu, sd);
}
///P(K <= k) where K is random variable.
double binomialCDF(ulong k, ulong n, double p) {
dstatsEnforce(k <= n, "k cannot be > n in binomial distribution.");
dstatsEnforce(p >= 0 && p <= 1, "p must be between 0, 1 in binomial distribution.");
if(k == n) {
return 1;
} else if(k == 0) {
return pow(1.0 - p, cast(double) n);
}
if(n > BINOM_APPROX) {
if(n * p < BINOM_POISSON) {
return poissonCDF(k, n * p);
} else if(n * (1 - p) < BINOM_POISSON) {
return poissonCDFR(n - k, n * (1 - p));
} else {
return normApproxBinomCDF(k, n, p);
}
}
return betaIncomplete(n - k, k + 1, 1.0 - p);
}
unittest {
assert(approxEqual(binomialCDF(10, 100, .11), 0.4528744401));
assert(approxEqual(binomialCDF(15, 100, .12), 0.8585510507));
assert(approxEqual(binomialCDF(50, 1000, .04), 0.95093595));
assert(approxEqual(binomialCDF(7600, 15000, .5), .9496193045414));
assert(approxEqual(binomialCDF(0, 10, 0.2), 0.1073742));
// Absurdly huge numbers:
{
ulong k = (1UL << 60) - 100_000_000;
ulong n = 1UL << 61;
assert(approxEqual(binomialCDF(k, n, 0.5L), 0.4476073));
}
// Test Poisson branch.
double poisAns = binomialCDF(85, 1UL << 26, 1.49e-6);
assert(approxEqual(poisAns, 0.07085327));
// Test poissonCDFR branch.
poisAns = binomialCDF( (1UL << 25) - 100, 1UL << 25, 0.9999975L);
assert(approxEqual(poisAns, 0.04713316));
// Make sure cutoff is reasonable: Just below it, we should get similar
// results for normal, exact.
for(ulong n = BINOM_APPROX / 2; n < BINOM_APPROX; n += 200_000) {
for(double p = 0.01; p <= 0.99; p += 0.05) {
long lowerK = roundTo!long( n * p * 0.99);
long upperK = roundTo!long( n * p / 0.99);
for(ulong k = lowerK; k <= min(n, upperK); k += 1_000) {
double normRes = normApproxBinomCDF(k, n, p);
double exactRes = binomialCDF(k, n, p);
assert(abs(normRes - exactRes) < 0.001,
text(normRes, '\t', exactRes));
}
}
}
}
// betaIncomplete is numerically unstable for huge values of n.
// Luckily this is exactly when the normal approximation becomes
// for all practical purposes exact.
private double normApproxBinomCDFR(ulong k, ulong n, double p)
in {
assert(k <= n);
assert(p >= 0 && p <= 1);
} body {
double mu = p * n;
double sd = sqrt( to!double(n) ) * sqrt(p) * sqrt(1 - p);
double xCC = k - 0.5L;
return normalCDFR(xCC, mu, sd);
}
///P(K >= k) where K is random variable.
double binomialCDFR(ulong k, ulong n, double p) {
dstatsEnforce(k <= n, "k cannot be > n in binomial distribution.");
dstatsEnforce(p >= 0 && p <= 1, "p must be between 0, 1 in binomial distribution.");
if(k == 0) {
return 1;
} else if(k == n) {
return pow(p, cast(double) n);
}
if(n > BINOM_APPROX) {
if(n * p < BINOM_POISSON) {
return poissonCDFR(k, n * p);
} else if(n * (1 - p) < BINOM_POISSON) {
return poissonCDF(n - k, n * (1 - p));
} else {
return normApproxBinomCDFR(k, n, p);
}
}
return betaIncomplete(k, n - k + 1, p);
}
unittest {
// Values from R, Maxima.
assert(approxEqual(binomialCDF(10, 100, .11), 1 -
binomialCDFR(11, 100, .11)));
assert(approxEqual(binomialCDF(15, 100, .12), 1 -
binomialCDFR(16, 100, .12)));
assert(approxEqual(binomialCDF(50, 1000, .04), 1 -
binomialCDFR(51, 1000, .04)));
assert(approxEqual(binomialCDF(7600, 15000, .5), 1 -
binomialCDFR(7601, 15000, .5)));
assert(approxEqual(binomialCDF(9, 10, 0.3), 1 -
binomialCDFR(10, 10, 0.3)));
// Absurdly huge numbers, test normal branch.
{
ulong k = (1UL << 60) - 100_000_000;
ulong n = 1UL << 61;
assert(approxEqual(binomialCDFR(k, n, 0.5L), 0.5523927));
}
// Test Poisson inversion branch.
double poisRes = binomialCDFR((1UL << 25) - 70, 1UL << 25, 0.9999975L);
assert(approxEqual(poisRes, 0.06883905));
// Test Poisson branch.
poisRes = binomialCDFR(350, 1UL << 25, 1e-5);
assert(approxEqual(poisRes, 0.2219235));
// Make sure cutoff is reasonable: Just below it, we should get similar
// results for normal, exact.
for(ulong n = BINOM_APPROX / 2; n < BINOM_APPROX; n += 200_000) {
for(double p = 0.01; p <= 0.99; p += 0.05) {
long lowerK = roundTo!long( n * p * 0.99);
long upperK = roundTo!long( n * p / 0.99);
for(ulong k = lowerK; k <= min(n, upperK); k += 1_000) {
double normRes = normApproxBinomCDFR(k, n, p);
double exactRes = binomialCDFR(k, n, p);
assert(abs(normRes - exactRes) < 0.001,
text(normRes, '\t', exactRes));
}
}
}
}
/**Returns the value of k for the given p-value, n and p. If p-value does
* not exactly map to a value of k, the value for which binomialCDF(k, n, p)
* is closest to pVal is used.*/
uint invBinomialCDF(double pVal, uint n, double p) {
dstatsEnforce(pVal >= 0 && pVal <= 1, "p-values must be between 0, 1.");
dstatsEnforce(p >= 0 && p <= 1, "p must be between 0, 1 in binomial distribution.");
// Use normal approximation to get approx answer, then brute force search.
// This works better than you think because for small n, there's not much
// search space and for large n, the normal approx. is doublely good.
uint guess = cast(uint) max(round(
invNormalCDF(pVal, n * p, sqrt(n * p * (1 - p)))) + 0.5, 0);
if(guess > n) {
if(pVal < 0.5) // Numerical issues/overflow.
guess = 0;
else guess = n;
}
double guessP = binomialCDF(guess, n, p);
if(guessP == pVal) {
return guess;
} else if(guessP < pVal) {
for(uint k = guess + 1; k <= n; k++) {
double newP = guessP + binomialPMF(k, n, p);
if(abs(newP - pVal) > abs(guessP - pVal)) {
return k - 1;
} else {
guessP = newP;
}
}
return n;
} else {
for(uint k = guess - 1; k != uint.max; k--) {
double newP = guessP - binomialPMF(k + 1, n, p);
if(abs(newP - pVal) > abs(guessP - pVal)) {
return k + 1;
} else {
guessP = newP;
}
}
return 0;
}
}
unittest {
Random gen = Random(unpredictableSeed);
foreach(i; 0..1_000) {
// Restricted variable ranges are because, in the tails, more than one
// value of k can map to the same p-value at machine precision.
// Obviously, this is one of those corner cases that nothing can be
// done about. Using small n's, moderate p's prevents this.
uint n = uniform(5U, 10U);
uint k = uniform(0U, n);
double p = uniform(0.1L, 0.9L);
double pVal = binomialCDF(k, n, p);
assert(invBinomialCDF(pVal, n, p) == k);
}
}
///
double hypergeometricPMF(long x, long n1, long n2, long n)
in {
assert(x <= n);
} body {
if(x > n1 || x < (n - n2)) {
return 0;
}
double result = logNcomb(n1, x) + logNcomb(n2, n - x) - logNcomb(n1 + n2, n);
return exp(result);
}
unittest {
assert(approxEqual(hypergeometricPMF(5, 10, 10, 10), .3437182));
assert(approxEqual(hypergeometricPMF(9, 12, 10, 15), .27089783));
assert(approxEqual(hypergeometricPMF(9, 100, 100, 15), .15500003));
}
/**P(X <= x), where X is random variable. Uses either direct summation,
* normal or binomial approximation depending on parameters.*/
// If anyone knows a better algorithm for this, feel free...
// I've read a decent amount about it, though, and getting hypergeometric
// CDFs that are both accurate and fast is just plain hard. This
// implementation attempts to strike a balance between the two, so that
// both speed and accuracy are "good enough" for most practical purposes.
double hypergeometricCDF(long x, long n1, long n2, long n) {
dstatsEnforce(x <= n, "x must be <= n in hypergeometric distribution.");
dstatsEnforce(n <= n1 + n2, "n must be <= n1 + n2 in hypergeometric distribution.");
dstatsEnforce(x >= 0, "x must be >= 0 in hypergeometric distribution.");
ulong expec = (n1 * n) / (n1 + n2);
long nComp = n1 + n2 - n, xComp = n2 + x - n;
// Try to reduce number of calculations using identities.
if(x >= n1 || x == n) {
return 1;
} else if(x > expec && x > n / 2) {
return 1 - hypergeometricCDF(n - x - 1, n2, n1, n);
} else if(xComp < x && xComp > 0) {
return hypergeometricCDF(xComp, n2, n1, nComp);
}
// Speed depends on x mostly, so always use exact for small x.
if(x <= 100) {
return hyperExact(x, n1, n2, n);
}
// Determine whether to use exact, normal approx or binomial approx.
// Using obviously arbitrary but relatively stringent standards
// for determining whether to approximate.
enum NEXACT = 50L;
double p = cast(double) n1 / (n1 + n2);
double pComp = cast(double) n2 / (n1 + n2);
double pMin = min(p, pComp);
if(min(n, nComp) * pMin >= 100) {
// Since high relative error in the lower tail is a significant problem,
// this is a hack to improve the normal approximation: Use the normal
// approximation, except calculate the last NEXACT elements exactly,
// since elements around the e.v. are where absolute error is highest.
// For large x, gives most of the accuracy of an exact calculation in
// only a small fraction of the time.
if(x <= expec + NEXACT / 2) {
return min(1, normApproxHyper(x - NEXACT, n1, n2, n) +
hyperExact(x, n1, n2, n, x - NEXACT + 1));
} else {
// Just use plain old normal approx. Since P is large, the
// relative error won't be so bad anyhow.
return normApproxHyper(x, n1, n2, n);
}
}
// Try to make n as small as possible by applying mathematically equivalent
// transformations so that binomial approx. works as well as possible.
ulong bSc1 = (n1 + n2) / n, bSc2 = (n1 + n2) / n1;
if(bSc1 >= 50 && bSc1 > bSc2) {
// Same hack as normal approximation for rel. acc. in lower tail.
if(x <= expec + NEXACT / 2) {
return min(1, binomialCDF(cast(uint) (x - NEXACT), cast(uint) n, p) +
hyperExact(x, n1, n2, n, x - NEXACT + 1));
} else {
return binomialCDF(cast(uint) x, cast(uint) n, p);
}
} else if(bSc2 >= 50 && bSc2 > bSc1) {
double p2 = cast(double) n / (n1 + n2);
if(x <= expec + NEXACT / 2) {
return min(1, binomialCDF(cast(uint) (x - NEXACT), cast(uint) n1, p2) +
hyperExact(x, n1, n2, n, x - NEXACT + 1));
} else {
return binomialCDF(cast(uint) x, cast(uint) n1, p2);
}
} else {
return hyperExact(x, n1, n2, n);
}
}
unittest {
// Values from R and the Maxima CAS.
// Test exact branch, including reversing, complementing.
assert(approxEqual(hypergeometricCDF(5, 10, 10, 10), 0.6718591));
assert(approxEqual(hypergeometricCDF(3, 11, 15, 10), 0.27745322));
assert(approxEqual(hypergeometricCDF(18, 27, 31, 35), 0.88271714));
assert(approxEqual(hypergeometricCDF(21, 29, 31, 35), 0.99229253));
// Normal branch.
assert(approxEqual(hypergeometricCDF(501, 2000, 1000, 800), 0.002767073));
assert(approxEqual(hypergeometricCDF(565, 2000, 1000, 800), 0.9977068));
assert(approxEqual(hypergeometricCDF(2700, 10000, 20000, 8000), 0.825652));
// Binomial branch. One for each transformation.
assert(approxEqual(hypergeometricCDF(110, 5000, 7000, 239), 0.9255627));
assert(approxEqual(hypergeometricCDF(19840, 2950998, 12624, 19933), 0.2020618));
assert(approxEqual(hypergeometricCDF(130, 24195, 52354, 295), 0.9999973));
assert(approxEqual(hypergeometricCDF(103, 901, 49014, 3522), 0.999999));
}
///P(X >= x), where X is random variable.
double hypergeometricCDFR(ulong x, ulong n1, ulong n2, ulong n) {
dstatsEnforce(x <= n, "x must be <= n in hypergeometric distribution.");
dstatsEnforce(n <= n1 + n2, "n must be <= n1 + n2 in hypergeometric distribution.");
dstatsEnforce(x >= 0, "x must be >= 0 in hypergeometric distribution.");
return hypergeometricCDF(n - x, n2, n1, n);
}
unittest {
//Reverses n1, n2 and subtracts x from n to get mirror image.
assert(approxEqual(hypergeometricCDF(5,10,10,10),
hypergeometricCDFR(5,10,10,10)));
assert(approxEqual(hypergeometricCDF(3, 11, 15, 10),
hypergeometricCDFR(7, 15, 11, 10)));
assert(approxEqual(hypergeometricCDF(18, 27, 31, 35),
hypergeometricCDFR(17, 31, 27, 35)));
assert(approxEqual(hypergeometricCDF(21, 29, 31, 35),
hypergeometricCDFR(14, 31, 29, 35)));
}
double hyperExact(ulong x, ulong n1, ulong n2, ulong n, ulong startAt = 0) {
dstatsEnforce(x <= n, "x must be <= n in hypergeometric distribution.");
dstatsEnforce(n <= n1 + n2, "n must be <= n1 + n2 in hypergeometric distribution.");
dstatsEnforce(x >= 0, "x must be >= 0 in hypergeometric distribution.");
immutable double constPart = logFactorial(n1) + logFactorial(n2) +
logFactorial(n) + logFactorial(n1 + n2 - n) - logFactorial(n1 + n2);
double sum = 0;
for(ulong i = x; i != startAt - 1; i--) {
double oldSum = sum;
sum += exp(constPart - logFactorial(i) - logFactorial(n1 - i) -
logFactorial(n2 + i - n) - logFactorial(n - i));
if(isIdentical(sum, oldSum)) { // At full machine precision.
break;
}
}
return sum;
}
double normApproxHyper(ulong x, ulong n1, ulong n2, ulong n) {
double p1 = cast(double) n1 / (n1 + n2);
double p2 = cast(double) n2 / (n1 + n2);
double numer = x + 0.5L - n * p1;
double denom = sqrt(n * p1 * p2 * (n1 + n2 - n) / (n1 + n2 - 1));
return normalCDF(numer / denom);
}
// Aliases for old names. Not documented because new names should be used.
deprecated {
alias chiSquareCDF chiSqrCDF;
alias chiSquareCDFR chiSqrCDFR;
alias invChiSquareCDFR invChiSqCDFR;
}
///
double chiSquarePDF(double x, double v) {
dstatsEnforce(x >= 0, "x must be >= 0 in chi-square distribution.");
dstatsEnforce(v >= 1.0, "Must have at least 1 degree of freedom for chi-square.");
// Calculate in log space for stability.
immutable logX = log(x);
immutable numerator = logX * (0.5 * v - 1) - 0.5 * x;
immutable denominator = LN2 * (0.5 * v) + logGamma(0.5 * v);
return exp(numerator - denominator);
}
unittest {
assert( approxEqual(chiSquarePDF(1, 2), 0.3032653));
assert( approxEqual(chiSquarePDF(2, 1), 0.1037769));
}
/**
* $(POWER χ,2) distribution function and its complement.
*
* Returns the area under the left hand tail (from 0 to x)
* of the Chi square probability density function with
* v degrees of freedom. The complement returns the area under
* the right hand tail (from x to ∞).
*
* chiSquareCDF(x | v) = ($(INTEGRATE 0, x)
* $(POWER t, v/2-1) $(POWER e, -t/2) dt )
* / $(POWER 2, v/2) $(GAMMA)(v/2)
*
* chiSquareCDFR(x | v) = ($(INTEGRATE x, ∞)
* $(POWER t, v/2-1) $(POWER e, -t/2) dt )
* / $(POWER 2, v/2) $(GAMMA)(v/2)
*
* Params:
* v = degrees of freedom. Must be positive.
* x = the $(POWER χ,2) variable. Must be positive.
*
*/
double chiSquareCDF(double x, double v) {
dstatsEnforce(x >= 0, "x must be >= 0 in chi-square distribution.");
dstatsEnforce(v >= 1.0, "Must have at least 1 degree of freedom for chi-square.");
// These are very common special cases where we can make the calculation
// a lot faster and/or more accurate.
if(v == 1) {
// Then it's the square of a normal(0, 1).
return 1.0L - erfc(sqrt(x) * SQRT1_2);
} else if(v == 2) {
// Then it's an exponential w/ lambda == 1/2.
return 1.0L - exp(-0.5 * x);
} else {
return gammaIncomplete(0.5 * v, 0.5 * x);
}
}
///
double chiSquareCDFR(double x, double v) {
dstatsEnforce(x >= 0, "x must be >= 0 in chi-square distribution.");
dstatsEnforce(v >= 1.0, "Must have at least 1 degree of freedom for chi-square.");
// These are very common special cases where we can make the calculation
// a lot faster and/or more accurate.
if(v == 1) {
// Then it's the square of a normal(0, 1).
return erfc(sqrt(x) * SQRT1_2);
} else if(v == 2) {
// Then it's an exponential w/ lambda == 1/2.
return exp(-0.5 * x);
} else {
return gammaIncompleteCompl(0.5 * v, 0.5 * x);
}
}
/**
* Inverse of complemented $(POWER χ, 2) distribution
*
* Finds the $(POWER χ, 2) argument x such that the integral
* from x to ∞ of the $(POWER χ, 2) density is equal
* to the given cumulative probability p.
*
* Params:
* p = Cumulative probability. 0<= p <=1.
* v = Degrees of freedom. Must be positive.
*
*/
double invChiSquareCDFR(double v, double p) {
dstatsEnforce(v >= 1.0, "Must have at least 1 degree of freedom for chi-square.");
dstatsEnforce(p >= 0 && p <= 1, "P-values must be between 0, 1.");
return 2.0 * gammaIncompleteComplInverse( 0.5*v, p);
}
unittest {
assert(feqrel(chiSquareCDFR(invChiSquareCDFR(3.5, 0.1), 3.5), 0.1)>=double.mant_dig-3);
assert(approxEqual(
chiSquareCDF(0.4L, 19.02L) + chiSquareCDFR(0.4L, 19.02L), 1.0L));
assert(ae( invChiSquareCDFR( 3, chiSquareCDFR(1, 3)), 1));
assert(ae(chiSquareCDFR(0.2, 1), 0.6547208));
assert(ae(chiSquareCDFR(0.2, 2), 0.9048374));
assert(ae(chiSquareCDFR(0.8, 1), 0.3710934));
assert(ae(chiSquareCDFR(0.8, 2), 0.67032));
assert(ae(chiSquareCDF(0.2, 1), 0.3452792));
assert(ae(chiSquareCDF(0.2, 2), 0.09516258));
assert(ae(chiSquareCDF(0.8, 1), 0.6289066));
assert(ae(chiSquareCDF(0.8, 2), 0.3296800));
}
///
double normalPDF(double x, double mean = 0, double sd = 1) {
dstatsEnforce(sd > 0, "Standard deviation must be > 0 for normal distribution.");
double dev = x - mean;
return exp(-(dev * dev) / (2 * sd * sd)) / (sd * SQ2PI);
}
unittest {
assert(approxEqual(normalPDF(3, 1, 2), 0.1209854));
}
///P(X < x) for normal distribution where X is random var.
double normalCDF(double x, double mean = 0, double stdev = 1) {
dstatsEnforce(stdev > 0, "Standard deviation must be > 0 for normal distribution.");
// Using a slightly non-obvious implementation in terms of erfc because
// it seems more accurate than erf for very small values of Z.
double Z = (-x + mean) / stdev;
return erfc(Z*SQRT1_2)/2;
}
unittest {
assert(approxEqual(normalCDF(2), .9772498));
assert(approxEqual(normalCDF(-2), .02275013));
assert(approxEqual(normalCDF(1.3), .90319951));
}
///P(X > x) for normal distribution where X is random var.
double normalCDFR(double x, double mean = 0, double stdev = 1) {
dstatsEnforce(stdev > 0, "Standard deviation must be > 0 for normal distribution.");
double Z = (x - mean) / stdev;
return erfc(Z * SQRT1_2) / 2;
}
unittest {
//Should be essentially a mirror image of normalCDF.
for(double i = -8; i < 8; i += .1) {
assert(approxEqual(normalCDF(i), normalCDFR(-i)));
}
}
private enum SQRT2PI = 0x1.40d931ff62705966p+1; // 2.5066282746310005024
private enum EXP_2 = 0.13533528323661269189L; /* exp(-2) */
/******************************
* Inverse of Normal distribution function
*
* Returns the argument, x, for which the area under the
* Normal probability density function (integrated from
* minus infinity to x) is equal to p.
*/
double invNormalCDF(double p, double mean = 0, double sd = 1) {
dstatsEnforce(p >= 0 && p <= 1, "P-values must be between 0, 1.");
dstatsEnforce(sd > 0, "Standard deviation must be > 0 for normal distribution.");
return normalDistributionInverse(p) * sd + mean;
}
unittest {
// The values below are from Excel 2003.
assert(fabs(invNormalCDF(0.001) - (-3.09023230616779))< 0.00000000000005);
assert(fabs(invNormalCDF(1e-50) - (-14.9333375347885))< 0.00000000000005);
assert(feqrel(invNormalCDF(0.999), -invNormalCDF(0.001))>double.mant_dig-6);
// Excel 2003 gets all the following values wrong!
assert(invNormalCDF(0.0)==-double.infinity);
assert(invNormalCDF(1.0)==double.infinity);
assert(invNormalCDF(0.5)==0);
// I don't know the correct result for low values
// (Excel 2003 returns norminv(p) = -30 for all p < 1e-200).
// The value tested here is the one the function returned in Jan 2006.
double unknown1 = invNormalCDF(1e-250L);
assert( fabs(unknown1 -(-33.79958617269L) ) < 0.00000005);
Random gen;
gen.seed(unpredictableSeed);
// normalCDF function trivial given ERF, unlikely to contain subtle bugs.
// Just make sure invNormalCDF works like it should as the inverse.
foreach(i; 0..1000) {
double x = uniform(0.0L, 1.0L);
double mean = uniform(0.0L, 100.0L);
double sd = uniform(1.0L, 3.0L);
double inv = invNormalCDF(x, mean, sd);
double rec = normalCDF(inv, mean, sd);
assert(approxEqual(x, rec));
}
}
///
double logNormalPDF(double x, double mu = 0, double sigma = 1) {
dstatsEnforce(sigma > 0, "sigma must be > 0 for log-normal distribution.");
immutable mulTerm = 1.0L / (x * sigma * SQ2PI);
double expTerm = log(x) - mu;
expTerm *= expTerm;
expTerm /= 2 * sigma * sigma;
return mulTerm * exp(-expTerm);