Saklı Markov Modelinin Deprem Verilerine Uygulanması: Ege Denizinde Bir Örnek Olay

Öz

Deprem verilerinin olasılıksal modellemesi ile ilgili çalışmalar son yıllarda giderek artmaktadır. Depremlerin yer altındaki gerilim düzeyindeki gözlenemeyen değişimler sonucu oluştukları bilinen bir gerçektir. Saklı Markov modelleri varsayımlarından dolayı deprem verilerini modellemek için uygun bir çerçeve sunar. Yeraltı gerilim düzeyindeki gözlenemeyen değişimleri göz önünde bulundurmak ve Ege denizinde bazı olasılıksal deprem tahmini yapmak için gizli Markov modeli sunmaktayız. Ege bölgesi, deprem oluşumu bakımından aktif bir bölge olduğu için seçilmiştir. Gizli durumları yeraltı stres düzeyi olan bir gizli Markov modeli tanımlanmıştır. Gizli durum sayıları farklı olan dört ayrı model tanımlanmış ve bu modeler Akaike ve Bayesian bilgi kriterlerine göre karşılaştırılmıştır. Önerilen model deprem büyüklükleri ve bölgelerine ilişkin kısa dönem olasılık tahminleri verebilmektedir. Model parametrelerini tahmin etmek için iteratif bir algoritma olan Baum-Welch algoritması kullanılmıştır. Geleneksel Baum-Welch algoritması iterasyonlarda yalnız bir gözlem değişkeni kullanır. Bu çalışmada, Baum-Welch algoritmasının birden fazla gözlem değişkeni bulunduğunda kullanımı için oldukça kolay ve anlaşılır bir bakış açısı önerilmiştir. Bu yaklaşımla çoklu gözlem değişkenlerinin marjinal ve ortak olasılık fonksiyonlarını elde etmek mümkün olmaktadır.

Anahtar Kelimeler

• Saklı Markov modeli
• Baum-Welch algoritması
• Çoklu gözlem
• Deprem
• Tahmin
• Ege Denizi

Fitting Hidden Markov Model to Earthquake Data: A Case Study in the Aegean Sea

Öz

Studies about stochastic modeling of earthquake data have increased considerably in recent years. It is a well-known fact that earthquakes occur as a result of unobservable changes in underground stress levels. The hidden Markov model provides a suitable framework for modeling earthquake data due to its assumptions. We present a hidden Markov model to examine hidden changes in the underground stress level and to make some probabilistic earthquake forecasts in the Aegean Sea. The Aegean region is selected for the modeling because of the active nature of earthquake occurrences. A hidden Markov chain is defined in which the corresponding states are stress levels of the ground. Four models with different numbers of hidden states are constructed and compared according to the Akaike and Bayesian information criteria. The proposed model is capable of forecasting the short-term probabilities of both earthquake magnitudes and also locations. Baum-Welch algorithm, which is an iterative expectation-maximization algorithm, is used for the estimation of model parameters. The traditional Baum-Welch algorithm considers only one variable as an observation for the iterations. In this paper, a naive and quite simple approach is used for the Baum-Welch algorithm to estimate the model parameters with more than one observation. It is possible to obtain the marginal and joint probability distributions of multiple observations with this approach.

Anahtar Kelimeler

• Hidden Markov model
• Baum-Welch algorithm
• Multiple observations
• Earthquake
• Forecasting
• Aegean Sea

Teşekkür

The authors would like to thank the anonymous referees for their invaluable and helpful comments for the improvement of the study.

Kaynakça

1. Akaike, H. 1974. A new look at the statistical model identification. IEEE Trans. Automat. Control, 19:716-723. https://doi.org/10.1007/978-1-4612-1694-0_16
2. Alvarez, EE. 2005. Estimation in stationary Markov renewal processes, with application to earthquake forecasting in Turkey. Methodol. Comput. Appl., 7:119-130. https://doi.org/10.1007/s11009-005-6658-2
3. Anagnos, T., Kiremidjian, AS. 1988. A review of earthquake occurrence models for seismic hazard analysis. Probab. Eng. Mech., 3:3-11. https://doi.org/10.1016/0266-8920(88)90002-1
4. Baum, LE., Petrie, T. 1966. Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Stat., 37:1554-1563. http://dx.doi.org/10.1214/aoms/1177699147
5. Baum, LE., Petrie, T., Soules, G., Weiss, N. 1970. A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Stat., 41:164-171. http://dx.doi.org/10.1214/aoms/1177697196
6. Chambers, DW., Baglivo, JA., Ebel, JE., Kafka, AL. 2012. Earthquake forecasting using hidden Markov models. Pure. Appl. Geophys.,169:625-639. https://doi.org/10.1007/s00024-011-0315-1
7. Chambers, DW., Ebel, JE., Kafka, AL., Baglivo, JA. 2003. A hidden Markov approach to modeling interevent earthquake times. Eos. Trans. AGU Fall. Meet. Suppl., 84(46), Abstract S52F-0179
8. Ebel, JE., Chambers, DW., Kafka, AL., Baglivo, JA. 2007. Non-Poissonian earthquake clustering and the hidden Markov model as bases for earthquake forecasting in California. Seismol. Res. Lett., 78:57-65. http://dx.doi.org/10.1785/gssrl.78.1.57

9. Granat, R., Donnellan, A. 2002. A hidden Markov model based tool for geophysical data exploration.. Pure. Appl. Geophys., 159:2271-2283. https://doi.org/10.1007/s00024-002-8735-6
10. Masala, G. 2012. Earthquakes occurrences estimation through a parametric semi-Markov approach. J. Appl. Stat., 39:81-96. https://doi.org/10.1080/02664763.2011.578617
11. Orfanogiannaki, K., Karlis, D., Papadopoulos, GA. 2010. Identifying seismicity levels via Poisson hidden Markov models. Pure. Appl. Geophys., 167:919-931. https://doi.org/10.1007/s00024-010-0088-y
12. Pertsinidou, CE., Tsaklidis, G., Papadimitriou, E., Limnios, N. 2017. Application of hidden semi-Markov models for the seismic hazard assessment of the North and South Aegean Sea, Greece. J. Appl. Stat., 44:1064-1085. http://dx.doi.org/10.1080/02664763.2016.1193724
13. Reid, H.F. 1910. The mechanics of the earthquake, the California earthquake of April 18, 1906. Report of the state investigation comission, vol 2. Carnegie Institution of Washington, Washington D.C. (reprinted in 1969).
14. Sadeghian, R. 2012. Forecasting time and place of earthquakes using a semi-Markov model (with case study in Tehran province). J. Ind. Eng. Int., 8:20. https://doi.org/10.1186/2251-712X-8-20
15. Schwarz, GE. 1978. Estimating the dimension of a model. Ann. Statist., 6:461-464. https://www.jstor.org/stable/2958889 Tiampo, KF., Shcherbakov, R. 2012. Seismicity-based earthquake forecasting techniques: Ten years of progress. Tectonophysics, 522-523:89-121.
16. Vere-Jones, D. 1978. Earthquake prediction: a statistician’s view. J. Phys. Earth., 26:129-146. https://doi.org/10.4294/jpe1952.26.129
17. Viterbi, AJ. 1967. Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. IEEE Trans. Inf. Theory., 13:260-269. DOI:10.1109/TIT.1967.1054010
18. Votsi, I., Limnios, N., Tsaklidis, G., Papadimitriou, E. 2012. Estimation of the expected number of earthquake occurrences based on semi Markov models. Methodol. Comput. Appl. Probab., 14:685-703. https://doi.org/10.1007/s11009-011-9257-4
19. Votsi, I., Limnios, N., Tsaklidis, G., Papadimitriou, E. 2013. Hidden Markov models revealing the stress field underlying the earthquake generation. Physica A, 392:2868-2885. https://doi.org/10.1016/j.physa.2012.12.043
20. Votsi, I., Limnios, N., Papadimitriou, E., Tsaklidis, G. 2019. Earthquake statistical analysis through multi-state modeling. Wiley, New Jersey.
21. Yip, CF., Ng, WL., Yau, CY. 2018. A hidden Markov model for earthquake prediction. Stoch. Environ. Res. Risk. Assess., 32:1415-1434. https://doi.org/10.1007/s00477-017-1457-1