Arthur Charpentier - Freakonometrics .fr

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake ...... distribution P(λ), with probability function ... Proposition If (εt) are Poisson random variables, then (Nt) will also be a ..... γ(0) with γ(0) solution of ..... Seismicity Levels via Poisson Hidden Markov Models", Pure and Applied.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Bivariate Count Processes for Earthquake Frequency Mathieu Boudreault & Arthur Charpentier Université du Québec à Montréal [email protected] http ://freakonometrics.blog.free.fr/

Séminaire GeoTop, January 2012

1

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Figure : Time and distance distribution (to 6,000 km) of large (56.5) in days

4

Number of earthquakes (magnitude >4) per 15 sec., average before=100

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

220

Number of earthquakes before and after a major one, magnitude of the main event, small events more than 4 ● ●

200

Main event, magnitude > 6.5 Main event, magnitude > 6.8 Main event, magnitude > 7.1 Main event, magnitude > 7.4





180

● ● ●

80

100

120

140

160

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−15

−10

−5

0

5

10

15

Time before and after a major eathquake (magnitude >6.5) in days

5

1000



● ●

800

Same techtonic plate as major one Different techtonic plate as major one ● ●



200

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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ●●● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ●●● ● ● ●●●●●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●●● ●●●●●● ●● ● ● ● ● ●● ● ●●● ● ●● ● ●● ●● ●●●● ● ● ●● ● ● ● ●●●●● ● ● ● ● ● ● ●

0

Number of earthquakes (magnitude >2) per 15 sec., average before=100

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

−15

−10

−5

0

5

10

15

Time before and after a major eathquake (magnitude >6.5) in days

6

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Shapefiles from http://www.colorado.edu/geography/foote/maps/assign/hotspots/hotspots.html

7

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

We look at seismic risks with the eyes of actuaries and statisticians...

8

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

9

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Agenda • Motivation (Parsons & Velasco (2011)) • Modeling dynamics ◦ AR(1) : Gaussian autoregressive processes (as a starting point) ◦ VAR(1) : multiple AR(1) processes, possible correlated ◦ INAR(1) : autoregressive processes for counting variates ◦ MINAR(1) : multiple counting processes • Application to earthquakes frequency ◦ counting earthquakes on tectonic plates ◦ causality between different tectonic plates ◦ counting earthquakes with different magnitudes

10

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Part 1

Modeling dynamics of counts

11

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

(ANSS) http://www.ncedc.org/cnss/catalog-search.html

350

Number of earthquakes (Magnitude ≥ 5) per month, worldwide



300





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250

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● ● ●



● ●

200



150











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● ●



● ●



50

100

● ●





1970

1980

1990

2000

2010

Number of earthquakes (Magn.>5) worldwide per month

12

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

(ANSS) http://www.ncedc.org/cnss/catalog-search.html Number of earthquakes (Magnitude ≥ 5) per month, in western U.S.

15







10



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● ● ●

5

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● ●





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0

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1950

1960

1970

1980

1990

2000

2010

Number of earthquakes (Magn.>5) in West−US per month

13

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

(Gaussian) Auto Regressive processes AR(1) Definition A time series (Xt )t∈N with values in R is called an AR(1) process if Xt = φ0 +φ1 Xt−1 + εt

(1)

for all t, for real-valued parameters φ0 and φ1 , and some i.i.d. random variables εt with values in R. It is common to assume that εt are independent variables, with a Gaussian distribution N (0, σ 2 ), with density   2 1 ε ϕ(ε) = √ exp − 2 , ε ∈ R. 2σ 2πσ Note that we assume also that εt is independent of X t−1 , i.e. past observations X0 , X1 , · · · , Xt−1 . Thus, (εt )t∈N is called the innovation process.

14

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Example : Xt = φ1 Xt−1 + εt with εt ∼ N (0, 1), i.i.d., and φ = 0.6

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● ● ●

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2





0



−2



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−4

● ●●

0

50

100

150

200

250

300

15

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Example : Xt = φ1 Xt−1 + εt with εt ∼ N (0, 1), i.i.d., and φ = 0.6

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2





0



−2



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● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●





−4

● ●●

0

50

100

150

200

250

300

16

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

0.4 0.2 0.0

ACF

0.6

0.8

1.0

Example : Xt = φ1 Xt−1 + εt : autocorrelation ρ(h) = corr(Xt , Xt−h ) = φh1

0

5

10

15

20

Lag

17

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Definition A time series (Xt )t∈N is said to be (weakly) stationary if • E(Xt ) is independent of t ( =: µ) • cov(Xt , Xt−h ) is independent of t (=: γ(h)), called autocovariance function

Remark As a consequence, var(Xt ) = E([Xt − E(Xt )]2 ) is independent of t (=: γ(0)). Define the autocorrelation function ρ(·) as ρ(h) := corr(Xt , Xt−h ) = p

cov(Xt , Xt−h ) var(Xt )var(Xt−h )

=

γ(h) , ∀h ∈ N. γ(0)

Proposition (Xt )t∈N is a stationary AR(1) time series if and only if φ1 ∈ (−1, 1). Remark If φ1 = 1, (Xt )t∈N is called a random walk. Proposition If (Xt )t∈N is a stationary AR(1) time series, ρ(h) = φh1 ,

∀h ∈ N.

18

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

From univariate to multivariate models Density of the Gaussian distribution

Univariate gaussian distribution N (0, σ 2 )   2 x 1 exp − 2 , for all x ∈ R ϕ(x) = √ 2σ 2πσ

0.20 0.15 0.10 0.05 0.00 3 2 1 0 −1 1

−2

2

3

Multivariate gaussian distribution N (0, Σ)   x0 Σ−1 x 1 , ϕ(x) = p exp − d 2 (2π) | det Σ| for all x ∈ Rd .

0 −1

−3

−2 −3

X = AZ where AA0 = Σ and Z ∼ N (0, I) (geometric interpretation)

19

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Vector (Gaussian) AutoRegressive processes V AR(1) Definition A time series (X t = (X1,t , · · · , Xd,t ))t∈N with values in Rd is called a VAR(1) process if   X1,t = φ1,1 X1,t−1 + φ1,2 X2,t−1 + · · · + φ1,d Xd,t−1 + ε1,t     X = φ X 2,t 2,1 1,t−1 + φ2,2 X2,t−1 + · · · + φ2,d Xd,t−1 + ε2,t (2)  ···     X = φ X d,t d,1 1,t−1 + φd,2 X2,t−1 + · · · + φd,d Xd,t−1 + εd,t or equivalently    φ X  1,t   1,1    X2,t  φ2,1     ..  =  ..  .   .    Xd,t | {z } Xt

φd,1

|

φ1,2

···

φ2,2 .. .

···

φd,2 · · · {z Φ

φ1,d









ε X   1,t−1   1,t      φ2,d  X2,t−1  ε2,t      ..   ..  +  ..   .   .  .      φd,d Xd,t−1 εd,t } | {z } | {z } X t−1

εt

20

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

for all t, for some real-valued d × d matrix Φ, and some i.i.d. random vectors εt with values in Rd . It is common to assume that εt are independent variables, with a Gaussian distribution N (0, Σ), with density  0 −1  εΣ ε 1 exp − , ∀ε ∈ Rd . ϕ(ε) = p 2 (2π)d | det Σ| Thus, independent means time independent, but can be dependent componentwise. Note that we assume also that εt is independent of X t−1 , i.e. past observations X 0 , X 1 , · · · , X t−1 . Thus, (εt )t∈N is called the innovation process. Definition A time series (X t )t∈N is said to be (weakly) stationary if • E(X t ) is independent of t (=: µ) • cov(X t , X t−h ) is independent of t (=: γ(h)), called autocovariance matrix 21

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Remark As a consequence, var(X t ) = E([X t − E(X t )]0 [X t − E(X t )]) is independent of t (=: γ(0)). Define finally the autocorrelation matrix, q  ρ(h) = ∆−1 γ(h)∆−1 , where ∆ = diag γi,i (0) . Proposition (X t )t∈N is a stationary AR(1) time series if and only if the d eignvalues of Φ should have a norm lower than 1. Proposition If (X t )t∈N is a stationary AR(1) time series, ρ(h) = Φh , h ∈ N.

22

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Statistical inference for AR(1) time series Consider a series of observations X1 , · · · , Xn . The likelihood is the joint distribution of the vectors X = (X1 , · · · , Xn ), which is not the product of marginal distribution, since consecutive observations are not independent (cov(Xt , Xt−h ) = φh ). Nevertheless L(φ, σ; (X0 , X )) =

n Y

πφ,σ (Xt |Xt−1 )

t=1

where πφ,σ (·|Xt−1 ) is a Gaussian density. Maximum likelihood estimators are bσ (φ, b) ∈ argmax log L(φ, σ; (X0 , X ))

23

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Poisson distribution - and process - for counts N as a Poisson distribution is P(N = k) = e

k −λ λ

k!

where k ∈ N.

If N ∼ P(λ), then E(N ) = λ. (Nt )t≥0 is an homogeneous Poisson process, with parameter λ ∈ R+ if • on time frame [t, t + h], (Nt+h − Nt ) ∼ P(λ · h) • on [t1 , t2 ] and [t3 , t4 ] counts are independent, if 0 ≤ t1 < t2 < t3 < t4 , (Nt2 − Nt1 ) ⊥⊥ (Nt4 − Nt3 )

24

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Poisson processes and counting models Earthquake count models are mostly based upon the Poisson process (see Utsu (1969), Gardner & Knopoff (1974), Lomnitz (1974), Kagan & Jackson (1991)), Cox process (self-exciting, cluster or branching processes, stress-release models (see Rathbun (2004) for a review), or Hidden Markov Models (HMM) (see Zucchini & MacDonald (2009) and Orfanogiannaki et al. (2010)). See also Vere-Jones (2010) for a summary of statistical and stochastic models in seismology. Recently, Shearer & Starkb (2012) and Beroza (2012) rejected homogeneous Poisson model,

25

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Thinning operator ◦ Steutel & van Harn (1979) defined a thinning operator as follows Definition Define operator ◦ as p ◦N = Y1 + · · · + YN if N 6= 0, and 0 otherwise, where N is a random variable with values in N, p ∈ [0, 1], and Y1 , Y2 , · · · are i.i.d. Bernoulli variables, independent of N , with P(Yi = 1) = p = 1 − P(Yi = 0). Thus p ◦ N is a compound sum of i.i.d. Bernoulli variables. Hence, given N , p ◦ N has a binomial distribution B(N, p). L

Note that p ◦ (q ◦ N ) = [pq] ◦ N for all p, q ∈ [0, 1]. Further E (p ◦ N ) = pE(N ) and var (p ◦ N ) = p2 var(N ) + p(1 − p)E(N ).

26

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

(Poisson) Integer AutoRegressive processes IN AR(1) Based on that thinning operator, Al-Osh & Alzaid (1987) and McKenzie (1985) defined the integer autoregressive process of order 1 : Definition A time series (Xt )t∈N with values in R is called an INAR(1) process if Xt = p ◦ Xt−1 + εt ,

(3)

where (εt ) is a sequence of i.i.d. integer valued random variables, i.e. Xt−1

Xt =

X

Yi + εt , where Yi0 s are i.i.d. B(p).

i=1

Such a process can be related to Galton-Watson processes with immigration, or physical branching model.

27

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Xt+1 =

Xt X

Yi + εt+1 , where Yi0 s are i.i.d. B(p)

i=1

28

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

E(εt ) pE(εt ) + var(εt ) Proposition E (Xt ) = , var (Xt ) = γ(0) = and 1−p 1 − p2 γ(h) = cov(Xt , Xt−h ) = ph . It is common to assume that εt are independent variables, with a Poisson distribution P(λ), with probability function P(εt = k) = e

k −λ λ

k!

, k ∈ N.

Proposition If (εt ) are Poisson random variables, then (Nt ) will also be a sequence of Poisson random variables. Note that we assume also that εt is independent of X t−1 , i.e. past observations X0 , X1 , · · · , Xt−1 . Thus, (εt )t∈N is called the innovation process. Proposition (Xt )t∈N is a stationary INAR(1) time series if and only if p ∈ [0, 1). Proposition If (Xt )t∈N is a stationary INAR(1) time series, (Xt )t∈N is an homogeneous Markov chain. 29

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

π(xt , xt−1 )

=

P(Xt = xt |Xt−1 = xt−1 ) =

xt X

P

xt−1

!

X

Yi = xt − k · P(ε = k) . | {z } {z } Poisson

i=1

k=0

|

Binomial

30

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Inference of Integer AutoRegressive processes IN AR(1) Consider a Poisson INAR(1) process, then the likelihood is " n #   X0 Y λ λ L(p, λ; X0 , X ) = ft (Xt ) · exp − X0 X ! (1 − p) 1−p 0 t=1 where min{Xt ,Xt−1 }

ft (y) = exp(−λ)

X i=0

  λy−i Yt−1 i p (1 − p)Yt−1 −y , for t = 1, · · · , n. (y − i)! i

Maximum likelihood estimators are b ∈ argmax log L(p, λ; (X0 , X )) (b p, λ)

31

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Multivariate Integer Autoregressive processes M IN AR(1) Let Nt := (N1,t , · · · , Nd,t ), denote a multivariate vector of counts. Definition Let P := [pi,j ] be a d × d matrix with entries in [0, 1]. If N = (N1 , · · · , Nd ) is a random vector with values in Nd , then P ◦ N is a d-dimensional random vector, with i-th component [P ◦ N ]i =

d X

pi,j ◦ Nj ,

j=1

for all i = 1, · · · , d, where all counting variates Y in pi,j ◦ Nj ’s are assumed to be independent. L

Note that P ◦ (Q ◦ N ) = [P Q] ◦ N . Further, E (P ◦ N ) = P E(N ), and E ((P ◦ N )(P ◦ N )0 ) = P E(N N 0 )P 0 + ∆, with ∆ := diag(V E(N )) where V is the d × d matrix with entries pi,j (1 − pi,j ). 32

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Definition A time series (X t ) with values in Nd is called a d-variate MINAR(1) process if X t = P ◦ X t−1 + εt

(4)

for all t, for some d × d matrix P with entries in [0, 1], and some i.i.d. random vectors εt with values in Nd . (X t ) is a Markov chain with states in Nd with transition probabilities π(xt , xt−1 ) = P(X t = xt |X t−1 = xt−1 )

(5)

satisfying π(xt , xt−1 ) =

xt X

P(P ◦ xt−1 = xt − k) · P(ε = k).

k=0

33

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Parameter inference for M IN AR(1) Proposition Let (X t ) be a d-variate MINAR(1) process satisfying stationary conditions, as well as technical assumptions (called C1-C6 in Franke & Subba b of θ = (P , Λ) Rao (1993)), then the conditional maximum likelihood estimate θ is asymptotically normal, √ L b − θ) → n(θ N (0, Σ−1 (θ)), as n → ∞. Further, L b 0 ) − log L(N , θ|N 0 )] → 2[log L(N , θ|N χ2 (d2 + dim(λ)), as n → ∞.

34

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Granger causality with BIN AR(1) (X1,t ) and (X2,t ) are instantaneously related if ε is a noncorrelated noise,g g g ggggggggggg





        X p p1,2 X ε ε λ  1,t  =  1,1  ◦  1,t−1  +  1,t , with var  1,t  =  1 X2,t p2,1 p2,2 X2,t−1 ε2,t ε2,t ϕ | {z } | {z } | {z } | {z } Xt



P

X t−1



ϕ λ2



εt

35

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Granger causality with BIN AR(1) 1. (X1 ) and (X2 ) are instantaneously related if ε is a noncorrelated noise, g g g g gggggggggg





        X p p1,2 X ε ε λ  1,t  =  1,1  ◦  1,t−1  +  1,t , with var  1,t  =  1 X2,t p2,1 p2,2 X2,t−1 ε2,t ε2,t ? | {z } | {z } | {z } | {z } Xt



P

X t−1



? λ2



εt

36

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Granger causality with BIN AR(1) 2. (X1 ) and (X2 ) are independent, (X1 )⊥(X2 ) if P is diagonal, i.e. p1,2 = p2,1 = 0, and ε1 and ε2 are independent, 



        X p 0 X ε ε λ  1,t  =  1,1  ◦  1,t−1  +  1,t , with var  1,t  =  1 X2,t 0 p2,2 X2,t−1 ε2,t ε2,t 0 | {z } | {z } | {z } | {z } Xt



P

X t−1



0 λ2



εt

37

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Granger causality with BIN AR(1) 3. (N1 ) causes (N2 ) but (N2 ) does not cause (X1 ), (X1 )→(X2 ), if P is a lower triangle matrix, i.e. p2,1 6= 0 while p1,2 = 0, 



        X p 0 X ε ε λ  1,t  =  1,1  ◦  1,t−1  +  1,t , with var  1,t  =  1 X2,t ? p2,2 X2,t−1 ε2,t ε2,t ϕ | {z } | {z } | {z } | {z } Xt



P

X t−1



ϕ λ2



εt

38

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Granger causality with BIN AR(1) 4. (N2 ) causes (N1 ) but (N1,t ) does not cause (N2 ), (N1 )←(N2,t ), if P is a upper triangle matrix, i.e. p1,2 6= 0 while p2,1 = 0, 



        X p ? X ε ε λ  1,t  =  1,1  ◦  1,t−1  +  1,t , with var  1,t  =  1 X2,t 0 p2,2 X2,t−1 ε2,t ε2,t ϕ | {z } | {z } | {z } | {z } Xt



P

X t−1



ϕ λ2



εt

39

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Granger causality with BIN AR(1) 5. (N1 ) causes (N2 ) and conversely, i.e. a feedback effect (N1 )↔(N2 ), if P is a full matrix, i.e. p1,2 , p2,1 6= 0 



        X p ? X ε ε λ  1,t  =  1,1  ◦  1,t−1  +  1,t , with var  1,t  =  1 X2,t ? p2,2 X2,t−1 ε2,t ε2,t ϕ | {z } | {z } | {z } | {z } Xt



P

X t−1



ϕ λ2



εt

40

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Bivariate Poisson BIN AR(1) A classical distribution for εt is the bivariate Poisson distribution, with one common shock, i.e.  ε = M + M 1,t 1,t 0,t ε2,t = M2,t + M0,t where M1,t , M2,t and M0,t are independent Poisson variates, with parameters λ1 − ϕ, λ2 − ϕ and ϕ, respectively. In that case, εt = (ε1,t , ε2,t ) has joint probability function     1 ,k2 }  k2 min{k k1 X (λ2 − ϕ) k1 k2 ϕ −[λ1 +λ2 −ϕ] (λ1 − ϕ) i! e k1 ! k2 ! i i [λ1 − ϕ][λ2 − ϕ] i=0 with λ1 , λ2 > 0, ϕ ∈ [0, min{λ1 , λ2 }].    λ1 λ1    λ= and Λ = λ2 ϕ

ϕ λ2

  41

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Bivariate Poisson BIN AR(1) and Granger causality For instantaneous causality, we test H0 : ϕ = 0 against H1 : ϕ 6= 0 b denote the conditional maximum likelihood estimate of Proposition Let λ λ = (λ1 , λ2 , ϕ) in the non-constrained MINAR(1) model, and λ⊥ denote the conditional maximum likelihood estimate of λ⊥ = (λ1 , λ2 , 0) in the constrained model (when innovation has independent margins), then under suitable conditions, ⊥ L b b 2[log L(N , λ|N 0 ) − log L(N , λ |N 0 )] → χ2 (1), as n → ∞, under H0 .

42

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Bivariate Poisson BIN AR(1) and Granger causality For lagged causality, we test H0 : P ∈ P against H1 : P ∈ / P, where P is a set of constrained shaped matrix, e.g. P is the set of d × d diagonal matrices for lagged independence, or a set of block triangular matrices for lagged causality. b denote the conditional maximum likelihood estimate of P in Proposition Let P c b the non-constrained MINAR(1) model, and P denote the conditional maximum likelihood estimate of P in the constrained model, then under suitable conditions, c L b b 2[log L(N , P |N 0 ) − log L(N , P |N 0 )] → χ2 (d2 − dim(P)), as n → ∞, under H0 .

Example Testing (N1,t )←(N2,t ) is testing whether p1,2 = 0, or not.

43

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Autocorrelation of M IN AR(1) processes Proposition Consider a MINAR(1) process with representation X t = P ◦ X t−1 + εt , where (εt ) is the innovation process, with λ := E(εt ) and Λ := var(εt ). Let µ := E(X t ) and γ(h) := cov(X t , X t−h ). Then µ = [I − P ]−1 λ and for all h ∈ Z, γ(h) = P h γ(0) with γ(0) solution of γ(0) = P γ(0)P 0 + (∆ + Λ).

44

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Part 2

Application to earthquakes

45

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Multivariate models ?

Shapefiles from http://www.colorado.edu/geography/foote/maps/assign/hotspots/hotspots.html

46

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

The dataset, and stationarity issues We work with 16 (17) tectonic plates, – Japan is at the limit of 4 tectonic plates (Pacific, Okhotsk, Philippine and Amur), – California is at the limit of the Pacific, North American and Juan de Fuca plates. Data were extracted from the Advanced National Seismic System database (ANSS) http://www.ncedc.org/cnss/catalog-search.html – 1965-2011 for magnitude M > 5 earthquakes (70,000 events) ; – 1992-2011 for M > 6 earthquakes (3,000 events) ; – To count the number of earthquakes, used time ranges of 3, 6, 12, 24, 36 and 48 hours ; – Approximately 8,500 to 135,000 periods of observation ;

47

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Multivariate models : comparing dynamics  p X  1,t  =  1,1 X2,t p2,1 



     ε ε X p1,2 λ  ◦  1,t−1  +  1,t  with var  1,t  =  1 ε2,t ϕ ε2,t X2,t−1 p2,2  





ϕ λ2



Complete model, with full dependence

48

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Multivariate models : comparing dynamics  p X  1,t  =  1,1 X2,t 0 



     λ ε ε X 0  ◦  1,t−1  +  1,t  with var  1,t  =  1 ϕ ε2,t ε2,t X2,t−1 p2,2  





ϕ λ2



Partial model, with diagonal thinning matrix, no-crossed lag correlation

49

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Multivariate models : comparing dynamics  p X  1,t  =  1,1 X2,t 0 



     λ ε ε X 0  ◦  1,t−1  +  1,t  with var  1,t  =  1 0 ε2,t ε2,t X2,t−1 p2,2  





0 λ2



Two independent INAR processes

50

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Multivariate models : comparing dynamics  p X  1,t  =  1,1 X2,t 0 



     λ ε ε X 0  ◦  1,t−1  +  1,t  with var  1,t  =  1 0 ε2,t ε2,t X2,t−1 p2,2  





0 λ2



Two independent INAR processes

51

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Multivariate models : comparing dynamics    X 0  1,t  =  0 X2,t

        X ε ε λ 0  ◦  1,t−1  +  1,t  with var  1,t  =  1 0 X2,t−1 ε2,t ε2,t ϕ

ϕ λ2

 

Two (possibly dependent) Poisson processes

52

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Multivariate models : tectonic plates interactions – For all pairs of tectonic plates, at all frequencies, autoregression in time is important (very high statistical significance) ; – Long sequence of zeros, then mainshocks and aftershocks ; – Rate of aftershocks decreases exponentially over time (Omori’s law) ; – For 7-13% of pairs of tectonic plates, diagonal BINAR has significant better fit than independent INARs ; – Contribution of dependence in noise ; – Spatial contagion of order 0 (within h hours) ; – Contiguous tectonic plates ; – For 7-9% of pairs of tectonic plates, proposed BINAR has significant better fit than diagonal BINAR ; – Contribution of spatial contagion of order 1 (in time interval [h, 2h]) ; – Contiguous tectonic plates ; – for approximately 90%, there is no significant spatial contagion for M > 5 earthquakes 53

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Granger causality N1 → N2 or N1 ← N2 1. North American Plate, 2. Eurasian Plate,3. Okhotsk Plate, 4. Pacific Plate (East), 5. Pacific Plate (West), 6. Amur Plate, 7. Indo-Australian Plate, 8. African Plate, 9. Indo-Chinese Plate, 10. Arabian Plate, 11. Philippine Plate, 12. Coca Plate, 13. Caribbean Plate, 14. Somali Plate, 15. South American Plate, 16. Nasca Plate, 17. Antarctic Plate

Granger Causality test, 3 hours

1

3

2

5

4

6

8

7

Granger Causality test, 6 hours

9 10 11 12 13 14 15 16 17

1



2



3



4



5



6



7



8



9



10



11



12



13



14



15



16 17

● ●

1



3

2

5

4

6

8

7

9 10 11 12 13 14 15 16 17

1 2



3



4



5



6



7



8



9



10



11



12



13



14



15



16 17





● ●

54

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Granger causality N1 → N2 or N1 ← N2 1. North American Plate, 2. Eurasian Plate,3. Okhotsk Plate, 4. Pacific Plate (East), 5. Pacific Plate (West), 6. Amur Plate, 7. Indo-Australian Plate, 8. African Plate, 9. Indo-Chinese Plate, 10. Arabian Plate, 11. Philippine Plate, 12. Coca Plate, 13. Caribbean Plate, 14. Somali Plate, 15. South American Plate, 16. Nasca Plate, 17. Antarctic Plate

Granger Causality test, 12 hours

1

3

2

5

4

6

8

7

Granger Causality test, 24 hours

9 10 11 12 13 14 15 16 17

1



2



3



4



5



6



7



8



9



10



11



12



13



14



15



16 17

● ●

1



3

2

5

4

6

8

7

9 10 11 12 13 14 15 16 17

1 2



3



4



5



6



7



8



9



10



11



12



13



14



15



16 17





● ●

55

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Granger causality N1 → N2 or N1 ← N2 1. North American Plate, 2. Eurasian Plate,3. Okhotsk Plate, 4. Pacific Plate (East), 5. Pacific Plate (West), 6. Amur Plate, 7. Indo-Australian Plate, 8. African Plate, 9. Indo-Chinese Plate, 10. Arabian Plate, 11. Philippine Plate, 12. Coca Plate, 13. Caribbean Plate, 14. Somali Plate, 15. South American Plate, 16. Nasca Plate, 17. Antarctic Plate

Granger Causality test, 36 hours

1

3

2

5

4

6

8

7

Granger Causality test, 48 hours

9 10 11 12 13 14 15 16 17

1



2



3



4



5



6



7



8



9



10



11



12



13



14



15



16 17

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2

5

4

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7

9 10 11 12 13 14 15 16 17

1 2



3



4



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12



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16 17





● ●

56

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Multivariate models : frequency versus magnitude X1,t =

X i=1

1(Ti ∈ [t, t + 1), Mi ≤ s) and X2,t =

X

1(Ti ∈ [t, t + 1), Mi > s)

i=1

Here we work on two sets of data : medium-size earthquakes (M ∈ (5, 6)) and large-size earthquakes (M > 6). – Investigate direction of relationship (which one causes the other, or both) ; – Pairs of tectonic plates : – Uni-directional causality : most common for contiguous plates (North American causes West Pacific, Okhotsk causes Amur) ; – Bi-directional causality : Okhotsk and West Pacific, South American and Nasca for example ; – Foreshocks and aftershocks : – Aftershocks much more significant than foreshocks (as expected) ; – Foreshocks announce arrival of larger-size earthquakes ; – Foreshocks significant for Okhotsk, West Pacific, Indo-Australian, Indo-Chinese, Philippine, South American ; 57

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Risk management issues – Interested in computing P

 T (N + N ) ≥ n F0 for various values of T 1,t 2,t t=1

P

(time horizons) and n (tail risk measure) ; – Total number of earthquakes on a set of two tectonic plates ; – 100 000 simulated paths of diagonal and proposed BINAR models ; – Use estimated parameters of both models ; – Pair : Okhotsk and West Pacific ; – Scenario : on a 12-hour period, 23 earthquakes on Okhotsk and 46 earthquakes on West Pacific (second half of March 10th, 2011) ;

58

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Diagonal model n / days

1 day

3 days

7 days

14 days

5

0.9680

0.9869

0.9978

0.9999

10

0.5650

0.7207

0.8972

0.9884

15

0.1027

0.2270

0.4978

0.8548

20

0.0067

0.0277

0.1308

0.4997

Proposed model n / days

1 day

3 days

7 days

14 days

5

0.9946

0.9977

0.9997

1.0000

10

0.8344

0.9064

0.9712

0.9970

15

0.3638

0.5288

0.7548

0.9479

20

0.0671

0.1573

0.3616

0.7256

59

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Some references Al-Osh, M.A. & A.A. Alzaid (1987) "First-order integer-valued autoregressive process", Journal of Time Series Analysis 8, 261-275. Beroza, G.C. (2012) How many great earthquakes should we expect ? PNAS, 109, 651-652. Dion, J.-P., G. Gauthier & A. Latour (1995), "Branching processes with immigration and integer-valued time series", Serdica Mathematical Journal 21, 123-136. Du, J.-G. & Y. Li (1991), "The integer-valued autoregressive (INAR(p)) model", Journal of Time Series Analysis 12, 129-142. Ferland, R.A., Latour, A. & Oraichi, D. (2006), Integer-valued GARCH process. Journal of Time Series Analysis 27, 923-942. Fokianos, K. (2011). Count time series models. to appear in Handbook of Time Series Analysis. 60

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Franke, J. & Subba Rao, T. (1993). Multivariae first-order integer-valued autoregressions. Forschung Universität Kaiserslautern, 95. Gourieroux, C. & J. Jasiak (2004) "Heterogeneous INAR(1) model with application to car insurance", Insurance : Mathematics & Economics 34, 177-192. Gardner, J.K. & I. Knopoff (1974) "Is the sequence of earthquakes in Southern California, with aftershocks removed, Poissonean ?" Bulletin of the Seismological Society of America 64, 1363-1367. Joe, H. (1996) Time series models with univariate margins in the convolution-closed infinitely divisible class. Journal of Time Series Analysis 104, 117-133. Johnson, N.L, Kotz, S. & Balakrishnan, N. (1997). Discrete Multivariate Distributions. Wiley Interscience. Kagan, Y.Y. & D.D. Jackson (1991) Long-term earthquake clustering, Geophysical Journal International 104, 117-133. 61

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Kocherlakota, S. & Kocherlakota K. (1992). Bivariate Discrete Distributions. CRC Press. Latour, A. (1997) "The multivariate GINAR(p) process", Advances in Applied Probability 29, 228-248. Latour, A. (1998) Existence and stochastic structure of a non-negative integer-valued autoregressive process, Journal of Time Series Analysis 19, 439-455. Lomnitz, C. (1974) "Global Tectonic and Earthquake Risk", Elsevier. Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Springer Verlag. Mahamunulu, D. M. (1967). A note on regression in the multivariate Poisson distribution. Journal of the American Statistical Association, 62, 25 1-258. McKenzie, E. (1985) Some simple models for discrete variate time series, Water Resources Bulletin 21, 645-650. 62

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Orfanogiannaki, K., D. Karlis & G.A. Papadopoulos (2010) "Identifying Seismicity Levels via Poisson Hidden Markov Models", Pure and Applied Geophysics 167, 919-931. Parsons, T., & A.A. Velasco (2011) "Absence of remotely triggered large earthquakes beyond the mainshock region", Nature Geoscience, March 27th, 2011. Pedeli, X. & Karlis, D. (2011) "A bivariate Poisson INAR(1) model with application", Statistical Modelling 11, 325-349. Pedeli, X. & Karlis, D. (2011). "On estimation for the bivariate Poisson INAR process", To appear in Communications in Statistics, Simulation and Computation. Rathbun, S.L. (2004), “Seismological modeling”, Encyclopedia of Environmetrics. Rosenblatt, M. (1971). Markov processes, Structure and Asymptotic Behavior. Springer-Verlag. 63

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency

Shearer, P.M. & Stark, P.B (2012). Global risk of big earthquakes has not recently increased. PNAS, 109, 717-721. Steutel, F. & K. van Harn (1979) "Discrete analogues of self-decomposability and stability", Annals of Probability 7, 893-899. Utsu, T. (1969) "Aftershocks and earthquake statistics (I) - Some parameters which characterize an aftershock sequence and their interrelations", Journal of the Faculty of Science of Hokkaido University, 121-195. Vere-Jones, D. (2010), "Foundations of Statistical Seismology", Pure and Applied Geophysics 167, 645-653. Weiβ, C.H. (2008) "Thinning operations for modeling time series of counts : a survey", Advances in Statistical Analysis 92, 319-341. Zucchini, W. & I.L. MacDonald (2009) "Hidden Markov Models for Time Series : An Introduction Using R", 2nd edition, Chapman & Hall

The article can be downloaded from http ://arxiv.org/abs/1112.0929 64