97:
4035:
Bogachev, Mikhail I.; Lyanova, Asya I.; Sinitca, Aleksandr M.; Pyko, Svetlana A.; Pyko, Nikita S.; Kuzmenko, Alexander V.; Romanov, Sergey A.; Brikova, Olga I.; Tsygankova, Margarita; Ivkin, Dmitry Y.; Okovityi, Sergey V.; Prikhodko, Veronika A.; Kaplun, Dmitrii I.; Sysoev, Yuri I.; Kayumov, Airat R.
4038:"Understanding the complex interplay of persistent and antipersistent regimes in animal movement trajectories as a prominent characteristic of their behavioral pattern profiles: Towards an automated and robust model based quantification of anxiety test data"
1002:
2229:
681:
3711:
1118:
268:
2542:
The DFA method has been applied to many systems, e.g. DNA sequences, neuronal oscillations, speech pathology detection, heartbeat fluctuation in different sleep stages, and animal behavior pattern analysis.
350:
495:
2453:
852:
867:
2630:
3757:
2683:
1908:
1842:
1288:
436:
195:
2892:
2337:
2283:
2721:
1775:
The standard DFA algorithm given above removes a linear trend in each segment. If we remove a degree-n polynomial trend in each segment, it is called DFAn, or higher order DFA.
2757:
1597:
1463:
2791:
1214:
2093:
1661:
1530:
2532:
2455:. Essentially, the scaling exponents need not be independent of the scale of the system. In particular, DFA measures the scaling-behavior of the second moment-fluctuations.
1502:
1427:
561:
528:
2491:
1561:
1363:
3205:
3170:
3018:
2983:
1632:
3132:
2945:
2366:
3249:
3062:
2822:
2581:
1748:
1693:
1386:
1242:
2842:
714:
2073:
2046:
2019:
1992:
1962:
1935:
1869:
1803:
741:
3803:
Hardstone, Richard; Poil, Simon-Shlomo; Schiavone, Giuseppina; Jansen, Rick; Nikulin, Vadim V.; Mansvelder, Huibert D.; Linkenkaer-Hansen, Klaus (1 January 2012).
1317:
3275:
2392:
3225:
3082:
3038:
1717:
1337:
1161:
1141:
761:
118:
4136:
Heneghan; et al. (2000). "Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes".
4220:
566:
3387:
Hardstone, Richard; Poil, Simon-Shlomo; Schiavone, Giuseppina; Jansen, Rick; Nikulin, Vadim; Mansvelder, Huibert; Linkenkaer-Hansen, Klaus (2012).
2458:
Kantelhardt et al. intended this scaling exponent as a generalization of the classical Hurst exponent. The classical Hurst exponent corresponds to
1163:, then one can either discard the remainder of the sequence, or repeat the procedure on the reversed sequence, then take their root-mean-square.)
1010:
3448:"Multifractal temporally weighted detrended fluctuation analysis and its application in the analysis of scaling behavior in temperature series"
1726:
Also, there are many scaling exponent-like quantities that can be measured for a self-similar time series, including the divider dimension and
96:
4228:
203:
84:
et al. introduced DFA in 1994 in a paper that has been cited over 3,000 times as of 2022 and represents an extension of the (ordinary)
3493:
Peng, C.K.; et al. (1994). "Quantification of scaling exponents and crossover phenomena in nonstationary heartbeat time series".
3960:
276:
3857:
Buldyrev; et al. (1995). "Long-Range
Correlation-Properties of Coding And Noncoding Dna-Sequences- Genbank Analysis".
4252:
1720:
66:, except that DFA may also be applied to signals whose underlying statistics (such as mean and variance) or dynamics are
3937:
441:
2798:
1937:(visible as short sections of "flat plateaus"). In this regard, DFA1 removes the mean from segments of the time series
2400:
3295:
3093:
2903:
4217:
1664:
774:
4268:
3654:
Kantelhardt J.W.; et al. (2001). "Detecting long-range correlations with detrended fluctuation analysis".
2586:
52:
3781:
Movahed, M. Sadegh; et al. (2006). "Multifractal detrended fluctuation analysis of sunspot time series".
3601:
Clauset, Aaron; Rohilla
Shalizi, Cosma; Newman, M. E. J. (2009). "Power-Law Distributions in Empirical Data".
2639:
3984:
Bunde A.; et al. (2000). "Correlated and uncorrelated regions in heart-rate fluctuations during sleep".
1874:
1808:
1251:
1723:(MLE), rather than least-squares has been shown to better approximate the scaling, or power-law, exponent.
377:
135:
2851:
2288:
2234:
1759:
85:
3905:
2691:
684:
2727:
2082:
removes constant trends in the original sequence and thus, in its detrending it is equivalent to DFA1.
1568:
1434:
3447:
2764:
1172:
48:
4145:
4094:
3993:
3866:
3733:
3673:
3620:
3557:
3502:
3343:
3305:
2560:
1637:
1509:
28:
20:
3900:
Bunde A, Havlin S (1996). "Fractals and
Disordered Systems, Springer, Berlin, Heidelberg, New York".
2496:
1473:
1398:
533:
500:
3290:
1755:
997:{\displaystyle F(n,i)={\sqrt {{\frac {1}{n}}\sum _{t=in+1}^{in+n}\left(X_{t}-Y_{t,n}\right)^{2}}}.}
2461:
1663:
would correspond to uncorrelated white noise. When the exponent is between 0 and 1, the result is
1540:
1342:
4187:
Taqqu, Murad S.; et al. (1995). "Estimators for long-range dependence: an empirical study".
4169:
4118:
4084:
4057:
4037:
4017:
3966:
3749:
3723:
3710:
H.E. Stanley, J.W. Kantelhardt; S.A. Zschiegner; E. Koscielny-Bunde; S. Havlin; A. Bunde (2002).
3689:
3663:
3636:
3610:
3526:
3475:
3369:
3175:
3137:
2988:
2950:
1613:
67:
3099:
2909:
2342:
4273:
4161:
4110:
4009:
3956:
3882:
3836:
3583:
3518:
3467:
3428:
3410:
3361:
1751:
75:
3234:
3047:
2807:
2566:
1733:
1678:
1371:
1227:
1167:
4196:
4153:
4102:
4049:
4001:
3948:
3874:
3826:
3816:
3741:
3681:
3628:
3573:
3565:
3510:
3459:
3418:
3400:
3351:
2827:
693:
3463:
2051:
2024:
1997:
1970:
1940:
1913:
1847:
1781:
719:
4224:
3918:
1696:
1293:
81:
71:
3254:
2371:
1695:
for any time series, it does not necessarily imply that the time series is self-similar.
4149:
4098:
3997:
3870:
3737:
3677:
3624:
3561:
3506:
3347:
2224:{\displaystyle F_{q}(n)=\left({\frac {1}{N/n}}\sum _{i=1}^{N/n}F(n,i)^{q}\right)^{1/q}.}
3945:
2006 IEEE International
Conference on Acoustics Speed and Signal Processing Proceedings
3831:
3804:
3578:
3545:
3423:
3388:
3310:
3228:
3210:
3067:
3041:
3023:
2845:
2633:
2079:
1727:
1702:
1600:
1389:
1322:
1146:
1126:
746:
103:
70:(changing with time). It is related to measures based upon spectral techniques such as
63:
3745:
3685:
4262:
4061:
3479:
3300:
1245:
764:
676:{\displaystyle \log(n_{2})-\log(n_{1})\approx \log(n_{3})-\log(n_{2})\approx \cdots }
40:
4173:
4021:
3970:
3753:
3693:
4122:
3640:
3373:
24:
4075:
Hu, K.; et al. (2001). "Effect of trends on detrended fluctuation analysis".
3530:
4005:
3952:
3933:
1607:
1466:
1392:, with the precise value giving information about the series self-correlations:
368:
364:
130:
44:
4106:
4053:
4200:
1533:
4240:
code for rapidly calculating the DFA scaling exponent on very large datasets.
4157:
3878:
3821:
3471:
3414:
3405:
3356:
3331:
3805:"Detrended Fluctuation Analysis: A Scale-Free View on Neuronal Oscillations"
3389:"Detrended Fluctuation Analysis: A Scale-Free View on Neuronal Oscillations"
56:
4243:
4165:
4114:
4013:
3840:
3587:
3522:
3432:
2021:. For example, DFA1 removes linear trends from segments of the time series
1113:{\displaystyle F(n)={\sqrt {{\frac {1}{N/n}}\sum _{i=1}^{N/n}F(n,i)^{2}}}.}
3886:
3712:"Multifractal detrended fluctuation analysis of nonstationary time series"
3365:
2801:. The relation of DFA to the power spectrum method has been well studied.
1699:
requires the log-log graph to be sufficiently linear over a wide range of
3668:
3278:
4089:
3728:
2048:
before quantifying the fluctuation, DFA1 removes parabolic trends from
3632:
3569:
43:
of a signal. It is useful for analysing time series that appear to be
4237:
4234:
3514:
361:
263:{\displaystyle \langle x\rangle ={\frac {1}{N}}\sum _{t=1}^{N}x_{t}}
4249:
3615:
95:
4255:
implementation of (Multifractal) Detrended
Fluctuation Analysis.
3281:
of FGN, thus, the exponents of their power spectra differ by 2.
563:, and the sequence is roughly distributed evenly in log-scale:
3938:"Nonlinear, Biophysically-Informed Speech Pathology Detection"
360:, of the original time series. For example, the profile of an
345:{\displaystyle X_{t}=\sum _{i=1}^{t}(x_{i}-\langle x\rangle )}
857:
Compute the root-mean-square deviation from the local trend (
100:
DFA on a
Brownian motion process, with increasing values of
4218:
Tutorial on how to calculate detrended fluctuation analysis
1675:
Though the DFA algorithm always produces a positive number
1758:
has, though in certain special cases it is related to the
1754:, and does not have certain desirable properties that the
2559:
In the case of power-law decaying auto-correlations, the
2550:
Relations to other methods, for specific types of signal
3783:
Journal of
Statistical Mechanics: Theory and Experiment
3452:
Journal of
Statistical Mechanics: Theory and Experiment
2086:
Generalization to different moments (multifractal DFA)
1771:
Generalization to polynomial trends (higher order DFA)
1642:
3277:. In this context, FBM is the cumulative sum or the
3257:
3237:
3213:
3178:
3140:
3102:
3070:
3050:
3026:
2991:
2953:
2912:
2854:
2830:
2810:
2767:
2730:
2694:
2642:
2589:
2569:
2499:
2464:
2403:
2374:
2345:
2291:
2237:
2096:
2054:
2027:
2000:
1973:
1943:
1916:
1877:
1850:
1811:
1784:
1736:
1719:. Furthermore, a combination of techniques including
1705:
1681:
1640:
1616:
1571:
1543:
1512:
1476:
1437:
1401:
1374:
1345:
1325:
1296:
1254:
1230:
1175:
1149:
1129:
1013:
1004:
And their root-mean-square is the total fluctuation:
870:
777:
749:
722:
696:
569:
536:
503:
444:
380:
279:
206:
138:
106:
2555:
For signals with power-law-decaying autocorrelation
4246:A good overview of DFA and C code to calculate it.
3269:
3243:
3219:
3199:
3164:
3126:
3076:
3056:
3032:
3012:
2977:
2939:
2886:
2836:
2816:
2785:
2751:
2715:
2677:
2624:
2575:
2526:
2485:
2447:
2386:
2360:
2331:
2277:
2223:
2067:
2040:
2013:
1986:
1956:
1929:
1902:
1863:
1836:
1797:
1742:
1711:
1687:
1655:
1626:
1591:
1555:
1524:
1496:
1457:
1421:
1380:
1357:
1331:
1311:
1282:
1236:
1208:
1155:
1135:
1112:
996:
846:
755:
735:
708:
675:
555:
522:
490:{\displaystyle n_{1}<n_{2}<\cdots <n_{k}}
489:
430:
344:
262:
189:
112:
2671:
2618:
2285:, If there is a strong linearity in the plot of
88:(FA), which is affected by non-stationarities.
3705:
3703:
2546:The effect of trends on DFA has been studied.
2448:{\displaystyle F_{q}(n)\propto n^{\alpha (q)}}
39:) is a method for determining the statistical
8:
3798:
3796:
3776:
3774:
1897:
1891:
1831:
1825:
1244:on the log-log plot indicates a statistical
425:
387:
336:
330:
213:
207:
3852:
3850:
3546:"Revisiting detrended fluctuation analysis"
2824:is tied to the slope of the power spectrum
847:{\displaystyle Y_{1,n},Y_{2,n},...,Y_{N,n}}
4088:
3947:. Vol. 2. pp. II-1080–II-1083.
3830:
3820:
3727:
3667:
3614:
3577:
3422:
3404:
3355:
3256:
3236:
3212:
3177:
3139:
3101:
3069:
3049:
3025:
2990:
2952:
2911:
2876:
2853:
2829:
2809:
2766:
2729:
2693:
2662:
2641:
2625:{\displaystyle C(L)\sim L^{-\gamma }\!\ }
2609:
2588:
2568:
2498:
2463:
2430:
2408:
2402:
2397:Multifractal systems scale as a function
2373:
2344:
2314:
2290:
2260:
2236:
2208:
2204:
2193:
2164:
2160:
2149:
2134:
2125:
2101:
2095:
2059:
2053:
2032:
2026:
2005:
1999:
1978:
1972:
1948:
1942:
1921:
1915:
1882:
1876:
1855:
1849:
1816:
1810:
1789:
1783:
1735:
1704:
1680:
1641:
1639:
1617:
1615:
1581:
1570:
1542:
1511:
1486:
1475:
1447:
1436:
1411:
1400:
1373:
1344:
1324:
1295:
1274:
1253:
1229:
1174:
1148:
1128:
1099:
1070:
1066:
1055:
1040:
1031:
1029:
1012:
983:
966:
953:
928:
908:
894:
892:
869:
832:
801:
782:
776:
748:
727:
721:
695:
658:
633:
608:
583:
568:
541:
535:
508:
502:
481:
462:
449:
443:
419:
394:
379:
321:
308:
297:
284:
278:
254:
244:
233:
219:
205:
181:
156:
143:
137:
105:
4042:Biomedical Signal Processing and Control
3332:"Mosaic organization of DNA nucleotides"
2678:{\displaystyle P(f)\sim f^{-\beta }\!\ }
1606:Because the expected displacement in an
683:. In other words, it is approximately a
62:The obtained exponent is similar to the
3322:
2797:The relations can be derived using the
1903:{\displaystyle x_{t}-\langle x\rangle }
1837:{\displaystyle x_{t}-\langle x\rangle }
1283:{\displaystyle F(n)\propto n^{\alpha }}
854:be the resulting piecewise-linear fit.
3914:
3903:
2685:. The three exponents are related by:
1730:. Therefore, the DFA scaling exponent
431:{\displaystyle T=\{n_{1},...,n_{k}\}}
190:{\displaystyle x_{1},x_{2},...,x_{N}}
7:
4229:Neurophysiological Biomarker Toolbox
2887:{\displaystyle \alpha =(\beta +1)/2}
2332:{\displaystyle \log n-\log F_{q}(n)}
2278:{\displaystyle \log n-\log F_{q}(n)}
1964:before quantifying the fluctuation.
743:into consecutive segments of length
3544:Bryce, R.M.; Sprague, K.B. (2012).
3446:Zhou, Yu; Leung, Yee (2010-06-21).
2090:DFA can be generalized by computing
763:. Within each segment, compute the
2716:{\displaystyle \gamma =2-2\alpha }
14:
2752:{\displaystyle \beta =2\alpha -1}
1592:{\displaystyle \alpha \simeq 3/2}
1458:{\displaystyle \alpha \simeq 1/2}
3464:10.1088/1742-5468/2010/06/P06021
3330:Peng, C.K.; et al. (1994).
2786:{\displaystyle \gamma =1-\beta }
2368:. DFA is the special case where
2231:then making the log-log plot of
1762:for the graph of a time series.
1209:{\displaystyle \log n-\log F(n)}
1967:Similarly, a degree n trend in
1910:, which is a constant trend in
1656:{\displaystyle {\tfrac {1}{2}}}
1525:{\displaystyle \alpha \simeq 1}
3159:
3147:
3121:
3109:
3088:For fractional Brownian motion
2972:
2960:
2934:
2919:
2873:
2861:
2652:
2646:
2599:
2593:
2527:{\displaystyle H=\alpha (2)-1}
2515:
2509:
2480:
2474:
2440:
2434:
2420:
2414:
2355:
2349:
2326:
2320:
2272:
2266:
2190:
2177:
2113:
2107:
1497:{\displaystyle \alpha >1/2}
1422:{\displaystyle \alpha <1/2}
1306:
1300:
1264:
1258:
1203:
1197:
1096:
1083:
1023:
1017:
886:
874:
664:
651:
639:
626:
614:
601:
589:
576:
556:{\displaystyle n_{k}\approx N}
523:{\displaystyle n_{1}\approx 4}
339:
314:
33:detrended fluctuation analysis
1:
3746:10.1016/s0378-4371(02)01383-3
3686:10.1016/s0378-4371(01)00144-3
2898:For fractional Gaussian noise
1721:maximum likelihood estimation
1319:monotonically increases with
2844:and is used to describe the
2486:{\displaystyle H=\alpha (2)}
1556:{\displaystyle \alpha >1}
1358:{\displaystyle \alpha >0}
4006:10.1103/physrevlett.85.3736
3953:10.1109/ICASSP.2006.1660534
3200:{\displaystyle \beta =2H+1}
3165:{\displaystyle \alpha \in }
3013:{\displaystyle \beta =2H-1}
2978:{\displaystyle \alpha \in }
1994:is a degree (n-1) trend in
1627:{\displaystyle {\sqrt {N}}}
1563:: non-stationary, unbounded
1388:is a generalization of the
4290:
4107:10.1103/physreve.64.011114
4054:10.1016/j.bspc.2022.104409
3932:Little, M.; McSharry, P.;
3296:Self-organized criticality
3127:{\displaystyle \beta \in }
3094:fractional Brownian motion
2940:{\displaystyle \beta \in }
2493:for stationary cases, and
2361:{\displaystyle \alpha (q)}
1671:Pitfalls in interpretation
200:Compute its average value
51:, e.g. power-law decaying
4201:10.1142/S0218348X95000692
2904:fractional Gaussian noise
2534:for nonstationary cases.
1665:fractional Gaussian noise
1224:A straight line of slope
4158:10.1103/physreve.62.6103
3879:10.1103/physreve.51.5084
3822:10.3389/fphys.2012.00450
3406:10.3389/fphys.2012.00450
3357:10.1103/physreve.49.1685
2563:decays with an exponent
1608:uncorrelated random walk
53:autocorrelation function
3809:Frontiers in Physiology
3393:Frontiers in Physiology
3244:{\displaystyle \alpha }
3057:{\displaystyle \alpha }
2817:{\displaystyle \alpha }
2799:Wiener–Khinchin theorem
2576:{\displaystyle \gamma }
1871:is a constant trend in
1805:is a cumulative sum of
1743:{\displaystyle \alpha }
1688:{\displaystyle \alpha }
1610:of length N grows like
1381:{\displaystyle \alpha }
1237:{\displaystyle \alpha }
767:straight-line fit (the
438:of integers, such that
3936:; Roberts, S. (2006).
3913:Cite journal requires
3271:
3245:
3221:
3201:
3166:
3128:
3078:
3058:
3034:
3014:
2979:
2941:
2888:
2848:by this relationship:
2838:
2837:{\displaystyle \beta }
2818:
2787:
2753:
2717:
2679:
2626:
2577:
2528:
2487:
2449:
2388:
2362:
2333:
2279:
2225:
2173:
2069:
2042:
2015:
1988:
1958:
1931:
1904:
1865:
1838:
1799:
1760:box-counting dimension
1744:
1713:
1689:
1657:
1628:
1593:
1557:
1526:
1498:
1459:
1423:
1382:
1359:
1333:
1313:
1284:
1238:
1210:
1157:
1137:
1114:
1079:
998:
942:
848:
757:
737:
716:, divide the sequence
710:
709:{\displaystyle n\in T}
677:
557:
524:
491:
432:
346:
313:
273:Sum it into a process
264:
249:
191:
121:
114:
3272:
3246:
3222:
3202:
3167:
3129:
3079:
3059:
3035:
3015:
2980:
2942:
2889:
2839:
2819:
2788:
2754:
2718:
2680:
2627:
2578:
2529:
2488:
2450:
2389:
2363:
2339:, then that slope is
2334:
2280:
2226:
2145:
2070:
2068:{\displaystyle x_{t}}
2043:
2041:{\displaystyle x_{t}}
2016:
2014:{\displaystyle x_{t}}
1989:
1987:{\displaystyle X_{t}}
1959:
1957:{\displaystyle x_{t}}
1932:
1930:{\displaystyle x_{t}}
1905:
1866:
1864:{\displaystyle X_{t}}
1839:
1800:
1798:{\displaystyle X_{t}}
1745:
1714:
1690:
1658:
1629:
1594:
1558:
1527:
1499:
1460:
1424:
1383:
1368:The scaling exponent
1360:
1334:
1314:
1285:
1239:
1211:
1158:
1138:
1115:
1051:
999:
904:
849:
758:
738:
736:{\displaystyle X_{t}}
711:
685:geometric progression
678:
558:
525:
492:
433:
347:
293:
265:
229:
192:
115:
99:
47:processes (diverging
4227:in Matlab using the
3306:Time series analysis
3255:
3251:for FBM is equal to
3235:
3211:
3176:
3138:
3100:
3068:
3064:for FGN is equal to
3048:
3024:
2989:
2951:
2910:
2852:
2828:
2808:
2765:
2728:
2692:
2640:
2587:
2567:
2561:correlation function
2497:
2462:
2401:
2372:
2343:
2289:
2235:
2094:
2052:
2025:
1998:
1971:
1941:
1914:
1875:
1848:
1844:, a linear trend in
1809:
1782:
1734:
1703:
1679:
1638:
1614:
1569:
1541:
1510:
1474:
1435:
1399:
1372:
1343:
1323:
1312:{\displaystyle F(n)}
1294:
1252:
1228:
1173:
1147:
1143:is not divisible by
1127:
1011:
868:
775:
747:
720:
694:
567:
534:
501:
442:
378:
277:
204:
136:
104:
86:fluctuation analysis
29:time series analysis
21:stochastic processes
4150:2000PhRvE..62.6103H
4099:2001PhRvE..64a1114H
3998:2000PhRvL..85.3736B
3871:1995PhRvE..51.5084B
3738:2002PhyA..316...87K
3678:2001PhyA..295..441K
3625:2009SIAMR..51..661C
3562:2012NatSR...2E.315B
3507:1995Chaos...5...82P
3348:1994PhRvE..49.1685P
3291:Multifractal system
3270:{\displaystyle H+1}
2387:{\displaystyle q=2}
1756:Hausdorff dimension
4223:2019-02-03 at the
3267:
3241:
3217:
3197:
3162:
3124:
3074:
3054:
3030:
3010:
2975:
2937:
2884:
2834:
2814:
2783:
2749:
2713:
2675:
2632:. In addition the
2622:
2573:
2524:
2483:
2445:
2384:
2358:
2329:
2275:
2221:
2065:
2038:
2011:
1984:
1954:
1927:
1900:
1861:
1834:
1795:
1740:
1709:
1685:
1653:
1651:
1624:
1589:
1553:
1522:
1494:
1455:
1419:
1378:
1355:
1329:
1309:
1280:
1234:
1206:
1153:
1133:
1110:
994:
844:
753:
733:
706:
673:
553:
520:
487:
428:
342:
260:
187:
122:
110:
3992:(17): 3736–3739.
3633:10.1137/070710111
3570:10.1038/srep00315
3220:{\displaystyle H}
3077:{\displaystyle H}
3033:{\displaystyle H}
2674:
2621:
2143:
1752:fractal dimension
1712:{\displaystyle n}
1650:
1634:, an exponent of
1622:
1429:: anti-correlated
1339:, we always have
1332:{\displaystyle n}
1156:{\displaystyle n}
1136:{\displaystyle N}
1105:
1049:
989:
902:
756:{\displaystyle n}
227:
113:{\displaystyle n}
76:Fourier transform
4281:
4205:
4204:
4184:
4178:
4177:
4144:(5): 6103–6110.
4133:
4127:
4126:
4092:
4072:
4066:
4065:
4032:
4026:
4025:
3981:
3975:
3974:
3942:
3929:
3923:
3922:
3916:
3911:
3909:
3901:
3897:
3891:
3890:
3865:(5): 5084–5091.
3854:
3845:
3844:
3834:
3824:
3800:
3791:
3790:
3778:
3769:
3768:
3766:
3765:
3756:. Archived from
3731:
3707:
3698:
3697:
3671:
3669:cond-mat/0102214
3662:(3–4): 441–454.
3651:
3645:
3644:
3618:
3598:
3592:
3591:
3581:
3541:
3535:
3534:
3515:10.1063/1.166141
3490:
3484:
3483:
3443:
3437:
3436:
3426:
3408:
3384:
3378:
3377:
3359:
3342:(2): 1685–1689.
3327:
3276:
3274:
3273:
3268:
3250:
3248:
3247:
3242:
3226:
3224:
3223:
3218:
3206:
3204:
3203:
3198:
3171:
3169:
3168:
3163:
3133:
3131:
3130:
3125:
3083:
3081:
3080:
3075:
3063:
3061:
3060:
3055:
3039:
3037:
3036:
3031:
3019:
3017:
3016:
3011:
2984:
2982:
2981:
2976:
2946:
2944:
2943:
2938:
2893:
2891:
2890:
2885:
2880:
2843:
2841:
2840:
2835:
2823:
2821:
2820:
2815:
2792:
2790:
2789:
2784:
2758:
2756:
2755:
2750:
2722:
2720:
2719:
2714:
2684:
2682:
2681:
2676:
2672:
2670:
2669:
2631:
2629:
2628:
2623:
2619:
2617:
2616:
2582:
2580:
2579:
2574:
2533:
2531:
2530:
2525:
2492:
2490:
2489:
2484:
2454:
2452:
2451:
2446:
2444:
2443:
2413:
2412:
2393:
2391:
2390:
2385:
2367:
2365:
2364:
2359:
2338:
2336:
2335:
2330:
2319:
2318:
2284:
2282:
2281:
2276:
2265:
2264:
2230:
2228:
2227:
2222:
2217:
2216:
2212:
2203:
2199:
2198:
2197:
2172:
2168:
2159:
2144:
2142:
2138:
2126:
2106:
2105:
2074:
2072:
2071:
2066:
2064:
2063:
2047:
2045:
2044:
2039:
2037:
2036:
2020:
2018:
2017:
2012:
2010:
2009:
1993:
1991:
1990:
1985:
1983:
1982:
1963:
1961:
1960:
1955:
1953:
1952:
1936:
1934:
1933:
1928:
1926:
1925:
1909:
1907:
1906:
1901:
1887:
1886:
1870:
1868:
1867:
1862:
1860:
1859:
1843:
1841:
1840:
1835:
1821:
1820:
1804:
1802:
1801:
1796:
1794:
1793:
1749:
1747:
1746:
1741:
1718:
1716:
1715:
1710:
1694:
1692:
1691:
1686:
1662:
1660:
1659:
1654:
1652:
1643:
1633:
1631:
1630:
1625:
1623:
1618:
1598:
1596:
1595:
1590:
1585:
1562:
1560:
1559:
1554:
1531:
1529:
1528:
1523:
1503:
1501:
1500:
1495:
1490:
1465:: uncorrelated,
1464:
1462:
1461:
1456:
1451:
1428:
1426:
1425:
1420:
1415:
1387:
1385:
1384:
1379:
1364:
1362:
1361:
1356:
1338:
1336:
1335:
1330:
1318:
1316:
1315:
1310:
1289:
1287:
1286:
1281:
1279:
1278:
1243:
1241:
1240:
1235:
1215:
1213:
1212:
1207:
1162:
1160:
1159:
1154:
1142:
1140:
1139:
1134:
1119:
1117:
1116:
1111:
1106:
1104:
1103:
1078:
1074:
1065:
1050:
1048:
1044:
1032:
1030:
1003:
1001:
1000:
995:
990:
988:
987:
982:
978:
977:
976:
958:
957:
941:
927:
903:
895:
893:
853:
851:
850:
845:
843:
842:
812:
811:
793:
792:
762:
760:
759:
754:
742:
740:
739:
734:
732:
731:
715:
713:
712:
707:
682:
680:
679:
674:
663:
662:
638:
637:
613:
612:
588:
587:
562:
560:
559:
554:
546:
545:
529:
527:
526:
521:
513:
512:
496:
494:
493:
488:
486:
485:
467:
466:
454:
453:
437:
435:
434:
429:
424:
423:
399:
398:
351:
349:
348:
343:
326:
325:
312:
307:
289:
288:
269:
267:
266:
261:
259:
258:
248:
243:
228:
220:
196:
194:
193:
188:
186:
185:
161:
160:
148:
147:
119:
117:
116:
111:
49:correlation time
16:Statistical term
4289:
4288:
4284:
4283:
4282:
4280:
4279:
4278:
4269:Autocorrelation
4259:
4258:
4225:Wayback Machine
4214:
4209:
4208:
4186:
4185:
4181:
4135:
4134:
4130:
4090:physics/0103018
4074:
4073:
4069:
4034:
4033:
4029:
3983:
3982:
3978:
3963:
3940:
3931:
3930:
3926:
3912:
3902:
3899:
3898:
3894:
3856:
3855:
3848:
3802:
3801:
3794:
3780:
3779:
3772:
3763:
3761:
3729:physics/0202070
3722:(1–4): 87–114.
3709:
3708:
3701:
3653:
3652:
3648:
3600:
3599:
3595:
3543:
3542:
3538:
3492:
3491:
3487:
3445:
3444:
3440:
3386:
3385:
3381:
3329:
3328:
3324:
3319:
3287:
3253:
3252:
3233:
3232:
3209:
3208:
3174:
3173:
3136:
3135:
3098:
3097:
3096:(FBM), we have
3090:
3066:
3065:
3046:
3045:
3022:
3021:
2987:
2986:
2949:
2948:
2908:
2907:
2906:(FGN), we have
2900:
2850:
2849:
2826:
2825:
2806:
2805:
2763:
2762:
2726:
2725:
2690:
2689:
2658:
2638:
2637:
2605:
2585:
2584:
2565:
2564:
2557:
2552:
2540:
2495:
2494:
2460:
2459:
2426:
2404:
2399:
2398:
2370:
2369:
2341:
2340:
2310:
2287:
2286:
2256:
2233:
2232:
2189:
2130:
2124:
2120:
2119:
2097:
2092:
2091:
2088:
2055:
2050:
2049:
2028:
2023:
2022:
2001:
1996:
1995:
1974:
1969:
1968:
1944:
1939:
1938:
1917:
1912:
1911:
1878:
1873:
1872:
1851:
1846:
1845:
1812:
1807:
1806:
1785:
1780:
1779:
1773:
1768:
1766:Generalizations
1732:
1731:
1701:
1700:
1697:Self-similarity
1677:
1676:
1673:
1636:
1635:
1612:
1611:
1567:
1566:
1539:
1538:
1508:
1507:
1472:
1471:
1433:
1432:
1397:
1396:
1370:
1369:
1341:
1340:
1321:
1320:
1292:
1291:
1270:
1250:
1249:
1226:
1225:
1222:
1171:
1170:
1145:
1144:
1125:
1124:
1095:
1036:
1009:
1008:
962:
949:
948:
944:
943:
866:
865:
828:
797:
778:
773:
772:
745:
744:
723:
718:
717:
692:
691:
654:
629:
604:
579:
565:
564:
537:
532:
531:
504:
499:
498:
497:, the smallest
477:
458:
445:
440:
439:
415:
390:
376:
375:
317:
280:
275:
274:
250:
202:
201:
177:
152:
139:
134:
133:
127:
102:
101:
94:
72:autocorrelation
17:
12:
11:
5:
4287:
4285:
4277:
4276:
4271:
4261:
4260:
4257:
4256:
4247:
4241:
4232:
4213:
4212:External links
4210:
4207:
4206:
4195:(4): 785–798.
4179:
4128:
4067:
4036:(March 2023).
4027:
3976:
3961:
3924:
3915:|journal=
3892:
3846:
3792:
3770:
3699:
3646:
3609:(4): 661–703.
3593:
3536:
3485:
3438:
3379:
3321:
3320:
3318:
3315:
3314:
3313:
3311:Hurst exponent
3308:
3303:
3298:
3293:
3286:
3283:
3266:
3263:
3260:
3240:
3229:Hurst exponent
3216:
3196:
3193:
3190:
3187:
3184:
3181:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3089:
3086:
3073:
3053:
3042:Hurst exponent
3029:
3009:
3006:
3003:
3000:
2997:
2994:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2899:
2896:
2883:
2879:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2846:color of noise
2833:
2813:
2795:
2794:
2782:
2779:
2776:
2773:
2770:
2760:
2748:
2745:
2742:
2739:
2736:
2733:
2723:
2712:
2709:
2706:
2703:
2700:
2697:
2668:
2665:
2661:
2657:
2654:
2651:
2648:
2645:
2634:power spectrum
2615:
2612:
2608:
2604:
2601:
2598:
2595:
2592:
2572:
2556:
2553:
2551:
2548:
2539:
2536:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2482:
2479:
2476:
2473:
2470:
2467:
2442:
2439:
2436:
2433:
2429:
2425:
2422:
2419:
2416:
2411:
2407:
2383:
2380:
2377:
2357:
2354:
2351:
2348:
2328:
2325:
2322:
2317:
2313:
2309:
2306:
2303:
2300:
2297:
2294:
2274:
2271:
2268:
2263:
2259:
2255:
2252:
2249:
2246:
2243:
2240:
2220:
2215:
2211:
2207:
2202:
2196:
2192:
2188:
2185:
2182:
2179:
2176:
2171:
2167:
2163:
2158:
2155:
2152:
2148:
2141:
2137:
2133:
2129:
2123:
2118:
2115:
2112:
2109:
2104:
2100:
2087:
2084:
2062:
2058:
2035:
2031:
2008:
2004:
1981:
1977:
1951:
1947:
1924:
1920:
1899:
1896:
1893:
1890:
1885:
1881:
1858:
1854:
1833:
1830:
1827:
1824:
1819:
1815:
1792:
1788:
1772:
1769:
1767:
1764:
1739:
1728:Hurst exponent
1708:
1684:
1672:
1669:
1649:
1646:
1621:
1604:
1603:
1601:Brownian noise
1588:
1584:
1580:
1577:
1574:
1564:
1552:
1549:
1546:
1536:
1521:
1518:
1515:
1505:
1493:
1489:
1485:
1482:
1479:
1469:
1454:
1450:
1446:
1443:
1440:
1430:
1418:
1414:
1410:
1407:
1404:
1390:Hurst exponent
1377:
1354:
1351:
1348:
1328:
1308:
1305:
1302:
1299:
1277:
1273:
1269:
1266:
1263:
1260:
1257:
1233:
1221:
1220:Interpretation
1218:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1152:
1132:
1121:
1120:
1109:
1102:
1098:
1094:
1091:
1088:
1085:
1082:
1077:
1073:
1069:
1064:
1061:
1058:
1054:
1047:
1043:
1039:
1035:
1028:
1025:
1022:
1019:
1016:
993:
986:
981:
975:
972:
969:
965:
961:
956:
952:
947:
940:
937:
934:
931:
926:
923:
920:
917:
914:
911:
907:
901:
898:
891:
888:
885:
882:
879:
876:
873:
841:
838:
835:
831:
827:
824:
821:
818:
815:
810:
807:
804:
800:
796:
791:
788:
785:
781:
752:
730:
726:
705:
702:
699:
672:
669:
666:
661:
657:
653:
650:
647:
644:
641:
636:
632:
628:
625:
622:
619:
616:
611:
607:
603:
600:
597:
594:
591:
586:
582:
578:
575:
572:
552:
549:
544:
540:
530:, the largest
519:
516:
511:
507:
484:
480:
476:
473:
470:
465:
461:
457:
452:
448:
427:
422:
418:
414:
411:
408:
405:
402:
397:
393:
389:
386:
383:
367:is a standard
354:cumulative sum
352:. This is the
341:
338:
335:
332:
329:
324:
320:
316:
311:
306:
303:
300:
296:
292:
287:
283:
257:
253:
247:
242:
239:
236:
232:
226:
223:
218:
215:
212:
209:
184:
180:
176:
173:
170:
167:
164:
159:
155:
151:
146:
142:
126:
123:
109:
93:
90:
68:non-stationary
64:Hurst exponent
15:
13:
10:
9:
6:
4:
3:
2:
4286:
4275:
4272:
4270:
4267:
4266:
4264:
4254:
4251:
4248:
4245:
4242:
4239:
4236:
4233:
4230:
4226:
4222:
4219:
4216:
4215:
4211:
4202:
4198:
4194:
4190:
4183:
4180:
4175:
4171:
4167:
4163:
4159:
4155:
4151:
4147:
4143:
4139:
4132:
4129:
4124:
4120:
4116:
4112:
4108:
4104:
4100:
4096:
4091:
4086:
4083:(1): 011114.
4082:
4078:
4071:
4068:
4063:
4059:
4055:
4051:
4047:
4043:
4039:
4031:
4028:
4023:
4019:
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3962:1-4244-0469-X
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3806:
3799:
3797:
3793:
3788:
3784:
3777:
3775:
3771:
3760:on 2018-08-28
3759:
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3337:
3333:
3326:
3323:
3316:
3312:
3309:
3307:
3304:
3302:
3301:Self-affinity
3299:
3297:
3294:
3292:
3289:
3288:
3284:
3282:
3280:
3264:
3261:
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3238:
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2075:, and so on.
2060:
2056:
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2002:
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1975:
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1918:
1894:
1888:
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1575:
1572:
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1532:: 1/f-noise,
1519:
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1483:
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1246:self-affinity
1231:
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1200:
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766:
765:least squares
750:
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700:
697:
688:
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670:
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505:
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446:
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416:
412:
409:
406:
403:
400:
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391:
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381:
374:Select a set
372:
370:
366:
363:
359:
355:
333:
327:
322:
318:
309:
304:
301:
298:
294:
290:
285:
281:
271:
255:
251:
245:
240:
237:
234:
230:
224:
221:
216:
210:
198:
182:
178:
174:
171:
168:
165:
162:
157:
153:
149:
144:
140:
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89:
87:
83:
79:
77:
73:
69:
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60:
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41:self-affinity
38:
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4192:
4188:
4182:
4141:
4138:Phys. Rev. E
4137:
4131:
4080:
4077:Phys. Rev. E
4076:
4070:
4045:
4041:
4030:
3989:
3986:Phys. Rev. E
3985:
3979:
3944:
3927:
3906:cite journal
3895:
3862:
3859:Phys. Rev. E
3858:
3812:
3808:
3786:
3782:
3762:. Retrieved
3758:the original
3719:
3715:
3659:
3655:
3649:
3606:
3602:
3596:
3553:
3549:
3539:
3501:(1): 82–87.
3498:
3494:
3488:
3455:
3451:
3441:
3396:
3392:
3382:
3339:
3336:Phys. Rev. E
3335:
3325:
3091:
2901:
2803:
2796:
2558:
2545:
2541:
2538:Applications
2457:
2396:
2089:
2080:R/S analysis
2077:
1966:
1777:
1774:
1725:
1674:
1605:
1504:: correlated
1367:
1223:
1168:log-log plot
1165:
1122:
861:
858:
856:
768:
689:
373:
357:
353:
272:
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25:chaos theory
18:
3603:SIAM Review
3134:, and thus
2947:, and thus
1467:white noise
862:fluctuation
769:local trend
369:random walk
365:white noise
131:time series
45:long-memory
4263:Categories
4048:: 104409.
3764:2011-07-20
3317:References
2636:decays as
2078:The Hurst
1534:pink noise
92:Definition
4244:Physionet
4062:254206934
3934:Moroz, I.
3716:Physica A
3656:Physica A
3616:0706.1062
3480:119901219
3472:1742-5468
3415:1664-042X
3239:α
3180:β
3145:∈
3142:α
3107:∈
3104:β
3052:α
3005:−
2993:β
2958:∈
2955:α
2923:−
2917:∈
2914:β
2865:β
2856:α
2832:β
2812:α
2781:β
2778:−
2769:γ
2744:−
2741:α
2732:β
2711:α
2705:−
2696:γ
2667:β
2664:−
2656:∼
2614:γ
2611:−
2603:∼
2571:γ
2519:−
2507:α
2472:α
2432:α
2424:∝
2347:α
2308:
2302:−
2296:
2254:
2248:−
2242:
2147:∑
1898:⟩
1892:⟨
1889:−
1832:⟩
1826:⟨
1823:−
1750:is not a
1738:α
1683:α
1576:≃
1573:α
1545:α
1517:≃
1514:α
1478:α
1442:≃
1439:α
1403:α
1376:α
1347:α
1276:α
1268:∝
1232:α
1192:
1186:−
1180:
1166:Make the
1053:∑
960:−
906:∑
701:∈
690:For each
671:⋯
668:≈
649:
643:−
624:
618:≈
599:
593:−
574:
548:≈
515:≈
472:⋯
337:⟩
331:⟨
328:−
295:∑
231:∑
214:⟩
208:⟨
129:Given: a
125:Algorithm
57:1/f noise
4274:Fractals
4221:Archived
4189:Fractals
4174:10791480
4166:11101940
4115:11461232
4022:21568275
4014:11030994
3971:11068261
3841:23226132
3754:18417413
3694:55151698
3588:22419991
3550:Sci. Rep
3523:11538314
3433:23226132
3285:See also
3279:integral
3207:, where
3020:, where
1290:. Since
1248:of form
4235:FastDFA
4146:Bibcode
4123:2524064
4095:Bibcode
3994:Bibcode
3887:9963221
3867:Bibcode
3832:3510427
3815:: 450.
3734:Bibcode
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3641:9155618
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3579:3303145
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3374:3498343
3366:9961383
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3227:is the
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358:profile
4253:Python
4238:MATLAB
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3172:, and
2985:, and
2804:Thus,
2673:
2620:
1778:Since
362:i.i.d.
4250:MFDFA
4170:S2CID
4119:S2CID
4085:arXiv
4058:S2CID
4018:S2CID
3967:S2CID
3941:(PDF)
3750:S2CID
3724:arXiv
3690:S2CID
3664:arXiv
3637:S2CID
3611:arXiv
3527:S2CID
3495:Chaos
3476:S2CID
3370:S2CID
859:local
356:, or
55:) or
4162:PMID
4111:PMID
4010:PMID
3957:ISBN
3919:help
3883:PMID
3837:PMID
3584:PMID
3519:PMID
3468:ISSN
3456:2010
3429:PMID
3411:ISSN
3362:PMID
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3231:.
3044:.
2759:and
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183:N
179:x
175:,
172:.
169:.
166:.
163:,
158:2
154:x
150:,
145:1
141:x
120:.
108:n
35:(
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