116:
996:
17:
172:
is multiplied element-wise by the signal to provide a windowed trial from which one estimates the power at each component frequency. As each taper is pairwise orthogonal to all other tapers, the window functions are uncorrelated with one another. The final spectrum is obtained by averaging over all
164:
is generally biased (with the exception of white noise) and the bias depends upon the length of each realization, not the number of realizations recorded. Applying a single taper reduces bias but at the cost of increased estimator variance due to attenuation of activity at the start and end of each
119:
The importance of averaging in (cross-)spectral density estimation. (a) Synthetically generated noisy signal with two coherent frequencies at 0.03 and 0.6 Hz. (b) Multitaper (MT) spectral density estimates. (c) Coherence squared estimates using
Slepian multitaper analysis (thick ine, unshaded) and
176:
This method is especially useful when a small number of trials is available as it reduces the estimator variance beyond what is possible with single taper methods. Moreover, even when many trials are available the multitaper approach is useful as it permits more rigorous control of the trade-off
98:
to extract spectral information from a signal, we assume that each
Fourier coefficient is a reliable representation of the amplitude and relative phase of the corresponding component frequency. This assumption, however, is not generally valid for empirical data. For instance, a single trial
851:
103:
i.e., it is bad practice to estimate qualities of a population using individuals or very small samples. Likewise, a single sample of a process does not necessarily provide a reliable estimate of its spectral properties. Moreover, the naive
1245:
Not limited to time series, the multitaper method is easily extensible to multiple
Cartesian dimenions using custom Slepian functions, and can be reformulated for spectral estimation on the sphere using Slepian functions constructed from
594:
120:
Welch overlapping segment analysis (WOSA) (thin line, shaded area). (d) Estimate of the phase of the cross-spectral density estimate using MT (solid) and WOSA (dashed). At 0.03 Hz the signals are in phase, while at 0.6 Hz they are
159:
to each trial. However, this method is unreliable with small data sets and undesirable when one does not wish to attenuate signal components that vary across trials. Furthermore, even when many trials are available the untapered
315:
990:
708:
1353:
Simons, F. J.; Korenaga, J.; Zuber, M. T. (2000). "Isostatic response of the
Australian lithosphere: Estimation of effective elastic thickness and anisotropy using multitaper analysis".
1045:
1102:
650:
445:
688:
361:
1217:
1164:
388:
149:
450:
The multitaper spectral estimator utilizes several different data tapers which are orthogonal to each other. The multitaper cross-spectral estimator between channel
173:
the tapered spectra thus recovering some of the information that is lost due to partial attenuation of the signal that results from applying individual tapers.
472:
180:
Thomson chose the
Slepian functions or discrete prolate spheroidal sequences as tapers since these vectors are mutually orthogonal and possess desirable
205:
1447:
E. Sejdić, M. Luccini, S. Primak, K. Baddour, T. Willink, “Channel estimation using modulated discrete prolate spheroidal sequences based frames,” in
99:
represents only one noisy realization of the underlying process of interest. A comparable situation arises in statistics when estimating measures of
1267:
862:
1406:
1338:
1719:
1709:
1111:
orthogonal data tapers such that each one provides a good protection against leakage. These are given by the
Slepian sequences, after
1258:
among others. An extensive treatment about the application of this method to analyze multi-trial, multi-channel data generated in
168:
The multitaper method partially obviates these problems by obtaining multiple independent estimates from the same sample. Each
1389:
Simons, F. J.; Plattner, A. (2015). "Scalar and Vector
Slepian Functions, Spherical Signal Estimation and Spectral Analysis".
846:{\displaystyle {\hat {S}}_{k}^{lm}(f)={\frac {1}{N\Delta t}}{\lbrack J_{k}^{l}(f)\rbrack }^{*}{\lbrack J_{k}^{m}(f)\rbrack },}
1171:
181:
999:
The three leading
Slepian sequences for T=1000 and 2WT=6. Note that each higher order sequence has an extra zero crossing.
1434:
Slepian, D. (1978) "Prolate spheroidal wave functions, Fourier analysis, and uncertainty – V: The discrete case."
24:(black) and multitaper estimate (red) of a single trial local field potential measurement. This estimate used 9 tapers.
37:
1231:
channel. In recent years, a dictionary based on modulated DPSS was proposed as an overcomplete alternative to DPSS.
1724:
1714:
75:
1685:
1011:
1054:
602:
1263:
105:
1534:
Wieczorek, M. A.; Simons, F. J. (2007). "Minimum-variance multitaper spectral estimation on the sphere".
397:
155:
These problems are often overcome by averaging over many realizations of the same event after applying a
1362:
1584:
1508:
1115:(also known in literature as discrete prolate spheroidal sequences or DPSS for short) with parameter
655:
335:
1177:
1489:
Simons, F. J.; Dahlen, F. A.; Wieczorek, M. A. (2006). "Spatiospectral
Concentration on a Sphere".
1247:
1574:
1498:
1271:
1137:
197:
61:
1683:
1449:
Proc. of IEEE International
Conference on Acoustics, Speech, and Signal Processing (ICASSP 2008)
1617:
C++/Octave libraries for the multitaper method, including adaptive weighting (hosted on GitHub)
1423:
Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques
1402:
1334:
391:
370:
95:
45:
29:
123:
1592:
1543:
1516:
1470:
1394:
185:
91:
41:
1461:
Simons, F. J.; Wang, D. V. (2011). "Spatiospectral concentration in the Cartesian plane".
1375:
1235:
325:
169:
156:
1620:
589:{\displaystyle {\hat {S}}^{lm}(f)={\frac {1}{K}}\sum _{k=0}^{K-1}{\hat {S}}_{k}^{lm}(f).}
1588:
1512:
458:
is the average of K direct cross-spectral estimators between the same pair of channels (
1132:
109:
68:
49:
1703:
1597:
1562:
1112:
1654:
1642:
1259:
1630:
1398:
310:{\displaystyle \mathbf {X} (t)={\lbrack X(1,t),X(2,t),\dots ,X(p,t)\rbrack }^{T}}
1666:
1325:
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 13.4.3.
1287:
161:
21:
985:{\displaystyle J_{k}^{l}(f)=\sum _{t=1}^{N}h_{t,k}X(l,t)e^{-i2\pi ft\Delta t}.}
115:
1691:
1622:
Documentation on the multitaper method from the SSA-MTM Toolkit implementation
1547:
1520:
1474:
1251:
188:
is often used to compensate for increased energy loss at higher order tapers.
16:
1255:
177:
between bias and variance than what is possible in the single taper case.
1625:
90:
The multitaper method overcomes some of the limitations of non-parametric
995:
1275:
363:
can represent simultaneous measurement of electrical activity of those
64:
1693:
1668:
1656:
1644:
1632:
1615:
1503:
1678:
184:
properties (see the section on Slepian sequences). In practice, a
1579:
994:
114:
15:
1451:, Las Vegas, Nevada, USA, March 31-April 04, 2008, pp. 2849-2852.
100:
1227:, we get the multitaper estimator for the auto-spectrum of the
1563:"Spectral estimation on a sphere in geophysics and cosmology"
1627:
Fortran 90 library with additional multivariate applications
367:
channels. Let the sampling interval between observations be
1333:(3rd ed.), New York: Cambridge University Press,
108:
obtained from the signal's raw Fourier transform is a
1180:
1140:
1057:
1014:
865:
711:
658:
605:
475:
400:
373:
338:
208:
126:
1331:Numerical Recipes: The Art of Scientific Computing
1211:
1158:
1096:
1039:
984:
845:
682:
644:
588:
439:
382:
355:
309:
143:
332:refers to the total number of channels and hence
694:direct cross spectral estimator between channel
1425:. Cambridge: Cambridge University Press, 1993.
8:
1536:Journal of Fourier Analysis and Applications
1463:GEM: International Journal on Geomathematics
1034:
1015:
836:
809:
798:
771:
297:
228:
1696:script to generate Slepian sequences (dpss)
1270:. This technique is currently used in the
1596:
1578:
1502:
1308:Spectrum estimation and harmonic analysis
1200:
1179:
1170:defines the resolution bandwidth for the
1139:
1127: − 1. The maximum order
1076:
1071:
1060:
1059:
1056:
1022:
1013:
952:
918:
908:
897:
875:
870:
864:
821:
816:
808:
802:
783:
778:
770:
751:
730:
725:
714:
713:
710:
657:
624:
619:
608:
607:
604:
565:
560:
549:
548:
535:
524:
510:
489:
478:
477:
474:
417:
405:
399:
372:
339:
337:
301:
227:
209:
207:
133:
125:
1327:Multitaper methods and Slepian functions
112:estimate of the true spectral content.
1299:
1040:{\displaystyle \lbrace h_{t,k}\rbrace }
1371:
1360:
1236:Window function:DPSS or Slepian window
1097:{\displaystyle {\hat {S}}_{k}^{lm}(f)}
645:{\displaystyle {\hat {S}}_{k}^{lm}(f)}
324:denotes the matrix transposition. In
1561:Dahlen, F. A.; Simons, F. J. (2008).
7:
1421:Percival, D. B., and A. T. Walden.
440:{\displaystyle f_{N}=1/(2\Delta t)}
196:Consider a p-dimensional zero mean
1150:
971:
760:
428:
374:
14:
1567:Geophysical Journal International
165:recorded segment of the signal.
1598:10.1111/j.1365-246X.2008.03854.x
1051:direct cross-spectral estimator
340:
210:
683:{\displaystyle 0\leq k\leq K-1}
356:{\displaystyle \mathbf {X} (t)}
1212:{\displaystyle W\in (0,f_{N})}
1206:
1187:
1172:spectral concentration problem
1131:is chosen to be less than the
1091:
1085:
1065:
945:
933:
887:
881:
833:
827:
795:
789:
745:
739:
719:
639:
633:
613:
580:
574:
554:
504:
498:
483:
434:
422:
350:
344:
294:
282:
267:
255:
246:
234:
220:
214:
1:
1436:Bell System Technical Journal
198:stationary stochastic process
1399:10.1007/978-3-642-54551-1_30
74:, given a finite contiguous
1266:and elsewhere can be found
1159:{\displaystyle 2NW\Delta t}
466:) and hence takes the form
38:spectral density estimation
1741:
1391:Handbook of Geomathematics
1104:and is chosen as follows:
1047:is the data taper for the
1710:Frequency-domain analysis
1548:10.1007/s00041-006-6904-1
1521:10.1137/S0036144504445765
1475:10.1007/s13137-011-0016-z
1686:R (programming language)
383:{\displaystyle \Delta t}
1720:Time–frequency analysis
1312:Proceedings of the IEEE
144:{\displaystyle -\pi /4}
40:technique developed by
1671:code base to generate
1659:code base to generate
1635:code base to generate
1393:. pp. 2563–2608.
1370:Cite journal requires
1306:Thomson, D. J. (1982)
1264:biomedical engineering
1213:
1160:
1098:
1041:
1000:
986:
913:
847:
684:
646:
590:
546:
441:
384:
357:
311:
182:spectral concentration
152:
145:
106:power spectral density
25:
1647:code base to perform
1438:, 57, 1371–1430
1314:, 70, 1055–1096
1214:
1161:
1099:
1042:
1004:The Slepian sequences
998:
987:
893:
848:
685:
647:
591:
520:
442:
385:
358:
312:
146:
118:
94:. When applying the
19:
1649:spherical multitaper
1250:for applications in
1178:
1138:
1055:
1012:
863:
709:
656:
603:
473:
398:
371:
336:
206:
124:
1589:2008GeoJI.174..774D
1513:2006SIAMR..48..504S
1248:spherical harmonics
1107:We choose a set of
1084:
880:
826:
788:
738:
632:
573:
1688:multitaper Package
1357:(B8): 19163-19184.
1209:
1156:
1123: = 0 to
1094:
1058:
1037:
1001:
982:
866:
843:
812:
774:
712:
680:
642:
606:
586:
547:
437:
380:
353:
307:
153:
141:
26:
1725:Signal estimation
1715:Signal processing
1675:Slepian functions
1663:Slepian functions
1639:Slepian functions
1408:978-3-642-54550-4
1340:978-0-521-88068-8
1272:spectral analysis
1068:
767:
722:
616:
557:
518:
486:
392:Nyquist frequency
96:Fourier transform
30:signal processing
1732:
1673:spherical vector
1637:spherical scalar
1603:
1602:
1600:
1582:
1558:
1552:
1551:
1531:
1525:
1524:
1506:
1486:
1480:
1478:
1458:
1452:
1445:
1439:
1432:
1426:
1419:
1413:
1412:
1386:
1380:
1379:
1373:
1368:
1366:
1358:
1350:
1344:
1343:
1321:
1315:
1304:
1218:
1216:
1215:
1210:
1205:
1204:
1166:. The quantity 2
1165:
1163:
1162:
1157:
1103:
1101:
1100:
1095:
1083:
1075:
1070:
1069:
1061:
1046:
1044:
1043:
1038:
1033:
1032:
991:
989:
988:
983:
978:
977:
929:
928:
912:
907:
879:
874:
852:
850:
849:
844:
839:
825:
820:
807:
806:
801:
787:
782:
768:
766:
752:
737:
729:
724:
723:
715:
702:and is given by
689:
687:
686:
681:
651:
649:
648:
643:
631:
623:
618:
617:
609:
595:
593:
592:
587:
572:
564:
559:
558:
550:
545:
534:
519:
511:
497:
496:
488:
487:
479:
446:
444:
443:
438:
421:
410:
409:
389:
387:
386:
381:
362:
360:
359:
354:
343:
316:
314:
313:
308:
306:
305:
300:
213:
186:weighted average
150:
148:
147:
142:
137:
101:central tendency
92:Fourier analysis
67:finite-variance
42:David J. Thomson
1740:
1739:
1735:
1734:
1733:
1731:
1730:
1729:
1700:
1699:
1612:
1607:
1606:
1560:
1559:
1555:
1533:
1532:
1528:
1488:
1487:
1483:
1460:
1459:
1455:
1446:
1442:
1433:
1429:
1420:
1416:
1409:
1388:
1387:
1383:
1369:
1359:
1352:
1351:
1347:
1341:
1324:
1322:
1318:
1305:
1301:
1296:
1284:
1243:
1196:
1176:
1175:
1136:
1135:
1053:
1052:
1018:
1010:
1009:
1006:
948:
914:
861:
860:
769:
756:
707:
706:
654:
653:
601:
600:
476:
471:
470:
401:
396:
395:
369:
368:
334:
333:
326:neurophysiology
226:
204:
203:
194:
122:
121:
88:
59:
12:
11:
5:
1738:
1736:
1728:
1727:
1722:
1717:
1712:
1702:
1701:
1698:
1697:
1689:
1681:
1676:
1664:
1652:
1640:
1628:
1623:
1618:
1611:
1610:External links
1608:
1605:
1604:
1553:
1526:
1497:(3): 504–536.
1481:
1453:
1440:
1427:
1414:
1407:
1381:
1372:|journal=
1345:
1339:
1316:
1298:
1297:
1295:
1292:
1291:
1290:
1283:
1280:
1242:
1239:
1208:
1203:
1199:
1195:
1192:
1189:
1186:
1183:
1155:
1152:
1149:
1146:
1143:
1133:Shannon number
1093:
1090:
1087:
1082:
1079:
1074:
1067:
1064:
1036:
1031:
1028:
1025:
1021:
1017:
1005:
1002:
993:
992:
981:
976:
973:
970:
967:
964:
961:
958:
955:
951:
947:
944:
941:
938:
935:
932:
927:
924:
921:
917:
911:
906:
903:
900:
896:
892:
889:
886:
883:
878:
873:
869:
854:
853:
842:
838:
835:
832:
829:
824:
819:
815:
811:
805:
800:
797:
794:
791:
786:
781:
777:
773:
765:
762:
759:
755:
750:
747:
744:
741:
736:
733:
728:
721:
718:
679:
676:
673:
670:
667:
664:
661:
641:
638:
635:
630:
627:
622:
615:
612:
597:
596:
585:
582:
579:
576:
571:
568:
563:
556:
553:
544:
541:
538:
533:
530:
527:
523:
517:
514:
509:
506:
503:
500:
495:
492:
485:
482:
436:
433:
430:
427:
424:
420:
416:
413:
408:
404:
390:, so that the
379:
376:
352:
349:
346:
342:
318:
317:
304:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
233:
230:
225:
222:
219:
216:
212:
193:
190:
151:out of phase.
140:
136:
132:
129:
87:
84:
69:random process
55:
50:power spectrum
36:analysis is a
20:Comparison of
13:
10:
9:
6:
4:
3:
2:
1737:
1726:
1723:
1721:
1718:
1716:
1713:
1711:
1708:
1707:
1705:
1695:
1692:
1690:
1687:
1684:
1682:
1680:Python module
1679:
1677:
1674:
1670:
1667:
1665:
1662:
1658:
1655:
1653:
1650:
1646:
1643:
1641:
1638:
1634:
1631:
1629:
1626:
1624:
1621:
1619:
1616:
1614:
1613:
1609:
1599:
1594:
1590:
1586:
1581:
1576:
1572:
1568:
1564:
1557:
1554:
1549:
1545:
1541:
1537:
1530:
1527:
1522:
1518:
1514:
1510:
1505:
1500:
1496:
1492:
1485:
1482:
1476:
1472:
1468:
1464:
1457:
1454:
1450:
1444:
1441:
1437:
1431:
1428:
1424:
1418:
1415:
1410:
1404:
1400:
1396:
1392:
1385:
1382:
1377:
1364:
1356:
1349:
1346:
1342:
1336:
1332:
1328:
1320:
1317:
1313:
1309:
1303:
1300:
1293:
1289:
1286:
1285:
1281:
1279:
1277:
1273:
1269:
1265:
1261:
1257:
1253:
1249:
1240:
1238:
1237:
1232:
1230:
1226:
1223: =
1222:
1201:
1197:
1193:
1190:
1184:
1181:
1173:
1169:
1153:
1147:
1144:
1141:
1134:
1130:
1126:
1122:
1118:
1114:
1113:David Slepian
1110:
1105:
1088:
1080:
1077:
1072:
1062:
1050:
1029:
1026:
1023:
1019:
1008:The sequence
1003:
997:
979:
974:
968:
965:
962:
959:
956:
953:
949:
942:
939:
936:
930:
925:
922:
919:
915:
909:
904:
901:
898:
894:
890:
884:
876:
871:
867:
859:
858:
857:
840:
830:
822:
817:
813:
803:
792:
784:
779:
775:
763:
757:
753:
748:
742:
734:
731:
726:
716:
705:
704:
703:
701:
697:
693:
677:
674:
671:
668:
665:
662:
659:
636:
628:
625:
620:
610:
583:
577:
569:
566:
561:
551:
542:
539:
536:
531:
528:
525:
521:
515:
512:
507:
501:
493:
490:
480:
469:
468:
467:
465:
461:
457:
453:
448:
431:
425:
418:
414:
411:
406:
402:
393:
377:
366:
347:
331:
328:for example,
327:
323:
302:
291:
288:
285:
279:
276:
273:
270:
264:
261:
258:
252:
249:
243:
240:
237:
231:
223:
217:
202:
201:
200:
199:
191:
189:
187:
183:
178:
174:
171:
166:
163:
158:
138:
134:
130:
127:
117:
113:
111:
107:
102:
97:
93:
85:
83:
81:
77:
73:
70:
66:
63:
58:
54:
51:
47:
43:
39:
35:
31:
23:
18:
1672:
1660:
1648:
1636:
1570:
1566:
1556:
1539:
1535:
1529:
1504:math/0408424
1494:
1490:
1484:
1466:
1462:
1456:
1448:
1443:
1435:
1430:
1422:
1417:
1390:
1384:
1363:cite journal
1354:
1348:
1330:
1326:
1319:
1311:
1307:
1302:
1260:neuroscience
1244:
1241:Applications
1233:
1228:
1224:
1220:
1167:
1128:
1124:
1120:
1116:
1108:
1106:
1048:
1007:
855:
699:
695:
691:
598:
463:
459:
455:
451:
449:
364:
329:
321:
319:
195:
179:
175:
167:
154:
89:
79:
71:
56:
52:
33:
27:
1491:SIAM Review
1288:Periodogram
1274:toolkit of
1119:and orders
192:Formulation
162:periodogram
76:realization
22:periodogram
1704:Categories
1573:(3): 774.
1542:(6): 665.
1294:References
1252:geophysics
170:data taper
86:Motivation
62:stationary
44:. It can
34:multitaper
1661:Cartesian
1580:0705.3083
1256:cosmology
1234:See also
1185:∈
1151:Δ
1066:^
972:Δ
963:π
954:−
895:∑
804:∗
761:Δ
720:^
690:) is the
675:−
669:≤
663:≤
614:^
555:^
540:−
522:∑
484:^
429:Δ
375:Δ
274:…
131:π
128:−
82:as data.
1651:analysis
1469:: 1–36.
1282:See also
46:estimate
1585:Bibcode
1509:Bibcode
1276:Chronux
1219:. When
65:ergodic
1694:S-Plus
1669:MATLAB
1657:MATLAB
1645:MATLAB
1633:MATLAB
1405:
1337:
856:where
599:Here,
110:biased
1575:arXiv
1499:arXiv
652:(for
320:Here
157:taper
60:of a
1403:ISBN
1376:help
1335:ISBN
1268:here
1254:and
1174:and
698:and
462:and
454:and
48:the
1593:doi
1571:174
1544:doi
1517:doi
1471:doi
1395:doi
1355:105
1329:",
394:is
78:of
28:In
1706::
1591:.
1583:.
1569:.
1565:.
1540:13
1538:.
1515:.
1507:.
1495:48
1493:.
1465:.
1401:.
1367::
1365:}}
1361:{{
1310:.
1278:.
1262:,
447:.
32:,
1601:.
1595::
1587::
1577::
1550:.
1546::
1523:.
1519::
1511::
1501::
1479:.
1477:.
1473::
1467:2
1411:.
1397::
1378:)
1374:(
1323:*
1229:l
1225:m
1221:l
1207:)
1202:N
1198:f
1194:,
1191:0
1188:(
1182:W
1168:W
1154:t
1148:W
1145:N
1142:2
1129:K
1125:K
1121:k
1117:W
1109:K
1092:)
1089:f
1086:(
1081:m
1078:l
1073:k
1063:S
1049:k
1035:}
1030:k
1027:,
1024:t
1020:h
1016:{
980:.
975:t
969:t
966:f
960:2
957:i
950:e
946:)
943:t
940:,
937:l
934:(
931:X
926:k
923:,
920:t
916:h
910:N
905:1
902:=
899:t
891:=
888:)
885:f
882:(
877:l
872:k
868:J
841:,
837:]
834:)
831:f
828:(
823:m
818:k
814:J
810:[
799:]
796:)
793:f
790:(
785:l
780:k
776:J
772:[
764:t
758:N
754:1
749:=
746:)
743:f
740:(
735:m
732:l
727:k
717:S
700:m
696:l
692:k
678:1
672:K
666:k
660:0
640:)
637:f
634:(
629:m
626:l
621:k
611:S
584:.
581:)
578:f
575:(
570:m
567:l
562:k
552:S
543:1
537:K
532:0
529:=
526:k
516:K
513:1
508:=
505:)
502:f
499:(
494:m
491:l
481:S
464:m
460:l
456:m
452:l
435:)
432:t
426:2
423:(
419:/
415:1
412:=
407:N
403:f
378:t
365:p
351:)
348:t
345:(
341:X
330:p
322:T
303:T
298:]
295:)
292:t
289:,
286:p
283:(
280:X
277:,
271:,
268:)
265:t
262:,
259:2
256:(
253:X
250:,
247:)
244:t
241:,
238:1
235:(
232:X
229:[
224:=
221:)
218:t
215:(
211:X
139:4
135:/
80:X
72:X
57:X
53:S
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
