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

Yıl 2021, Cilt: 11 Sayı: 1, 44 - 53, 09.06.2021

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.

Teşekkür

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

Kaynakça

  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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).
  • 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
  • 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.
  • Vere-Jones, D. 1978. Earthquake prediction: a statistician’s view. J. Phys. Earth., 26:129-146. https://doi.org/10.4294/jpe1952.26.129
  • 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
  • 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
  • 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
  • Votsi, I., Limnios, N., Papadimitriou, E., Tsaklidis, G. 2019. Earthquake statistical analysis through multi-state modeling. Wiley, New Jersey.
  • 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

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

Yıl 2021, Cilt: 11 Sayı: 1, 44 - 53, 09.06.2021

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.

Kaynakça

  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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).
  • 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
  • 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.
  • Vere-Jones, D. 1978. Earthquake prediction: a statistician’s view. J. Phys. Earth., 26:129-146. https://doi.org/10.4294/jpe1952.26.129
  • 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
  • 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
  • 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
  • Votsi, I., Limnios, N., Papadimitriou, E., Tsaklidis, G. 2019. Earthquake statistical analysis through multi-state modeling. Wiley, New Jersey.
  • 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
Toplam 21 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Araştırma Makaleleri
Yazarlar

Özgür Danışman 0000-0001-5855-6413

Umay Kocer 0000-0001-5044-6236

Yayımlanma Tarihi 9 Haziran 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 11 Sayı: 1

Kaynak Göster

APA Danışman, Ö., & Kocer, U. (2021). Fitting Hidden Markov Model to Earthquake Data: A Case Study in the Aegean Sea. Karaelmas Fen Ve Mühendislik Dergisi, 11(1), 44-53.
AMA Danışman Ö, Kocer U. Fitting Hidden Markov Model to Earthquake Data: A Case Study in the Aegean Sea. Karaelmas Fen ve Mühendislik Dergisi. Haziran 2021;11(1):44-53.
Chicago Danışman, Özgür, ve Umay Kocer. “Fitting Hidden Markov Model to Earthquake Data: A Case Study in the Aegean Sea”. Karaelmas Fen Ve Mühendislik Dergisi 11, sy. 1 (Haziran 2021): 44-53.
EndNote Danışman Ö, Kocer U (01 Haziran 2021) Fitting Hidden Markov Model to Earthquake Data: A Case Study in the Aegean Sea. Karaelmas Fen ve Mühendislik Dergisi 11 1 44–53.
IEEE Ö. Danışman ve U. Kocer, “Fitting Hidden Markov Model to Earthquake Data: A Case Study in the Aegean Sea”, Karaelmas Fen ve Mühendislik Dergisi, c. 11, sy. 1, ss. 44–53, 2021.
ISNAD Danışman, Özgür - Kocer, Umay. “Fitting Hidden Markov Model to Earthquake Data: A Case Study in the Aegean Sea”. Karaelmas Fen ve Mühendislik Dergisi 11/1 (Haziran 2021), 44-53.
JAMA Danışman Ö, Kocer U. Fitting Hidden Markov Model to Earthquake Data: A Case Study in the Aegean Sea. Karaelmas Fen ve Mühendislik Dergisi. 2021;11:44–53.
MLA Danışman, Özgür ve Umay Kocer. “Fitting Hidden Markov Model to Earthquake Data: A Case Study in the Aegean Sea”. Karaelmas Fen Ve Mühendislik Dergisi, c. 11, sy. 1, 2021, ss. 44-53.
Vancouver Danışman Ö, Kocer U. Fitting Hidden Markov Model to Earthquake Data: A Case Study in the Aegean Sea. Karaelmas Fen ve Mühendislik Dergisi. 2021;11(1):44-53.