-
Notifications
You must be signed in to change notification settings - Fork 18
/
core.cljc
704 lines (642 loc) · 23.7 KB
/
core.cljc
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
(ns kixi.stats.core
(:refer-clojure :exclude [count min max])
(:require [kixi.stats.digest :as digest]
#?(:clj [kixi.stats.distribution :as d])
[kixi.stats.estimate :as e]
[kixi.stats.math :refer [sq sqrt pow root infinity negative-infinity infinite?]]
[kixi.stats.protocols :as p]
[kixi.stats.test :as t]
[redux.core :refer [fuse-matrix]]))
(defn ^:no-doc somef
[f & args]
(fn [x]
(when-not (nil? x)
(apply f x args))))
(defn monoid
"Add 0-arity returning `init` to `f`."
[f init]
(fn
([] init)
([acc] (f acc))
([acc x] (f acc x))))
(defn ^:no-doc post-complete
[rf f]
(completing rf #(f (rf %))))
#?(:clj
(def histogram
"Calculates a histogram of numeric inputs using the t-digest with default arguments."
(digest/t-digest {:compression 100})))
#?(:clj
(def median
"Calculates the median of numeric inputs."
(post-complete histogram d/median)))
#?(:clj
(def iqr
"Calculates the interquartile range of numeric inputs."
(post-complete histogram d/iqr)))
#?(:clj
(def summary
"Calculates the five number summary of numeric inputs."
(post-complete histogram d/summary)))
(defn cross-tabulate
"Given a sequence of n functions, each of which returns a categorical value
(e.g. keyword or string) of a factor, calculates an n-dimensional contingency table
implementing PContingencyTable. This can be passed to kixi.stats.test/chi-squared-test
to determine if the relationship between factors is significant.
See also: kixi.stats.core/chi-squared-test"
[& fxs]
(let [f (apply juxt fxs)
k (clojure.core/count fxs)
inc (fnil inc 0)]
(fn
([] (vector {} (vec (repeat k {})) 0))
([[cells margins n] x]
[(update cells (f x) inc)
(first (reduce (fn [[margins i] fx]
[(update-in margins [i (fx x)] inc) (inc i)])
[margins 0]
fxs))
(inc n)])
([[cells margins n]]
(reify p/PContingencyTable
(cell [_ coordinates]
(get cells coordinates 0))
(grand-total [_] n)
(margin-totals [_] margins)
(size [_]
(mapv clojure.core/count margins)))))))
(def count
"Calculates the count of inputs."
(fn
([] 0)
([n _] (inc n))
([n] n)))
(def arithmetic-mean
"Calculates the arithmetic mean of numeric inputs."
(fn
([] [0.0 0])
([[^double s ^long c :as acc] e]
(if (nil? e)
acc
(let [e (double e)]
[(+ s e) (inc c)])))
([[s c]]
(when-not (zero? c)
(/ s c)))))
(def mean
"Alias for arithmetic-mean."
arithmetic-mean)
(def geometric-mean
"Calculates the geometric mean of numeric inputs. Defined only for positive numbers."
(fn
([] [1 0])
([[s c :as acc] e]
(cond
(nil? e) acc
(neg? e) (reduced [nil 0])
:else [(* s e) (inc c)]))
([[s c]]
(when-not (zero? c)
(if (zero? s)
0.0 (root s c))))))
(def harmonic-mean
"Calculates the harmonic mean of numeric inputs."
(fn
([] [0 0])
([[s c :as acc] e]
(cond
(nil? e) acc
(zero? e) (reduced [0 (inc c)])
:else [(+ s (/ 1 e)) (inc c)]))
([[s c]]
(when-not (zero? c)
(if (zero? s)
0.0 (/ c s))))))
(def variance-s
"Estimates an unbiased variance of numeric inputs."
(fn
([] [0 0.0 0.0])
([[^long c ^double m ^double ss :as acc] e]
(if (nil? e)
acc
(let [e (double e)
c' (inc c)
m' (+ m (/ (- e m) c'))]
[c' m' (+ ss (* (- e m') (- e m)))])))
([[c _m ss]]
(when-not (zero? c)
(let [c' (dec c)]
(if (pos? c')
(/ ss c') 0))))))
(def variance
"Alias for variance-s."
variance-s)
(def variance-p
"Calculates the population variance of numeric inputs."
(completing variance-s (fn [[c _ ss]]
(when-not (zero? c)
(/ ss c)))))
(def standard-deviation-s
"Estimates the sample standard deviation of numeric inputs."
(post-complete variance-s (somef sqrt)))
(def standard-deviation
"Alias for standard-deviation-s."
standard-deviation-s)
(def standard-deviation-p
"Calculates the population standard deviation of numeric inputs."
(post-complete variance-p (somef sqrt)))
(def standard-error-s
"Calculates the standard error of sample means."
(completing standard-deviation-s
(fn [[c _ ss]]
(when-not (zero? c)
(let [c' (dec c)]
(if (pos? c')
(sqrt (/ ss c' c)) 0))))))
(def standard-error
"Alias for standard-error-s"
standard-error-s)
(def skewness-s
"Estimates the sample skewness of numeric inputs.
See https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance."
(fn
([] [0.0 0.0 0.0 0.0])
([[^double c ^double m1 ^double m2 ^double m3 :as acc] e]
(if (nil? e)
acc
(let [e (double e)
c' (inc c)
d (- e m1)
dc (/ d c')
m1' (+ m1 dc)
m2' (+ m2 (* (sq d) (/ c c')))
m3' (+ m3
(/ (* (pow d 3) (- c' 1) (- c' 2)) (sq c'))
(* -3 m2 dc))]
[c' m1' m2' m3'])))
([[c _ m2 m3]]
(let [d (* (pow m2 1.5) (- c 2))]
(when-not (zero? d)
(/ (* (sqrt (dec c)) m3 c) d))))))
(def skewness
"Alias for skewness-s."
skewness-s)
(def skewness-p
"Calculates the population skewness of numeric inputs.
See: http://www.real-statistics.com/descriptive-statistics/symmetry-skewness-kurtosis."
(completing skewness-s
(fn [[c _ m2 m3]]
(let [d (pow m2 1.5)]
(when-not (zero? d)
(/ (* (sqrt c) m3) d))))))
(def kurtosis-s
"Estimates the sample kurtosis of numeric inputs.
See https://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
and http://www.real-statistics.com/descriptive-statistics/symmetry-skewness-kurtosis."
(fn
([] [0.0 0.0 0.0 0.0 0.0])
([[^double c ^double m1 ^double m2 ^double m3 ^double m4 :as acc] e]
(if (nil? e)
acc
(let [e (double e)
c' (inc c)
d (- e m1)
dc (/ d c')
m1' (+ m1 dc)
m2' (+ m2 (* (sq d) (/ c c')))
m3' (+ m3
(/ (* (pow d 3) (- c' 1) (- c' 2)) (sq c'))
(* -3 m2 dc))
m4' (+ m4
(/ (* (pow d 4) (- c' 1) (+ (sq c') (* -3 c') 3))
(pow c' 3))
(* 6 m2 (sq dc))
(* -4 m3 dc))]
[c' m1' m2' m3' m4'])))
([[c _ m2 _ m4]]
(when-not (or (zero? m2) (< c 4))
(let [v (/ m2 (dec c))]
(- (/ (* c (inc c) m4)
(* (- c 1) (- c 2) (- c 3) (sq v)))
(/ (* 3 (sq (dec c)))
(* (- c 2) (- c 3)))))))))
(def kurtosis
"Alias for kurtosis-s."
kurtosis-s)
(def kurtosis-p
"Calculates the population kurtosis of numeric inputs.
See http://www.macroption.com/kurtosis-formula/"
(completing kurtosis-s (fn [[c _ m2 _ m4]]
(when-not (zero? m2)
(- (/ (* c m4)
(sq m2)) 3)))))
(defn covariance-s
"Given two functions of an input `(fx input)` and `(fy input)`, each of which
returns a number, estimates the unbiased covariance of those functions over
inputs.
Ignores any inputs where `(fx input)` or `(fy input)` are nil. If no
inputs have both x and y, returns nil."
[fx fy]
(fn
([] [0.0 0.0 0.0 0.0])
([[^double c ^double mx my ss :as acc] e]
(let [x (fx e)
y (fy e)]
(if (or (nil? x) (nil? y))
acc
(let [x (double x)
y (double y)
c' (inc c)
mx' (+ mx (/ (- x mx) c'))
my' (+ my (/ (- y my) c'))]
[c' mx' my'
(+ ss (* (- x mx') (- y my)))]))))
([[c _ _ ss]]
(when-not (zero? c)
(let [c' (dec c)]
(if (pos? c')
(/ ss c') 0))))))
(def covariance
"Alias for covariance-s"
covariance-s)
(defn covariance-p
"Given two functions of an input `(fx input)` and `(fy input)`, each of which
returns a number, estimates the population covariance of those functions over
inputs.
Ignores any inputs where `(fx input)` or `(fy input)` are nil. If no
inputs have both x and y, returns nil."
[fx fy]
(completing (covariance-s fx fy)
(fn [[c _ _ ss]]
(when-not (zero? c)
(/ ss c)))))
(defn covariance-matrix
"Given a map of key names to functions that extract values for those keys
from an input, computes the covariance for each of the n^2 key pairs.
For example:
(covariance-matrix {:name-length #(.length (:name %))
:age :age
:num-cats (comp count :cats)})
will, when reduced, return a map like:
{[:name-length :age] 0.56
[:name-length :num-cats] 0.95
...}"
[kvs]
(fuse-matrix covariance kvs))
(defn correlation
"Given two functions: (fx input) and (fy input), each of which returns a
number, estimates the unbiased linear correlation coefficient between fx and
fy over inputs. Ignores any records where fx or fy are nil. If there are no
records with values for fx and fy, the correlation is nil. See
http://mathworld.wolfram.com/CorrelationCoefficient.html."
[fx fy]
(fn
([] [0.0 0.0 0.0 0.0 0.0 0.0])
([[^double c ^double mx ^double my ^double ssx ^double ssy ^double ssxy :as acc] e]
(let [x (fx e)
y (fy e)]
(if (or (nil? x) (nil? y))
acc
(let [x (double x)
y (double y)
c' (inc c)
mx' (+ mx (/ (- x mx) c'))
my' (+ my (/ (- y my) c'))]
[c' mx' my'
(+ ssx (* (- x mx') (- x mx)))
(+ ssy (* (- y my') (- y my)))
(+ ssxy (* (- x mx') (- y my)))]))))
([[_c _mx _my ssx ssy ssxy]]
(let [d (sqrt (* ssx ssy))]
(when-not (zero? d)
(/ ssxy d))))))
(defn r-squared
"Given two functions: (fŷ input) and (fy input), returning
the predicted and actual values of y respectively, estimates
the coefficient of determination R^2.
This is the fraction of variance in y explained by the model."
[fy-hat fy]
(fn
([] [0.0 0.0 0.0 0.0])
([[^double c ^double my ^double ssr ^double ssy :as acc] e]
(let [y-hat (fy-hat e)
y (fy e)]
(if (or (nil? y-hat) (nil? y))
acc
(let [r (double (- y y-hat)) ;; Residual
y (double y)
c' (inc c)
my' (+ my (/ (- y my) c'))]
[c' my'
(+ ssr (* r r))
(+ ssy (* (- y my') (- y my)))]))))
([[c _my ssr ssy]]
(when-not (or (zero? c) (zero? ssy))
(- 1 (/ ssr ssy))))))
(defn adjusted-r-squared
"Given two functions: (fŷ input) and (fy input), returning
the predicted and actual values of y respectively, and a constant k
equal to the number of terms in the model, estimates the adjusted
coefficient of determination R^2 using Wherry's Formula-1.
This is the fraction of variance in y explained by the model,
adjusted for the number of terms in the model.
https://stats.stackexchange.com/questions/48703/what-is-the-adjusted-r-squared-formula-in-lm-in-r-and-how-should-it-be-interpret"
[fy-hat fy k]
(completing (r-squared fy-hat fy)
(fn [[c _my ssr ssy]]
(when (and (pos? ssy)
(pos? (- c k 1)))
(- 1 (/ (* (/ ssr ssy) (dec c))
(- c k 1)))))))
(defn mse
"Given two functions: (fŷ input) and (fy input), returning
the predicted and actual values of y respectively, calculates
the mean squared error of the estimate."
[fy-hat fy]
(fn
([] [0.0 0.0])
([[^double c ^double mse :as acc] e]
(let [y-hat (fy-hat e)
y (fy e)]
(if (or (nil? y-hat) (nil? y))
acc
(let [se (sq (- y y-hat))
c' (inc c)]
[c' (+ mse (/ (- se mse) c'))]))))
([[c mse]]
(when (pos? c)
mse))))
(defn rmse
"Given two functions: (fŷ input) and (fy input), returning
the predicted and actual values of y respectively, calculates
the root mean squared error of the estimate."
[fy-hat fy]
(post-complete (mse fy-hat fy) (somef sqrt)))
(defn correlation-matrix
"Given a map of key names to functions that extract values for those keys
from an input, computes the correlation for each of the n^2 key pairs.
For example:
(correlation-matrix {:name-length #(.length (:name %))
:age :age
:num-cats (comp count :cats)})
will, when reduced, return a map like:
{[:name-length :age] 0.56
[:name-length :num-cats] 0.95
...}"
[kvs]
(fuse-matrix correlation kvs))
(defn cramers-v
"Cramer's Phi is the intercorrelation of two discrete variables and may be used with variables having two or more levels. It gives a value between 0 and +1 (inclusive).
Given two functions: (fx input) and (fy input), each of which returns a the relevant discrete value."
[fx fy]
(fn
([] [{} {} {} 0])
([[f1 f2 f12 n] row]
(let [k1 (fx row)
k2 (fy row)
k12 [k1 k2]
increment-count (fn [m k] (update m k (fnil inc 0)))
f1' (increment-count f1 k1)
f2' (increment-count f2 k2)
f12' (increment-count f12 k12)
n' (inc n)]
[f1' f2' f12' n']))
([[f1 f2 f12 n]]
(let [r (clojure.core/count f1)
r-tilde (when (> n 1) (- r (/ (sq (dec r)) (- n 1))))
k (clojure.core/count f2)
k-tilde (when (> n 1) (- k (/ (sq (dec k)) (- n 1))))
chi-squared (reduce-kv (fn [acc k v]
(let [n1 (get f1 (first k))
n2 (get f2 (last k))
n12 v]
(+ acc (/ (sq (- n12 (/ (* n1 n2) n)))
(/ (* n1 n2) n)))))
0
f12)]
(when (and r-tilde k-tilde (> r-tilde 1) (> k-tilde 1))
(sqrt (/ (/ chi-squared n) (clojure.core/min (- r-tilde 1) (- k-tilde 1)))))))))
(def sum-squares digest/sum-squares)
(defn simple-linear-regression
"Given two functions: (fx input) and (fy input), each of which returns a
number, calculates a least squares linear model between fx and fy over inputs.
Returns a reified kixi.stats.protocols/PParamaterised.
Ignores any records with fx or fy are nil. If there are no records with
values for fx and fy, the linear relationship is nil. See
https://en.wikipedia.org/wiki/Simple_linear_regression."
[fx fy]
(post-complete (sum-squares fx fy) e/simple-linear-regression))
(defn regression-standard-error
"Given two functions: (fx input) and (fy input), each of which returns a
number, and an x value, calculates the standard error of the least
squares linear model of fx and fy over inputs.
Returns a reified kixi.stats.protocols/PDependent.
Ignores any records with fx or fy are nil. If there are no records with
values for fx and fy, the standard error of the estimate is nil."
([fx fy]
(post-complete (sum-squares fx fy)
(fn [sum-squares]
(reify p/PDependent
(measure [_ x]
(e/regression-standard-error sum-squares x))))))
([fx fy x]
(post-complete (sum-squares fx fy) #(e/regression-standard-error % x))))
(defn regression-confidence-interval
"Given two functions: (fx input) and (fy input), each of which returns a
number, and an x value, calculates the standard error of the least
squares linear model of fx and fy over inputs.
Returns a reified kixi.stats.protocols/PDependent if alpha is supplied,
or a reified kixi.stats.protocols/PDependentWithSignificance otherwise.
Ignores any records with fx or fy are nil. If there are no records with
values for fx and fy, the standard error of the estimate is nil."
([fx fy]
(post-complete (sum-squares fx fy)
(fn [sum-squares]
(reify p/PDependentWithSignificance
(measure-with-significance [_ x alpha]
(e/regression-confidence-interval sum-squares x alpha))))))
([fx fy alpha]
(post-complete (sum-squares fx fy)
(fn [sum-squares]
(reify p/PDependent
(measure [_ x]
(e/regression-confidence-interval sum-squares x alpha))))))
([fx fy alpha x]
(post-complete (sum-squares fx fy)
#(e/regression-confidence-interval % x alpha))))
(defn regression-prediction-standard-error
"Given two functions: (fx input) and (fy input), each of which returns a
number, and an x value, calculates the standard error of the least
squares linear model of fx and fy over inputs.
Returns a reified kixi.stats.protocols/PDependent.
Ignores any records with fx or fy are nil. If there are no records with
values for fx and fy, the standard error of the estimate is nil."
([fx fy]
(post-complete (sum-squares fx fy)
(fn [sum-squares]
(when sum-squares
(reify p/PDependent
(measure [_ x]
(e/regression-prediction-standard-error sum-squares x)))))))
([fx fy x]
(post-complete (sum-squares fx fy)
#(e/regression-prediction-standard-error % x))))
(defn regression-prediction-confidence-interval
"Given two functions: (fx input) and (fy input), each of which returns a
number, and an x value, calculates the standard error of the least
squares linear model of fx and fy over inputs.
Returns a reified kixi.stats.protocols/PDependent if alpha is supplied,
or a reified kixi.stats.protocols/PDependentWithSignificance otherwise.
Ignores any records with fx or fy are nil. If there are no records with
values for fx and fy, the standard error of the estimate is nil."
([fx fy]
(post-complete (sum-squares fx fy)
(fn [sum-squares]
(reify p/PDependentWithSignificance
(measure-with-significance [_ x alpha]
(e/regression-prediction-interval sum-squares x alpha))))))
([fx fy alpha]
(post-complete (sum-squares fx fy)
(fn [sum-squares]
(reify p/PDependent
(measure [_ x]
(e/regression-prediction-interval sum-squares x alpha))))))
([fx fy alpha x]
(post-complete (sum-squares fx fy)
#(e/regression-prediction-interval % x alpha))))
(defn chi-squared-test
"Given a sequence of functions, each of which returns the categorical value
(e.g. keyword or string) of a factor, performs a X^2 test of independence."
[& fxs]
(post-complete (apply cross-tabulate fxs) t/chi-squared-test))
(defn simple-t-test
"Performs a simple t test against a specified population mean
and standard deviation. The standard deviation is optional,
if not provided, a 'plug-in' test using the sample's sd
will be performed instead.
mu: the population mean
sd: (optional) the population standard deviation"
[{:keys [mu sd]}]
(if sd
(completing mean
(fn [[s c]]
(when-not (zero? c)
(t/simple-t-test {:mu mu :sd sd}
{:mean (/ s c) :n c}))))
(completing variance
(fn [[c m ss]]
(when-not (zero? c)
(let [c' (dec c)
var (if (pos? c') (/ ss c') 0)]
(t/simple-t-test {:mu mu :sd (sqrt var)}
{:mean m :n c})))))))
(defn t-test
"Given two functions of an input `(fx input)` and `(fy input)`, each of which
returns a number, performs the t test of mean significance of those functions over
inputs.
Ignores only inputs where both `(fx input)` and `(fy input)` are nil."
[fx fy]
(fn
([] [0.0 0.0 0.0 0.0 0.0 0.0])
([[^double cx ^double cy ^double mx ^double my ^double ssx ^double ssy :as acc] e]
(let [x (some-> (fx e) double)
y (some-> (fy e) double)]
(if (and (nil? x) (nil? y))
acc
(let [cx' (cond-> cx x inc)
cy' (cond-> cy y inc)
mx' (cond-> mx x (+ (/ (- x mx) cx')))
my' (cond-> my y (+ (/ (- y my) cy')))
ssx' (cond-> ssx x (+ (* (- x mx') (- x mx))))
ssy' (cond-> ssy y (+ (* (- y my') (- y my))))]
[cx' cy' mx' my' ssx' ssy']))))
([[cx cy mx my ssx ssy]]
(let [cx' (dec cx) cy' (dec cy)]
(when (and (pos? cx') (pos? cy'))
(t/t-test {:mean mx :sd (sqrt (/ ssx cx')) :n cx}
{:mean my :sd (sqrt (/ ssy cy')) :n cy}))))))
(defn simple-z-test
"Performs a simple z test against a specified population mean
and standard deviation. The standard deviation is optional,
if not provided, a 'plug-in' test using the sample's sd
will be performed instead.
mu: the population mean
sd: (optional) the population standard deviation"
[{:keys [mu sd]}]
(if sd
(completing mean
(fn [[s c]]
(when-not (zero? c)
(t/simple-z-test {:mu mu :sd sd}
{:mean (/ s c) :n c}))))
(completing variance
(fn [[c m ss]]
(when-not (zero? c)
(let [c' (dec c)
var (if (pos? c') (/ ss c') 0)]
(t/simple-z-test {:mu mu :sd (sqrt var)}
{:mean m :n c})))))))
(defn z-test
"Given two functions of an input `(fx input)` and `(fy input)`, each of which
returns a number, performs the z-test of mean significance of those functions over
inputs.
Ignores only inputs where both `(fx input)` and `(fy input)` are nil."
[fx fy]
(fn
([] [0.0 0.0 0.0 0.0 0.0 0.0])
([[^double cx ^double cy ^double mx ^double my ^double ssx ^double ssy :as acc] e]
(let [x (some-> (fx e) double)
y (some-> (fy e) double)]
(if (and (nil? x) (nil? y))
acc
(let [cx' (cond-> cx x inc)
cy' (cond-> cy y inc)
mx' (cond-> mx x (+ (/ (- x mx) cx')))
my' (cond-> my y (+ (/ (- y my) cy')))
ssx' (cond-> ssx x (+ (* (- x mx') (- x mx))))
ssy' (cond-> ssy y (+ (* (- y my') (- y my))))]
[cx' cy' mx' my' ssx' ssy']))))
([[cx cy mx my ssx ssy]]
(let [cx' (dec cx) cy' (dec cy)]
(when (and (pos? cx') (pos? cy'))
(t/z-test {:mean mx :sd (sqrt (/ ssx cx')) :n cx}
{:mean my :sd (sqrt (/ ssy cy')) :n cy}))))))
(defn proportion
"Calculate the proportion of inputs for which `pred` returns true."
[pred]
(fn
([] [0 0])
([[n d] e]
(vector (cond-> n
(pred e) inc)
(inc d)))
([[n d]]
(when (pos? d)
(double (/ n d))))))
(def share
"Alias for proportion"
proportion)
(def min
"Like clojure.core/min, but transducer and nil-friendly."
(fn
([] infinity)
([acc]
(when-not (infinite? acc)
acc))
([^double acc e]
(if (nil? e)
acc
(let [e (double e)]
(clojure.core/min acc e))))))
(def max
"Like clojure.core/max, but transducer and nil-friendly."
(fn
([] negative-infinity)
([acc]
(when-not (infinite? acc)
acc))
([^double acc e]
(if (nil? e)
acc
(let [e (double e)]
(clojure.core/max acc e))))))