Prediction and validation of short- to long-term earthquake probabilities in inland Japan
using the hierarchical space-time ETAS and space-time Poisson process models
Yosihiko Ogata (The Institute of Statistical Mathematics)
Training data

Spatial probability density score

Forecasting data

1923－2018, M≧4

2019－2021.9.30, M≧4(a)ˆ( x , y , M )
λ

iii
ˆ
log ∏ f ( xi , yi , M i ) = log ∏  ∞ˆ ( x, y, M ) dx dydM {i ; M i ≥ M c }{i ; M i ≥ M c } λ ∫M c ∫∫Inland(b)log10 λˆ ( x, y )Log (M≧4 / deg^2 / day)
10I will discuss the prediction ofearthquakes directly inland.
-- The Bayesian models such as
intensity functions for spatial
point configurations discussed
in this presentation are definedby piecewise linear functions
defined on Delaunay triangular
networks made from earthquake
configurations, which are linearlyinterpolated. In this regard.
-- We have recently published the software and manual(see Ogata et al. 2021 in the abstract).

Hierarchical space-time ETAS (HIST-ETAS) model
BG rateAftershock Productivity| H t ) µ ( x, y ) + ∑λ (t , x, y={ j :t j < t }K0 ( x j , y j )(t − t j + c)

(c)

(d)
Mw≧5.4log10 µˆ µ K ( x, y )Log10(M≧4 / deg^2 / day)log10 µˆ 5 pa ( x, y )Log10(M≧4 / deg^2 / day)Figure caption. Poisson spatio-temporal models for the long-term seismicity In
the inland Japan, all of which are obtained by the respective MAP estimate thatminimize the Akaike Bayesian Iinformation Criterion: (a) inland uniform
Poisson process model, (b) non-uniform spatial Poisson process model, (c)background ¥mu intensities of the HIST-ETAS-mK model and (d) that of theHIST-ETAS-5pa model, respectively. The colors and contours are inlogarithmic scale.

×p( x , y ) x − xj (x − x j , y − y j ) S j yy−j +d
α (xj , y j )M je







jj−q( x j , y j )Predicted number ofM≥4 earthquakes/day/square degree,where H t is the historical information of occurrences
before time t

Intensity rate :HIST-ETAS-5pa model：if α, p , and q depend on the position
coordinates in the above equation (the weights of theconstraints are 5-dimensional vectors)
HIST-ETAS-µk model：if α, p, and q are constants in the above
equation (the weights of the constraints are 2-dim. vectors)
Time-independent Poisson process models:λ (t , x=, y | H t ) λ ( x, y ) , 0 ≤ t ≤ T

Forecast ofmagnitude range(a) Uniform Poisson model for inland area
λˆinland if ( x, y ) ∈ inlandλ ( x, y ) =０otherwise(b) Nonhomogeneous Poisson modelsλ ( x, y ) = λˆ ( x, y ); optimal MAP estimate(c) Background rate of HIST-ETAS-µK modelλ ( x, y ) ∝ µˆ µ K ( x, y )(d) Background rate of HIST-ETAS-5pa modelλ ( x, y ) ∝ µˆ 5 pa ( x, y )The number is the "log-likelihood score" which compares the performance
relative to the score for the inland uniform Poisson process, which is set at 0.
-- The larger the score, the better the score, with red being the best. -- HISTETAS-5pa is the best at predicting magnitudes from 4.0 to 5.0, followed byHIST-ETAS-µK.

Allevents

M≥
5.5

M≥
6.0

M≥
6.5

M≥
7.0M≥
7.5

number of events 131

12911457253(a) uniform in inland n0.0

0.0

0.0

0.0

0.0

0.0

(b) non-homogeneous

-7.8

-11.3

-15.7 -16.3 -12.4-1.2(c) HIST-ETAS-mK BG

32.9

28.1

24.0

9.8

4.0

1.1

(d) HIST-ETAS-5pa BG

36.7

31.8

29.4

12.5

5.5

1.6