U“ pUkqkPN backward recurrence times of the semi-Markov chain pJ,Sq “ pJn, SnqnPN Markov renewal chain. P“ ppijqi,jPE transition matrix of the (embedded) Markov chain f “ pfijpkqqi,jPE,kPN conditional residence time distribution.
Motivation and Definition of the Subject
For the purposes of the thesis, we consider a prediction to be a specification that an earthquake will or will not occur at a given location, during a given time interval and magnitude. On the other hand, prediction is considered to be a specification of the likelihood or probability of an earthquake occurring in a given location, during a given time window, within a given magnitude range.
Seismic Hazard Assessment
Reiter p. 1990q affirms that 'the deterministic approach provides a clear and traceable method for calculating seismic hazard, the assumptions of which are easily understood. According to Reiterp1990q, a probabilistic approach is capable of incorporating a wide range of information and uncertainty into a flexible framework.
Earthquake Occurrence Models
- Stress Release Models
- Renewal Models
- Stress-Based Models
- Markov and Semi-Markov Models
- Hidden Markov and Hidden Semi-Markov Models
In addition to the average frequency (or recurrence time) of earthquakes, these models require additional information on the variability of the event frequency (change or standard deviation) and the time of the last event. This is due to the memoryless properties of the geometric distribution (the discrete-time case) and the exponential distribution (the continuous-time case).
Contributions to the Research Field
This classification was introduced to improve the predictability of the semi-Markov model. In section 2.3.3 the stationary distribution of the semi-Markov process is described and estimated in a non-parametric manner.
Continuous-Time Semi-Markov Framework
In section 2.4, a new classification of states of the semi-Markov model is proposed, which further includes the spatial component. The empirical estimator of the semi-Markov kernel is strongly consistent and asymptotically normal.
Modeling Earthquakes in Northern Aegean Sea
- Data Selection
- Classification of States
- Stationary Distribution of the SMP
- Hitting Times for Earthquake Occurrences, Expected
- Estimating Earthquake Occurrence Rates
The distribution of the impact time for an earthquake of the third state is expressed by the formulaWptq “ 1´ř. The empirical estimator of the total earthquake occurrence rate is given by the formula (Limnios and Opri ¸san, 2001).
New Classification of States
For each fault orientation class, the fault type is represented by mean values of the strike, rake and dip angles. By incorporating the seismotectonic criteria into the classification of states, the strike time for an earthquake.
Conclusions
The stationary distribution along with the mean return times for each of the states were calculated. In addition, the transition probabilities and the distributions of the strike times to the strongest earthquakes were calculated. In section 3.4.4, different HMMs are compared with each other based on the Akaike and Bayesian information criteria.
Discrete-Time Hidden Markov Framework
There is no known analytical solution to the problem that maximizes the likelihood of a sequence of observations. Given the model parameters, θ, and a sequence of observations, Y0M, we want to reveal the sequence of states that most likely produced a particular sequence of observations. Starting from the Y0M observation sequence, we need to estimate the characteristics of the MC as well as the conditional distribution of Y.
Seismotectonic Properties of the Study Area and Data Used . 51
- Three Observation Types and Two Dimensional State
- Three Observation Types and Three Dimensional State
- Optimal Model and Related Results
- Number of Steps for the First Occurrence of an Antici-
The second level of the actual stress field emits earthquakes belonging to the second type, i.e. the estimated emission probability matrix, which is the result of the application of the Baum-Welch algorithm, turns out to be. New datasets (of size 500 and 1000) with 116 data points each (same size as the original dataset) are generated using the MLEs of the parameters.
Second Approach
- Three Observation Types and Two Dimensional State
- Three Observation Types and Three Dimensional State
- Optimal Model and Related Results
- Time for the First Occurrence of an Anticipated Earth-
The total variance of the first transition time to the set of states D is defined by V arpTq “ ør. As a result, the HMC "remains" at the first stage of the actual stress field during the aforementioned periods. In addition, the period 1952-1957 is dominated exclusively by emissions from the third stage of the actual voltage field.
Conclusions
The use of hidden semi-Markov models (HSMM) (Barbu and Limnios, 2008) could improve the description of the problem by allowing any distribution (beyond geometric) for the residence times in the different states of the hidden chain. As for the asymptotic properties of maximum likelihood estimators (MLEs) for nonparametric HSMMs, they were first studied by Barbu and Limnios (2006). Section 4.4.1 describes the stationary distribution of the underlying SMC, which is estimated nonparametrically.
Discrete-Time Semi-Markov Framework
The empirical estimator of the cumulative state residence time distribution iPE,Hpipk, Mq,k PN˚, kďM has the form. For i, j PE two fixed states, the empirical estimator of the conditional residence time distribution pXnqnPN is defined as. Barbu and Limnios (2006a) proved that the empirical semi-Markov kernel estimator (4.2) is uniformly strongly consistent and asymptotically normal.
Introduction of Hidden Semi-Markov Models
At this point we prove the asymptotic normality of the estimatorqppnqij pk, Mq, i, j P E,k, nPN, kďM. They do not allow any other distribution for the residence times in the states of the hidden process than the geometric one. The knowledge of the Markov chain pJn, XnqnPN provides the knowledge of the MRC pJn, SnqnPN, through the association between the jump times and the dwell times.
Modeling Earthquakes in Greece
Stationary Distribution of the SMC
We propose the following estimator for the stationary distribution of the semi-Markov chain. where pνipMq is an estimator of the stationary distribution of EMC and mpipMq is an estimator of the mean residence time in statei. In particular, we will use the empirical estimator of the stationary distribution of the nested Markov chain, defined by . whereas the estimator of the average residence time in state in P E is defined by. 4.5) In addition, the empirical estimator of the mean repetition time in mode iPE is given by the formula.
Hitting Times for Earthquake Occurrences and Expected
In Table4.1, the empirical estimator of the quantities given in equations p4.4qenp4.5q as well as the estimated mean recurrence times for any state visited by the SMC are exhibited. The distribution of the strike time for an earthquake emitted from the second hidden state in which we are more interested is expressed by the formula. The estimated distribution of the strike time of an earthquake emitted from the second hidden state is presented in Figure 4.5.
Earthquake Occurrence Rates
Given that the last earthquake was emitted from hidden state i and at least one time interval of length ∆ has already passed, the probability that an earthquake will be emitted from hidden state j in the next time interval of length denoted by Pk|∆ pigs. The term Instantaneous Earthquake Occurrence Rate in the next step conditional on the initial condition is used to describe the probability Pk|∆pijq, which is expressed by semi-Markov clustered kernels with the formula. It provides prediction results for an earthquake occurrence in the next time interval of length ∆, knowing that the last earthquake occurred at least six months ago and assuming different values for years).
Conclusions
The core of this chapter is the discrete-time intensity of stroke time (DTIHT). It is the discrete analog of the continuous time rate of occurrence of failure (ROCOF) denoted by roptq, t P R`. Here, the study of DTIHT is addressed for the first time for semi-Markov chains and hidden Markov renewal chains.
DTIHT for Semi-Markov Chains
Statistical Estimation of the DTIHT for a SMC
In the second case, a single realization of the process is observed and the asymptotic properties are obtained as the time M of observation increases. In this study we follow the second procedure and we consider a single realization of the ergodic MRC pJ,Sq, censored at a fixed arbitrary timeM P N,. Then the empirical estimator of the transition probability matrix of the MC pZ,Uqis is defined in terms of the aforementioned estimators by.
Asymptotic Properties of DTIHT Estimator for a SMC . 101
In our case, since one trajectory is considered, the estimate of the initial distribution p. Billingsley, 1960; Sadek and Limnios, 2002). According to the continuous mapping theorem (Billingsley, 1968), the vectorF converges in distribution to the centered normal distribution with covariance matrix Γ, where Γ “ p. The asymptotic consistency of the DTIHT estimator appears to be verified (Fig. 5.2), that is, as the length of the trajectory increases, the estimated values tend towards the true one.
DTIHT for Hidden Markov Renewal Chains
Statistical Estimation of the DTIHT for an HMRC
The estimation procedure concerns a single realization of the HMRCpJ,S,Yq in a time intervalr0, Ms,. We focus our interest on the estimation of the average number of transitions from the HMRC to the subsetCat timek, which the subsetC refers to. In particular, SetC includes HMRC visits to the subset of observations A1.
Conclusions
We are now interested in estimating the average number of HMRC visits to subset C at time k, where C corresponds to HMRC visits to subset observations A1. The true value of DTIHT and its estimator for the trajectory of length M” 200 are shown in Figure 5.6. Second, we observe the impact of the differences by calculating the mean and variance of the number of steps required by the Markov chain (HMM case) and EMC (HSMM case) to visit a given state for the first time.
Introduction
Unlike HSMMs, HMMs are not expected to adopt the behavior of the original model under a semi-Markov framework. Section 6.2 concerns the comparison between an HMM and an HSMM in a discrete-time Markov framework as well as in a discrete-time semi-Markov framework for the general case where residence times are considered to follow specific distributions. In particular, section 6.2.1 includes the comparison of the models under a Markov environment, whereas in section 6.2.2 a semi-Markov environment is built, and a comparison of the models is made further.
General Case
Discrete-time Markov framework
Successively visited states of the Markov chain, up to the first M steps, pZ0, .., ZMq;. The variance of the first transition time from the i´th state U to the set of states D is defined by V aripTDq “ Vpiq ´ pL1piqq2, where V “ pI´ P11q´1“. A discrete-time HMM is now applied to each of the observation sequences included in the trajectories.
Discrete-time semi-Markov framework
We should note here that in a Markov environment both the HMM and the HSMM are well adjusted, giving small values of the norm (||M||1,8 ă0.1). The algorithm is the direct analogue of the algorithm for generating a sample path of a given semi-Markov chain in the time interval r0, Me. Here we should note that the values of the norm in this case are greater than the corresponding values in the Markov framework.
Real-Data Case
Hidden Markov Model
Therefore, since the HMM is not well fitted, the application of the HSMM is necessary to adopt the behavior of the original model.
Hidden Semi-Markov Model
HSMM is more appropriate to be applied than HMM in the case of real data.
Conclusions
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