889:
36:
777:
460:
798:
can also be used directly as basis splines for regression analysis involving positive response data (constraining the regression coefficients to be non-negative).
930:
816:
518:
954:
923:
949:
814:
Curry, H.B.; Schoenberg, I.J. (1966). "On Polya frequency functions. IV. The fundamental spline functions and their limits".
374:
916:
29:
896:
866:
21:
900:
856:
825:
943:
888:
17:
861:
844:
35:
791:
870:
829:
361:
can be efficiently and stably computed using the following recursions:
772:{\displaystyle M_{i}(x|k,t)={\frac {k\left}{(k-1)(t_{i+k}-t_{i})}}.}
790:
can be integrated to produce a family of monotone splines called
307: − 2 continuous derivatives at interior knots
904:
455:{\displaystyle M_{i}(x|1,t)={\frac {1}{t_{i+1}-t_{i}}}}
521:
377:
771:
454:
43:family of order three with four interior knots.
924:
64:free parameters is defined by a set of knots
8:
208:) has the following mathematical properties
931:
917:
860:
754:
735:
683:
665:
640:
607:
595:
582:
558:
538:
526:
520:
443:
424:
414:
394:
382:
376:
34:
845:"Monotone Regression Splines in Action"
806:
7:
885:
883:
903:. You can help Knowledge (XXG) by
14:
887:
78: ≤ ... ≤
817:Journal d'Analyse Mathématique
760:
728:
725:
713:
703:
684:
677:
658:
633:
627:
608:
601:
588:
569:
552:
539:
532:
408:
395:
388:
1:
121: = ... =
101: = ... =
505:) = 0 otherwise.
971:
882:
955:Applied mathematics stubs
899:-related article is a
773:
456:
44:
950:Splines (mathematics)
862:10.1214/ss/1177012761
843:Ramsay, J.O. (1988).
774:
457:
38:
519:
375:
164:The family includes
897:applied mathematics
849:Statistical Science
512: > 1,
172: = 1,...,
168:members indexed by
56:functions of order
830:10.1007/BF02788653
769:
452:
45:
28:is a non-negative
22:numerical analysis
912:
911:
764:
450:
349:) integrates to 1
255:) is zero unless
232:) is non-negative
962:
933:
926:
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884:
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864:
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587:
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461:
459:
458:
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448:
447:
435:
434:
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398:
387:
386:
368: = 1,
268: <
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970:
969:
965:
964:
963:
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960:
959:
940:
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938:
937:
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837:
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804:
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750:
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636:
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568:
564:
560:
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516:
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487:
473:
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420:
419:
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373:
372:
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336:
326:
316:
290:
280:
263:
242:
219:
195:
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109:
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50:
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958:
957:
952:
942:
941:
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935:
928:
921:
913:
910:
909:
892:
877:
876:
855:(4): 425–441.
835:
805:
803:
800:
784:
781:
780:
779:
768:
762:
757:
753:
749:
744:
741:
738:
734:
730:
727:
724:
721:
718:
715:
709:
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702:
699:
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693:
690:
686:
682:
679:
674:
671:
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664:
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654:
649:
646:
643:
639:
635:
632:
629:
626:
623:
620:
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614:
610:
606:
603:
598:
594:
590:
585:
581:
577:
574:
571:
567:
563:
557:
554:
551:
548:
545:
541:
537:
534:
529:
525:
492:
482:
469:
463:
462:
446:
442:
438:
433:
430:
427:
423:
418:
413:
410:
407:
404:
401:
397:
393:
390:
385:
381:
355:
352:
351:
350:
332:
327:
321:
311:
286:
281:
272:
259:
238:
233:
215:
191:
181:
178:
162:
161:
148:
139:
134:
125:
115:
110:
105:
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920:
915:
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908:
906:
902:
898:
893:
890:
886:
881:
872:
868:
863:
858:
854:
850:
846:
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823:
819:
818:
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807:
801:
799:
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793:
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782:
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751:
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742:
739:
736:
732:
722:
719:
716:
707:
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697:
694:
691:
688:
680:
672:
669:
666:
662:
655:
652:
647:
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641:
637:
630:
624:
621:
618:
615:
612:
604:
596:
592:
583:
579:
575:
572:
565:
561:
555:
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546:
543:
535:
527:
523:
515:
514:
513:
511:
506:
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500:
495:
491:
485:
481:
477:
474: ≤
472:
468:
444:
440:
436:
431:
428:
425:
421:
416:
411:
405:
402:
399:
391:
383:
379:
371:
370:
369:
367:
362:
360:
353:
348:
344:
340:
335:
331:
328:
324:
320:
314:
310:
306:
302:
298:
294:
289:
285:
282:
279:
275:
271:
267:
264: ≤
262:
258:
254:
250:
246:
241:
237:
234:
231:
227:
223:
218:
214:
211:
210:
209:
207:
203:
199:
194:
190:
187:
179:
177:
175:
171:
167:
160:
155:
151:
147:
142:
138:
135:
132:
128:
124:
118:
114:
111:
108:
104:
97:
94:
93:
92:
89:
85:
81:
74:
71: ≤
67:
63:
59:
55:
47:
42:
37:
33:
31:
27:
23:
19:
905:expanding it
894:
879:
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848:
838:
821:
815:
809:
795:
787:
786:
783:Applications
509:
507:
502:
498:
493:
489:
483:
479:
475:
470:
466:
464:
365:
363:
358:
357:
346:
342:
338:
333:
329:
322:
318:
312:
308:
304:
300:
296:
292:
287:
283:
277:
273:
269:
265:
260:
256:
252:
248:
244:
239:
235:
229:
225:
221:
216:
212:
205:
201:
197:
192:
188:
185:
183:
173:
169:
165:
163:
158:
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149:
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140:
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126:
122:
116:
112:
106:
102:
95:
87:
83:
79:
72:
65:
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57:
53:
52:A family of
51:
40:
25:
20:subfield of
18:mathematical
15:
354:Computation
944:Categories
824:: 71–107.
802:References
180:Properties
91:such that
48:Definition
32:function.
796:M-splines
792:I-splines
788:M-splines
748:−
720:−
692:−
653:−
616:−
576:−
437:−
359:M-splines
325:−1
186:M-spline
157:for all
54:M-spline
41:M-spline
26:M-spline
871:2245395
345:,
317:, ...,
299:,
251:,
228:,
204:,
16:In the
869:
488:, and
303:) has
30:spline
895:This
867:JSTOR
60:with
24:, an
901:stub
508:For
364:For
857:doi
826:doi
794:.
501:|1,
465:if
184:An
39:An
946::
865:.
851:.
847:.
822:17
820:.
486:+1
315:+1
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873:.
859::
853:3
832:.
828::
767:.
761:)
756:i
752:t
743:k
740:+
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726:)
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717:k
714:(
708:]
704:)
701:t
698:,
695:1
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681:x
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673:1
670:+
667:i
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656:x
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638:t
634:(
631:+
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625:t
622:,
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613:k
609:|
605:x
602:(
597:i
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570:(
566:[
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550:t
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426:i
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406:t
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400:1
396:|
392:x
389:(
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