-
Notifications
You must be signed in to change notification settings - Fork 49
/
ml.q
1191 lines (990 loc) · 45.5 KB
/
ml.q
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
\d .ml
/ returns boolean indicating preference not to flip matrices
noflip:{system"g"} / redefine to customize behavior
/ apply (f)unction (in parallel) to the 2nd dimension of (X)
f2nd:{[f;X]$[noflip[];(f value::) peach flip (count[X]#`)!X;f peach flip X]}
/ matrix primitives
mm:{$[;y] peach x} / X * Y
mmt:{(y$) peach x} / X * Y'
mtm:{f2nd[$[;y];x]} / X' * Y
minv:inv / X**-1
mlsq:lsq / least squares
dot:$ / dot product
diag:{$[0h>t:type x;x;@[n#t$0;;:;]'[til n:count x;x]]}
eye:{diag x#1f}
mdet:{[X] / determinant
if[2>n:count X;:X];
if[2=n;:(X[0;0]*X[1;1])-X[0;1]*X[1;0]];
d:dot[X 0;(n#1 -1)*(.z.s (X _ 0)_\:) each til n];
d}
mchol:{[X] / Cholesky decomposition
m:count X;
L:(m;m)#0f;
i:-1;
while[m>i+:1;
L[i;i]:sqrt X[i;i]-dot[L i;L i];
j:i;
while[m>j+:1;
L[j;i]:(X[j;i]-dot[L i;L j])%L[i;i];
];
];
L}
/ tensor variant of where
twhere:{
if[type x;:enlist where x];
x:(,'/) til[count x] {((1;count y 0)#x),y}' .z.s each x;
x}
/ returns true if all values are exactly equal
identical:{min first[x]~':x}
/ returns true if x is a matrix as defined by q
ismatrix:{
if[type x;:0b];
if[not all 9h=type each x;:0b];
b:identical count each x;
b}
/ basic utilities
/ find row indices of each atom/vec y in matrix/flipped table x
mfind:{{[x;i;j;y]?[y=x;i&j;i]}[y]/[count[first x]#n;til n:count x;x]}
/ return first index of atom/vec y in vec/dict/matrix/flipped table x
find:{$[0h>type first x;?;type x;key[x]mfind::;mfind][x;y]}
imax:{find[x;max x]} / index of max element
imin:{find[x;min x]} / index of min element
/ (pre|ap)pend n rows of repeated x to matri(X)
pend:{[n;x;X]$[n>0;,[;X];X,](abs n;count X 0)#x}
prepend:pend[1] / prepend 1 row of repeated x to matri(X)
append:pend[-1] / append 1 row of repeated x to matri(X)
/ where not any null
wnan:{$[all type each x;where not any null x;::]}
/ norm primitives
mnorm:sum abs:: / Manhattan (taxicab) norm
enorm2:{x wsum x} / Euclidean norm squared
enorm:sqrt enorm2:: / Euclidean norm
pnorm:{[p;x]sum[abs[x] xexp p] xexp 1f%p} / parameterized norm
/ distance primitives
hdist:sum (<>):: / Hamming distance
mdist:mnorm (-):: / Manhattan (taxicab) distance
edist2:enorm2 (-):: / Euclidean distance squared
edist:enorm (-):: / Euclidean distance
pedist2:{enorm2[x]+/:enorm2[y]+-2f*mtm["f"$y;"f"$x]} / pairwise edist2
mkdist:{[p;x;y]pnorm[p] x-y} / Minkowski distance
hmean:1f%avg 1f% / harmonic mean
cossim:{sum[x*y]%enorm[x i]*enorm y i:wnan(x;y)} / cosine similarity
cosdist:1f-cossim:: / cosine distance
cordist:1f-(cor):: / correlation distance
/ Spearman's rank (tied values get averaged rank)
srank:{@[x;g;:;avg each (x:"f"$rank x) g@:where 1<count each g:group x]}
scor:{srank[x i] cor srank y i:wnan(x;y)} / Spearman's rank correlation
scordist:1f-scor:: / Spearman's rank correlation distance
gower:{ / Gower distance
if[0h=t:min abs type each (x;y);:navg .z.s'[x;y]]; / iterate
if[1h=t;:not[all (x;y)]+0n 0 any (x;y)]; / asymmetric binary
if[(t > 19h)|t in 2 10 11h;:"f"$not x=y]; / nominal
d:abs[x-y]%(-) . (max;min) revo\: (x;y); / ordinal and continuous
d}
/ null-aware primitives (account for nulls in matrices)
ncount:{count[x]-$[type x;sum null x;0i {x+null y}/ x]}
nsum:{$[type x;sum x;0i {x+0i^y}/ x]}
navg:{$[type x;avg x;nsum[x]%ncount x]}
nwavg:{[w;x]$[type x;w wavg x;(%/){x+y*(0f^z;not null z)}/[0 0f;w;x]]}
nvar:{$[type x;var x;navg[x*x]-m*m:navg x]}
ndev:sqrt nvar::
nsvar:{$[type x;svar x;(n*nvar x)%-1+n:ncount x]}
nsdev:sqrt nsvar::
/ normalization primitives
/ return a function that applies (d)yadic function to the result of
/ (a)ggregating vector/matrix/dictionary/table x
daxf:{[d;a;x]$[0h>type first x; d[;a x]; d[;a x] peach]}
/ apply (d)yadic function to the result of (a)ggregating
/ vector/matrix/dictionary/table x
dax:{[d;a;x]daxf[d;a;x] x}
/ apply the result of f[x] to x
fxx:{[f;x]f[x] x}
/ normalize each vector to unit length
normalize:dax[%;enorm]
/ centered
demean:fxx demeanf:daxf[-;navg]
/ feature normalization (centered/unit variance)
zscore:fxx zscoref:{daxf[%;nsdev;x] demeanf[x]::}
/ feature normalization (scale values to [0,1])
minmax:fxx minmaxf:{daxf[%;{max[x]-min x};x] daxf[-;min;x]::}
/ decimal precision (scale values to [0,1])
decprec:fxx decprecf:{daxf[%;{10 xexp ceiling 10 xlog max[x]|neg min x};x]}
/ convert densities into probabilities
prb:dax[%;sum]
/ identify the minimum values with 1b
ismin:dax[=;min]
/ given (g)rouped dictionary, compute the odds
odds:{[g]prb count each g}
/ given (w)eight vector and (g)rouped dictionary, compute the weighted odds
wodds:{[w;g]prb sum each w g}
/ frequency and mode primitives
/ given a (w)eight atom or vector and data (x), return a dictionary (sorted
/ by key) mapping the distinct items to their weighted count
wfreq:{[w;x]x!@[count[x]#0*first w;(x:asc distinct x)?x;+;w]}
freq:wfreq[1]
/ given a (w)eight atom or vector and data (x), return x with maximum
/ weighted frequency
wmode:imax wfreq:: / weighted mode
mode:wmode[1] / standard mode
/ weighted average or mode
isord:{type[x] in 0 8 9h} / is ordered
aom:{$[isord x;avg;mode]x} / average or mode
waom:{[w;x]$[isord x;nwavg;wmode][w;x]} / weighted average or mode
/ binary classification evaluation metrics (summary statistics)
/ given actual values (y) and (p)redicted values, compute (tp;tn;fp;fn)
tptnfpfn:{[y;p]tp,(("i"$count y)-tp+sum f),f:(sum p;sum y)-/:tp:sum p&y}
/ aka Rand measure (William M. Rand 1971)
accuracy:{[tp;tn;fp;fn](tp+tn)%tp+tn+fp+fn}
precision:{[tp;tn;fp;fn]tp%tp+fp}
recall:sensitivity:hitrate:tpr:{[tp;tn;fp;fn]tp%tp+fn} / true positive rate
selectivity:specificity:tnr:{[tp;tn;fp;fn]tn%tn+fp} / true negative rate
fallout:fpr:{[tp;tn;fp;fn]fp%fp+tn} / false positive rate
missrate:fnr:{[tp;tn;fp;fn]fn%fn+tp} / false negative rate
dice:{[tp;tn;fp;fn]tp%fp+fn+tp*:2} / dice coefficient
/ receiver operating characteristic
roc:{[y;p]
r:(til[count s]-s;s:0f,sums y i:idesc p); / (fp;tp)
r:(r%last each r),enlist 0w,p i; / (fpr;tpr;threshold)
r:r@\:where reverse differ reverse r 2; / filter duplicate thresholds
r}
auc:{[x;y] .5*sum (x-prev x)*y+prev y} / area under the curve
/ f measure: given (b)eta and tp,tn,fp,fn compute the harmonic mean of
/ precision and recall
f:{[b;tp;tn;fp;fn]
f:1+b2:b*b;
f*:r:recall[tp;tn;fp;fn];
f*:p:precision[tp;tn;fp;fn];
f%:r+p*b2;
f}
f1:f[1]
/ Fowlkes–Mallows index (E. B. Fowlkes & C. L. Mallows 1983)
/ geometric mean of precision and recall
fmi:{[tp;tn;fp;fn]tp%sqrt(tp+fp)*tp+fn}
/ returns a number between 0 and 1 indicating the similarity of two datasets
jaccard:{[tp;tn;fp;fn]tp%tp+fp+fn}
/ Matthews correlation coefficient
/ correlation coefficient between the observed and predicted
/ -1 0 1 (none right, same as random prediction, all right)
mcc:{[tp;tn;fp;fn]((tp*tn)-fp*fn)%prd sqrt(tp;tp;tn;tn)+(fp;fn;fp;fn)}
/ regression evaluation metrics
/ given true (y) and (p)redicted values return the r^2
r2:{[y;p]1f-edist2[y;p]%edist2[y;avg y]}
/ given true labels y and predicted labels p, return a confusion matrix
cm:{[y;p]
n:count u:asc distinct y,p;
m:./[(n;n)#0;flip (u?p;u?y);1+];
t:([]y:u)!flip (`$string u)!m;
t}
/ given group (I)ndices, return list of (train;test) splits (where each split
/ is used for validation and the remaining k-1 folds are for training)
kfold:{[I]flip ((raze I _) each til count I;I)}
/ given group (I)ndices, return list of time-series (train;test) splits
/ (where each split is used for validation and the prior folds are for
/ training). (tr)ai(n) (f)unction and (t)e(st) (f)unction can be used to
/ customize the folds.
tsfold:{[trnf;tstf;I]
trn:(trnf raze #[;I]::) each 1_til count I;
tst:tstf each 1_I;
flip (trn;tst)}
/ index vector or second dimension of matrix
at:{[x;i]$[type x;x i;x[;i]]}
/ use (train;test) (i)ndices to fit a model using the (f)itting (f)unction
/ with training subset of x. return predictions obtained from using the
/ (p)rediction (f)unction on the test subset. x can be a table or (y|Y;x|X)
/ pair -- corresponding to ff arguments.
cv:{[ff;pf;x;i] / cross validate
if[not type i 0;:.z.s[ff;pf;x] peach i]; / iterate over folds
tt:$[type x;x i;raze[x at\:/:i] _ 2]; / handle table vs (y|Y;x|X)
m:ff . -1 _ tt; / fit model on train set
p:pf[m] last tt; / make predictions
p}
/ use all (f)old(s) (except the (i)th) to fit a model using the (f)itting
/ (f)unction and then use (p)rediction (f)unction on fs[i]. fs can be a list
/ of tables or (y;X) pairs -- corresponding to ff arguments.
xv:{[ff;pf;fs;i] / cross validate
v:fs i;fs _: i; / split training and validation sets
a:$[type v;enlist raze fs;[v@:1;(raze;,'/)@'flip fs]]; / build ff arguments
m:ff . a; / fit model on training set
p:pf[m] v; / make predictions on validation set
p}
/ k nearest neighbors
/ find (k) smallest values from (d)istance vector (or matrix) and use
/ (w)eighting (f)unction to return the best estimate of y
knn:{[wf;k;y;d]
if[not type d;:.z.s[wf;k;y] peach d]; / recurse for matrix d
if[any n:null d;d@:i:where not n; y@:i]; / filter null distances
p:(waom . (wf d::;y)@\:#[;iasc d]::) peach k&count d; / make predictions
p}
/ given (w)eighting (f)unction, (d)istance (f)unction, atom or vector of (k)
/ values, a (y) vector and matri(X), 'fit' a knn 'model'
fknn:{[wf;df;k;y;X] knn[wf;k;y] df[X]::}
pknn:@ / predict knn by applying model returned from fknn to X
/ partitional clustering initialization methods
/ generate (k) centroids by randomly choosing (k) samples from matri(X)
forgy:{[k;X]neg[k]?/:X} / Forgy method
/ generate (k) centroids by applying (c)entroid (f)unction to (k) random
/ partitions of matri(X)
rpart:{[cf;k;X](cf'') X@\:value group count[X 0]?k} / random partition
/ return the index of n (w)eighted samples
iwrand:{[n;w]s binr n?last s:sums w}
/ find n (w)eighted samples of x
wrand:{[n;w;x]x iwrand[n] w}
/ k-means++ initialization algorithm
/ using (d)istance (f)unction and matri(X), append the next centroid to the
/ min centroid (d)istance and all (C)entroids
kpp:{[df;X;d;C]
if[not count C;:(0w;X@\:1?count X 0)]; / first centroid
if[count[X 0]=n:count C 0;:(d;C)]; / no more centroids
d&:df[X] C[;n-1]; / update distance vector
C:C,'X@\: first iwrand[1] d; / pick next centroid
(d;C)}
kmeanspp:kpp[edist2] / k-means++ initialization
kmedianspp:kpp[mdist] / k-medians++ initialization
/ partitional clustering algorithms
/ using the (d)istance (f)unction, group matri(X) based on the closest
/ (C)entroid and return the cluster indices
cgroup:{[df;X;C]value group imin f2nd[df X] C}
/ Stuart Lloyd's algorithm. uses (d)istance (f)unction to assign the
/ matri(X) to the nearest (C)entroid and then uses the (c)entroid (f)unction
/ to update the centroid location.
lloyd:{[df;cf;X;C]cf X@\: cgroup[df;X;C]}
/ use (r)esponsibility (f)unction David Mackay's Information Theory..(pg289)
lloyds:{[df;cf;rf;X;C] cf[rf f2nd[df X] C;X]} / soft assignment
kmeans:lloyd[edist2;avg''] / k-means
kmedians:lloyd[mdist;med''] / k-medians
khmeans:lloyd[edist2;hmean''] / k harmonic means
skmeans:lloyd[cosdist;normalize (avg'')::] / spherical k-means
kmeanss:lloyds[edist2;wavg\:/:;ismin] / k-means using Lloyd with rf
/ v1 David Mackay using stiffness parameter (b)eta. 1%sqrt b represents the
/ sigma (or radius) of the cluster
kmeanssmax:{[b;X]lloyds[edist2;wavg\:/:;ssoftmax neg[b]*;X]}
/ using (d)istance (f)unction, find the medoid in matri(X)
medoid:{[df;X]X@\:imin f2nd[sum df[X]::] X}
/ given a (d)istance (f)unction, return a new function that finds a medoid
/ during the "update" step of lloyd's algorithm
pam:{[df]lloyd[df;flip f2nd[medoid df]::]} / partitioning around medoids
/ cluster purity primitives
/ given matri(X) compute the sum of squared errors (distortion)
sse:{[X]sum edist2[X] avg each X}
/ given matri(X) and cluster (I)ndices, compute within-cluster sse
ssw:{[X;I]sum (sse X@\:) peach I}
/ given matri(X) and cluster (I)ndices, compute between-cluster sse
ssb:{[X;I]count'[I] wsum edist2[(avg '')G] (avg raze::) each G:X@\:I}
/ using (d)istance (f)unction, matri(X) and (C)entroids, compute total
/ cluster distortion
distortion:{[X;C]ssw[X] cgroup[edist2;X] C}
/ given (d)istance (f)unction, matri(X), and cluster (I)ndices, compute the
/ silhouette statistic. group I if not already grouped
silhouette:{[df;X;I]
if[type I;s:.z.s[df;X]I:value group I;:raze[s] iasc raze I];
if[1=n:count I;:count[I 0]#0f]; / special case a single cluster
a:{[df;X](1f%-1+count X 0)*sum f2nd[df X] X}[df] peach G:X@\:/:I;
b:{[df;G;i]min{f2nd[avg x[z]::]y}[df;G i]'[G _ i]}[df;G] peach til n;
s:0f^(b-a)%a|b; / 0 fill to handle single point clusters
s}
/ hierarchical agglomerative clustering
/ Lance-Williams algorithm linkage functions. can be either a vector of four
/ floats or a function that accepts the cluster counts of i, j and list of
/ all cluster counts
lw.single:.5 .5 0 -.5
lw.complete:.5 .5 0 .5
lw.average:{(x%sum x _:2),0 0f}
lw.weighted:.5 .5 0 0
lw.centroid:{((x,neg prd[x]%s)%s:sum x _:2),0f}
lw.median:.5 .5 -.25 0
lw.ward:{((k+/:x 0 1),(neg k:x 2;0f))%\:sum x}
/ implementation of Lance-Williams algorithm for performing hierarchical
/ agglomerative clustering. given (l)inkage (f)unction to determine distance
/ between new and remaining clusters, (D)issimilarity matrix, cluster
/ (a)ssignments and (L)inkage stats: (j;i). returns updated (D;a;L)
lancewilliams:{[lf;D;a;L]
n:count D;
d:D@'di:imin peach D; / find closest distances
if[null d@:i:imin d;:(D;a;L)]; j:di i; / find closest clusters
c:$[9h=type lf;lf;lf(freq a)@/:(i;j;til n)]; / determine coefficients
nd:sum c*nd,(d;abs(-/)nd:D (i;j)); / calc new distances
D[;i]:D[i]:nd; / update distances
D[;j]:D[j]:n#0n; / erase j
a[where j=a]:i; / all elements in cluster j are now in i
L:L,'(j;i); / append linkage stats
(D;a;L)}
/ given a (l)inkage (f)unction and (D)issimilarity matrix, run the
/ Lance-Williams linkage algorithm for hierarchical agglomerative clustering
/ and return the linkage stats: (from index j;to index i)
link:{[lf;D]
D:@'[D;a:til count D;:;0n]; / define cluster assignments and ignore loops
if[-11h=type lf;lf:get lf]; / dereference lf
L:last .[lancewilliams[lf]] over (D;a;2#()); / obtain linkage stats
L}
/ use (L)inkage stats to create (k) clusters
clust:{[L;k]
if[0h>type k;:first .z.s[L] k,()]; / special case atoms
c:1 cut til 1+count L 0; / initial clusters
k@:i:idesc k; / sort k descending
fl:(1-mk:last k)_ flip L; / drop unwanted links
fls:(0,-1_count[c]-k) cut fl; / list of flipped link stats
c:{[c;fl]{x[y 1],:x y 0;x[y 0]:();x}/[c;fl]}\[c;fls]; / link into k clusters
c:c except\: enlist (); / remove empty clusters
c:c iasc i; / reorder based on original k
c}
/ random variate primitives
pi:acos -1f
twopi:2f*pi
logtwopi:log twopi
/ Box-Muller
bm:{
if[count[x] mod 2;:-1_.z.s x,rand 1f];
x:raze (sqrt -2f*log first x)*/:(cos;sin)@\:twopi*last x:2 0N#x;
x}
/ random number generators
/ generate (n) uniform distribution variates
runif:{[n]n?1f}
/ generate (n) Bernoulli distribution variates with (p)robability of success
rbern:{[n;p]p>runif n}
/ generate (n) binomial distribution (sum of Bernoulli) variates with (k)
/ trials and (p)robability
rbinom:{[n;k;p](sum rbern[k]::) each n#p}
/ generate (n) multinomial distribution variate-vectors with (k) trials and
/ (p)robability vector defined for each class
rmultinom:{[n;k;p](sum til[count p]=/:sums[p] binr runif::) each n#k}
/ generate (n) normal distribution variates with mean (mu) and standard
/ deviation (sigma)
rnorm:{[n;mu;sigma]mu+sigma*bm runif n}
/ C(n,k) or n choose k
choose:{[n;k](%). prd each(n-k;0)+\:1f+til k&:n-k}
/ P(n,k) or n permute k
permute:{[n;k]prd(1f+n-k)+til k}
/ [log]likelihood and maximum likelihood estimator (mle)
/ binomial likelihood (without the binomial coefficient nCk)
binl:{[n;p;k](p xexp k)*(1f-p) xexp n-k}
/ binomial log likelihood
binll:{[n;p;k](k*log p)+(n-k)*log 1f-p}
/binl:exp binll:: / more numerically stable
/ binomial mle with Dirichlet smoothing (a)
binmle:{[n;a;x]enlist avg a+x%n}
/ weighted binomial mle with Dirichlet smoothing (a)
wbinmle:{[n;a;w;x]enlist w wavg a+x%n}
/ binomial density
bind:{[n;p;k] choose[n;k]*binl[n;p;k]}
/ binomial mixture model likelihood
bmml:prd binl::
/ binomial mixture model log likelihood
bmmll:sum binll::
/bmml:exp bmmll:: / more numerically stable
/ binomial mixture model mle with Dirichlet smoothing (a)
bmmmle:{[n;a;x]enlist avg each a+x%n}
/ weighted binomial mixture model mle with Dirichlet smoothing (a)
wbmmmle:{[n;a;w;x]enlist w wavg/: a+x%n}
/ multinomial likelihood approximation (without the multinomial coefficient)
multil:{[p;k]p xexp k}
/ multinomial log likelihood
multill:{[p;k]k*log p}
/ multinomial mle with (a)dditive smoothing
multimle:{[a;x]enlist each prb a+sum each x}
/ weighted multinomial mle with (a)dditive smoothing
wmultimle:{[a;w;x]enlist each prb a+w wsum/: x}
/ multinomial mixture model likelihood
mmml:prd multil::
/ multinomial mixture model log likelihood
mmmll:sum multill::
/mmml:exp mmmll:: / more numerically stable
/ multinomial mixture model mle with Dirichlet smoothing (a)
mmmmle:{[n;a;x]enlist avg each a+x%n}
/ weighted multinomial mixture model mle with Dirichlet smoothing (a)
wmmmmle:{[n;a;w;x]enlist w wavg/: a+x%n}
/ Gaussian kernel
gaussk:{[mu;sigma;x] exp (enorm2 x-mu)%-2f*sigma}
/ Gaussian likelihood
gaussl:{[mu;sigma;x] exp[(x*x-:mu)%-2f*sigma]%sqrt sigma*twopi}
/ Gaussian log likelihood
gaussll:{[mu;sigma;x] -.5*logtwopi+log[sigma]+(x*x-:mu)%sigma}
/ Gaussian mle
gaussmle:{[x](mu;avg x*x-:mu:avg x)}
/ weighted Gaussian mle
wgaussmle:{[w;x](mu;w wavg x*x-:mu:w wavg x)}
/ Gaussian multivariate
gaussmvl:{[mu;SIGMA;X]
if[type SIGMA;SIGMA:diag count[X]#SIGMA];
p:exp -.5*sum X*mm[minv SIGMA;X-:mu];
p%:sqrt mdet[SIGMA]*twopi xexp count X;
p}
/ Gaussian multivariate log likelihood
gaussmvll:{[mu;SIGMA;X]
if[type SIGMA;SIGMA:diag count[X]#SIGMA];
p:sum X*mm[minv SIGMA;X-:mu];
p+:log[mdet SIGMA]+logtwopi*count X;
p*:-.5;
p}
/ Gaussian multivariate mle
gaussmvmle:{[X](mu;avg X (*\:/:)' X:flip X-mu:avg each X)}
/ weighted Gaussian multivariate mle
wgaussmvmle:{[w;X](mu;w wavg X (*\:/:)' X:flip X-mu:w wavg/: X)}
/ expectation maximization
likelihood:{[l;lf;X;phi;THETA]
p:(@[;X]lf .) peach THETA; / compute [log] probability densities
p:$[l;p+log phi;p*phi]; / apply prior probabilities
p}
/ using (l)ikelihood (f)unction, (w)eighted (m)aximum likelihood estimator
/ (f)unction with prior probabilities (p)hi and distribution parameters
/ (THETA), optionally (f)fit (p)hi and perform expectation maximization
em:{[fp;lf;wmf;X;phi;THETA]
W:prb likelihood[0b;lf;X;phi;THETA]; / weights (responsibilities)
if[fp;phi:avg each W]; / new phi estimates
THETA:wmf[;X] peach W; / new THETA estimates
(phi;THETA)}
/ term frequency primitives
/ term document matrix built from (c)orpus and (v)ocabulary
tdm:{[c;v](-1_@[(1+count v)#0;;+;1]::) each v?c}
lntf:log 1f+ / log normalized term frequency
dntf:{[k;x]k+(1f-k)*x% max each x} / double normalized term frequency
idf: {log count[x]%sum 0<x} / inverse document frequency
idfs:{log 1f+count[x]%sum 0<x} / inverse document frequency smooth
idfm:{log 1f+max[x]%x:sum 0<x} / inverse document frequency max
pidf:{log (max[x]-x)%x:sum 0<x} / probabilistic inverse document frequency
tfidf:{[tff;idff;x]tff[x]*\:idff x}
/ naive Bayes
/ fit parameters given (w)eighted (m)aximization (f)unction returns a
/ dictionary with prior and conditional likelihoods
fnb:{[wmf;w;y;X]
if[(::)~w;w:count[y]#1f]; / handle unassigned weight
pT:(odds g; (wmf . (w;X@\:) @\:) peach g:group y);
pT}
/ using a [log](l)ikelihood (f)unction and prior probabilities (p)hi and
/ distribution parameters (T)HETA, perform naive Bayes classification
pnb:{[l;lf;pT;X]
d:{(x . z) y}[lf]'[X] peach pT[1]; / compute probability densities
c:imax $[l;log[pT 0]+sum flip d;pT[0]*prd flip d];
c}
/ decision trees
/ classification impurity functions
misc:{1f-avg x=mode x} / misclassification
wmisc:{[w;x]1f-avg x=wmode[w;x]} / weighted misclassification
gini:{1f-enorm2 odds group x} / Gini
wgini:{[w;x]1f-enorm2 wodds[w] group x} / weighted Gini
entropy:{neg sum x*log x:odds group x} / entropy
wentropy:{[w;x]neg sum x*log x:wodds[w] group x} / weighted entropy
/ regression impurity functions
mse:{enorm2[x-avg x]%count x} / mean squared error
wmse:{[w;x]enorm2[x-w wavg x]%count x} / weighted mean squared error
mae:{avg abs x-avg x} / mean absolute error
wmae:{[w;x]avg abs x-w wavg x} / weighted mean absolute error
rms:{sqrt avg x*x} / root mean square error
/ combinations of length x (or all lengths if null x) from count (or list) y
cmb:{
if[not 0>type y;:y .z.s[x] count y]; / list y
if[null x;:raze .z.s[;y] each 1+til y]; / null x = all lengths
c:flip enlist flip enlist til y-:x-:1;
c:raze c {(x+z){raze x,''y}'x#\:y}[1+til y]/til x;
c}
/ use (i)m(p)urity (f)unction to compute the (w)eighted information gain of
/ x after splitting on y
ig:{[ipf;w;x;y] / information gain
g:ipf[w] x;
g-:sum wodds[w;gy]*(not null key gy)*w[gy] ipf' x gy:group y;
(g;::;gy)}
/ use (i)m(p)urity (f)unction to compute the (w)eighted gain ratio of x
/ after splitting on y
gr:{[ipf;w;x;y] / gain ratio
g:ig[ipf;w;x;y]; / first compute information gain
g:@[g;0;%[;ipf[w;y]]]; / then divide by splitinfo
g}
/ use (i)m(p)urity (f)unction to pick the maximum (w)eighted information
/ gain of x after splitting across all sets of distinct y
sig:{[ipf;w;x;y] / set information gain
c:raze cmb[;u] peach 1+til 1|count[u:distinct y] div 2; / combinations of y
g:(ig[ipf;w;x] y in) peach c; / all gains
g@:i:imax g[;0]; / highest gain
g[1]:in[;c i]; / replace split func
g}
/ use (i)m(p)urity (f)unction to pick the maximum (w)eighted information
/ gain of x after splitting across all values of y
oig:{[ipf;w;x;y] / ordered information gain
g:(ig[ipf;w;x] y >) peach u:asc distinct y; / all gains
g@:i:imax g[;0]; / highest gain (not gain ratio)
g[1]:>[;avg u i+0 1]; / replace split func
g}
/ use (i)m(p)urity (f)unction to pick the maximum (w)eighted gain ratio of x
/ after splitting across all values of y
ogr:{[ipf;w;x;y] / ordered gain ratio
g:oig[ipf;w;x;y]; / first compute information gain
g:@[g;0;%[;ipf[w;g[1] y]]]; / then divide by splitinfo
g}
/ given a vector of (w)eights (or ::) and a (t)able of features where the
/ first column is the target attribute, create a decision tree using the
/ (c)ategorical (g)ain (f)unction and (o)rdered (g)ain (f)unction. the
/ (i)m(p)urity (f)unction determines which statistic to minimize. a dict of
/ (opt)ions specify the (max) (d)epth, (min)imum # of (s)amples required to
/ (s)plit, (min)imum # of (s)amples at each (l)eaf, (min)imum (g)ain and the
/ (max)imum (f)eature (f)unction used to sub sample features for random
/ forests. defaults are: opt:`maxd`minss`minsl`ming`maxff!(0N;2;1;0;::)
dt:{[cgf;ogf;ipf;opt;w;t]
if[(::)~w;w:n#1f%n:count t]; / compute default weight vector
if[1=count d:flip t;:(w;first d)]; / no features to test
opt:(`maxd`minss`minsl`ming`maxff!(0N;2;1;0;::)),opt; / default options
if[0=opt`maxd;:(w;first d)]; / check if we've reached max depth
if[identical a:first d;:(w;a)]; / check if all values are equal
if[opt[`minss]>count a;:(w;a)]; / check if insufficient samples
d:((neg floor opt[`maxff] count d)?key d)#d:1 _d; / sub-select features
d:{.[x isord z;y] z}[(cgf;ogf);(ipf;w;a)] peach d; / compute gains
d:(where (any opt[`minsl]>count each last::) each d) _ d; / filter on minsl
if[0=count d;:(w;a)]; / check if all leaves have < minsl samples
if[opt[`ming]>=first b:d bf:imax d[;0];:(w;a)]; / check gain of best feature
c:count k:key g:last b; / grab subtrees, feature names and count
/ distribute nulls down each branch with reduced weight
if[c>ni:null[k]?1b;w:@[w;n:g nk:k ni;%;c-1];g:(nk _g),\:n];
if[(::)~b 1;t:(1#bf)_t]; / don't reuse exhausted features
b[2]:.z.s[cgf;ogf;ipf;@[opt;`maxd;-;1]]'[w g;t g]; / split sub-trees
bf,1_b}
/ use decision (tr)ee to make predictions for (d)ictionary
pdt:{[tr;d]
if[98h=type d;:.z.s[tr] peach d]; / iterate on a table
p:waom . pdtr[tr;d];
p}
/ use decision (tr)ee to recursively find leaf/leaves for (d)ictionary
pdtr:{[tr;d]
if[2=count tr;:tr]; / (w;a)
if[not null k:d tr 0;if[(a:tr[1][k]) in key tr[2];:.z.s[tr[2] a;d]]];
v:(,'/) tr[2] .z.s\: d; / dig deeper for null values
v}
/ decision tree pruning primitives
/ Wilson score - binary confidence interval (Edwin Bidwell Wilson)
wscore:{[z;f;n](f+(.5*z2n)+-1 1f*z*sqrt((.25*z2n)+f-f*f)%n)%1f+z2n:z*z%n}
/ pessimistic error
perr:{[z;w;x]last wscore[z;wmisc[w;x];count x]}
/ use (e)rror (f)unction to post-prune (tr)ee
prune:{[ef;tr]
if[2=count tr;:tr]; / (w;a)
b:value tr[2]:.z.s[ef] each tr 2; / prune subtree
if[any 3=count each b;:tr]; / can't prune
e:ef . wa:(,'/) b; / pruned error
if[e<((sum first::) each b) wavg (ef .) each b;:wa];
tr}
/ return the leaves of (tr)ee
leaves:{[tr]$[2=count tr;enlist tr;raze .z.s each last tr]}
/ using (e)rror (f)unction, return the decision (tr)ee's risk R(T) and
/ number of terminal nodes |T|
dtriskn:{[ef;tr](sum'[l[;0]] wsum ef ./: l;count l:leaves tr)}
/ using (e)rror (f)unction and regularization coefficient a, compute cost
/ complexity for (tr)ee
dtcc:{[ef;a;tr](1f;a) wsum dtriskn[ef;tr]}
/ given a decision (tr)ee, return all the subtrees sharing the same root
subtrees:{[tr]
if[2=count tr;:enlist tr];
str:tr 2; / subtree
if[all l:2=count each str;:enlist (,'/) str]; / prune
strs:(@[str;;:;].) each raze flip each flip (i;.z.s each str i:where not l);
trs:@[tr;2;:;] each strs;
trs,:enlist (,'/) leaves tr; / collapse this node too
trs}
/ given an (i)m(p)urity function and the pair of values (a;tr), return the
/ minimum (a)lpha and its associated sub(tr)ee.
dtmina:{[ipf;atr]
if[2=count tr:last atr;:atr];
en:dtriskn[ipf;tr];
ens:dtriskn[ipf] peach trs:subtrees tr;
a:neg (%) . en - flip ens;
atr:(a;trs)@\:i imin a i:idesc ens[;1]; / sort descending # nodes
atr}
/ given an (e)rror function, a cost parameter (a)lpha and decision (tr)ee,
/ return the subtree that minimizes the cost complexity
dtmincc:{[ef;tr;a]
if[2=count tr;:tr];
strs:subtrees tr;
strs@:iasc (count leaves::) each strs; / prefer smaller trees
str:strs imin dtcc[ef;a] each strs;
str}
/ k-fold cross validate (i)th table in (t)able(s) using (d)ecision (t)ree
/ (f)unction, (a)lphas and misclassification (e)rror (f)unction
dtxv:{[dtf;ef;a;ts]xv[dtmincc[ef]\[;a]dtf::;pdt\:/:;ts]}
/ use (train;test) (i)ndices to cross validate (t)able using (d)ecision
/ (t)ree (f)unction, (a)lphas and misclassification (e)rror (f)unction
dtcv:{[dtf;ef;a;t;i]cv[dtmincc[ef]\[;a]dtf::;pdt\:/:;t;i]}
/ decision tree utilities
/ print leaf: prediction followed by classification error% or regression sse
pleaf:{[w;x]
v:waom[w;x]; / value
e:$[isord x;string sum e*e:v-x;string[.1*"i"$1e3*1f-avg x = v],"%"];
s:string[v], " (n = ", string[count x],", err = ",e, ")";
s}
/ print (tr)ee with i(n)dent
ptree:{[n;tr]
if[not n;:(pleaf . first xs),last xs:.z.s[n+1;tr]];
if[2=count tr;:(tr;"")];
s:1#"\n";
s,:raze[(n)#enlist "| "],raze string[tr 0 1],\:" ";
s:s,/:string k:asc key tr 2;
c:.z.s[n+1] each tr[2]k; / child
x:first each c;
s:s,'": ",/:(pleaf .) each x;
s:raze s,'last each c;
x:(,'/) x;
(x;s)}
/ given (p)arent id, (n)ode id, label and (tr)ee print Graphviz node
pnode:{[p;n;l;tr]
s:n," [label = \""; / label
st:$[b:2=count tr;();tr 2]; / sub tree
cn:n,/:"0"^(neg max count each cn)$ cn:string til count st; / child node ids
c:$[b;enlist (tr;st);.z.s[n]'[cn;key st;value st]]; / children
s,:pleaf . x:(,'/) first each c; / error stats
if[not b;s,:"\\n",raze string[2#tr],\: " "]; / node title
s:enlist s,"\"]";
if[count p;s,:enlist p," -> ",n," [label = \"",string[l],"\"]"]; / edge
s,:raze last each c;
(x;s)}
/ print graph text for use with the 'dot' Graphviz command, graph-easy or
/ http://webgraphviz.com
pgraph:{[tr]
s:enlist "digraph Tree {";
s,:enlist "node [shape = box]";
s,:last pnode["";"0";`;tr];
s,:1#"}";
s}
/ decision tree projections
/ given a (t)able of classifiers and labels where the first column is target
/ attribute, create a decision tree
aid:dt[sig;oig;wmse] / automatic interaction detection
thaid:dt[sig;oig;wmisc] / theta automatic interaction detection
id3:dt[ig;ig;wentropy] / iterative dichotomizer 3
q45:dt[gr;ogr;wentropy] / like c4.5
ct:dt[oig;oig;wgini] / classification tree
rt:dt[oig;oig;wmse] / regression tree
/ random forest
/ generate (n) decision trees by applying (f) to a resampled (with
/ replacement) (t)able
bag:{[n;f;t](f ?[;t]::) peach n#count t} / (b)ootstrap (ag)gregating
/ given an atom or list (k), and bootstrap aggregating (m)odel, make
/ prediction on (d)ictionary
pbag:{[k;m;d]
if[count[m]<max k;'`length];
if[98h=type d;:.z.s[k;m] peach d]; / iterate on a table
p:k {aom x#y}\: pdt[;d] peach m;
p}
/ discrete adaptive boosting
/ given (t)rain (f)unction, discrete (c)lassifier (f)unction, initial
/ (w)eights, and (t)able with -1 1 discrete target class values in first
/ column, return ((m)odel;(a)lpha;new (w)eights)
adaboost:{[tf;cf;w;t]
if[(::)~w;w:n#1f%n:count t]; / initialize weights
m:tf[w] t; / train model
p:cf[m] t; / make predictions
e:sum w*not p=y:first flip t; / compute weighted error
a:.5*log (c:1f-e)%e; / compute alpha (minimize exponential loss)
/ w*:exp neg a*y*p; / increase/decrease weights
/ w%:sum w; / normalize weights
w%:2f*?[y=p;c;e]; / increase/decrease and normalize weights
(m;a;w)}
/ given an atom or list (k), (t)rain (f)unction, discrete (c)lassifier
/ (f)unction, and (t)able perform max(k) iterations of adaboost
fab:{[k;tf;cf;t] 1_max[k] (adaboost[tf;cf;;t] last::)\ (::)}
/ given an atom or list (k), discrete (c)lassifier function, adaboost
/ (m)odel, make prediction on (d)ictionary
pab:{[k;cf;m;d]
if[count[m]<mx:max k;'`length];
if[98h=type d;:.z.s[k;cf;m] peach d]; / iterate on a table
p:m[;1] * cf[;d] peach m[;0];
p:signum $[0h>type k;sum k#p;sums[mx#p] k-1];
p}
/ regularization primitives
/ reverse of over (start deep and end shallow)
revo:{[f;x]$[type x;f x;type first x;f f peach x;f .z.s[f] peach x]}
/ given l1 regularization (l)ambda and size of dimension (m), return two
/ function compositions that compute the cost and gradient
l1:{[l;m]((l%m)*revo[sum] abs::;(l%m)*signum::)}
/ given l2 regularization (l)ambda and size of dimension (m), return two
/ function compositions that compute the cost and gradient
l2:{[l;m]((.5*l%m)*revo[sum] {x*x}::;(l%m)*)}
/ given (a)lpha and (l)ambda (r)atio elastic net parameters, convert them
/ into l1 and l2 units and return a pair of l1 and l2 projections
enet:{[a;lr](l1 a*lr;l2 a*1f-lr)}
/ gradient descent utilities
/ accumulate cost by calling (c)ost (f)unction on the result of applying
/ (m)inimization (f)unction to THETA. return (THETA;new cost vector)
acccost:{[cf;mf;THETA;c] (THETA;c,cf THETA:mf THETA)}
/ print # of iterations, current (c)ost and % decrease to (h)andle, return a
/ continuation boolean: % decrease > float (p) or iterations < integer (p)
continue:{[h;p;c]
pct:$[2>n:count c;0w;1f-(%/)c n-1 2];
b:$[-8h<type p;p>n;p<pct];
s:" | " sv ("iter: ";"cost: ";"pct: ") ,' string (n;last c;pct);
if[not null h; h s,"\n\r" b];
b}
/ keep calling (m)inimization (f)unction on (THETA) and logging status to
/ (h)andle until the % decrease in the (c)ost (f)unction is less than
/ (p). return (cost vector;THETA)
iter:{[h;p;cf;mf;THETA](continue[h;p]last::)acccost[cf;mf]//(::;cf)@\:THETA}
/ (a)lpha: learning rate, gf: gradient function
gd:{[a;gf;THETA] THETA-a*gf THETA} / gradient descent
/ optimize (THETA) by using gradient descent with learning rate (a) and
/ (g)radient (f)unction over (n) subsamples of (X) and (Y) generated with
/ (s)ampling (f)unction: til = no shuffle, 0N? = shuffle, {x?x} = bootstrap
sgd:{[a;gf;sf;n;Y;X;THETA] / stochastic gradient descent
I:(n;0N)#sf count X 0;
THETA:THETA (gd[a] . (gf .;::)@'{(x[;;z];y)}[(Y;X)]::)/ I;
THETA}
/ linear regression
/ given target matrix Y and data matri(X), return the THETA matrix resulting
/ from minimizing sum of squared residuals
normeq:{[Y;X]mm[mmt[Y;X]] minv mmt[X;X]} / normal equations ols
/ given (l2) regularization parameter, target matrix Y and data matri(X),
/ return the THETA matrix resulting from performing ridge regression
ridge:{[l2;Y;X]mm[mmt[Y;X]] minv mmt[X;X]+diag count[X]#l2}
/ given (l2) regularization parameter, target vector y and data matri(X),
/ return theta vector resulting from performing weighted ridge regression by
/ scaling the regularization parameter by the count of non-null values
wridge:{[l2;y;X]first ridge[l2*count i;enlist y i;X[;i:where not null y]]}
/ linearly predict Y values by prepending matri(X) with a vector of 1s and
/ multiplying the result to (THETA) coefficients
plin:{[X;THETA]mm[THETA] prepend[1f] X}
/ linear regression cost
lincost:{[rf;Y;X;THETA]
J:(.5%m:count X 0)*revo[sum] E*E:plin[X;THETA]-Y; / cost
if[count rf,:();THETA[;0]:0f; J+:sum rf[;m][;0][;THETA]]; / regularization
J}
/ linear regression gradient
lingrad:{[rf;Y;X;THETA]
G:(1f%m:count X 0)*mmt[0f^mm[THETA;X]-Y] X:prepend[1f] X; / gradient
if[count rf,:();THETA[;0]:0f; G+:sum rf[;m][;1][;THETA]]; / regularization
G}
/ linear cost & gradient
lincostgrad:{[rf;Y;X;theta]
THETA:(count Y;0N)#theta; X:prepend[1f] X; / unroll theta
J:(.5%m:count X 0)*revo[sum] E*E:0f^mm[THETA;X]-Y; / cost
G:(1f%m)*mmt[E] X; / gradient
if[count rf,:();THETA[;0]:0f;JG:rf[;m][;;THETA];J+:sum JG@'0;G+:sum JG@'1];
(J;raze G)}
/ activation primitives (derivatives optionally accept `z`a!(z;a) dict)
linear:(::) / linear
dlinear:{1f+0f*$[99h=type x;x`z;x]} / linear gradient
sigmoid:1f%1f+exp neg:: / sigmoid
dsigmoid:{x*1f-x:$[99h=type x;x`a;sigmoid x]} / sigmoid gradient
tanh:1f-2f%1f+exp 2f* / hyperbolic tangent
dtanh:{1f-x*x:$[99h=type x;x`a;tanh x]} / hyperbolic tangent gradient
relu:0f| / rectified linear unit
drelu:{"f"$0f<=$[99h=type x;x`z;x]} / rectified linear unit gradient
lrelu:{x*1 .01@0f>x} / leaky rectified linear unit
dlrelu:{1 .01@0f>$[99h=type x;x`z;x]} / leaky rectified linear unit gradient
softmax:prb exp:: / softmax
ssoftmax:softmax dax[-;max]:: / stable softmax
dsoftmax:{diag[x] - x*\:/:x:softmax x} / softmax gradient
/ loss primitives
/ given true (y) and (p)redicted values return the log loss
logloss:{[y;p]neg (y*log 1e-15|p)+(1f-y)*log 1e-15|1f-p}
/ given true (y) and (p)redicted values return the cross entropy loss
celoss:{[y;p]neg sum y*log 1e-15|p}
/ given true (y) and (p)redicted values return the mean squared error loss
mseloss:{[y;p].5*y*y-:p}
/ logistic regression
/ logistic regression predict
plog:sigmoid plin::
/ logistic regression cost
logcost:{[rf;Y;X;THETA]
J:(1f%m:count X 0)*revo[sum] logloss[Y] plog[X;THETA]; / cost
if[count rf,:();THETA[;0]:0f; J+:sum rf[;m][;0][;THETA]]; / regularization
J}
/ logistic regression gradient
loggrad:{[rf;Y;X;THETA]
G:(1f%m:count X 0)*mmt[sigmoid[mm[THETA;X]]-Y] X:prepend[1f] X; / gradient
if[count rf,:();THETA[;0]:0f; G+:sum rf[;m][;1][;THETA]]; / regularization
G}
logcostgrad:{[rf;Y;X;theta]
THETA:(count Y;0N)#theta; X:prepend[1f] X; / unroll theta
J:(1f%m:count X 0)*revo[sum] logloss[Y] P:sigmoid mm[THETA] X; / cost
G:(1f%m)*mmt[P-Y] X; / gradient
if[count rf,:();THETA[;0]:0f;JG:rf[;m][;;THETA];J+:sum JG@'0;G+:sum JG@'1];
(J;raze G)}
logcostgradf:{[rf;Y;X]
Jf:logcost[rf;Y;X]enlist::;
Gf:loggrad[rf;Y;X]enlist::;
(Jf;Gf)}
/ one vs all
/ given binary classification fitting (f)unction, fit a one-vs.-all model
/ against Y for each unique (lbls)
fova:{[f;Y;lbls] (f "f"$Y=) peach lbls}
/ neural network matrix initialization primitives
/ Xavier Glorot and Yoshua Bengio (2010) initialization
/ given the number of (i)nput and (o)utput nodes, initialize THETA matrix
glorotu:{[i;o]sqrt[6f%i+o]*-1f+i?/:o#2f} / uniform
glorotn:{[i;o]rnorm'[o#i;0f;sqrt 2f%i+o]} / normal
/ Kaiming He, Xiangyu Zhang, Shaoqing Ren, Jian Sun (2015) initialization
/ given the number of (i)nput and (o)utput nodes, initialize THETA matrix
heu:{[i;o]sqrt[6f%i]*-1f+i?/:o#2f} / uniform
hen:{[i;o]rnorm'[o#i;0f;sqrt 2f%i]} / normal
/ neural network primitives
/ use (h)idden and (o)utput layer functions to predict neural network Y
pnn:{[hof;X;THETA]
X:X (hof[`h] plin::)/ -1_THETA;
Y:hof[`o] plin[X] last THETA;
Y}
/ (r)egularization (f)unction, holf: (h)idden (o)utput (l)oss functions
nncost:{[rf;holf;Y;X;THETA]
J:(1f%m:count X 0)*revo[sum] holf[`l][Y] pnn[holf;X] THETA; / cost
if[count rf,:();THETA[;;0]:0f;J+:sum rf[;m][;0][;THETA]]; / regularization
J}
/ (r)egularization (f)unction, hgof: (h)idden (g)radient (o)utput functions
nngrad:{[rf;hgof;Y;X;THETA]
ZA:enlist[(X;X)],(X;X) {(z;x z:plin[y 1;z])}[hgof`h]\ -1_THETA;
P:hgof[`o] plin[last[ZA]1;last THETA]; / prediction
G:hgof[`g]@'`z`a!/:1_ZA; / activation gradient
D:reverse{[D;THETA;G]G*1_mtm[THETA;D]}\[E:P-Y;reverse 1_THETA;reverse G];
G:(1%m:count X 0)*(D,enlist E) mmt' prepend[1f] each ZA[;1]; / full grad
if[count rf,:();THETA[;;0]:0f; G+:sum rf[;m][;1][;THETA]]; / regularization
G}
/ neural network cut
nncut:{[n;x]n cut' sums[prev[n+:1]*n:-1_n] cut x}