( )ij i j M ) ~( , - ism.ac.jp · 6) Varini, E. (2005) Sequential estimation methods in...

1
~ Gamma(mean=0.4, var=0.03) ~ Normal(mean=-5, var=3) ~ Gamma(mean=0.01, var=0.00005) ~ Gamma(mean=1.5, var=0.7) (1+q 11 , q 12 , q 13 ) ~ Dirichlet(35, 2, 0.5) At each time t , a PARTICLE SET is sampled from the Posterior distribution where is the weight of the sample . Let Y = be a time-magnitude sequence of earthquakes occurred in a region ( , ). We consider the following marked point processes for seismic sequences: ( are parameters to estimate) Poisson model (Pm) : random time occurrence of the earthquakes. Stress Release model (Srm) : some elastic stress gradually accumulates and is suddenly released when the stress exceeds the strength of the medium. Etas model (Em) : tendency of the earthquakes to occur in clusters. ( ) ) ( - 2 ) ( t R t e t ρ β α λ + = Particle filtering of a state-space model for seismic sequences Varini Elisa (2) , Chopin Nicolas (1) and Rotondi Renata (2) (1) Department of Mathematics - University of Bristol (UK) (2) Institute for Applied Mathematics and Information Technology – CNR, Milan (Italy) N i i i i m t y 1 } ) , ( { = = ] , 0 [ T t i 0 M m i μ λ = ) ( 1 t : ) - ( 3 ) - ( ) ( 0 t t i p i M m i i c t t e k t < + = γ λ : ) - ( 75 . 0 0 10 ) ( t t i M m i i t R < = p c k , , , , , , , γ ρ β α μ Remark : each of these models is considered as reasonable for a specific subset of earthquakes, that is, the best fitting model changes in different time intervals. This leads to devise a STATE - SPACE MODEL ( X , Y ) STATE PROCESS X = : Homogeneous pure jump Markov process such that and the unknown generator is , where for all and . OBSERVATION PROCESS Y = : Marked point process such that . X and Y have no common jumps. 0 } { t t Y 0 } { t t X ) ( ) ( t t t X λ λ = { } 3 , 2 , 1 = S X t ( ) S j i ij q Q = , 0 > ij q j i 0 = S j ij q time t ) (t λ We are interested in 1) ESTIMATE OF THE PARAMETERS 2) FILTERING DISTRIBUTION , for and the past observed history, since it allows the prediction (t > s): We apply a ) , | ( t t j X P H θ = Pm Srm Em Srm Pm Em 1 τ 2 τ 3 τ 4 τ 5 τ . ) , | ( ) , , | ( ] , | [ s s S j s s t s t i X P i X j X P j X P H H H θ θ θ = = = = = ) , , , , , , , , ( Q p c k γ ρ β α μ θ = BAYESIAN PARTICLE FILTERING PROCEDURE It is an iterative procedure providing a discrete approximation of the posterior distribution , where is a realization of the state process in [0,t] . It is based on the importance sampling principle : when it is difficult to draw a sample from the target distribution p, an easy-to sample distribution , called importance distribution, is considered so that , where . ) | , ( : 0 t t x p H θ t x : 0 m i i t i t i t x ,..., 1 ) ( ) ( : 0 ) ( } ) , , ( { = ω θ ) , ( ) ( : 0 ) ( i t i t x θ ) (i t ω ) | , ( : 0 t t x p H θ For i = 1,…, m - at time t = 0 , - at time t = t n-1 - at time t = t n where ) , ( ~ ) , , ( 0 ) ( 0 ) ( 0 ) ( 0 x p x i i i θ ω θ ~ ~ ~ ) | , ( ~ ) , , ( 1 - 1 - : 0 ) ( 1 - ) ( 1 - : 0 ) ( 1 - n n i n i n i n x p x H θ ω θ M ) | , ( ~ ) , , ( : 0 ) ( ) ( : 0 ) ( n n i n i n i n x p x H θ ω θ ) ( 1 - ) ( i n i n θ θ = ) , ( ) ( : 1 - ) ( 1 - : 0 ) ( : 0 i n n i n i n x x x = ) , | ( ~ 1 - 1 - : 1 - ) ( : 1 - n n n n i n n x x H θ π ) , | ( : 0 : 1 - ) ( 1 - ) ( n n n n i n i n x y p H ω ω 1 1 ) ( = = m i i n ω Likelihood in ( t n-1 , t n ] Importance distribution (we choose it equal to the prior distribution) ) , ( ) , ( x d x dp θ π ω θ = ) , ( / ) , ( x x p θ π θ ω = ABSTRACT A state-space model is proposed in order to analyse a sequence of earthquakes; the basic assumption is that, at each time, the physical process is in an unobservable state, chosen in a finite set, and that each state is characterized by the occurrence of events following a specific marked point process. We consider three possible states corresponding to the following point processes for seismic sequences: the Poisson model, the stress release model and the Etas model. Statistical inference is carried out by exploiting a Bayesian sequential Monte Carlo (or particle filtering) method. This recent statistical methodology allows us to estimate both the model parameters and the filtering probabilities (the probability that one of the considered marked point process is active at time t). Moreover, it allows us to update the present estimates as new information comes in. S j t H so that where M Prior distribution In order to test the methodology, we study a SIMULATED DATA SET We set the model parameters as follows: and we perform the simulation of the proposed state-space process by using the inverse transform method and the thinning method. The magnitude of each event is simulated from a truncated exponential distribution following the Gutenberg- Richter law. S j j X P Q p c k all for 3 / 1 ) ( 06 . 0 - 006 . 0 054 . 0 0135 . 0 015 . 0 - 0015 . 0 002 . 0 018 . 0 02 . 0 - 1 , 5 . 0 , 6 . 0 , 1 . 0 , 8 . 2 , 02 . 0 , 5 . 4 - , 5 . 0 0 = = = = = = = = = = = γ ρ β α μ REFERENCES 1) Carpenter, J., Clifford, P., and Fearnhead, P. (1999), An improved particle filter for non–linear problems, Radar, Sonar and Navigation, IEE Proceedings, 146, 1,2-7. 2) Doucet, A., De Freitas, N., and Gordon, N., Sequential Monte Carlo methods in practice (Springer-Verlag, New York, 2001). 3) Kitagawa, G. (1998) A self-organizing state-space model, Journal of the American Statistical Association, 93, 443, 1203-1215. 4) Lewis, P.A., Shedler G.S. (1976) Simulation of nonhomogeneous Poisson processes with log linear rate function, Biometrika, 63, 3, 501-505. 5) Ogata, Y. (1999) Seismicity analysis through point-process modeling: a review, Pure and Applied Geophysics, 155, 471-507. 6) Varini, E. (2005) Sequential estimation methods in continuous-time state-space models, PhD thesis, University Bocconi of Milan, Italy. 7) Zheng, X.G., Vere-Jones, D. (1991) Application of stress release models to historical earthquakes from North China, Pure and Applied Geophysics, 135, 4, 559-576. ANALYSIS AND RESULTS Table - Summary of the prior and importance distributions. They are chosen to be equal, so that we get an easy formula to update the weights . ) (i n ω Figure 1-top : Conditional intensity function related to the simulation of the state-space model (black line) and mixture of the plug-in estimates of the conditional intensity functions (violet line). The horizontal coloured line represents the simulated states active at each instant. Figure 1-bottom : the simulated time-magnitude data set. magnitude (t) time (in years) k ~ Gamma(mean=0.5, var=0.7) ~ Gamma(mean=0.7, var=0.2) c ~ Gamma(mean=0.3, var=0.02) p ~ Gamma(mean=1, var=0.005) A uniform initial distribution. Figure 2 – The prior particle set (violet), the posterior particle set (ruled histogram) and the value used in the simulation (green) are represented for the parameters , p, q 23 respectively. ρ 23 q p Future research: in the light of the complexity of the model, the results are quite satisfactory and the methodology seems promising. The 67% of the states corresponding to the maximum filtering probabilities matches the states of the simulated data set; a refinement of the parameter estimation could lead to further improvement. m i / 1 ) ( 0 = ω

Transcript of ( )ij i j M ) ~( , - ism.ac.jp · 6) Varini, E. (2005) Sequential estimation methods in...

Page 1: ( )ij i j M ) ~( , - ism.ac.jp · 6) Varini, E. (2005) Sequential estimation methods in continuous-time state-space models, PhD thesis, University Bocconi of Milan, Italy. 7) Zheng,

� ~ Gamma(mean=0.4, var=0.03)α ~ Normal(mean=-5, var=3) β~ Gamma(mean=0.01, var=0.00005)ρ ~ Gamma(mean=1.5, var=0.7)

(1+q11, q12, q13) ~ Dirichlet(35, 2, 0.5)

At each time t ,

a PARTICLE SET is sampled from the Posterior distribution

where is the weight of the sample .

Let Y = be a time-magnitude sequence of earthquakes

occurred in a region ( , ).

We consider the following marked point processes for seismic sequences:

( are parameters to estimate)

Poisson model (Pm) : random time occurrence of the earthquakes.

Stress Release model (Srm) : some elastic stress gradually accumulates and is suddenly released when the stress exceeds thestrength of the medium.

Etas model (Em) : tendency of the earthquakes to occur in clusters.

( ))(-2 )( tRtet ρβαλ +=

Particle filtering of a state-space model for seismic sequencesVarini Elisa (2), Chopin Nicolas (1) and Rotondi Renata (2)

(1) Department of Mathematics - University of Brist ol (UK)

(2) Institute for Applied Mathematics and Informati on Technology – CNR, Milan (Italy)

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Remark : each of these models is considered as reasonable for a specific subset of earthquakes, that is, the best fitting model changes in different time intervals. This leads to devise a

STATE - SPACE MODEL ( X , Y )

STATE PROCESS X = : Homogeneous pure jump Markov process

such that and the unknown generator is ,

where for all and .

OBSERVATION PROCESSY= : Marked point process

such that .

X and Y have no common jumps.

0≥}{ ttY

0≥}{ ttX

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ijq

time t

)(tλ

We are interested in

1) ESTIMATE OF THE PARAMETERS

2) FILTERING DISTRIBUTION , for and the past observed history,

since it allows the prediction (t > s):

We apply a

),|( tt jXP Hθ=

Pm Srm Em Srm Pm Em

1τ 2τ 3τ 4τ 5τ

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ssSj

sstst iXPiXjXPjXP HHH θθθ =====

),,,,,,,,( Qpck γρβαµθ =

BAYESIAN PARTICLE FILTERING PROCEDURE

• It is an iterative procedure providing a discrete approximation of the posterior distribution , where is a realization of the state process in [0,t] .

• It is based on the importance sampling principle : when it is difficult to draw a sample from the target distribution p, an easy-to sample distribution π , called importance distribution, is considered so that , where .

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- at time t = 0 ,

- at time t = tn-1

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Likelihood in ( tn-1 , tn ]

Importance distribution

(we choose it equal to the prior distribution)

),(),( xdxdp θπωθ = ),(/),( xxp θπθω =

ABSTRACT

A state-space model is proposed in order to analyse a sequence of earthquakes; the basic assumption is that, at each time, the physical process is in an unobservable state, chosen in a finite set, and that each state is characterized by the occurrence of events following a specific marked point process. We consider three possible states corresponding to the following point processes for seismic sequences: the Poisson model, the stress release model and the Etas model. Statistical inference is carried out by exploiting a Bayesian sequential Monte Carlo (or particle filtering) method. This recent statistical methodology allows us to estimate both the model parameters and the filtering probabilities (the probability that one of the considered marked point process is active at time t). Moreover, it allows us to update the present estimates as new information comes in.

Sj ∈∈∈∈ tH

so that

where

M

Prior distribution

In order to test the methodology, we study a

SIMULATED DATA SET

We set the model parameters as follows:

and we perform the simulation of the proposed state-space process by using the inverse transform method and the thinning method. The magnitude of each event is simulated from a truncated exponential distribution following the Gutenberg-Richter law.

SjjXPQ

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REFERENCES1) Carpenter, J., Clifford, P., and Fearnhead, P. (1999), An improved particle filter for non–linear problems, Radar, Sonar and Navigation, IEE Proceedings, 146, 1,2-7.2) Doucet, A., De Freitas, N., and Gordon, N., Sequential Monte Carlo methods in practice(Springer-Verlag, New York, 2001).3) Kitagawa, G. (1998) A self-organizing state-space model, Journal of the American Statistical Association, 93, 443, 1203-1215. 4) Lewis, P.A., Shedler G.S. (1976) Simulation of nonhomogeneous Poisson processes with log linear rate function, Biometrika, 63, 3, 501-505. 5) Ogata, Y. (1999) Seismicity analysis through point-process modeling: a review, Pure and Applied Geophysics, 155, 471-507.6) Varini, E. (2005) Sequential estimation methods in continuous-time state-space models, PhD thesis, University Bocconi of Milan, Italy.7) Zheng, X.G., Vere-Jones, D. (1991) Application of stress release models to historical earthquakes from North China, Pure and Applied Geophysics, 135, 4, 559-576.

ANALYSIS AND RESULTS

Table - Summary of the prior and importance distributions. They are chosen to be equal, so that we get an easy formula to update the weights .)(i

Figure 1-top : Conditional intensity function related to the simulation of the state-space model (black line) and mixture of the plug-in estimates of the conditional intensity functions (violet line). The horizontal coloured line represents the simulated states active at each instant.

Figure 1-bottom : the simulated time-magnitude data set.

mag

nitu

de

λ (t)

time (in years)

k ~ Gamma(mean=0.5, var=0.7)γ ~ Gamma(mean=0.7, var=0.2)

c ~ Gamma(mean=0.3, var=0.02)

p ~ Gamma(mean=1, var=0.005)

A uniform initial distribution.

Figure 2 – The prior particle set (violet), the posterior particle set (ruled histogram) and the value used in the simulation (green) are represented for the parameters

ρ

, p, q23 respectively.

ρ23qp

Future research: in the light of the complexity of the model, the results are quite satisfactory and the methodology seems promising. The 67% of the states corresponding to the maximum filtering probabilities matches the states of the simulated data set; a refinement of the parameter estimation could lead to further improvement.

mi /1)(0 =ω