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These algorithms start with the assumption that there are no steps and introduce possible candidate steps one at a time, testing each candidate to find the one that minimizes some criteria (such as the least-squares fit of the estimated, underlying piecewise constant signal). An example is the
217:
By considering a small "window" of the signal, these algorithms look for evidence of a step occurring within the window. The window "slides" across the time series, one time step at a time. The evidence for a step is tested by statistical procedures, for example, by use of the two-sample
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All the algorithms mentioned above have certain advantages and disadvantages in particular circumstances, yet, a surprisingly large number of these step detection algorithms are special cases of a more general algorithm. This algorithm involves the minimization of a global functional:
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Many algorithms explicitly fit 0-degree splines to the noisy signal in order to detect steps (including stepwise jump placement methods), but there are other popular algorithms that can also be seen to be spline fitting methods after some transformation, for example
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Bottom-up algorithms take the "opposite" approach to top-down methods, first assuming that there is a step in between every sample in the digital signal, and then successively merging steps based on some criteria tested for every candidate merge.
2041:
In: Large-Scale
Disturbances (Regime Shifts) and Recovery in Aquatic Ecosystems: Challenges for Management Toward Sustainability, V. Velikova and N. Chipev (Eds.), UNESCO-ROSTE/BAS Workshop on Regime Shifts, 14–16 June 2005, Varna, Bulgaria,
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298:. For this reason, step detection can be profitably viewed as the problem of recovering a piecewise constant signal corrupted by noise. There are two complementary models for piecewise constant signals: as
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Mumford, D., & Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on pure and applied mathematics, 42(5), 577-685.
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Sowa, Y.; Rowe, A. D.; Leake, M. C.; Yakushi, T.; Homma, M.; Ishijima, A.; Berry, R. M. (2005). "Direct observation of steps in rotation of the bacterial flagellar motor".
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93:, and this makes the problem challenging because the step may be hidden by the noise. Therefore, statistical and/or signal processing algorithms are often required.
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algorithms are applied to the data potentially long after it has been received. Most offline algorithms for step detection in digital data can be categorised as
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Kerssemakers, J.W.J.; Munteanu, E.L.; Laan, L.; Noetzel, T.L.; Janson, M.E.; Dogterom, M. (2006). "Assembly dynamics of microtubules at molecular resolution".
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238:
Global algorithms consider the entire signal in one go, and attempt to find the steps in the signal by some kind of optimization procedure. Algorithms include
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with a few unique levels. Many algorithms for step detection are therefore best understood as either 0-degree spline fitting, or level set recovery, methods.
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are appropriate. These techniques are best understood as methods for finding a level set description of the underlying piecewise constant signal.
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Because the aim of step detection is to find a series of instantaneous jumps in the mean of a signal, the wanted, underlying, mean signal is
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are also often used (a popular approach in the biophysics community). When there are only a few unique values of the mean, then
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A classical variational method for step detection is the Potts model. It is given by the non-convex optimization problem
1253:{\displaystyle \Lambda ={\frac {1}{2}}\left|x_{i}-m_{j}\right|^{2}I(i-j=0)+\gamma \left|m_{i}-m_{j}\right|^{0}I(i-j=1)}
97:
1266:
1887:
Gill, D. (1970). "Application of a statistical zonation method to reservoir evaluation and digitized log analysis".
2462:
2424:-Potts functional for robust jump-sparse reconstruction." SIAM Journal on Numerical Analysis, 53(1):644-673 (2015).
751:{\displaystyle \Lambda ={\frac {1}{2}}\left|x_{i}-m_{j}\right|^{2}I(i-j=0)+\gamma \left|m_{i}-m_{j}\right|I(i-j=1)}
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2304:
Mrazek, P.; Weickert, J.; Bruhn, A. (2006). "On robust estimation and smoothing with spatial and tonal kernels".
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is applied to the signal. Filters such as these attempt to remove the noise whilst preserving the abrupt steps.
1823:
2256:
Chan, D.; T. Chan (2003). "Edge-preserving and scale-dependent properties of total variation regularization".
2199:"Generalized methods and solvers for noise removal from piecewise constant signals: Part I. Background theory"
1914:
Snijders, A.M.; et al. (2001). "Assembly of microarrays for genome-wide measurement of DNA copy number".
1061:
1343:. Still others are non-convex but a range of algorithms for minimizing these functionals have been devised.
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907: > 0 is a parameter that determines the support of the mean shift kernel. Another example is:
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or change point detection. Often, the step is small and the time series is corrupted by some kind of
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specifying a group of algorithms that attempt to greedily fit 0-degree splines to the signal. Here,
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algorithm, when using an adaptive step size Euler integrator initialized with the input signal
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algorithm, first studied in geophysical problems, that has found recent uses in modern biophysics.
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The step detection problem occurs in multiple scientific and engineering contexts, for example in
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1044:{\displaystyle \Lambda ={\frac {1-\exp(-\beta |m_{i}-m_{j}|^{2}/2)}{\beta }}\cdot I(|i-j|\leq W)}
512:{\displaystyle H=\sum _{i=1}^{N}\sum _{j=1}^{N}\Lambda (x_{i}-m_{j},m_{i}-m_{j},x_{i}-x_{j},i-j)}
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PWCTools: Flexible Matlab and Python software for step detection by piecewise constant denoising
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approaches to step detection generally do not use classical smoothing techniques such as the
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or signal. It is usually considered as a special case of the statistical method known as
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25:
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Friedrich; et al. (2008). "Complexity penalized M-estimation: fast computation".
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Mallat, S.; Hwang, W.L. (1992). "Singularity detection and processing with wavelets".
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E.S. Page (1955). "A test for a change in a parameter occurring at an unknown point".
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is piecewise constant the steps are given by the non-zero locations of the gradient
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Pottslab: Matlab toolbox for piecewise constant estimation based on the Potts model
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When there are only a few unique values of the mean, clustering techniques such as
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there are fast algorithms which give an exact solution of the Potts problem in
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Rodionov, S.N., 2005a: A brief overview of the regime shift detection methods.
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When the step detection must be performed as and when the data arrives, then
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as recorded in time-position traces). For 2D signals, the related problem of
2147:"Analysis of Single-Molecule FRET Trajectories Using Hidden Markov Modeling"
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1986:
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Winkler, G.; Liebscher, V. (2002). "Smoothers for discontinuous signals".
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43:, and (d) red pixel intensity from a single scan line of a digital image.
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1665:. The parameter γ > 0 controls the tradeoff between regularity and
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286:. Instead, most algorithms are explicitly nonlinear or time-varying.
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Linear versus nonlinear signal processing methods for step detection
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Examples of signals that may contain steps corrupted by noise. (a)
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Because steps and (independent) noise have theoretically infinite
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is the signal output from the algorithm. The goal is to minimize
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is the spatial kernel support. Yet another special case is:
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determines the particular algorithm. For example, choosing:
1546:{\displaystyle \|\nabla u\|_{0}=\#\{i:u_{i}\neq u_{i+1}\}}
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Generalized step detection by piecewise constant denoising
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being the most directly related method), in exploration
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are usually used, and it becomes a special case of
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Detection of Abrupt
Changes: Theory and Application
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1339:: they can be minimized using methods from
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2010:Image analysis and mathematical morphology
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777:algorithm with regularization parameter
160:. Such algorithms include the classical
2110:IEEE Transactions on Information Theory
2023:Basseville, M.; I.V. Nikiforov (1993).
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569:with respect to the output signal
250:. Where the steps can be modelled as a
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1328:{\displaystyle \scriptstyle \Lambda }
1307:Many of the functionals in equation (
588:{\displaystyle \scriptstyle \Lambda }
36:, (c) rotation speed against time of
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1347:Step detection using the Potts model
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310:Step detection as level set recovery
2473:Feature detection (computer vision)
2330:Journal of Nonparametric Statistics
2012:. London; New York: Academic Press.
1304: = 0, and one otherwise.
1086:is the tonal kernel parameter, and
164:method applied to changes in mean.
108:(where the problem is to segment a
2204:Proceedings of the Royal Society A
2197:Little, M.A.; Jones, N.S. (2011).
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2468:Statistical signal processing
2308:. Berlin, Germany: Springer.
1719:{\displaystyle \nabla u^{*}}
2163:10.1529/biophysj.106.082487
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573:. The form of the function
120:(the problem of separating
98:statistical process control
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2278:10.1088/0266-5611/19/6/059
1661:measures fidelity to data
1857:10.1093/biomet/42.3-4.523
775:total variation denoising
333:total variation denoising
244:total variation denoising
2377:10.1198/106186008x285591
1824:Time-series segmentation
1807:{\displaystyle O(N^{2})}
302:with a few knots, or as
246:which uses methods from
790:{\displaystyle \gamma }
198:stepwise jump placement
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2342:10.1080/10485250211388
2225:10.1098/rspa.2010.0671
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1669:. Since the minimizer
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256:Hidden Markov Models
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2211:(2135): 3088–3114.
2151:Biophysical Journal
2079:10.1038/nature04928
2071:2006Natur.442..709K
1979:10.1038/nature04003
1971:2005Natur.437..916S
1771:{\displaystyle p=1}
1745:{\displaystyle p=2}
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1341:convex optimization
248:convex optimization
222:. Alternatively, a
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114:stratigraphic zones
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124:data into similar
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1965:(7060): 916–919.
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134:molecular machine
128:regimes), and in
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1498:
1493:
1489:
1485:
1482:
1479:
1468:
1467:
1454:
1449:
1445:
1441:
1438:
1435:
1432:
1429:
1424:
1420:
1416:
1413:
1410:
1407:
1400:
1395:
1390:
1387:
1383:
1379:
1376:
1373:
1368:
1364:
1348:
1345:
1323:
1286:
1281:
1278:
1275:
1261:
1260:
1249:
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1237:
1234:
1231:
1228:
1223:
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1208:
1204:
1199:
1195:
1190:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1153:
1148:
1142:
1138:
1134:
1129:
1125:
1120:
1113:
1110:
1105:
1102:
1074:
1071:
1068:
1052:
1051:
1040:
1037:
1034:
1030:
1026:
1023:
1020:
1016:
1012:
1009:
1006:
1001:
997:
994:
990:
984:
979:
972:
968:
964:
959:
955:
950:
946:
943:
940:
937:
934:
931:
928:
922:
919:
893:
892:
880:
876:
873:
868:
863:
857:
853:
849:
844:
840:
835:
828:
825:
819:
815:
812:
809:
786:
759:
758:
747:
744:
741:
738:
735:
732:
729:
726:
722:
716:
712:
708:
703:
699:
694:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
658:
653:
647:
643:
639:
634:
630:
625:
618:
615:
610:
607:
583:
560:
539:
531:
530:
521:
519:
508:
505:
502:
499:
496:
491:
487:
483:
478:
474:
470:
465:
461:
457:
452:
448:
444:
439:
435:
431:
426:
422:
418:
415:
410:
405:
402:
399:
395:
389:
384:
381:
378:
374:
370:
367:
364:
361:
358:
340:
337:
327:
324:
311:
308:
291:
288:
267:
264:
235:
232:
214:
213:Sliding window
211:
205:
202:
192:
189:
181:sliding window
149:
146:
138:edge detection
78:edge detection
73:jump detection
65:step filtering
61:step smoothing
57:step detection
38:R. Sphaeroides
13:
10:
9:
6:
4:
3:
2:
2485:
2474:
2471:
2469:
2466:
2464:
2461:
2459:
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2453:
2444:
2441:
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2394:
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2382:
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2374:
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2366:
2359:
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2279:
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2263:
2259:
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2249:
2244:
2240:
2235:
2230:
2226:
2222:
2218:
2214:
2210:
2206:
2205:
2200:
2193:
2191:
2187:
2182:
2178:
2173:
2168:
2164:
2160:
2156:
2152:
2148:
2141:
2138:
2133:
2129:
2124:
2119:
2115:
2111:
2104:
2101:
2096:
2092:
2088:
2084:
2080:
2076:
2072:
2068:
2064:
2060:
2053:
2051:
2049:
2045:
2040:
2034:
2031:
2026:
2019:
2016:
2011:
2004:
2001:
1996:
1992:
1988:
1984:
1980:
1976:
1972:
1968:
1964:
1960:
1953:
1950:
1945:
1941:
1937:
1933:
1929:
1928:10.1038/ng754
1925:
1921:
1917:
1910:
1907:
1902:
1898:
1894:
1890:
1883:
1881:
1879:
1877:
1873:
1867:
1862:
1858:
1854:
1850:
1846:
1839:
1836:
1829:
1825:
1822:
1821:
1817:
1815:
1796:
1792:
1785:
1765:
1762:
1759:
1739:
1736:
1733:
1711:
1707:
1681:
1677:
1668:
1667:data fidelity
1664:
1646:
1634:
1630:
1626:
1621:
1617:
1606:
1601:
1598:
1595:
1591:
1587:
1582:
1577:
1569:
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1563:
1535:
1532:
1529:
1525:
1521:
1516:
1512:
1508:
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1491:
1483:
1452:
1447:
1439:
1436:
1433:
1427:
1422:
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1398:
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1377:
1374:
1371:
1366:
1362:
1354:
1353:
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1342:
1338:
1312:
1311:
1305:
1303:
1284:
1279:
1276:
1273:
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1241:
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1235:
1232:
1226:
1221:
1216:
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1206:
1202:
1197:
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1183:
1180:
1174:
1171:
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1165:
1162:
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1136:
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1118:
1111:
1108:
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1093:
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1032:
1024:
1021:
1018:
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926:
920:
910:
909:
908:
906:
902:
898:
895:leads to the
878:
874:
871:
866:
861:
855:
851:
847:
842:
838:
833:
826:
823:
817:
810:
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797:. Similarly:
784:
776:
772:
768:
764:
742:
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736:
733:
730:
724:
720:
714:
710:
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701:
697:
692:
688:
685:
679:
676:
673:
670:
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661:
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637:
632:
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623:
616:
613:
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551:
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542:
538:
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522:
520:
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500:
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476:
472:
468:
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455:
450:
446:
442:
437:
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429:
424:
420:
408:
403:
400:
397:
393:
387:
382:
379:
376:
372:
368:
362:
356:
349:
348:
345:
338:
336:
334:
325:
323:
321:
317:
309:
307:
305:
301:
297:
289:
287:
285:
281:
277:
276:Fourier basis
273:
265:
263:
261:
257:
253:
249:
245:
242:methods, and
241:
233:
231:
229:
228:median filter
225:
221:
212:
210:
203:
201:
199:
190:
188:
186:
182:
178:
174:
170:
167:By contrast,
165:
163:
159:
155:
147:
145:
143:
139:
135:
131:
127:
123:
119:
115:
111:
107:
103:
102:control chart
99:
94:
92:
88:
84:
80:
79:
74:
70:
66:
62:
58:
54:
50:
42:
39:
35:
31:
27:
23:
18:
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2368:
2364:
2358:
2333:
2329:
2323:
2314:
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2261:
2257:
2251:
2208:
2202:
2154:
2150:
2140:
2113:
2109:
2103:
2062:
2058:
2033:
2024:
2018:
2009:
2003:
1962:
1958:
1952:
1919:
1915:
1909:
1892:
1888:
1848:
1844:
1838:
1662:
1469:
1350:
1308:
1306:
1301:
1262:
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1053:
904:
900:
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760:
570:
566:
561:
557:
553:
549:
545:
540:
536:
534:
523:
342:
329:
313:
293:
269:
252:Markov chain
237:
226:such as the
216:
207:
197:
194:
184:
180:
176:
172:
168:
166:
151:
95:
76:
72:
68:
64:
60:
56:
46:
2039:link to PDF
1895:: 719–729.
126:copy-number
83:time series
2452:Categories
1845:Biometrika
1830:References
897:mean shift
320:mean-shift
304:level sets
148:Algorithms
130:biophysics
122:microarray
106:geophysics
49:statistics
30:cosmic ray
28:data, (b)
26:microarray
2385:117951377
2350:119562495
2118:CiteSeerX
1995:262329024
1712:∗
1704:∇
1682:∗
1627:−
1592:∑
1574:‖
1567:−
1561:‖
1522:≠
1500:#
1488:‖
1481:∇
1478:‖
1470:The term
1444:‖
1437:−
1431:‖
1419:‖
1412:∇
1409:‖
1406:γ
1389:∈
1378:
1367:∗
1322:Λ
1236:−
1203:−
1184:γ
1166:−
1133:−
1101:Λ
1067:β
1033:≤
1022:−
1005:⋅
1000:β
963:−
945:β
942:−
936:
930:−
918:Λ
848:−
808:Λ
785:γ
734:−
707:−
689:γ
671:−
638:−
606:Λ
582:Λ
501:−
482:−
456:−
430:−
414:Λ
394:∑
373:∑
272:bandwidth
204:Bottom-up
187:methods.
177:bottom-up
2286:30704800
2243:22003312
2181:16766620
2087:16799566
1987:16208378
1944:19460203
1936:11687795
1818:See also
1058:, where
191:Top-down
173:top-down
118:genetics
110:well-log
2266:Bibcode
2234:3191861
2213:Bibcode
2172:1544307
2095:4359681
2067:Bibcode
1967:Bibcode
903:. Here
254:, then
240:wavelet
169:offline
2383:
2348:
2284:
2241:
2231:
2179:
2169:
2120:
2093:
2085:
2059:Nature
2042:17-24.
1993:
1985:
1959:Nature
1942:
1934:
1726:. For
1337:convex
761:where
556:, and
535:Here,
234:Global
185:global
116:), in
2381:S2CID
2346:S2CID
2282:S2CID
2091:S2CID
1991:S2CID
1940:S2CID
183:, or
162:CUSUM
100:(the
91:noise
24:from
2239:PMID
2177:PMID
2083:PMID
1983:PMID
1932:PMID
1752:and
1335:are
1070:>
544:for
51:and
2373:doi
2338:doi
2274:doi
2229:PMC
2221:doi
2209:467
2167:PMC
2159:doi
2128:doi
2075:doi
2063:442
1975:doi
1963:437
1924:doi
1897:doi
1861:hdl
1853:doi
1814:.
1382:min
1375:arg
933:exp
814:min
318:or
75:or
47:In
2454::
2379:.
2369:17
2367:.
2344:.
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2294:^
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2061:.
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1973:.
1961:.
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1930:.
1920:29
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2130::
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2077::
2069::
1997:.
1977::
1969::
1946:.
1926::
1903:.
1899::
1869:.
1863::
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1802:)
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
