UNLV Retrospective Theses & Dissertations 1-1-2006 Arima models for forecasting Poisson process observations: Application to the volcanoes worldwide Heng-Wei Cheng University of Nevada, Las Vegas Follow this and additional works at: https://digitalscholarship.unlv.edu/rtds Repository Citation Cheng, Heng-Wei, "Arima models for forecasting Poisson process observations: Application to the volcanoes worldwide" (2006). UNLV Retrospective Theses & Dissertations. 2094. http://dx.doi.org/10.25669/1ru9-piwg This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Thesis has been accepted for inclusion in UNLV Retrospective Theses & Dissertations by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. ARIMA MODELS FOR FORECASTING POISSON PROCESS OBSERVATIONS: APPLICATION TO THE VOLCANOES WORLDWIDE by Heng-Wei Cheng Bachelor of Arts University of Nevada, Las Vegas 2004 A thesis submitted in partial fulfillment of the requirements for the Master of Science Degree in Mathematical Sciences Department of Mathematical Sciences College of Sciences Graduate College University of Nevada, Las Vegas May 2007 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 1443742 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. UMI UMI Microform 1443742 Copyright 2007 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thesis Approval ITNTV The Graduate College University of Nevada, Las Vegas March 2 , ^20. 0 7 The Thesis prepared by Heng-Wei Cheng Entitled ARIMA Models for Forecasting Poisson Process Observations; Application to the Volcanoes Worldwide is approved in partial fulfillment of the requirements for the degree of M a sters Examination Committee Chair Dean of the Graduate College Examination Committee Member — Examination Committee Member Graduate College Faculty Representative It Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT ARIMA Models for Forecasting Poisson Process Observations: Application to the Volcanoes Worldwide by Heng-Wei Cheng Dr. Chih-Hsiang Ho, Examination Committee Chair Professor of Mathematical Sciences University of Nevada, Las Vegas A nonhomogeneous Poisson process (NHPP) is often appropriate for the modeling of a series of events that occur over time in a nonstationary fashion. A major difficulty with the NHPP is that it has infinitely many forms for the intensity function. In this thesis, we propose a linking bridge between a point process and the classical time series via a sequence of the empirical recurrence rates (ERR), calculated sequentially at equidistant time intervals. The distinctive technique is demonstrated with an ERR-plot, designed to fingerprint the temporal pattern of a point process. Moreover, Autoregressive Integrated Moving Average (ARIMA) models are presented to find the best fitting model to forecast the intensity associated with the underlying NHPP. Valuable modeling and computing techniques are demonstrated using volcanic data. Specifically, we split each time series data set into two groups. The first set, called the training sample, is used to develop the candidate models. The remaining data, called prediction set, is used to further 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. evaluate the reasonableness and predictive ability of the candidate models. The information of how the candidate models considered in the model selection phase fare with the new data (prediction set), will be evaluated to conclude the “best” model. IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ABSTRACT.......................................................................................................................................... iii LIST OF TABLES.............................................................................................................................. vi LIST OF FIGURES...........................................................................................................................vii ACKNOWLEDGEMENTS............................................................................................................viii CHAPTER 1 INTRODUCTION .................................................................................................... 1 CHAPTER 2 THEORIES AND METHODS.............................................................................. 3 2.1 Nonhomogeneous Poisson Process .............................................................................. 3 2.2 Empirieal Recurrence Rates ...........................................................................................4 2.3 ARIMA Model .................................................................................................................. 5 2.3.1 ARIMA Definition ........................................................................................ 5 2.3.2 Splitting Data and ERR Plotting ................................................................ 7 2.3.3 Choosing Appropriate Data: Volcanoes Example .................................9 2.3.4 Obtaining ARIMA Model ..........................................................................10 CHAPTER 3 APPLICATION .......................................................................................................14 3.1 A vachinsky ......................................................................................................................15 3.2 Vesuvius ........................................................................................................................... 25 3.3 Yucca Mountain ............................................................................................................. 36 3.4 E tn a ................................................................................................................................... 42 CHAPTER 4 CONCLUSIONS....................................................................................................46 A PPEND IX ...........................................................................................................................................48 REFERENCE.................. 74 VITA .......................................................................................................................................................75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF TABLES Table AT Predictions of Avachinsky (Training Sample) .................................................23 Table A.2 Predicted number of events in the next 25 years ............................................ 25 Table A.3-4 Calculation of probability of the next 25 years ...............................................25 Table B.l Comparison of predicted values and actual values for Vesuvius ................ 33 Table B.2 Predictions of Vesuvius (Training Sample) by using AR(1) ....................... 35 Table B.3 Predicted number of events in the next 50 years ............................................. 36 Table C.l Prediction of Yucca Mountain ........................................................................... 41 Table C.2 Calculation of m{t) for Yucca Mountain (h = 0.1 m .y) ................................ 41 Table D Predictions of Etna (Training Sample) by using AR(6) ................................ 45 VI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES Figure 1 Dot plot of raw data...........................................................................................16 Figure 2(a) ERR plots of Avachinsky with different time step units .........................20 Figure 2(b) Training sample and prediction set of Avachinsky ..................................21 Figure 2(c) Residual ACF after fitting MA(1)................................................................. 21 Figure 2(d) Residual PACE after fitting MA(1) ..............................................................22 Figure 2(e) Prediction by using MA(1) .............................................................................23 Figure 2(f) An illustration of