# Natural time analysis

Natural time analysis is a statistical method applied to analyze complex time series and critical phenomena, based on event counts as a measure of "time" rather than the clock time.[1][2] Natural time concept was introduced by P. Varotsos, N. Sarlis and E. Skordas in 2001.[3] Natural time analysis has been primarily applied to earthquake prediction[1][2] / nowcasting[4][5][6][7][8][9][10][11][12][13] and secondarily to sudden cardiac death[14] / heart failure[15][16] and financial markets.[17] Natural time characteristics are considered to be unique.[9]

## Etymology

"Natural time" is a new view of time introduced in 2001[3] that is not continuous, in contrast to conventional time which is in the continuum of real numbers, but instead its values form countable sets as natural numbers.[18]

## Definition

In natural time domain each event is characterized by two terms, the "natural time" χ, and the energy Qk. χ is defined as k/N, where k is a natural number (the k-th event) and N is the total number of events in the time sequence of data. A related term, pk, is the ratio Qk / Qtotal, which describes the fractional energy released. The term κ1 is the variance in natural time:[19]

$\displaystyle{ \kappa_1=\sum_{k=1}^N p_k(\chi_k)^2 - \bigl(\sum_{k=1}^N p_k\chi_k\bigr)^2 }$
where $\displaystyle{ \textstyle\chi_k=k/N }$ and $\displaystyle{ \textstyle\ p_k=\frac{Q_k}{\sum_{n=1}^N Q_n} }$

## Time reversal

Time reversal, in contrast to clock time, is applicable when studying the approach of a system to criticality with natural time analysis. Living systems for example are considered to operate far from equilibrium as there is flow of energy crossing their boundaries, in contrast to deceased organisms where inner driving forces are absent. While time irreversibility is a fundamental property of a living system, the state of death is more time reversible by means of energy flow across the system's boundaries. Thus a critical state of a system can be estimated by applying natural time analysis upon calculating the entropy upon both normal time flow and time reversal and studying the difference of the two results.[15][14][16]

(a) ECG in which the RR distances are marked
(b) the same ECG plotted in (a) but read in natural time analysis
(c) ECG at conventional time upon time reversal
(d) ECG upon time reversal in natural time analysis
The length between RR distances in conventional time is approximately considered as the energy of each pulse (event) in natural time analysis.

## Applications

### Seismology

#### Earthquake prediction

Main page: Earthquake prediction

Natural time analysis has been initially applied to VAN method in order to improve the accuracy of the estimation of the time of a forthcoming earthquake that has been indicated to occur by seismic electric signals (SES). The method deems SES valid when κ1 = 0.070. Once the SES are deemed valid, a second NT analysis is started in which the subsequent seismic (rather than electric) events are noted, and the region is divided up as a Venn diagram with at least two seismic events per overlapping rectangle. When κ1 approaches the value κ1 = 0.070 for the candidate region, a critical seismic event is considered imminent, i.e. it will occur in a few days to one week or so.[20]

#### Earthquake nowcasting

In seismology, nowcasting is the estimate of the current dynamic state of a seismological system.[4][7] It differs from forecasting which aims to estimate the probability of a future event[12] but it is also considered a potential base for forecasting.[8][4] Nowcasting is based on the earthquake cycle model, a recurring cycle between pairs of large earthquakes in a geographical area, upon which the system is evaluated using natural time.[4] Nowcasting calculations produce the "earthquake potential score", an estimation of the current level of seismic progress.[9]

When applied to seismicity, natural time has the following advantages:[4]

1. Declustering of the aftershocks is not necessary as natural time count is evenly valid in any case of aftershock or backgroung seismicity.
2. Natural time statistics do not depend on the level of seismicity, given that the b value does not significantly vary.

Typical applications are: great global earthquakes and tsunamis,[5] aftershocks and induced seismicity,[8][13] induced seismicity at gas fields,[10] seismic risk to global megacities,[12] studying of clustering of large global earthquakes,[11] etc.

### Cardiology

Natural time analysis has been experimentally used for the diagnosis of heart failure syndrome[15][16] as well as identifying patients at high risk for sudden cardiac death,[14] even when measuring solely the heart rate, either using electrocardiography or far more inexpensive and portable equipment (i.e. oximeter).[16]

### Economy

Due to similarities of the dynamic characteristics between earthquakes and financial markets, natural time analysis, which is primarily used in seismology, was chosen to assist in developing winning strategies in financial markets, with encouraging results.[17]

## References

1. Varotsos, P. A.; Sarlis, N. V.; Skordas, E. S. (2002). "Long-range correlations in the electric signals that precede rupture". Physical Review E 66 (1 Pt 1): 011902. doi:10.1103/PhysRevE.66.011902. ISSN 1539-3755. PMID 12241379. Bibcode2002PhRvE..66a1902V.
2. Varotsos, Sarlis & Skordas 2011 (book), preface and chapter 2
3. P. Varotsos, N. Sarlis, E. Skordas (2001). "Spatio-temporal complexity aspects on the interrelation between seismic electric signals and seismicity". Practica of Athens Academy 76: 294–321.
4. Rundle, J. B.; Turcotte, D. L.; Donnellan, A.; Ludwig, L. Grant; Luginbuhl, M.; Gong, G. (2016). "Nowcasting earthquakes" (in en). Earth and Space Science 3 (11): 480–486. doi:10.1002/2016EA000185. ISSN 2333-5084. Bibcode2016E&SS....3..480R.
5. Rundle, John B.; Luginbuhl, Molly; Khapikova, Polina; Turcotte, Donald L.; Donnellan, Andrea; McKim, Grayson (2020-01-01). "Nowcasting Great Global Earthquake and Tsunami Sources" (in en). Pure and Applied Geophysics 177 (1): 359–368. doi:10.1007/s00024-018-2039-y. ISSN 1420-9136.
6. Williams, Charles A., ed (2019). "Introduction" (in en). Earthquakes and Multi-hazards Around the Pacific Rim, Vol. II. Pageoph Topical Volumes. Birkhäuser Basel. ISBN 978-3-319-92296-6.
7. Rundle, John B.; Giguere, Alexis; Turcotte, Donald L.; Crutchfield, James P.; Donnellan, Andrea (2019). "Global Seismic Nowcasting With Shannon Information Entropy" (in en). Earth and Space Science 6 (1): 191–197. doi:10.1029/2018EA000464. ISSN 2333-5084. PMID 30854411. Bibcode2019E&SS....6..191R.
8. Luginbuhl, Molly; Rundle, John B.; Turcotte, Donald L. (2019-01-14). "Statistical physics models for aftershocks and induced seismicity". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377 (2136): 20170397. doi:10.1098/rsta.2017.0397. PMID 30478209. Bibcode2019RSPTA.37770397L.
9. Pasari, Sumanta (2019-04-01). "Nowcasting Earthquakes in the Bay of Bengal Region" (in en). Pure and Applied Geophysics 176 (4): 1417–1432. doi:10.1007/s00024-018-2037-0. ISSN 1420-9136. Bibcode2019PApGe.176.1417P.
10. Luginbuhl, Molly; Rundle, John B.; Turcotte, Donald L. (2018-11-01). "Natural time and nowcasting induced seismicity at the Groningen gas field in the Netherlands" (in en). Geophysical Journal International 215 (2): 753–759. doi:10.1093/gji/ggy315. ISSN 0956-540X. Bibcode2018GeoJI.215..753L.
11. Luginbuhl, Molly; Rundle, John B.; Turcotte, Donald L. (2018-02-01). "Natural Time and Nowcasting Earthquakes: Are Large Global Earthquakes Temporally Clustered?" (in en). Pure and Applied Geophysics 175 (2): 661–670. doi:10.1007/s00024-018-1778-0. ISSN 1420-9136. Bibcode2018PApGe.175..661L.
12. Rundle, John B.; Luginbuhl, Molly; Giguere, Alexis; Turcotte, Donald L. (2018-02-01). "Natural Time, Nowcasting and the Physics of Earthquakes: Estimation of Seismic Risk to Global Megacities" (in en). Pure and Applied Geophysics 175 (2): 647–660. doi:10.1007/s00024-017-1720-x. ISSN 1420-9136. Bibcode2018PApGe.175..647R.
13. Luginbuhl, Molly; Rundle, John B.; Hawkins, Angela; Turcotte, Donald L. (2018-01-01). "Nowcasting Earthquakes: A Comparison of Induced Earthquakes in Oklahoma and at the Geysers, California" (in en). Pure and Applied Geophysics 175 (1): 49–65. doi:10.1007/s00024-017-1678-8. ISSN 1420-9136. Bibcode2018PApGe.175...49L.
14. Varotsos, P. A.; Sarlis, N. V.; Skordas, E. S.; Lazaridou, M. S. (2007-08-06). "Identifying sudden cardiac death risk and specifying its occurrence time by analyzing electrocardiograms in natural time". Applied Physics Letters 91 (6): 064106. doi:10.1063/1.2768928. ISSN 0003-6951. Bibcode2007ApPhL..91f4106V.
15. Sarlis, N. V.; Skordas, E. S.; Varotsos, P. A. (2009-07-01). "Heart rate variability in natural time and 1/f "noise"" (in en). EPL 87 (1): 18003. doi:10.1209/0295-5075/87/18003. ISSN 0295-5075. Bibcode2009EL.....8718003S.
16. Baldoumas, George; Peschos, Dimitrios; Tatsis, Giorgos; Chronopoulos, Spyridon K.; Christofilakis, Vasilis; Kostarakis, Panos; Varotsos, Panayiotis; Sarlis, Nicholas V. et al. (2019-11-05). "A Prototype Photoplethysmography Electronic Device that Distinguishes Congestive Heart Failure from Healthy Individuals by Applying Natural Time Analysis" (in en). Electronics 8 (11): 1288. doi:10.3390/electronics8111288.
17. Mintzelas, A.; Kiriakopoulos, K. (2016-01-01). "Natural time analysis in financial markets" (in en). Algorithmic Finance 5 (1–2): 37–46. doi:10.3233/AF-160057. ISSN 2158-5571.
18. Varotsos, Sarlis & Skordas 2011 (book), preface
19. Varotsos, Sarlis & Skordas 2011 (book), pages 121 & 131
20. Varotsos, Sarlis & Skordas 2011 (book), chapter 7

## Bibliography

• Varotsos, Panayiotis A.; Sarlis, Nicholas V.; Skordas, Efthimios S. (2011). Natural time analysis : the new view of time; Precursory seismic electric signals, earthquakes and other complex time series. Berlin: Springer. ISBN 978-3-642-16449-1. OCLC 755081829.