jja ca giancang zhuaua gngds.rois.ac.jp/archive/tric/human/issi2011/media/e-1.pdf · outlines 1....

N d E h k F G d bNext‐day Earthquake Forecasts Generated by the ETAS Model and performance evaluationp

Jiancang ZhuangJ a ca g ua g

Institute of Statistical Mathematics TokyoInstitute of Statistical Mathematics, Tokyo

OutlinesOutlines

1. ETAS model in CSEP projects

2. “Offline optimization, online forecasting” R li ti f ti ETAS d l i thRealization of space‐time ETAS model in the CSEP projects.

3 Output examples of the ETAS model in the3. Output examples of the ETAS model in the CSEP project.

Space-Time Epidemic Type Aftershock Sequence (ETAS) model(ETAS) model

Seismicity rate = "background" + “Triggered seismicity":y g gg y

:( , , ) ( , ) ( ) ( ) ( , )

i

i i i ii t t

t x y x y m g t t f x x y yλ μ κ<

= + − − −∑

Time distribution: the Omori Utsu law 011)( >⎟

⎞⎜⎛ +

− −

ttptp

the Omori-Utsu law

Spatial location distribution of children:

0 ,1)( >⎟⎠⎞

⎜⎝⎛ += t

ccptg

Spatial location distribution of children:

1 ,11);,( )(

22

)( >⎟⎟⎠

⎞⎜⎜⎝

⎛ ++

−=

−

qD

yxD

qmyxfq

mmmm γγ

productivity: mean number of children

,);,( )()( ⎟⎠

⎜⎝

−− qDeDe

yf mmmm cc γγπ

cmm mmAem c ≥= − ,)( )(ακ

Space time ETAS modelSpace‐time ETAS model

• Conditional intensity

:( , , ) ( , ) ( ) ( ) ( , )

i

i i i ii t t

t x y x y m g t t f x x y y mλ μ κ<

= + − − −∑ ;

• Likelihood function

0( , , ) [0, ]

ln ln ( , , ) ( , , )i i i

T

i i it x y T A A

L t x y t x y dtdxdyλ λ∈ ×

= −∑ ∫ ∫∫


• Time varying seismicity rate (conditional intensity or stochastic intensity)

:( , , ) ( , ) ( ) ( ) ( , , )

i

i i i it

ii t


= + − − −∑i

Contribution from background seismicitybackground seismicity

( , )( , , )

x yt x y

μλ


• Time varying seismicity rate (conditional intensity or stochastic intensity) at event j

:( , , ) ( , ) ( ) ( ) ( , )

i

j j j j j i j i j i j ii t t

t x y x y m g t t f x x y yλ μ κ<

= + − − −∑: ii t t<

Contribution from background seismicity Pr{event j is from background}

( )

background seismicity { j g }( , )

( )j j

j

x yt x yμλ

ϕ =( , )

( , , )j j

j j j

x yt x yμλ

( , , )j j jt x yλ

Estimation problemsEstimation problems


:( , , ) ( , ) ( ) ( ) ( , , )i i i i

ti

i tt x y x y m g t t f x x y y mλ μ κ

<

= + − − −∑: i ti t <

Contribution from background seismicity Clustering partbackground seismicity Clustering part



:( , , ) ( , ) ( ) ( ) ( , , )

i

i i i it

ii t


= + − − −∑

How to estimate background

How to estimate l

How to estimate time independent background

seismicity?clustering parameters?

total seismicity

?),( yxλ



:( , , ) ( , ) ( ) ( ) ( , , )

i

i i i it

ii t


= + − − −∑


How to estimate clustering

How to estimate time‐free total background


seismicity

?),( yxλ

Kernel, spline, tessellationtessellation, histogram, …



:( , , ) ( , ) ( ) ( ) ( , , )

i

i i i it

ii t


= + − − −∑





seismicity

?),( yxλ

Kernel functions, spline tessellation

Maximum likelihood estimate if background

spline, tessellation, histogram, …

seismicity μ is known



:( , , ) ( , ) ( ) ( ) ( , , )

i

i i i it

ii t


= + − − −∑





seismicity

?),( yxλ

Kernel functions, spline tessellation

Maximum likelihood estimate if background ?spline, tessellation,

histogram, …seismicity μ is known

?



:( , , ) ( , ) ( ) ( ) ( , , )

i

i i i it

ii t


= + − − −∑





seismicity

?),( yxλ

Kernel function, spline, Kernel function, spline,

tessellation histogram ith

Pr{event j is from background}( , )

( , , )j j

j j jj

x yt x yμλ

ϕ =

tessellation, histogram, …

tessellation, histogram, …, with each event weighted by jϕ



:( , , ) ( , ) ( ) ( ) ( , , )

i

i i i it

ii t


= + − − −∑Kernel functions with each event weighted by

Kernel functions jϕfunctions j

∑ dh )(1)(ˆ

∑ −−= ii dyyxxhT

yx );,(1),(λ̂∑ −−=

jjjj dyyxxh

Tyx );,(),( ϕμ

iT

Solution—estimating parameters and background rate simultaneously

It ti l ithIterative algorithm:1. Assume an initial background rate.2 Using MLE to estimate parameters in the clustering2. Using MLE to estimate parameters in the clustering

structures.3. Using the assumed background and estimated clustering g g g

parameters to evaluate .4. Using to get a better background rate.5 U d t th b k d t b thi b tt

jϕ

jϕ5. Update the background rate by this better one.6. Go to Steps 2 to 5 until results converge.

: Estimate of probability that event j is of backgroundjϕ

Simulation (forecast) algorithmSimulation (forecast) algorithm

G t th b k d t l ith th ti t d• Generate the background catalog with the estimated background rate, recorded as Generation 0.

• For each event, in the last simulated generation, generate its children, with their occurrence times, glocations and magnitude from the p.d.f.s as assumed in the model, where the number of children is a Poisson random variable with a mean of the productivityrandom variable with a mean of the productivity function.

• Repeat last step until no more new event is generated. Return with all the events in all generations

Diagram of CSEP ETAS implementation

Input module

Estimation Simulation

Online forecasting

Earthquake catalog+

Historical catalog

Estimation Estimating background

probabilitieswithout optimization

Simulation Simulation seismicity

In the forecasting regionand time interval

Kernel Smoothing

Input controls:1 P l i

Providing model parameters for online forecasting1. Polygon region

for model fitting2. Magnitude threshold3. Bandwidths for

for online forecasting

Outputs

estimating background rates

4. Bandwidths for smoothing in f i

EstimationEstimating

model parametersvia optimization (MLE)

CSEP Common scoringforecasting

5.Time intervals for training and forecast

scoring procedures

Offline optimization

Forecasting Landers aftershocksForecasting Landers aftershocks

ETAS model Poisson model

Unit: events/day/deg^2

Forecasting for an aseismic period(2007/01/01)

ETAS model Poisson model


Example 2: Retrospective forecast of the Tokachi‐Oki EarthquakeEarthquake

lData: JMA catalog, 121‐155E , 21‐48N, Depth 0‐100km, MJ≥4.0 , 1965 1 1 t 2003 9 231965‐1‐1 to 2003‐9‐23

Example 2: Retrospective forecast of the Tokachi‐Oki Earthquake

Time independent total seismicity ratetotal seismicity rate

Time free

Background seismicity rate



Unit: events/day/deg^2Unit: events/day/deg^2

Time free

Typical scoring methods (1)Typical scoring methods (1)(1) Contingency‐table based test for Deterministic:

Yes/No prediction :Yes/No prediction :

Eq obs. No eq. obse Row sumq q

Predict eqs a b a+b

Predict no eq d c c+d

Hanssen Kuiper skill score or R score namely

Column sum a+d b+c N=a+b+c+d

Hanssen‐Kuiper skill score, or R‐score, namely

a bR =HKRa d b c

= −+ +

S i l i l l t fScoring alarming‐level type of predictionspredictions

Molchan’s error diagram (Molchan 1990, 1991, 1997, 2003; ROC curve): )

usually for alarming‐level based forecastsa: number of successful forecasts of occurrence; b: number of false alarms; c: number of successful forecastsf

(Molchan, 1995)

of non‐occurrence; d: number of failures to predict.

~d a ba d a b c d

ν τ += =

+ + + +a d a b c d+ + + +

Gambling score (1)Gambling score (1)

Each time the forecaster make a prediction, he bets 1 point of his reputations. If he fails, p p ,he looses this point; if he wins, he should be rewarded fairlybe rewarded fairly.

Question: How to fairly reward the forecaster for a success?for a success?

Gambling score (2)Gambling score (2)Question: How to reward the forecaster for a success

fairly?

Answer: 0 0(1 ) /G p p= −

Return for each prediction

0 : prob. given by the reference model that the prediction is correctp

Return for each prediction

Earthquake occurrence Yes No

Forecaster predicts Yes Gyes -1

Forecaster predicts No -1 GForecaster predicts No 1 Gno

Forecaster predicts Yes with prob. p

Gyesp-(1-p) (1-p)Gno-pwith prob. p

scoring methods for probability scoring methods for probability forecasts Information score (likelihood ratio, Vere‐Jones

1998):1998):

p 1 p⎡ ⎤i ii i(0) (0)

i i i

p 1-pI= X log +(1-X )logp 1-p

⎡ ⎤⎢ ⎥⎣ ⎦

∑

i(0)

p :prob. of earthquake occurrence given by model;

b i b b li d l(0)i

i

p : prob. given by baseline model;X : =0,if no event occurs;=1 otherwise.i


observed daily numbers

forecasted daily numbers

5%‐95% confidence bands

Time free

Temporal variation of forecasted daily numbers of earthquake during p y q g2003‐09‐24 to 2003‐10‐22.


fTime free

Daily information gains against the Poisson model

ConclusionsConclusions• The space‐time ETAS model has been implemented as an “off‐lineThe space time ETAS model has been implemented as an off line

optimization and online forecasting” scheme in the Japan and SCEC CSEP projects. It consists of four components: (1) off‐line optimization; (2) a simulation procedure; (3) a smoothing procedure. p ; ( ) g p

• Using the ETAS model, I have made retrospective experiments on 1‐day forecasts of earthquake probabilities in the Japan region before and afterforecasts of earthquake probabilities in the Japan region before and after the Tokachi‐Oki earthquake in September 2003, in the format of contour images.

• These forecasts were test against the reference model, the Poisson process which is stationary in time but spatially inhomogeneous. As

t d th f t b d th ETAS d l t h th t l dexpected, the forecasts based on the ETAS model catch the temporal and spatial features of the aftershock sequence, and the ETAS model performs better than the Poisson model.

This presentation is partially based on Zhuang, 2011, EPS

Thank you.

jja ca giancang zhuaua gngds.rois.ac.jp/archive/tric/human/issi2011/media/e-1.pdf · outlines 1....

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