669:
262:
664:{\displaystyle \tan 2\theta _{0}={\frac {K\sum _{k=0}^{K-1}\sin 2\omega t_{k}-2\left(\sum _{k=0}^{K-1}\cos \omega t_{k}\right)\left(\sum _{k=0}^{K-1}\sin \omega t_{k}\right)}{K\sum _{k=0}^{K-1}\cos 2\omega t_{k}-{\big (}\sum _{k=0}^{K-1}\cos \omega t_{k}{\big )}^{2}+{\big (}\sum _{k=0}^{K-1}\sin \omega t_{k}{\big )}^{2}}},}
1442:
2183:
M. Gruberbauer; K. Kolenberg; J. F. Rowe; D. Huber; J. M. Matthews; P. Reegen; R. Kuschnig; C. Cameron; T. Kallinger; W. W. Weiss; D. B. Guenther; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker (2007). "MOST photometry of the RRdLyrae variable AQLeo: two radial modes, 32 combination
904:
1139:
1834:
2247:
M. Gruberbauer; H. Saio; D. Huber; T. Kallinger; W. W. Weiss; D. B. Guenther; R. Kuschnig; J. M. Matthews; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker (2008). "MOST photometry and modeling of the rapidly oscillating (roAp) star γ Equulei".
2669:
K. Zwintz; T. Kallinger; D. B. Guenther; M. Gruberbauer; D. Huber; J. Rowe; R. Kuschnig; W. W. Weiss; J. M. Matthews; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker; M. P. Casey (2009). "MOST photometry of the enigmatic PMS pulsator HD 142666".
2374:
D. Huber; H. Saio; M. Gruberbauer; W. W. Weiss; J. F. Rowe; M. Hareter; T. Kallinger; P. Reegen; J. M. Matthews; R. Kuschnig; D. B. Guenther; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker (2008). "MOST photometry of the roAp star 10 Aquilae".
1297:
1933:
1974:
and to classify stellar pulsation (see references below). The fact that this method incorporates the properties of the time-domain sampling appropriately makes it a valuable tool for typical astronomical measurements containing data gaps.
2422:
T. Kallinger; D. B. Guenther; J. M. Matthews; W. W. Weiss; D. Huber; R. Kuschnig; A. F. J. Moffat; S. M. Rucinski; D. Sasselov (2008). "Nonradial p-modes in the G9.5 giant ε Ophiuchi? Pulsation model fits to MOST photometry".
1308:
2717:
K. Zwintz; M. Hareter; R. Kuschnig; P. J. Amado; N. Nesvacil; E. Rodriguez; D. Diaz-Fraile; W. W. Weiss; T. Pribulla; D. B. Guenther; J. M. Matthews; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker (2009).
677:
912:
73:, appropriately incorporating a time series to be averaged to zero before applying the DFT, which is done in many practical applications. When a zero-mean corrected dataset has to be statistically compared to a
1622:
1732:
1170:
1845:
1658:
157:
254:
200:
2186:
227:
1162:
1692:
1956:
1724:
111:
1437:{\displaystyle \phi (A)={\frac {KA\cdot \operatorname {sock} }{2<x^{2}>}}\exp \left(-{\frac {KA^{2}}{4<x^{2}>}}\cdot \operatorname {sock} \right),}
62:. It represents the answer to the question, “What would be the chance to obtain an amplitude like the measured one or higher, if the analysed time series were
2295:
D. B. Guenther; T. Kallinger; P. Reegen; W. W. Weiss; J. M. Matthews; R. Kuschnig; A. F. J. Moffat; S. M. Rucinski; D. Sasselov; G. A. H. Walker (2007).
2622:
M. Zechmeister; M. Kuerster (2009). "The gemeralised Lomb-Scargle periodogram. A new formalism for the floating-mean and
Keplerian periodograms".
899:{\displaystyle \alpha _{0}={\sqrt {{\frac {2}{K^{2}}}\left(K\sum _{k=0}^{K-1}\cos ^{2}\left(\omega t_{k}-\theta _{0}\right)-\left^{2}\right)}},}
2782:
2239:
1134:{\displaystyle \beta _{0}={\sqrt {{\frac {2}{K^{2}}}\left(K\sum _{k=0}^{K-1}\sin ^{2}\left(\omega t_{k}-\theta _{0}\right)-\left^{2}\right)}}.}
2102:
J. D. Scargle (1982). "Studies in astronomical time series analysis. II. Statistical aspects of spectral analysis of unevenly spaced data".
2470:
T. Kallinger; P. Reegen; W. W. Weiss (2008). "A heuristic derivation of the uncertainty for frequency determination in time series data".
2540:
P. Reegen; M. Gruberbauer; L. Schneider; W. W. Weiss (2008). "Cinderella - Comparison of INDEpendent RELative Least-squares
Amplitudes".
1665:
2792:
2530:
78:
1453:
26:) is a statistical technique to provide the reliability of periodicities in a measured (noisy and not necessarily equidistant)
1829:{\displaystyle \Phi _{\operatorname {FA} }(A)=\exp \left(-{\frac {KA^{2}}{4<x^{2}>}}\cdot \operatorname {sock} \right).}
160:
1984:
2006:
35:
2331:
2787:
70:
1292:{\displaystyle \tan \theta ={\frac {\sum _{k=0}^{K-1}\sin \omega t_{k}}{\sum _{k=0}^{K-1}\cos \omega t_{k}}},}
1928:{\displaystyle \operatorname {sig} (A)={\frac {KA^{2}\log e}{4<x^{2}>}}\cdot \operatorname {sock} .}
2233:
1938:
It returns the number of random time series one would have to examine to obtain one amplitude exceeding
2767:
2731:
2689:
2641:
2600:
2559:
2489:
2442:
2394:
2346:
2308:
2267:
2205:
2155:
2111:
2068:
2025:
1630:
116:
2749:
2705:
2679:
2657:
2631:
2575:
2549:
2505:
2479:
2458:
2432:
2410:
2384:
2362:
2283:
2257:
2221:
2195:
2171:
2145:
2084:
2041:
2015:
232:
178:
205:
2526:
2004:
P. Reegen (2007). "SigSpec - I. Frequency- and phase-resolved significance in
Fourier space".
1839:
The sig is defined as the negative logarithm of the false-alarm probability and evaluates to
1147:
2739:
2735:
2697:
2693:
2649:
2645:
2608:
2567:
2563:
2497:
2493:
2450:
2446:
2402:
2398:
2354:
2316:
2275:
2271:
2213:
2163:
2119:
2076:
2072:
2033:
2029:
1967:
31:
1670:
2517:
P. Reegen (2005). ""SigSpec - reliable computation of significance in
Fourier space", in
2604:
2350:
2312:
2209:
2159:
2115:
1941:
1709:
96:
2776:
2753:
2217:
2175:
1971:
172:
164:
74:
2709:
2661:
2579:
2509:
2462:
2366:
2287:
2225:
2045:
2414:
2136:
M. Breger; S. M. Rucinski; P. Reegen (2007). "The
Pulsation of EE Camelopardalis".
2088:
59:
2744:
2719:
2701:
2653:
1702:
Integration of the pdf yields the false-alarm probability that white noise in the
2571:
2501:
2454:
2406:
2279:
2037:
2297:"Searching for p-modes in η Bootis & Procyon using MOST satellite data"
2059:
N. R. Lomb (1976). "Least-squares frequency analysis of unequally spaced data".
1703:
55:
27:
168:
51:
47:
54:
measure of the probability that the given amplitude level would be seen in
1661:
2330:
D. B. Guenther; T. Kallinger; K. Zwintz; W. W. Weiss; J. Tanner (2007).
2020:
2080:
2613:
2588:
2321:
2296:
63:
2358:
2167:
2123:
2684:
2636:
2554:
2484:
2437:
2389:
2262:
2200:
2150:
89:
Probability density function (pdf) of white noise in
Fourier space
82:
2525:. Cambridge, UK: Cambridge University Press. pp. 791–798.
2768:
Website with further information on SigSpec calculation, etc.
1617:{\displaystyle \operatorname {sock} (\omega ,\theta )=\left}
2521:, eds. J. Zverko, J. Ziznovsky, S.J. Adelman, W.W. Weiss".
93:
Considering a time series to be represented by a set of
2720:"MOST observations of the young open cluster NGC 2264"
1944:
1848:
1735:
1712:
1673:
1633:
1456:
1311:
1173:
1150:
915:
680:
265:
235:
208:
181:
175:
angle may be described in terms of three parameters,
119:
99:
69:
SigSpec may be considered a formal extension to the
2523:
The A-Star Puzzle, Proceedings of IAU Symposium 224
2332:"Seismology of Pre-Main-Sequence Stars in NGC 6530"
1950:
1927:
1828:
1718:
1686:
1652:
1616:
1436:
1302:the probability density of amplitudes is given by
1291:
1156:
1133:
898:
663:
248:
221:
194:
151:
105:
2589:"A new slowly pulsating subdwarf-B star: HD 4539"
2187:Monthly Notices of the Royal Astronomical Society
1698:False-alarm probability and spectral significance
256:, defining the “sampling profile”, according to
1144:In terms of the phase angle in Fourier space,
644:
590:
574:
520:
8:
2519:The A-Star Puzzle, Proceedings IAU Symp. 224
2743:
2683:
2635:
2612:
2553:
2483:
2436:
2388:
2320:
2261:
2199:
2149:
2019:
1943:
1904:
1877:
1867:
1847:
1800:
1782:
1772:
1740:
1734:
1711:
1678:
1672:
1641:
1632:
1601:
1596:
1580:
1556:
1549:
1538:
1533:
1517:
1493:
1486:
1455:
1408:
1390:
1380:
1354:
1327:
1310:
1277:
1252:
1241:
1229:
1204:
1193:
1186:
1172:
1149:
1115:
1099:
1086:
1056:
1045:
1021:
1008:
987:
971:
960:
940:
931:
929:
920:
914:
880:
864:
851:
821:
810:
786:
773:
752:
736:
725:
705:
696:
694:
685:
679:
649:
643:
642:
635:
610:
599:
589:
588:
579:
573:
572:
565:
540:
529:
519:
518:
509:
481:
470:
450:
425:
414:
394:
369:
358:
337:
309:
298:
288:
279:
264:
240:
234:
213:
207:
186:
180:
140:
127:
118:
98:
2587:C. Schoenaers; A. E. Lynas-Gray (2007).
38:(DFT) and assigns a quantity called the
1996:
2231:
1447:where the sock function is defined by
7:
2593:Communications in Asteroseismology
2301:Communications in Asteroseismology
2238:: CS1 maint: unflagged free DOI (
1958:at the given frequency and phase.
1737:
1706:produces an amplitude of at least
14:
2218:10.1111/j.1365-2966.2007.12042.x
2061:Astrophysics and Space Science
1861:
1855:
1752:
1746:
1475:
1463:
1321:
1315:
146:
120:
1:
2783:Statistical signal processing
1966:SigSpec is primarily used in
1653:{\displaystyle <x^{2}>}
152:{\displaystyle (t_{k},x_{k})}
30:. It relies on the amplitude
42:(frequently abbreviated by “
2745:10.1051/0004-6361/200911863
2702:10.1051/0004-6361:200811116
2654:10.1051/0004-6361:200811296
1985:Spectral density estimation
249:{\displaystyle \theta _{0}}
195:{\displaystyle \alpha _{0}}
2809:
2724:Astronomy and Astrophysics
2672:Astronomy and Astrophysics
2624:Astronomy and Astrophysics
2572:10.1051/0004-6361:20078855
2542:Astronomy and Astrophysics
2502:10.1051/0004-6361:20077559
2472:Astronomy and Astrophysics
2455:10.1051/0004-6361:20078171
2425:Astronomy and Astrophysics
2407:10.1051/0004-6361:20079220
2377:Astronomy and Astrophysics
2280:10.1051/0004-6361:20078830
2250:Astronomy and Astrophysics
2038:10.1051/0004-6361:20066597
2007:Astronomy and Astrophysics
222:{\displaystyle \beta _{0}}
36:Discrete Fourier transform
2793:Digital signal processing
2339:The Astrophysical Journal
2184:frequencies and beyond".
2104:The Astrophysical Journal
2138:The Astronomical Journal
71:Lomb-Scargle periodogram
2736:2009A&A...502..239Z
2694:2009A&A...494.1031Z
2646:2009A&A...496..577Z
2564:2008A&A...484..601R
2494:2008A&A...481..571K
2447:2008A&A...478..497K
2399:2008A&A...483..239H
2272:2008A&A...480..223G
2073:1976Ap&SS..39..447L
2030:2007A&A...467.1353R
1157:{\displaystyle \theta }
1952:
1929:
1830:
1720:
1688:
1654:
1618:
1438:
1293:
1263:
1215:
1158:
1135:
1067:
982:
900:
832:
747:
665:
621:
551:
492:
436:
380:
320:
250:
223:
196:
153:
107:
85:only) has to be zero.
1953:
1930:
1831:
1721:
1689:
1687:{\displaystyle x_{k}}
1655:
1619:
1439:
1294:
1237:
1189:
1159:
1136:
1041:
956:
901:
806:
721:
666:
595:
525:
466:
410:
354:
294:
251:
224:
197:
154:
108:
50:. This quantity is a
40:spectral significance
24:SIGnificance SPECtrum
16:Statistical technique
1942:
1846:
1733:
1710:
1671:
1631:
1454:
1309:
1171:
1148:
913:
678:
263:
233:
206:
179:
117:
97:
58:, in the sense of a
2605:2007CoAst.151...67S
2351:2007ApJ...671..581G
2313:2007CoAst.151....5G
2210:2007MNRAS.379.1498G
2160:2007AJ....134.1994B
2116:1982ApJ...263..835S
1606:
1543:
2081:10.1007/BF00648343
1948:
1925:
1826:
1716:
1684:
1666:dependent variable
1650:
1614:
1592:
1529:
1434:
1289:
1154:
1131:
896:
661:
246:
219:
192:
163:of white noise in
149:
103:
2614:10.1553/cia151s67
1951:{\displaystyle A}
1914:
1810:
1719:{\displaystyle A}
1607:
1544:
1418:
1364:
1284:
1126:
946:
891:
711:
656:
106:{\displaystyle K}
81:(rather than the
2800:
2788:Fourier analysis
2757:
2747:
2713:
2687:
2678:(3): 1031–1040.
2665:
2639:
2618:
2616:
2583:
2557:
2536:
2513:
2487:
2466:
2440:
2418:
2392:
2370:
2336:
2326:
2324:
2322:10.1553/cia151s5
2291:
2265:
2243:
2237:
2229:
2203:
2194:(4): 1498–1506.
2179:
2153:
2144:(5): 1994–1998.
2128:
2127:
2099:
2093:
2092:
2056:
2050:
2049:
2023:
2014:(3): 1353–1371.
2001:
1968:asteroseismology
1957:
1955:
1954:
1949:
1934:
1932:
1931:
1926:
1915:
1913:
1909:
1908:
1892:
1882:
1881:
1868:
1835:
1833:
1832:
1827:
1822:
1818:
1811:
1809:
1805:
1804:
1788:
1787:
1786:
1773:
1745:
1744:
1725:
1723:
1722:
1717:
1693:
1691:
1690:
1685:
1683:
1682:
1659:
1657:
1656:
1651:
1646:
1645:
1623:
1621:
1620:
1615:
1613:
1609:
1608:
1605:
1600:
1591:
1590:
1586:
1585:
1584:
1561:
1560:
1550:
1545:
1542:
1537:
1528:
1527:
1523:
1522:
1521:
1498:
1497:
1487:
1443:
1441:
1440:
1435:
1430:
1426:
1419:
1417:
1413:
1412:
1396:
1395:
1394:
1381:
1365:
1363:
1359:
1358:
1342:
1328:
1298:
1296:
1295:
1290:
1285:
1283:
1282:
1281:
1262:
1251:
1235:
1234:
1233:
1214:
1203:
1187:
1163:
1161:
1160:
1155:
1140:
1138:
1137:
1132:
1127:
1125:
1121:
1120:
1119:
1114:
1110:
1109:
1105:
1104:
1103:
1091:
1090:
1066:
1055:
1031:
1027:
1026:
1025:
1013:
1012:
992:
991:
981:
970:
947:
945:
944:
932:
930:
925:
924:
905:
903:
902:
897:
892:
890:
886:
885:
884:
879:
875:
874:
870:
869:
868:
856:
855:
831:
820:
796:
792:
791:
790:
778:
777:
757:
756:
746:
735:
712:
710:
709:
697:
695:
690:
689:
670:
668:
667:
662:
657:
655:
654:
653:
648:
647:
640:
639:
620:
609:
594:
593:
584:
583:
578:
577:
570:
569:
550:
539:
524:
523:
514:
513:
491:
480:
461:
460:
456:
455:
454:
435:
424:
404:
400:
399:
398:
379:
368:
342:
341:
319:
308:
289:
284:
283:
255:
253:
252:
247:
245:
244:
228:
226:
225:
220:
218:
217:
201:
199:
198:
193:
191:
190:
159:, the amplitude
158:
156:
155:
150:
145:
144:
132:
131:
112:
110:
109:
104:
34:obtained by the
2808:
2807:
2803:
2802:
2801:
2799:
2798:
2797:
2773:
2772:
2764:
2730:(1): 1239–252.
2716:
2668:
2621:
2586:
2539:
2533:
2516:
2469:
2421:
2373:
2334:
2329:
2294:
2246:
2230:
2182:
2135:
2132:
2131:
2101:
2100:
2096:
2058:
2057:
2053:
2021:physics/0703160
2003:
2002:
1998:
1993:
1981:
1964:
1940:
1939:
1900:
1893:
1873:
1869:
1844:
1843:
1796:
1789:
1778:
1774:
1768:
1764:
1736:
1731:
1730:
1708:
1707:
1700:
1674:
1669:
1668:
1637:
1629:
1628:
1576:
1569:
1565:
1552:
1551:
1513:
1506:
1502:
1489:
1488:
1485:
1481:
1452:
1451:
1404:
1397:
1386:
1382:
1376:
1372:
1350:
1343:
1329:
1307:
1306:
1273:
1236:
1225:
1188:
1169:
1168:
1146:
1145:
1095:
1082:
1078:
1074:
1040:
1036:
1035:
1017:
1004:
1000:
996:
983:
952:
948:
936:
916:
911:
910:
860:
847:
843:
839:
805:
801:
800:
782:
769:
765:
761:
748:
717:
713:
701:
681:
676:
675:
641:
631:
571:
561:
505:
462:
446:
409:
405:
390:
353:
349:
333:
290:
275:
261:
260:
236:
231:
230:
209:
204:
203:
182:
177:
176:
167:, depending on
136:
123:
115:
114:
95:
94:
91:
83:population mean
17:
12:
11:
5:
2806:
2804:
2796:
2795:
2790:
2785:
2775:
2774:
2771:
2770:
2763:
2762:External links
2760:
2759:
2758:
2714:
2666:
2630:(2): 577–584.
2619:
2584:
2548:(2): 601–608.
2537:
2531:
2514:
2478:(2): 571–574.
2467:
2431:(2): 497–505.
2419:
2383:(1): 239–248.
2371:
2359:10.1086/522880
2345:(1): 581–591.
2327:
2292:
2256:(1): 223–232.
2244:
2180:
2168:10.1086/522795
2130:
2129:
2124:10.1086/160554
2094:
2067:(2): 447–462.
2051:
1995:
1994:
1992:
1989:
1988:
1987:
1980:
1977:
1972:variable stars
1963:
1960:
1947:
1936:
1935:
1924:
1921:
1918:
1912:
1907:
1903:
1899:
1896:
1891:
1888:
1885:
1880:
1876:
1872:
1866:
1863:
1860:
1857:
1854:
1851:
1837:
1836:
1825:
1821:
1817:
1814:
1808:
1803:
1799:
1795:
1792:
1785:
1781:
1777:
1771:
1767:
1763:
1760:
1757:
1754:
1751:
1748:
1743:
1739:
1715:
1699:
1696:
1681:
1677:
1649:
1644:
1640:
1636:
1625:
1624:
1612:
1604:
1599:
1595:
1589:
1583:
1579:
1575:
1572:
1568:
1564:
1559:
1555:
1548:
1541:
1536:
1532:
1526:
1520:
1516:
1512:
1509:
1505:
1501:
1496:
1492:
1484:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1445:
1444:
1433:
1429:
1425:
1422:
1416:
1411:
1407:
1403:
1400:
1393:
1389:
1385:
1379:
1375:
1371:
1368:
1362:
1357:
1353:
1349:
1346:
1341:
1338:
1335:
1332:
1326:
1323:
1320:
1317:
1314:
1300:
1299:
1288:
1280:
1276:
1272:
1269:
1266:
1261:
1258:
1255:
1250:
1247:
1244:
1240:
1232:
1228:
1224:
1221:
1218:
1213:
1210:
1207:
1202:
1199:
1196:
1192:
1185:
1182:
1179:
1176:
1153:
1142:
1141:
1130:
1124:
1118:
1113:
1108:
1102:
1098:
1094:
1089:
1085:
1081:
1077:
1073:
1070:
1065:
1062:
1059:
1054:
1051:
1048:
1044:
1039:
1034:
1030:
1024:
1020:
1016:
1011:
1007:
1003:
999:
995:
990:
986:
980:
977:
974:
969:
966:
963:
959:
955:
951:
943:
939:
935:
928:
923:
919:
907:
906:
895:
889:
883:
878:
873:
867:
863:
859:
854:
850:
846:
842:
838:
835:
830:
827:
824:
819:
816:
813:
809:
804:
799:
795:
789:
785:
781:
776:
772:
768:
764:
760:
755:
751:
745:
742:
739:
734:
731:
728:
724:
720:
716:
708:
704:
700:
693:
688:
684:
672:
671:
660:
652:
646:
638:
634:
630:
627:
624:
619:
616:
613:
608:
605:
602:
598:
592:
587:
582:
576:
568:
564:
560:
557:
554:
549:
546:
543:
538:
535:
532:
528:
522:
517:
512:
508:
504:
501:
498:
495:
490:
487:
484:
479:
476:
473:
469:
465:
459:
453:
449:
445:
442:
439:
434:
431:
428:
423:
420:
417:
413:
408:
403:
397:
393:
389:
386:
383:
378:
375:
372:
367:
364:
361:
357:
352:
348:
345:
340:
336:
332:
329:
326:
323:
318:
315:
312:
307:
304:
301:
297:
293:
287:
282:
278:
274:
271:
268:
243:
239:
216:
212:
189:
185:
148:
143:
139:
135:
130:
126:
122:
102:
90:
87:
15:
13:
10:
9:
6:
4:
3:
2:
2805:
2794:
2791:
2789:
2786:
2784:
2781:
2780:
2778:
2769:
2766:
2765:
2761:
2755:
2751:
2746:
2741:
2737:
2733:
2729:
2725:
2721:
2715:
2711:
2707:
2703:
2699:
2695:
2691:
2686:
2681:
2677:
2673:
2667:
2663:
2659:
2655:
2651:
2647:
2643:
2638:
2633:
2629:
2625:
2620:
2615:
2610:
2606:
2602:
2598:
2594:
2590:
2585:
2581:
2577:
2573:
2569:
2565:
2561:
2556:
2551:
2547:
2543:
2538:
2534:
2532:0-521-85018-5
2528:
2524:
2520:
2515:
2511:
2507:
2503:
2499:
2495:
2491:
2486:
2481:
2477:
2473:
2468:
2464:
2460:
2456:
2452:
2448:
2444:
2439:
2434:
2430:
2426:
2420:
2416:
2412:
2408:
2404:
2400:
2396:
2391:
2386:
2382:
2378:
2372:
2368:
2364:
2360:
2356:
2352:
2348:
2344:
2340:
2333:
2328:
2323:
2318:
2314:
2310:
2306:
2302:
2298:
2293:
2289:
2285:
2281:
2277:
2273:
2269:
2264:
2259:
2255:
2251:
2245:
2241:
2235:
2227:
2223:
2219:
2215:
2211:
2207:
2202:
2197:
2193:
2189:
2188:
2181:
2177:
2173:
2169:
2165:
2161:
2157:
2152:
2147:
2143:
2139:
2134:
2133:
2125:
2121:
2117:
2113:
2109:
2105:
2098:
2095:
2090:
2086:
2082:
2078:
2074:
2070:
2066:
2062:
2055:
2052:
2047:
2043:
2039:
2035:
2031:
2027:
2022:
2017:
2013:
2009:
2008:
2000:
1997:
1990:
1986:
1983:
1982:
1978:
1976:
1973:
1969:
1961:
1959:
1945:
1922:
1919:
1916:
1910:
1905:
1901:
1897:
1894:
1889:
1886:
1883:
1878:
1874:
1870:
1864:
1858:
1852:
1849:
1842:
1841:
1840:
1823:
1819:
1815:
1812:
1806:
1801:
1797:
1793:
1790:
1783:
1779:
1775:
1769:
1765:
1761:
1758:
1755:
1749:
1741:
1729:
1728:
1727:
1713:
1705:
1697:
1695:
1679:
1675:
1667:
1663:
1647:
1642:
1638:
1634:
1610:
1602:
1597:
1593:
1587:
1581:
1577:
1573:
1570:
1566:
1562:
1557:
1553:
1546:
1539:
1534:
1530:
1524:
1518:
1514:
1510:
1507:
1503:
1499:
1494:
1490:
1482:
1478:
1472:
1469:
1466:
1460:
1457:
1450:
1449:
1448:
1431:
1427:
1423:
1420:
1414:
1409:
1405:
1401:
1398:
1391:
1387:
1383:
1377:
1373:
1369:
1366:
1360:
1355:
1351:
1347:
1344:
1339:
1336:
1333:
1330:
1324:
1318:
1312:
1305:
1304:
1303:
1286:
1278:
1274:
1270:
1267:
1264:
1259:
1256:
1253:
1248:
1245:
1242:
1238:
1230:
1226:
1222:
1219:
1216:
1211:
1208:
1205:
1200:
1197:
1194:
1190:
1183:
1180:
1177:
1174:
1167:
1166:
1165:
1151:
1128:
1122:
1116:
1111:
1106:
1100:
1096:
1092:
1087:
1083:
1079:
1075:
1071:
1068:
1063:
1060:
1057:
1052:
1049:
1046:
1042:
1037:
1032:
1028:
1022:
1018:
1014:
1009:
1005:
1001:
997:
993:
988:
984:
978:
975:
972:
967:
964:
961:
957:
953:
949:
941:
937:
933:
926:
921:
917:
909:
908:
893:
887:
881:
876:
871:
865:
861:
857:
852:
848:
844:
840:
836:
833:
828:
825:
822:
817:
814:
811:
807:
802:
797:
793:
787:
783:
779:
774:
770:
766:
762:
758:
753:
749:
743:
740:
737:
732:
729:
726:
722:
718:
714:
706:
702:
698:
691:
686:
682:
674:
673:
658:
650:
636:
632:
628:
625:
622:
617:
614:
611:
606:
603:
600:
596:
585:
580:
566:
562:
558:
555:
552:
547:
544:
541:
536:
533:
530:
526:
515:
510:
506:
502:
499:
496:
493:
488:
485:
482:
477:
474:
471:
467:
463:
457:
451:
447:
443:
440:
437:
432:
429:
426:
421:
418:
415:
411:
406:
401:
395:
391:
387:
384:
381:
376:
373:
370:
365:
362:
359:
355:
350:
346:
343:
338:
334:
330:
327:
324:
321:
316:
313:
310:
305:
302:
299:
295:
291:
285:
280:
276:
272:
269:
266:
259:
258:
257:
241:
237:
214:
210:
187:
183:
174:
170:
166:
165:Fourier space
162:
141:
137:
133:
128:
124:
100:
88:
86:
84:
80:
76:
75:random sample
72:
67:
65:
61:
57:
53:
49:
45:
41:
37:
33:
29:
25:
21:
2727:
2723:
2675:
2671:
2627:
2623:
2596:
2592:
2545:
2541:
2522:
2518:
2475:
2471:
2428:
2424:
2380:
2376:
2342:
2338:
2304:
2300:
2253:
2249:
2234:cite journal
2191:
2185:
2141:
2137:
2107:
2103:
2097:
2064:
2060:
2054:
2011:
2005:
1999:
1970:to identify
1965:
1962:Applications
1937:
1838:
1701:
1660:denotes the
1626:
1446:
1301:
1143:
92:
68:
60:type I error
43:
39:
23:
22:(acronym of
19:
18:
2110:: 835–853.
1704:time domain
79:sample mean
56:white noise
52:logarithmic
46:”) to each
28:time series
2777:Categories
1991:References
2754:123505620
2685:0812.1960
2637:0901.2573
2599:: 67–76.
2555:0710.2963
2485:0801.0683
2438:0711.0837
2390:0803.1721
2263:0801.0863
2201:0705.4603
2176:120843648
2151:0709.3393
1917:⋅
1887:
1853:
1813:⋅
1770:−
1762:
1738:Φ
1594:β
1578:θ
1574:−
1571:θ
1563:
1531:α
1515:θ
1511:−
1508:θ
1500:
1473:θ
1467:ω
1461:
1421:⋅
1378:−
1370:
1337:⋅
1313:ϕ
1271:ω
1268:
1257:−
1239:∑
1223:ω
1220:
1209:−
1191:∑
1181:θ
1178:
1152:θ
1097:θ
1093:−
1080:ω
1072:
1061:−
1043:∑
1033:−
1019:θ
1015:−
1002:ω
994:
976:−
958:∑
918:β
862:θ
858:−
845:ω
837:
826:−
808:∑
798:−
784:θ
780:−
767:ω
759:
741:−
723:∑
683:α
629:ω
626:
615:−
597:∑
559:ω
556:
545:−
527:∑
516:−
503:ω
497:
486:−
468:∑
444:ω
441:
430:−
412:∑
388:ω
385:
374:−
356:∑
344:−
331:ω
325:
314:−
296:∑
277:θ
270:
238:θ
211:β
184:α
169:frequency
48:amplitude
2710:54503935
2662:10408194
2580:11390524
2510:18481860
2463:18201762
2367:54866017
2307:: 5–25.
2288:54726017
2226:55678660
2046:15076973
1979:See also
1662:variance
32:spectrum
2732:Bibcode
2690:Bibcode
2642:Bibcode
2601:Bibcode
2560:Bibcode
2490:Bibcode
2443:Bibcode
2415:3032930
2395:Bibcode
2347:Bibcode
2309:Bibcode
2268:Bibcode
2206:Bibcode
2156:Bibcode
2112:Bibcode
2089:2671466
2069:Bibcode
2026:Bibcode
1664:of the
1164:, with
20:SigSpec
2752:
2708:
2660:
2578:
2529:
2508:
2461:
2413:
2365:
2286:
2224:
2174:
2087:
2044:
113:pairs
77:, the
64:random
2750:S2CID
2706:S2CID
2680:arXiv
2658:S2CID
2632:arXiv
2576:S2CID
2550:arXiv
2506:S2CID
2480:arXiv
2459:S2CID
2433:arXiv
2411:S2CID
2385:arXiv
2363:S2CID
2335:(PDF)
2284:S2CID
2258:arXiv
2222:S2CID
2196:arXiv
2172:S2CID
2146:arXiv
2085:S2CID
2042:S2CID
2016:arXiv
173:phase
2527:ISBN
2240:link
1920:sock
1911:>
1898:<
1816:sock
1807:>
1794:<
1648:>
1635:<
1627:and
1458:sock
1424:sock
1415:>
1402:<
1361:>
1348:<
1340:sock
171:and
2740:doi
2728:502
2698:doi
2676:494
2650:doi
2628:496
2609:doi
2597:151
2568:doi
2546:484
2498:doi
2476:481
2451:doi
2429:478
2403:doi
2381:483
2355:doi
2343:671
2317:doi
2305:151
2276:doi
2254:480
2214:doi
2192:379
2164:doi
2142:134
2120:doi
2108:263
2077:doi
2034:doi
2012:467
1884:log
1850:sig
1759:exp
1554:sin
1491:cos
1367:exp
1265:cos
1217:sin
1175:tan
1069:sin
985:sin
834:cos
750:cos
623:sin
553:cos
494:cos
438:sin
382:cos
322:sin
267:tan
161:pdf
66:?”
44:sig
2779::
2748:.
2738:.
2726:.
2722:.
2704:.
2696:.
2688:.
2674:.
2656:.
2648:.
2640:.
2626:.
2607:.
2595:.
2591:.
2574:.
2566:.
2558:.
2544:.
2504:.
2496:.
2488:.
2474:.
2457:.
2449:.
2441:.
2427:.
2409:.
2401:.
2393:.
2379:.
2361:.
2353:.
2341:.
2337:.
2315:.
2303:.
2299:.
2282:.
2274:.
2266:.
2252:.
2236:}}
2232:{{
2220:.
2212:.
2204:.
2190:.
2170:.
2162:.
2154:.
2140:.
2118:.
2106:.
2083:.
2075:.
2065:39
2063:.
2040:.
2032:.
2024:.
2010:.
1742:FA
1726:,
1694:.
229:,
202:,
2756:.
2742::
2734::
2712:.
2700::
2692::
2682::
2664:.
2652::
2644::
2634::
2617:.
2611::
2603::
2582:.
2570::
2562::
2552::
2535:.
2512:.
2500::
2492::
2482::
2465:.
2453::
2445::
2435::
2417:.
2405::
2397::
2387::
2369:.
2357::
2349::
2325:.
2319::
2311::
2290:.
2278::
2270::
2260::
2242:)
2228:.
2216::
2208::
2198::
2178:.
2166::
2158::
2148::
2126:.
2122::
2114::
2091:.
2079::
2071::
2048:.
2036::
2028::
2018::
1946:A
1923:.
1906:2
1902:x
1895:4
1890:e
1879:2
1875:A
1871:K
1865:=
1862:)
1859:A
1856:(
1824:.
1820:)
1802:2
1798:x
1791:4
1784:2
1780:A
1776:K
1766:(
1756:=
1753:)
1750:A
1747:(
1714:A
1680:k
1676:x
1643:2
1639:x
1611:]
1603:2
1598:0
1588:)
1582:0
1567:(
1558:2
1547:+
1540:2
1535:0
1525:)
1519:0
1504:(
1495:2
1483:[
1479:=
1476:)
1470:,
1464:(
1432:,
1428:)
1410:2
1406:x
1399:4
1392:2
1388:A
1384:K
1374:(
1356:2
1352:x
1345:2
1334:A
1331:K
1325:=
1322:)
1319:A
1316:(
1287:,
1279:k
1275:t
1260:1
1254:K
1249:0
1246:=
1243:k
1231:k
1227:t
1212:1
1206:K
1201:0
1198:=
1195:k
1184:=
1129:.
1123:)
1117:2
1112:]
1107:)
1101:0
1088:k
1084:t
1076:(
1064:1
1058:K
1053:0
1050:=
1047:l
1038:[
1029:)
1023:0
1010:k
1006:t
998:(
989:2
979:1
973:K
968:0
965:=
962:k
954:K
950:(
942:2
938:K
934:2
927:=
922:0
894:,
888:)
882:2
877:]
872:)
866:0
853:k
849:t
841:(
829:1
823:K
818:0
815:=
812:l
803:[
794:)
788:0
775:k
771:t
763:(
754:2
744:1
738:K
733:0
730:=
727:k
719:K
715:(
707:2
703:K
699:2
692:=
687:0
659:,
651:2
645:)
637:k
633:t
618:1
612:K
607:0
604:=
601:k
591:(
586:+
581:2
575:)
567:k
563:t
548:1
542:K
537:0
534:=
531:k
521:(
511:k
507:t
500:2
489:1
483:K
478:0
475:=
472:k
464:K
458:)
452:k
448:t
433:1
427:K
422:0
419:=
416:k
407:(
402:)
396:k
392:t
377:1
371:K
366:0
363:=
360:k
351:(
347:2
339:k
335:t
328:2
317:1
311:K
306:0
303:=
300:k
292:K
286:=
281:0
273:2
242:0
215:0
188:0
147:)
142:k
138:x
134:,
129:k
125:t
121:(
101:K
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.