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One of the most important and interesting issues associated with the earthquakes is the long-term trend of the extreme events. Extreme value theory provides methods for analysis of the most extreme parts of data. We estimated the annual maximum magnitude of earthquakes in Japan by extreme value theory using earthquake data between 1900 and 2019. Generalized extreme value (GEV) distribution was applied to fit the extreme indices. The distribution was used to estimate the probability of extreme values in specified time periods. The various diagnostic plots for assessing the accuracy of the GEV model fitted to the magnitude of maximum earthquakes data in Japan gave the validity of the GEV model. The extreme value index,
*ξ* was evaluated as
−0.163, with a 95% confidence interval of [
−0.260,
−0.0174] by the use of profile likelihood. Hence, the annual maximum magnitude of earthquakes has a finite upper limit. We obtained the maximum return level for the return periods of 10, 20, 50, 100 and 500 years along with their respective 95% confidence interval. Further, to get a more accurate confidence interval, we estimated the profile log-likelihood. The return level estimate was obtained as 7.83, 8.60 and 8.99, with a 95% confidence interval of [7.67, 8.06], [8.32, 9.21] and [8.61, 10.0] for the 10-, 100- and 500-year return periods, respectively. Hence, the 2011 off the Pacific coast of Tohoku Earthquake, which was the largest in the observation history of Japan, had a magnitude of 9.0, and it was a phenomenon that occurs once every 500 year.

Extreme value theory has emerged as one of the most important statistical disciplines for the applied science. Using the extreme value theory, the theoretical distribution and its population parameter that the maximum value follows are estimated from long-term observation data. And the maximum value or a large value that occurs once every 100 years can be predicted based on the estimated result. Extreme value techniques are also becoming widely used for portfolio adjustment in the insurance industry, risk assessment on financial markets, and traffic prediction in telecommunications [

Statistical approaches focused on extreme values have shown promising results in forecasting unusual events in earth sciences, genetics and finance. For instance, Extreme Value Theory (EVT) was developed in the 1920s [

Applications of extreme value statistics in geology can be found in the magnitudes of and losses from earthquakes [

We used the annual maximum magnitude of earthquakes in Japan for 1900-2019 by the Japan Meteorological Agency.

When data are taken to be the maxima (or minima) over certain blocks of time (such as annual maximum precipitation), then it is appropriate to use the Generalized Extreme Value (GEV) distribution:

G ( z ) = { exp { − [ 1 + ξ ( z − μ σ ) ] − 1 / ξ } , f o r ξ ≠ 0 exp { − exp [ − ( z − μ σ ) ] } , f o r ξ = 0 , (1)

where μ is a location parameter; σ a scale parameter; and ξ a shape parameter. G is defined for all z such that (1 + ξ (z − μ)/σ) > 0 for ξ ≠ 0 and all z for ξ = 0. Three families of GEV distributions are defined depending on the value of ξ. For ξ > 0 we get the Fréchet distribution with heavy tail, ξ = 0, the Gumbel distribution with lighter tail and ξ < 0 the Weibull distribution with finite tail.

A method for modelling the extremes of a stationary time series is the method of block maxima, in which consecutive observations are grouped into non-overlapping blocks of length n, generating a series of m block maxima, Mn, 1, …, Mn, m, say, to which the GEV distribution can be fitted for some large value of n. The usual approach is to consider blocks of a given time length, thus yielding maxima at regular intervals [

Once a GEV distribution is fitted to empirical observations, it becomes possible to estimate the probability of an event that has not been observed yet. Estimates of extreme quantiles of the annual maximum distribution are obtained by inverting Equation (1):

z p = { μ − σ ξ [ 1 − { − log ( 1 − p ) } − ξ ] , f o r ξ ≠ 0 μ − σ log { − log ( 1 − p ) } , f o r ξ = 0 , (2)

where G(z_{p}) = 1 − p. The return level z_{p} is associated with the return period 1/p, since to a reasonable degree of accuracy, the level z_{p} is expected to be exceeded on average once every 1/p years. More precisely, z_{p} is exceeded by the annual maximum in any particular year with probability p [

Modeling was performed using the ismev package in R for GEV calculations.

The annual maximum magnitude of earthquakes in Japan was shown in

The various diagnostic plots for assessing the accuracy of the GEV model fitted to the magnitude of maximum earthquakes data in Japan are shown in

μ | σ | ξ | |
---|---|---|---|

Parameter estimate | 6.78 | 0.561 | −0.159 |

Standard errors | 0.0622 | 0.0437 | 0.0611 |

95% CI | [6.65, 6.90] | [0.475, 0.646] | [−0.279, −0.0397] |

Return period (year) | 10 | 20 | 50 | 100 | 500 |
---|---|---|---|---|---|

Return level | 7.84 | 8.10 | 8.41 | 8.60 | 8.99 |

Standard errors | 0.0950 | 0.117 | 0.159 | 0.197 | 0.300 |

95% CI | [7.65, 8.22] | [7.87, 8.33] | [8.09, 8.72] | [8.22, 8.99] | [8.40, 9.58] |

To get a more accurate confidence interval, we estimated the profile log-likelihood for the 10-, 100- and 500-year return periods in the annual magnitude of maximum earthquakes in Japan. From

We estimated the annual maximum magnitude of earthquakes in Japan by extreme value theory using earthquake data between 1900 and 2019. The GEV distribution was applied to fit the extreme indices. Our results are summarized as follows:

1) The various diagnostic plots for assessing the accuracy of the GEV model fitted to the magnitude of maximum earthquakes data in Japan gave the validity of the GEV model.

2) The extreme value index, ξ was evaluated as −0.163, with a 95% confidence interval of [−0.260, −0.0174] by the use of profile likelihood. Hence, the annual maximum magnitude of earthquakes has a finite upper limit.

3) We obtained the maximum return level for the return periods of 10, 20, 50, 100 and 500 years along with their respective 95% confidence interval.

4) To get a more accurate confidence interval, we estimated the profile log-likelihood. For the 10-, 100- and 500-year return periods, the return level estimate was obtained as 7.83, 8.60 and 8.99, with a 95% confidence interval of [7.67, 8.06], [8.32, 9.21] and [8.61, 10.0], respectively. Hence, the 2011 off the Pacific coast of Tohoku Earthquake, which was the largest in the observation history of Japan, had a magnitude of 9.0, and it was a phenomenon that occurs once every 500 year.

The author declares no conflicts of interest regarding the publication of this paper.

Maruyama, F. (2020) Analyzing the Annual Maximum Magnitude of Earthquakes in Japan by Extreme Value Theory. Open Journal of Applied Sciences, 10, 817-824. https://doi.org/10.4236/ojapps.2020.1012057