576:
639:
1195:
193:
25:
1187:
390:
1236:
up to ECe = 8 dS/m (ECe is the electric conductivity of an extract of a saturated soil sample), while beyond that value the crop production reduces. The graph is made with the method of partial regression to find the longest range of "no effect", i.e. where the line is horizontal. The two segments
411:
values (in the above example −3, 0, and 3) where the slope changes are typically called breakpoints, changepoints, threshold values or knots. As in many applications, this function is also continuous. The graph of a continuous piecewise linear function on a compact interval is a
976:
206:
1176:
610:
can be used to choose optimal separation points. However efficient computation and joint estimation of all model parameters (including the breakpoints) may be obtained by an iterative procedure currently implemented in the package
602:
can be performed independently on these partitions. However, continuity is not preserved in that case, and also there is no unique reference model underlying the observed data. A stable algorithm with this case has been derived.
583:
An approximation to a known curve can be found by sampling the curve and interpolating linearly between the points. An algorithm for computing the most significant points subject to a given error tolerance has been published.
840:
1056:
770:
680:-valued, or it may take values from a vector space, an affine space, a piecewise linear manifold, or a simplicial complex. (In these contexts, the term “linear” does not refer solely to
851:
471:
385:{\displaystyle f(x)={\begin{cases}-x-3&{\text{if }}x\leq -3\\x+3&{\text{if }}-3<x<0\\-2x+3&{\text{if }}0\leq x<3\\0.5x-4.5&{\text{if }}x\geq 3\end{cases}}}
1210:
of measured data is used to detect the range over which growth factors affect the yield and the range over which the crop is not sensitive to changes in these factors.
506:
1067:
999:
705:
generalize piecewise linear functions to higher-order polynomials, which are in turn contained in the category of piecewise-differentiable functions,
778:
627:
180:, there needs to be a finite collection of such intervals; if the domain is not compact, it may either be required to be finite or to be
89:
42:
395:
is piecewise linear with four pieces. The graph of this function is shown to the right. Since the graph of an affine(*) function is a
61:
1007:
646:
The notion of a piecewise linear function makes sense in several different contexts. Piecewise linear functions may be defined on
68:
1217:
the yield declines, whereas at deeper (> 7 dm) watertables the yield is unaffected. The graph is made using the method of
75:
732:
1312:
57:
1391:
122:
108:
1491:
971:{\displaystyle f({\vec {x}})=\min _{\Sigma \in \Pi }\max _{({\vec {a}},b)\in \Sigma }{\vec {a}}\cdot {\vec {x}}+b.}
426:
46:
642:
A piecewise linear function of two arguments (top) and the convex polytopes on which it is linear (bottom)
1632:
82:
1357:
665:
181:
699:. This guarantees that the graph of the function will be composed of polygonal or polytopal pieces.
230:
616:
1229:
607:
552:
35:
1627:
623:
161:
1237:
need not join at the same point. Only for the second segment method of least squares is used.
543:, a function composed of constant sub-functions, so also called a piecewise constant function
1274:
685:
169:
1171:{\displaystyle f({\vec {x}})=\max _{({\vec {a}},b)\in \Sigma }{\vec {a}}\cdot {\vec {x}}+b.}
476:
1583:
1412:
Vieth, E. (1989). "Fitting piecewise linear regression functions to biological responses".
1251:
1246:
702:
593:
141:
8:
1207:
718:
575:
564:
145:
691:
In dimensions higher than one, it is common to require the domain of each piece to be a
1569:
1542:
1472:
1332:
984:
669:
1464:
1429:
1376:
1308:
1256:
599:
404:
1560:
Ovchinnikov, Sergei (2002). "Max-min representation of piecewise linear functions".
1476:
1546:
1532:
1456:
1447:
Muggeo, V. M. R. (2003). "Estimating regression models with unknown break-points".
1421:
1372:
168:, such that there is a collection of intervals on each of which the function is an
1579:
1517:
1492:"Segmented: an R package to fit regression models with broken-line relationships"
1261:
722:
653:
546:
510:
413:
396:
173:
1425:
673:
535:
525:
1537:
1194:
638:
1621:
1233:
1218:
557:
540:
530:
177:
192:
160:
A piecewise linear function is a function defined on a (possibly unbounded)
1468:
661:
657:
400:
149:
1433:
835:{\displaystyle \Pi \in {\mathcal {P}}({\mathcal {P}}(\mathbb {R} ^{n+1}))}
1203:
677:
165:
129:
1608:
1596:
1214:
681:
420:
1574:
1460:
647:
24:
1222:
696:
579:
A function (blue) and a piecewise linear approximation to it (red)
172:. (Thus "piecewise linear" is actually defined to mean "piecewise
692:
717:
Important sub-classes of piecewise linear functions include the
1358:"Data point selection for piecewise linear curve approximation"
1392:"Least-squares Fit of a Continuous Piecewise Linear Function"
1051:{\displaystyle \Sigma \in {\mathcal {P}}(\mathbb {R} ^{n+1})}
706:
378:
1186:
598:
If partitions, and then breakpoints, are already known,
521:
There are other examples of piecewise linear functions:
399:, the graph of a piecewise linear function consists of
1070:
1010:
987:
854:
781:
735:
479:
429:
209:
49:. Unsourced material may be challenged and removed.
1170:
1050:
993:
970:
834:
765:{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
764:
729:-dimensional continuous piecewise linear function
725:piecewise linear functions. In general, for every
500:
465:
384:
1515:
1619:
1228:The graph on the right reveals that crop yields
1096:
896:
880:
1305:Technical Analysis And Applications With Matlab
508:; functions whose graph is a straight line are
1213:The image on the left shows that at shallow
1559:
1273:Apps, P., Long, N., & Rees, R. (2014).
1001:is convex and continuous, then there is a
1516:Landwehr, N.; Hall, M.; Frank, E. (2005).
1383:
1355:
121:For other uses of "piecewise linear", see
1573:
1536:
1198:Example of crop response to soil salinity
1029:
810:
758:
744:
466:{\displaystyle f(\lambda x)=\lambda f(x)}
109:Learn how and when to remove this message
1275:Optimal piecewise linear income taxation
1193:
1190:Crop response to depth of the watertable
1185:
637:
574:
191:
1389:
1302:
1620:
1489:
1446:
196:A continuous piecewise linear function
1597:A calculator for piecewise regression
1411:
1330:
676:). In each case, the function may be
570:
176:".) If the domain of the function is
1326:
1324:
47:adding citations to reliable sources
18:
1609:A calculator for partial regression
721:piecewise linear functions and the
630:learns piecewise linear functions.
13:
1562:Beiträge zur Algebra und Geometrie
1267:
1221:to find the two segments with the
1127:
1019:
1011:
927:
890:
884:
800:
790:
782:
712:
633:
587:
14:
1644:
1321:
1307:. Cengage Learning. p. 143.
1279:Journal of Public Economic Theory
606:If partitions are not known, the
123:Piecewise linear (disambiguation)
1356:Hamann, B.; Chen, J. L. (1994).
23:
1602:
1590:
1365:Computer Aided Geometric Design
1181:
34:needs additional citations for
1553:
1509:
1483:
1440:
1405:
1349:
1296:
1153:
1138:
1121:
1109:
1100:
1089:
1083:
1074:
1045:
1024:
953:
938:
921:
909:
900:
873:
867:
858:
829:
826:
805:
795:
754:
489:
483:
460:
454:
442:
433:
219:
213:
1:
1414:Journal of Applied Physiology
1289:
155:
16:Type of mathematical function
1377:10.1016/0167-8396(94)90004-3
1303:Stanley, William D. (2004).
473:and therefore in particular
7:
1426:10.1152/jappl.1989.67.1.390
1240:
187:
58:"Piecewise linear function"
10:
1649:
666:piecewise linear manifolds
591:
144:of a real variable, whose
120:
1538:10.1007/s10994-005-0466-3
1490:Muggeo, V. M. R. (2008).
656:, or more generally any
423:satisfies by definition
200:The function defined by
148:is composed of straight-
608:residual sum of squares
553:Heaviside step function
1518:"Logistic Model Trees"
1449:Statistics in Medicine
1390:Golovchenko, Nikolai.
1199:
1191:
1172:
1052:
995:
972:
836:
766:
684:, but to more general
682:linear transformations
643:
624:decision tree learning
580:
502:
501:{\displaystyle f(0)=0}
467:
386:
197:
1337:mathworld.wolfram.com
1197:
1189:
1173:
1053:
996:
973:
837:
767:
641:
578:
503:
468:
387:
195:
1333:"Piecewise Function"
1252:Spline interpolation
1247:Linear interpolation
1068:
1008:
985:
852:
779:
733:
670:simplicial complexes
594:Segmented regression
477:
427:
207:
142:real-valued function
43:improve this article
1331:Weisstein, Eric W.
1208:regression analysis
565:Triangular function
1633:Types of functions
1200:
1192:
1168:
1131:
1048:
991:
968:
931:
894:
832:
762:
644:
581:
571:Fitting to a curve
498:
463:
382:
377:
198:
138:segmented function
1455:(19): 3055–3071.
1257:Tropical geometry
1156:
1141:
1112:
1095:
1086:
994:{\displaystyle f}
956:
941:
912:
895:
879:
870:
600:linear regression
531:Sawtooth function
364:
326:
282:
250:
119:
118:
111:
93:
1640:
1612:
1606:
1600:
1594:
1588:
1587:
1577:
1557:
1551:
1550:
1540:
1531:(1–2): 161–205.
1525:Machine Learning
1522:
1513:
1507:
1506:
1496:
1487:
1481:
1480:
1461:10.1002/sim.1545
1444:
1438:
1437:
1409:
1403:
1402:
1400:
1398:
1387:
1381:
1380:
1362:
1353:
1347:
1346:
1344:
1343:
1328:
1319:
1318:
1300:
1177:
1175:
1174:
1169:
1158:
1157:
1149:
1143:
1142:
1134:
1130:
1114:
1113:
1105:
1088:
1087:
1079:
1057:
1055:
1054:
1049:
1044:
1043:
1032:
1023:
1022:
1000:
998:
997:
992:
977:
975:
974:
969:
958:
957:
949:
943:
942:
934:
930:
914:
913:
905:
893:
872:
871:
863:
841:
839:
838:
833:
825:
824:
813:
804:
803:
794:
793:
771:
769:
768:
763:
761:
753:
752:
747:
664:, as well as on
614:
507:
505:
504:
499:
472:
470:
469:
464:
391:
389:
388:
383:
381:
380:
365:
362:
327:
324:
283:
280:
251:
248:
134:piecewise linear
114:
107:
103:
100:
94:
92:
51:
27:
19:
1648:
1647:
1643:
1642:
1641:
1639:
1638:
1637:
1618:
1617:
1616:
1615:
1607:
1603:
1595:
1591:
1558:
1554:
1520:
1514:
1510:
1494:
1488:
1484:
1445:
1441:
1410:
1406:
1396:
1394:
1388:
1384:
1360:
1354:
1350:
1341:
1339:
1329:
1322:
1315:
1301:
1297:
1292:
1270:
1268:Further reading
1262:Polygonal chain
1243:
1184:
1148:
1147:
1133:
1132:
1104:
1103:
1099:
1078:
1077:
1069:
1066:
1065:
1033:
1028:
1027:
1018:
1017:
1009:
1006:
1005:
986:
983:
982:
948:
947:
933:
932:
904:
903:
899:
883:
862:
861:
853:
850:
849:
814:
809:
808:
799:
798:
789:
788:
780:
777:
776:
757:
748:
743:
742:
734:
731:
730:
715:
713:Specializations
654:Euclidean space
636:
634:Generalizations
612:
596:
590:
588:Fitting to data
573:
547:Boxcar function
478:
475:
474:
428:
425:
424:
421:linear function
414:polygonal chain
376:
375:
361:
359:
344:
343:
323:
321:
303:
302:
279:
277:
265:
264:
247:
245:
226:
225:
208:
205:
204:
190:
170:affine function
158:
126:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
1646:
1636:
1635:
1630:
1614:
1613:
1601:
1589:
1568:(1): 297–302.
1552:
1508:
1482:
1439:
1420:(1): 390–396.
1404:
1382:
1348:
1320:
1314:978-1401864811
1313:
1294:
1293:
1291:
1288:
1287:
1286:
1269:
1266:
1265:
1264:
1259:
1254:
1249:
1242:
1239:
1183:
1180:
1179:
1178:
1167:
1164:
1161:
1155:
1152:
1146:
1140:
1137:
1129:
1126:
1123:
1120:
1117:
1111:
1108:
1102:
1098:
1094:
1091:
1085:
1082:
1076:
1073:
1059:
1058:
1047:
1042:
1039:
1036:
1031:
1026:
1021:
1016:
1013:
990:
979:
978:
967:
964:
961:
955:
952:
946:
940:
937:
929:
926:
923:
920:
917:
911:
908:
902:
898:
892:
889:
886:
882:
878:
875:
869:
866:
860:
857:
843:
842:
831:
828:
823:
820:
817:
812:
807:
802:
797:
792:
787:
784:
772:, there is a
760:
756:
751:
746:
741:
738:
714:
711:
674:simplicial map
635:
632:
592:Main article:
589:
586:
572:
569:
568:
567:
562:
561:
560:
555:
550:
538:
536:Floor function
533:
528:
526:Absolute value
497:
494:
491:
488:
485:
482:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
393:
392:
379:
374:
371:
368:
360:
358:
355:
352:
349:
346:
345:
342:
339:
336:
333:
330:
322:
320:
317:
314:
311:
308:
305:
304:
301:
298:
295:
292:
289:
286:
278:
276:
273:
270:
267:
266:
263:
260:
257:
254:
246:
244:
241:
238:
235:
232:
231:
229:
224:
221:
218:
215:
212:
189:
186:
184:in the reals.
182:locally finite
157:
154:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
1645:
1634:
1631:
1629:
1628:Real analysis
1626:
1625:
1623:
1610:
1605:
1598:
1593:
1585:
1581:
1576:
1571:
1567:
1563:
1556:
1548:
1544:
1539:
1534:
1530:
1526:
1519:
1512:
1504:
1500:
1493:
1486:
1478:
1474:
1470:
1466:
1462:
1458:
1454:
1450:
1443:
1435:
1431:
1427:
1423:
1419:
1415:
1408:
1393:
1386:
1378:
1374:
1370:
1366:
1359:
1352:
1338:
1334:
1327:
1325:
1316:
1310:
1306:
1299:
1295:
1285:(4), 523–545.
1284:
1280:
1276:
1272:
1271:
1263:
1260:
1258:
1255:
1253:
1250:
1248:
1245:
1244:
1238:
1235:
1234:soil salinity
1231:
1226:
1224:
1220:
1219:least squares
1216:
1211:
1209:
1205:
1196:
1188:
1165:
1162:
1159:
1150:
1144:
1135:
1124:
1118:
1115:
1106:
1092:
1080:
1071:
1064:
1063:
1062:
1040:
1037:
1034:
1014:
1004:
1003:
1002:
988:
965:
962:
959:
950:
944:
935:
924:
918:
915:
906:
887:
876:
864:
855:
848:
847:
846:
821:
818:
815:
785:
775:
774:
773:
749:
739:
736:
728:
724:
720:
710:
708:
704:
700:
698:
694:
689:
687:
686:affine linear
683:
679:
675:
671:
667:
663:
659:
655:
652:
650:
640:
631:
629:
625:
622:A variant of
620:
618:
609:
604:
601:
595:
585:
577:
566:
563:
559:
558:Sign function
556:
554:
551:
548:
545:
544:
542:
541:Step function
539:
537:
534:
532:
529:
527:
524:
523:
522:
519:
517:
513:
512:
495:
492:
486:
480:
457:
451:
448:
445:
439:
436:
430:
422:
417:
415:
410:
406:
402:
401:line segments
398:
372:
369:
366:
356:
353:
350:
347:
340:
337:
334:
331:
328:
318:
315:
312:
309:
306:
299:
296:
293:
290:
287:
284:
274:
271:
268:
261:
258:
255:
252:
242:
239:
236:
233:
227:
222:
216:
210:
203:
202:
201:
194:
185:
183:
179:
175:
171:
167:
163:
153:
151:
150:line segments
147:
143:
139:
135:
131:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1604:
1592:
1575:math/0009026
1565:
1561:
1555:
1528:
1524:
1511:
1502:
1498:
1485:
1452:
1448:
1442:
1417:
1413:
1407:
1395:. Retrieved
1385:
1368:
1364:
1351:
1340:. Retrieved
1336:
1304:
1298:
1282:
1278:
1227:
1212:
1201:
1182:Applications
1060:
980:
844:
726:
716:
701:
690:
688:functions.)
662:affine space
658:vector space
651:-dimensional
648:
645:
621:
605:
597:
582:
520:
515:
514:rather than
509:
418:
408:
394:
199:
166:real numbers
159:
137:
133:
127:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1215:watertables
1204:agriculture
628:model trees
130:mathematics
1622:Categories
1371:(3): 289.
1342:2020-08-24
1290:References
1206:piecewise
1061:such that
845:such that
719:continuous
617:R language
156:Definition
99:March 2013
69:newspapers
1154:→
1145:⋅
1139:→
1128:Σ
1125:∈
1110:→
1084:→
1015:∈
1012:Σ
954:→
945:⋅
939:→
928:Σ
925:∈
910:→
891:Π
888:∈
885:Σ
868:→
786:∈
783:Π
755:→
613:segmented
449:λ
437:λ
370:≥
354:−
332:≤
307:−
285:−
259:−
256:≤
240:−
234:−
1505:: 20–25.
1477:36264047
1469:12973787
1241:See also
1230:tolerate
1223:best fit
697:polytope
615:for the
363:if
325:if
281:if
249:if
188:Examples
162:interval
1584:1913786
1547:6306536
1434:2759968
703:Splines
693:polygon
626:called
178:compact
83:scholar
1582:
1545:
1499:R News
1475:
1467:
1432:
1311:
723:convex
516:linear
511:affine
419:(*) A
407:. The
174:affine
85:
78:
71:
64:
56:
1570:arXiv
1543:S2CID
1521:(PDF)
1495:(PDF)
1473:S2CID
1397:6 Dec
1361:(PDF)
707:PDIFF
672:(see
146:graph
140:is a
90:JSTOR
76:books
1465:PMID
1430:PMID
1399:2012
1309:ISBN
678:real
668:and
405:rays
403:and
397:line
338:<
297:<
291:<
132:, a
62:news
1533:doi
1457:doi
1422:doi
1373:doi
1202:In
1097:max
981:If
897:max
881:min
695:or
660:or
619:.
357:4.5
348:0.5
164:of
136:or
128:In
45:by
1624::
1580:MR
1578:.
1566:43
1564:.
1541:.
1529:59
1527:.
1523:.
1501:.
1497:.
1471:.
1463:.
1453:22
1451:.
1428:.
1418:67
1416:.
1369:11
1367:.
1363:.
1335:.
1323:^
1283:16
1281:,
1277:.
1232:a
1225:.
709:.
518:.
416:.
152:.
1611:.
1599:.
1586:.
1572::
1549:.
1535::
1503:8
1479:.
1459::
1436:.
1424::
1401:.
1379:.
1375::
1345:.
1317:.
1166:.
1163:b
1160:+
1151:x
1136:a
1122:)
1119:b
1116:,
1107:a
1101:(
1093:=
1090:)
1081:x
1075:(
1072:f
1046:)
1041:1
1038:+
1035:n
1030:R
1025:(
1020:P
989:f
966:.
963:b
960:+
951:x
936:a
922:)
919:b
916:,
907:a
901:(
877:=
874:)
865:x
859:(
856:f
830:)
827:)
822:1
819:+
816:n
811:R
806:(
801:P
796:(
791:P
759:R
750:n
745:R
740::
737:f
727:n
649:n
549:,
496:0
493:=
490:)
487:0
484:(
481:f
461:)
458:x
455:(
452:f
446:=
443:)
440:x
434:(
431:f
409:x
373:3
367:x
351:x
341:3
335:x
329:0
319:3
316:+
313:x
310:2
300:0
294:x
288:3
275:3
272:+
269:x
262:3
253:x
243:3
237:x
228:{
223:=
220:)
217:x
214:(
211:f
125:.
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
