11:00 AM - 1:00 PM
[SSS12-P07] Comparisons of the Brownian Passage Time renewal model parameter estimates by different methods for paleo-earthquake sequences
Keywords:Paleo-earthquake sequences, Renewal process model, Brownian Passage Time distribution, Geometric mean of likelihood, Uncertain origin times
There is no established method for estimating the parameters of a renewal process model for paleo-earthquake sequences with uncertain origin times. Using a sampling method, we generate a large number of sets of time sequences by a Monte Carlo procedure, and obtain distributions of the model parameters and values of interest. In some cases, sets of the model parameters are determined from time sequences with middles of uncertain ranges as origin times (referred to as the middle point method), as not much is known about a bias of the shape parameter toward a smaller value than an average value. Ogata (1999) formulated a likelihood function in a multiproduct form of a probability density function integrated with respect to origin times over uncertain ranges. An additional condition implicitly introduced by this method results in a bias of the shape parameter toward an extraordinarily small value. To remove this bias, Imoto et al. (2021) proposed to use the geometric mean of likelihood over uncertain ranges rather than the arithmetic mean used by Ogata (1999).
In this paper, we estimate the model parameters of the Brownian Passage Time (BPT) distribution by using the sampling method, Ogata’s (1999) equation, Imoto et al.’s (2021) equation, and the middle point method for 14 paleo-earthquake sequences. By comparing the different values of the shape parameter, we investigate the characteristic features of the methods based on features such as the bias of the extraordinary small shape parameter and others.
First, we use the Monte Carlo method to obtain 10,000 sets of earthquake sequences. For each earthquake sequence, the model parameters (μi,αi,i=1,10000)are determined by the maximum likelihood method. The median and mean of the 10,000 shape parameters are denoted as αmed and αmean, respectively. The shape parameter determined by the middle point method is denoted as αMi. Ogata’s (1999) multiple product form (Table 1(1)) is approximated here by the multiple products of the Monte Carlo series (Table 1(2)). The grid search method is used to find a set of parameters (μAr,αAr) that maximizes Equation (2) in Table 1. The equation proposed by Imoto et al. (2021) (Table 1(3)) is approximated by Table 1(4). The maximum likelihood estimates in Table 1(4) are denoted as (μGe,αGe).
Figures 1-a and 1-b show the comparison between αmean and the values obtained by other methods. The abscissa indicates αmean, and the ordinate indicates values to be compared. In Fig. 1-a, αmed, αMi and αAr are marked with - (red), × (blue), and a squre (black), respectively. The value of αmed is always slightly smaller than that of αmean. It should be noted that αMi is even smaller and has not been found to exceed αmed. Extraordinary small values of 0.01 to 0.03 for αAr were found in five cases, in each of which earthquake sequences of periodic recurrence could be assumed in the respective set of uncertain ranges.
In Fig. 1-b, αmed and αGe are marked with - (red) and a circle (blue-green), respectively. The αGe value is slightly larger than that of αmean. When the αi value varies widely, extremely larger values contribute to the deviations of αmean and αGe from αmed. The biases of the shape parameter toward smaller values of αMi and αAr are due to their analysis forms, while the fact that αGe is determined to be larger than αmean reflects the situation of the data used, where extremely short earthquake intervals are allowed.
By comparing the shape parameters obtained by the different methods, the following results were obtained. In the cases of the middle point method αMi and αAr based on the Ogata (1999) equation, the biases toward smaller values are remarkable, and it is not appropriate to use these parameters as representative shape parameters of the paleo-earthquake sequences. It is reasonable to consider the set of parameters (μGe,αGe) as the representative set of the paleo-earthquake sequences since the (μGe,αGe) values are determined by using all the earthquake intervals at the same time.
In this paper, we estimate the model parameters of the Brownian Passage Time (BPT) distribution by using the sampling method, Ogata’s (1999) equation, Imoto et al.’s (2021) equation, and the middle point method for 14 paleo-earthquake sequences. By comparing the different values of the shape parameter, we investigate the characteristic features of the methods based on features such as the bias of the extraordinary small shape parameter and others.
First, we use the Monte Carlo method to obtain 10,000 sets of earthquake sequences. For each earthquake sequence, the model parameters (μi,αi,i=1,10000)are determined by the maximum likelihood method. The median and mean of the 10,000 shape parameters are denoted as αmed and αmean, respectively. The shape parameter determined by the middle point method is denoted as αMi. Ogata’s (1999) multiple product form (Table 1(1)) is approximated here by the multiple products of the Monte Carlo series (Table 1(2)). The grid search method is used to find a set of parameters (μAr,αAr) that maximizes Equation (2) in Table 1. The equation proposed by Imoto et al. (2021) (Table 1(3)) is approximated by Table 1(4). The maximum likelihood estimates in Table 1(4) are denoted as (μGe,αGe).
Figures 1-a and 1-b show the comparison between αmean and the values obtained by other methods. The abscissa indicates αmean, and the ordinate indicates values to be compared. In Fig. 1-a, αmed, αMi and αAr are marked with - (red), × (blue), and a squre (black), respectively. The value of αmed is always slightly smaller than that of αmean. It should be noted that αMi is even smaller and has not been found to exceed αmed. Extraordinary small values of 0.01 to 0.03 for αAr were found in five cases, in each of which earthquake sequences of periodic recurrence could be assumed in the respective set of uncertain ranges.
In Fig. 1-b, αmed and αGe are marked with - (red) and a circle (blue-green), respectively. The αGe value is slightly larger than that of αmean. When the αi value varies widely, extremely larger values contribute to the deviations of αmean and αGe from αmed. The biases of the shape parameter toward smaller values of αMi and αAr are due to their analysis forms, while the fact that αGe is determined to be larger than αmean reflects the situation of the data used, where extremely short earthquake intervals are allowed.
By comparing the shape parameters obtained by the different methods, the following results were obtained. In the cases of the middle point method αMi and αAr based on the Ogata (1999) equation, the biases toward smaller values are remarkable, and it is not appropriate to use these parameters as representative shape parameters of the paleo-earthquake sequences. It is reasonable to consider the set of parameters (μGe,αGe) as the representative set of the paleo-earthquake sequences since the (μGe,αGe) values are determined by using all the earthquake intervals at the same time.