Abstract
This paper intends, on the example of Urla station, to explore unit root nonstationary behavior of daily mean SP time series driven by infinite variance α-stable white noise. The two complete daily mean SP time series datasets, each of two years long, derived from 10-min digital recordings at the two channels of Urla station are used. It is found that both time series data exhibit a first-order differencing (d = 1) and well can be described by an Autoregressive-Integrated Moving Average (ARIMA(2,1,0)) model. Numerical results of the Autoregressive unit root tests of Augmented Dickey-Fuller (ADF) and Phillips-Perron (PP) tests showed that the null hypothesis of a unit root in the SP time series cannot be rejected. This issue is further checked based upon the Modified Log-periodogram Regression (LPR) and Local Whittle (LW) estimators of the differencing parameter d. Non-stability of the SP data is assessed through four diagnostic checking procedures. Stability (or tail) indexes estimated using Nolan’s Maximum Likelihood (ML) procedure were ranging around 1.12–1.13 for the first differences and 1.15–1.16 for the ARIMA(2,1,0) residuals. Numerical results of this study revealed that the SP time series observed at Urla station can be considered as realizations from an order-one integrated autoregressive process driven by nearly symmetrical, infinite variance (α-stable) white noise. It is recommended to take into account for these two crucial properties of the SP time series data in multivariate statistical models where the SP data will be used as a precursory covariate.
Similar content being viewed by others
References
AFAD (2020) https://deprem.afad.gov.tr/depremkatalogu Accessed 07 September 2020
Arltová M, Fedoravá D (2016) Selection of unit root test on the basis of length of the time series and value of AR(1) parameter. Statistika 96(3):47–64
Box GEP, Jenkins G (1976) Time series analysis, forecasting and control. Holden Day, San Francisco
Brunetti MT, Guzzetti F, Rossi M (2009) Probability distribution of landslide volumes. Nonlinear Process Geophys 16:179–188. https://doi.org/10.5194/npg-16-179-2009
Caner M (1998) Tests for cointegration with infinite variance errors. J Econ 86:155–175
Caner M, Kilian L (2001) Size distortions of tests of the null hypothesis of stationarity: evidence and implications for the PPP debate. J International Money and Finance 20:639–657
Cavaliere G, Georgiev I, Taylor AMR (2018) Unit root inference for nonstationary linear processes driven by infinite variance innovations. Econometric Theory 34(2):302–348
Chan NH, Tran LT (1989) On the first order autoregressive process with infinite variance. Econometric Theory 5:354–362
Colangelo G, Balasco M, Lapenna V, Telesca L (2004) Design and installation of a monitoring network to investigate the correlations between geoelectrical fluctuations and seismicity of basilica region. Phys Chem Earth 29:313–320. https://doi.org/10.1016/j.pce.2003.08.060
Cuomo V, Lapenna V, Macchiato M, Serio C, Telesca L (1998) Linear and nonlinear dynamics in electrical precursory time series: implications for earthquake prediction. Tectonophysics 287:279–298 https://doi.org/10.1016/S0040-1951(98)80074-7
Davis RA, Knight K, Liu J (1992) M-estimation for autoregressions with infinite variance. Stoch Process Appl 40:145–180. https://doi.org/10.1016/0304-4149(92)90142-D
Dickey DA, Fuller WA (1979) Distribution of the estimates for autoregressive time series with a unit root. J Am Stat Assoc 74:427–431. https://doi.org/10.2307/2286348
Dickey DA, Fuller WA (1981) Likelihood ratio test statistics for autoregressive time series with a unit root. Econometrica 49:1057–1072. https://doi.org/10.2307/1912517
Finizola A, Lénat JF, Macedo O, Ramos D, Thouret JC, Sortino F (2004) Fluid circulation and structural discontinuities inside Misti volcano (Peru) inferred from self-potential measurements. J of Volcanology and Geothermal Research 135:343–360. https://doi.org/10.1016/j.jvolgeores.2004.03.009
Fofack H, Nolan JP (1999) Tail behavior, modes and other characteristics of stable distributions. Extremes 2(1):39–58. https://doi.org/10.1023/A:1009908026279
Fuller WA (1976) Introduction to statistical time series. Wiley, New York
Geweke J, Porter-Hudak S (1983) The estimation and application of long-memory time series models. J Time Series Anal 4:221–228. https://doi.org/10.1111/j.1467-9892.1983.tb00371.x
Hamilton JD (1994) Time series analysis. Princeton University Press, New Jersey
Hill BM (1975) A simple general approach to inference about the tail of a distribution. Ann Stat 3(5):1163–1173. https://doi.org/10.1214/aos/1176343247
Kawase T, Uyeda S, Uyeshima M, Kinoshita M (1993) Possible correlation between geoelectric potential change in Izu-Oshima Island and earthquake swarm off the East Izu peninsula Japan. Tectonophysics 224:83–93. https://doi.org/10.1016/0040-1951(93)90059-S
McCulloch JH (1986) Simple consistent estimators of stable distribution parameters. Comm Statist: Simulation 15(4):1109–1136
Michael JR (1983) The stabilized probability plot. Biometrika 70:11–17. https://doi.org/10.1093/biomet/70.1.11
Nolan JP (1997) Numerical computation of stable densities and distribution functions. Comm. Statist. Stoch Model 13(4):759–774. https://doi.org/10.1080/15326349708807450
Nolan JP (2001) Maximum likelihood estimation and diagnostics for stable distributions. In: Barndorff-Nielson OE, Mikosch T, Reisnick SI (eds) Lévy processes: theory and applications. M A pp, Birkhäuser, Boston, pp 379–400
Nur A (1972) Dilatancy pore fluids and premonitory variations of ts/sp travel times. Bull Seismic Soc Am 62:217–222
Parasnis DS (1986) Principles of applied geophysics. Chapman & Hall, London
Phillips PCB (1990) Time series regression with a unit root and infinite variance errors. Econometric Theory 6:44–62
Phillips PCB, Perron P (1988) Testing for unit roots in time series regression. Biometrika 75:335–346. https://doi.org/10.2307/2336182
Phillips PCB (1999) Unit root log periodogram regression. Discussion paper no:1244, Cowles Foundation, Yale University
Phillips PCB, Shimotsu K (2004) Local whittle estimation in the nonstationary and unit root cases. Ann Stat 32(2):656–692. https://doi.org/10.1214/009053604000000139
Pickands J III (1975) Statistical inference using extreme order statistics. Ann Stat 3:119–131. https://doi.org/10.1214/aos/1176343003
Pisarenko VF, Sornette D (2003) Characterization of the frequency of extreme earthquake events by the generalized Pareto distribution. Pure Appl Geophys 160:2343–2364. https://doi.org/10.1007/s00024-003-2397-x
Revil A, Pezard PA, Glover PWJ (1999) Streaming potential in porous media: 1. Theory of the zeta-potential, J Geophys Res 104:20,021–20,031
Revil A, Saracco G, Labazuy P (2003) The volcano-electric effect. J Geophys Res 108(B5):2251. https://doi.org/10.1029/2002JB001835
Robinson PM (1995) Log-periodogram regression of time series with long memory dependence. Ann Stat 23:1048–1072. https://doi.org/10.1214/aos/1176324636
Said SE, Dickey D (1984) Testing for unit roots in autoregressive-moving-average models of unknown order. Biometrika 71:599–607. https://doi.org/10.2307/2336570
Sailhac P, Darnet M, Marquis G (2004) Electrical streaming potential measured at the ground surface: forward modeling and inversion issues for monitoring infiltration and characterizing the vadose zone. Vadose Zone J 3:1200–1206
Samarakoon M, Knight K (2009) A note on unit root tests with infinite variance noise. Econ Rev 28(4):314–334. https://doi.org/10.1080/07474930802458638
Samorodnitsky G, Taqqu MS (1994) Stable non-Gaussian random processes: stochastic models with infinite variance. Chapman and Hall, New York
Sharma PS (1997) Environmental and engineering geophysics. Cambridge University Press, Cambridge
Sornette D, Knopoff L (1997) The paradox of the expected time until the next earthquake. Bull Seismic Soc Am 87:789–798
Taqqu MS, Teverovsky V (1998) On estimating the intensity of long-range dependence in finite and infinite variance time series. In: R J Adler, RE Feldman, MS Taqqu (ed) a practical guide to heavy tails: statistical techniques and applications. Birkhäuser, Boston, pp 177–217
Telesca L, Balasco M, Colangelo G, Lapenna V, Macchiato M (2004a) Investigating the multifractal properties of geoelectrical signals measured in southern Italy. Phys Chem Earth 29:295–303. https://doi.org/10.1016/j.pce.2003.09.015
Telesca L, Colangelo G, Hattori K, Lapenna V (2004b) Principal component analysis of geoelectrical signals measured in the seismically active area of Basilicata region (southern Italy). Nat. Hazards and Earth Syst Sci 4:663–667. https://doi.org/10.5194/nhess-4-663-2004
Telesca L, Colangelo G, Lapenna V, Macchiato M (2003) Monofractal and multifractal characterization of geoelectrical signals measured in southern Italy. Chaos Solutions and Fractals 18:385–399. https://doi.org/10.1016/S0960-0779(02)00655-0
Telford WM, Geldart LP, Sheriff RE (2004) Applied geophysics. Cambridge University Press, Cambridge
Theoharatos C, Ifantis A, Laskaris NA, Economou G (2008) Charting of geoelectric potential signal dynamics via geometrical techniques and its possible relation to significant earthquakes in Western Greece. Comput Geosci 34:625–634. https://doi.org/10.1016/j.cageo.2007.06.006
The MathWorks I (2019) Econometrics toolbox. Econometric Modeler app, Natick, Massachusetts, United State https://www.mathworks.com/products/econometrics.html#econometric-app .
Timashev SF, Polyakov YS, Misurkin PI, Lakeev SG (2010) Anomalous diffusion as a stochastic component in the dynamics of complex processes. Amer Phys Rev E81 Issue 4:041428. https://doi.org/10.1103/PhysRevE.81.04112
Tokat Y, Rachev ST, Schwartz ES (2003) The stable non-Gaussian asset allocation: a comparison with the classical Gaussian approach. J Economic Dynamics and Control 27(6):937–969. https://doi.org/10.1016/S0165-1889(02)00050-7
Uchaikin VV, Zolatarev VM (1999) Chance and stability: stable distributions and their applications. De Gruyter, Utrecht
Uzelli T, Baba A, Mungan GG, Dirik RK, Sözbilir H (2017) Conceptual model of the Gülbahçe geothermal system, Western Anatolia, Turkey: based on structural and hydro geochemical data. Geothermics 68:67–85. https://doi.org/10.1016/j.geothermics.2017.03.003
Voit J (2005) The statistical mechanics of financial markets. Springer, Berlin
Zaliapin I, Kagan YY, Schoenberg FP (2005) Approximating the distribution of Pareto sums. Pure Appl Geophys 162(6–7):1187–1228. https://doi.org/10.1007/s00024-004-2666-3
Zivot E, Wang J (2006) Modeling financial time series with S-Plus 7.0, second Ed., springer-Verlag, NewYork
Acknowledgments
The author gratefully acknowledges the Research Fund Council of Dokuz Eylul University for their financial support (project no: 09.KB.FEN.13) and the administration of Izmir Institute of Technology (IZTECH) for their permission and help in installing the SP monitoring station at the northern border of the campus area. The author also would like to thank the referees of the paper for their valuable reviews and suggestions significantly improved the quality of the presented paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: H. Babaie
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sindirgi, P. On the unit root nonstationary behavior of daily self-potential (SP) time series with infinite variance noise: an example from Urla, Izmir-Turkey. Earth Sci Inform 14, 1185–1196 (2021). https://doi.org/10.1007/s12145-021-00626-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12145-021-00626-1