The Negative Binomial Distribution as a Renewal Model for the Recurrence of Large Earthquakes

Abstract

The negative binomial distribution is presented as the waiting time distribution of a cyclic Markov model. This cycle simulates the seismic cycle in a fault. As an example, this model, which can describe recurrences with aperiodicities between 0 and 0.5, is used to fit the Parkfield, California earthquake series in the San Andreas Fault. The performance of the model in the forecasting is expressed in terms of error diagrams and compared with other recurrence models from literature.

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Correspondence to Javier B. Gómez.

Appendix: Asymptotic Behavior of the Hazard Rate Function

Appendix: Asymptotic Behavior of the Hazard Rate Function

Recall that the N-step Markov-cycle distribution, Eq. (7), collapses to a NBD when all transition probabilities are equal, \(a = a_{1} = a_{2} = \cdots = a_{N}\):

$$P_{N,a} (n) = (1 - a)^{N} a^{n - N} \left( {\begin{array}{*{20}c} {n - 1} \\ {N - 1} \\ \end{array} } \right) = \left( {\frac{1 - a}{a}} \right)^{N} a^{n} \frac{(n - 1) \ldots (n - N + 1)}{(N - 1)!}.$$
(19)

Using the definition of hazard rate for a discrete distribution, Eq. (15) we can write

$$h_{N,a} (n) = \frac{{P_{N,a} (n)}}{{\mathop \sum \nolimits_{i = n}^{\infty } P_{N,a} (i)}} = \frac{{a^{n} (n - 1) \ldots (n - N + 1)}}{{\mathop \sum \nolimits_{i = n}^{\infty } a^{i} (i - 1) \ldots (i - N + 1)}} = \frac{1}{{\mathop \sum \nolimits_{i = 1}^{\infty } a^{i - n} \frac{i - 1}{n - 1} \ldots \frac{i - N + 1}{n - N + 1}}}.$$
(20)

To proceed further, we make the following change of variable:

$$i - n = m.$$
(21)

With this change of variable, the hazard rate of the general, two-parameter NBD, Eq. (20), can be written as

$$h_{N,a}^{ - 1} = \mathop \sum \limits_{m = 0}^{\infty } a^{m} \left( {1 + \frac{m}{n - 1}} \right) \ldots \left( {1 + \frac{m}{n - N + 1}} \right).$$
(22)

In the long-time limit, i.e., when n tends to infinity, we have

$$\mathop {\lim }\limits_{n \to \infty } h_{N,a}^{ - 1} = \mathop \sum \limits_{m = 0}^{\infty } a^{m} \left( {1 \times 1 \times 1 \times \cdots \times 1} \right) = \mathop \sum \limits_{m = 0}^{\infty } a^{m} = \frac{1}{1 - a}.$$
(23)

So, in the general, two-parameter NBD the asymptotic limit of the hazard rate is:

$$\mathop {\lim }\limits_{n \to \infty } h_{N,a} = 1 - a.$$
(24)

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Tejedor, A., Gómez, J.B. & Pacheco, A.F. The Negative Binomial Distribution as a Renewal Model for the Recurrence of Large Earthquakes. Pure Appl. Geophys. 172, 23–31 (2015). https://doi.org/10.1007/s00024-014-0871-2

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Key words

  • Negative binomial distribution
  • renewal process
  • seismic cycle
  • earthquake forecasting