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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Bivariate Count Processesfor Earthquake Frequency
Mathieu Boudreault & Arthur Charpentier
Université du Québec à Montréal
http ://freakonometrics.blog.free.fr/
Séminaire GeoTop, January 2012
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Figure : Time and distance distribution (to 6,000 km) of large (5<M<7)aftershocks from 205 M≥7 mainshocks (in sec. and h.).
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Motivation“Large earthquakes are known to trigger earthquakes elsewhere. Damaging largeaftershocks occur close to the mainshock and microearthquakes are triggered bypassing seismic waves at significant distances from the mainshock. It is unclear,however, whether bigger, more damaging earthquakes are routinely triggered atdistances far from the mainshock, heightening the global seismic hazard afterevery large earthquake. Here we assemble a catalogue of all possible earthquakesgreater than M5 that might have been triggered by every M7 or larger mainshockduring the past 30 years. [...] We observe a significant increase in the rate ofseismic activity at distances confined to within two to three rupture lengths of themainshock. Thus, we conclude that the regional hazard of larger earthquakes isincreased after a mainshock, but the global hazard is not.” Parsons & Velasco(2011)
Figure : Number of earthquakes (magnitude exceeding 2.0, per 15 sec.) followinga large earthquake (of magnitude 6.5), normalized so that the expected numberof earthquakes before and after is 100.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
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Time before and after a major eathquake (magnitude >6.5) in days
Num
ber
of e
arth
quak
es (
mag
nitu
de >
2) p
er 1
5 se
c., a
vera
ge b
efor
e=10
0
−15 −10 −5 0 5 10 15
8010
012
014
016
018
0
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
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Time before and after a major eathquake (magnitude >6.5) in days
Num
ber
of e
arth
quak
es (
mag
nitu
de >
4) p
er 1
5 se
c., a
vera
ge b
efor
e=10
0
−15 −10 −5 0 5 10 15
8010
012
014
016
018
020
022
0Main event, magnitude > 6.5Main event, magnitude > 6.8Main event, magnitude > 7.1Main event, magnitude > 7.4
Number of earthquakes before and after a major one, magnitude of the main event, small events more than 4
5
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
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Time before and after a major eathquake (magnitude >6.5) in days
Num
ber
of e
arth
quak
es (
mag
nitu
de >
2) p
er 1
5 se
c., a
vera
ge b
efor
e=10
0
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−15 −10 −5 0 5 10 15
020
040
060
080
010
00
Same techtonic plate as major oneDifferent techtonic plate as major one
6
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Shapefiles fromhttp://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
Number of earthquakes (Magnitude ≥ 5) per month, worldwide
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1970 1980 1990 2000 2010
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0
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.
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1950 1960 1970 1980 1990 2000 2010
05
1015
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+φ1Xt−1 + εt (1)
for all t, for real-valued parameters φ0 and φ1, and some i.i.d. random variablesεt with values in R.
It is common to assume that εt are independent variables, with a Gaussiandistribution N (0, σ2), with density
ϕ(ε) = 1√2πσ
exp(− ε2
2σ2
), ε ∈ R.
Note that we assume also that εt is independent of Xt−1, i.e. past observationsX0, 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 = φ1Xt−1 + εt with εt ∼ N (0, 1), i.i.d., and φ = 0.6
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0 50 100 150 200 250 300
−4
−2
02
15
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Example : Xt = φ1Xt−1 + εt with εt ∼ N (0, 1), i.i.d., and φ = 0.6
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0 50 100 150 200 250 300
−4
−2
02
16
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Example : Xt = φ1Xt−1 + εt : autocorrelation ρ(h) = corr(Xt, Xt−h) = φh1
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
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) = cov(Xt, Xt−h)√var(Xt)var(Xt−h)
= γ(h)γ(0) , ∀h ∈ N.
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
−3−2
−10
12
3
−3
−2
−1
0
1
23
0.00
0.05
0.10
0.15
0.20
Density of the Gaussian distribution Univariate gaussian distribution N (0, σ2)
ϕ(x) = 1√2πσ
exp(− x2
2σ2
), for all x ∈ R
Multivariate gaussian distribution N (0,Σ)
ϕ(x) = 1√(2π)d|det Σ|
exp(−x′Σ−1x
2
),
for all x ∈ Rd.
X = AZ where AA′ = 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 (Xt = (X1,t, · · · , Xd,t))t∈N with values in Rd is called aVAR(1) process if
X1,t = φ1,1X1,t−1 + φ1,2X2,t−1 + · · ·+ φ1,dXd,t−1 + ε1,t
X2,t = φ2,1X1,t−1 + φ2,2X2,t−1 + · · ·+ φ2,dXd,t−1 + ε2,t
· · ·Xd,t = φd,1X1,t−1 + φd,2X2,t−1 + · · ·+ φd,dXd,t−1 + εd,t
(2)
or equivalentlyX1,t
X2,t...
Xd,t
︸ ︷︷ ︸
Xt
=
φ1,1 φ1,2 · · · φ1,d
φ2,1 φ2,2 · · · φ2,d...
......
φd,1 φd,2 · · · φd,d
︸ ︷︷ ︸
Φ
X1,t−1
X2,t−1...
Xd,t−1
︸ ︷︷ ︸
Xt−1
+
ε1,t
ε2,t...εd,t
︸ ︷︷ ︸
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 εtwith values in Rd.
It is common to assume that εt are independent variables, with a Gaussiandistribution N (0,Σ), with density
ϕ(ε) = 1√(2π)d|det Σ|
exp(−ε′Σ−1ε
2
), ∀ε ∈ Rd.
Thus, independent means time independent, but can be dependentcomponentwise.
Note that we assume also that εt is independent of Xt−1, i.e. past observationsX0,X1, · · · ,Xt−1. Thus, (εt)t∈N is called the innovation process.
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 matrix
21
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Remark As a consequence, var(Xt) = E([Xt − E(Xt)]′[Xt − E(Xt)]) isindependent of t (=: γ(0)). Define finally the autocorrelation matrix,
ρ(h) = ∆−1γ(h)∆−1, where ∆ = diag(√
γi,i(0)).
Proposition (Xt)t∈N is a stationary AR(1) time series if and only if the deignvalues of Φ should have a norm lower than 1.
Proposition If (Xt)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 seriesConsider a series of observations X1, · · · , Xn. The likelihood is the jointdistribution of the vectors X = (X1, · · · , Xn), which is not the product ofmarginal distribution, since consecutive observations are not independent(cov(Xt, Xt−h) = φh). Nevertheless
L(φ, σ; (X0,X )) =n∏t=1
πφ,σ(Xt|Xt−1)
where πφ,σ(·|Xt−1) is a Gaussian density.
Maximum likelihood estimators are
(φ̂, σ̂) ∈ argmax logL(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−λλk
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 modelsEarthquake 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-releasemodels (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 modelsin 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). Thusp ◦N is a compound sum of i.i.d. Bernoulli variables.
Hence, given N , p ◦N has a binomial distribution B(N, p).
Note that p ◦ (q ◦N) L= [pq] ◦N for all p, q ∈ [0, 1].
Further
E (p ◦N) = pE(N) and var (p ◦N) = p2var(N) + p(1− p)E(N).
26
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
(Poisson) Integer AutoRegressive processes INAR(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 =Xt−1∑i=1
Yi + εt, where Y ′i s are i.i.d. B(p).
Such a process can be related to Galton-Watson processes with immigration, orphysical branching model.
27
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Xt+1 =Xt∑i=1
Yi + εt+1, where Y ′i s are i.i.d. B(p)
28
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Proposition E (Xt) = E(εt)1− p , var (Xt) = γ(0) = pE(εt) + var(εt)
1− p2 and
γ(h) = cov(Xt, Xt−h) = ph.
It is common to assume that εt are independent variables, with a Poissondistribution P(λ), with probability function
P(εt = k) = e−λλk
k! , k ∈ N.
Proposition If (εt) are Poisson random variables, then (Nt) will also be asequence of Poisson random variables.
Note that we assume also that εt is independent of Xt−1, i.e. past observationsX0, 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 anhomogeneous 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∑k=0
P
(xt−1∑i=1
Yi = xt − k
)︸ ︷︷ ︸
Binomial
·P(ε = k)︸ ︷︷ ︸Poisson
.
30
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Inference of Integer AutoRegressive processes INAR(1)Consider a Poisson INAR(1) process, then the likelihood is
L(p, λ;X0,X ) =[n∏t=1
ft(Xt)]· λX0
(1− p)X0X0! exp(− λ
1− p
)where
ft(y) = exp(−λ)min{Xt,Xt−1}∑
i=0
λy−i
(y − i)!
(Yt−1
i
)pi(1− p)Yt−1−y, for t = 1, · · · , n.
Maximum likelihood estimators are
(p̂, λ̂) ∈ argmax logL(p, λ; (X0,X ))
31
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate Integer Autoregressive processes MINAR(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]. IfN = (N1, · · · , Nd) is a random vector with values in Nd, then P ◦N is ad-dimensional random vector, with i-th component
[P ◦N ]i =d∑j=1
pi,j ◦Nj ,
for all i = 1, · · · , d, where all counting variates Y in pi,j ◦Nj ’s are assumed to beindependent.
Note that P ◦ (Q ◦N) L= [PQ] ◦N .
Further, E (P ◦N) = PE(N), and
E ((P ◦N)(P ◦N)′) = PE(NN ′)P ′ + ∆,
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 (Xt) with values in Nd is called a d-variate MINAR(1)process if
Xt = P ◦Xt−1 + εt (4)
for all t, for some d× d matrix P with entries in [0, 1], and some i.i.d. randomvectors εt with values in Nd.
(Xt) is a Markov chain with states in Nd with transition probabilities
π(xt,xt−1) = P(Xt = xt|Xt−1 = xt−1) (5)
satisfying
π(xt,xt−1) =xt∑
k=0P(P ◦ xt−1 = xt − k) · P(ε = k).
33
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Parameter inference for MINAR(1)Proposition Let (Xt) be a d-variate MINAR(1) process satisfying stationaryconditions, as well as technical assumptions (called C1-C6 in Franke & SubbaRao (1993)), then the conditional maximum likelihood estimate θ̂ of θ = (P ,Λ)is asymptotically normal,
√n(θ̂ − θ) L→ N (0,Σ−1(θ)), as n→∞.
Further,
2[logL(N , θ̂|N0)− logL(N ,θ|N0)] L→ χ2(d2 + dim(λ)), as n→∞.
34
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Granger causality with BINAR(1)(X1,t) and (X2,t) are instantaneously related if ε is a noncorrelated noise,g g gg g g g g g g g g g g
X1,t
X2,t
︸ ︷︷ ︸
Xt
=
p1,1 p1,2
p2,1 p2,2
︸ ︷︷ ︸
P
◦
X1,t−1
X2,t−1
︸ ︷︷ ︸
Xt−1
+
ε1,t
ε2,t
︸ ︷︷ ︸
εt
, with var
ε1,t
ε2,t
=
λ1 ϕ
ϕ λ2
35
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Granger causality with BINAR(1)1. (X1) and (X2) are instantaneously related if ε is a noncorrelated noise, g g g g
g g g g g g g g g g
X1,t
X2,t
︸ ︷︷ ︸
Xt
=
p1,1 p1,2
p2,1 p2,2
︸ ︷︷ ︸
P
◦
X1,t−1
X2,t−1
︸ ︷︷ ︸
Xt−1
+
ε1,t
ε2,t
︸ ︷︷ ︸
εt
, with var
ε1,t
ε2,t
=
λ1 ?
? λ2
36
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Granger causality with BINAR(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,
X1,t
X2,t
︸ ︷︷ ︸
Xt
=
p1,1 00 p2,2
︸ ︷︷ ︸
P
◦
X1,t−1
X2,t−1
︸ ︷︷ ︸
Xt−1
+
ε1,t
ε2,t
︸ ︷︷ ︸
εt
, with var
ε1,t
ε2,t
=
λ1 00 λ2
37
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Granger causality with BINAR(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,
X1,t
X2,t
︸ ︷︷ ︸
Xt
=
p1,1 0? p2,2
︸ ︷︷ ︸
P
◦
X1,t−1
X2,t−1
︸ ︷︷ ︸
Xt−1
+
ε1,t
ε2,t
︸ ︷︷ ︸
εt
, with var
ε1,t
ε2,t
=
λ1 ϕ
ϕ λ2
38
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Granger causality with BINAR(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,
X1,t
X2,t
︸ ︷︷ ︸
Xt
=
p1,1 ?
0 p2,2
︸ ︷︷ ︸
P
◦
X1,t−1
X2,t−1
︸ ︷︷ ︸
Xt−1
+
ε1,t
ε2,t
︸ ︷︷ ︸
εt
, with var
ε1,t
ε2,t
=
λ1 ϕ
ϕ λ2
39
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Granger causality with BINAR(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
X1,t
X2,t
︸ ︷︷ ︸
Xt
=
p1,1 ?
? p2,2
︸ ︷︷ ︸
P
◦
X1,t−1
X2,t−1
︸ ︷︷ ︸
Xt−1
+
ε1,t
ε2,t
︸ ︷︷ ︸
εt
, with var
ε1,t
ε2,t
=
λ1 ϕ
ϕ λ2
40
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Bivariate Poisson BINAR(1)A classical distribution for εt is the bivariate Poisson distribution, with onecommon shock, i.e. ε1,t = M1,t +M0,t
ε2,t = M2,t +M0,t
where M1,t, M2,t and M0,t are independent Poisson variates, with parametersλ1 − ϕ, λ2 − ϕ and ϕ, respectively. In that case, εt = (ε1,t, ε2,t) has jointprobability function
e−[λ1+λ2−ϕ] (λ1 − ϕ)k1
k1!(λ2 − ϕ)k2
k2!
min{k1,k2}∑i=0
(k1
i
)(k2
i
)i!(
ϕ
[λ1 − ϕ][λ2 − ϕ]
)with λ1, λ2 > 0, ϕ ∈ [0,min{λ1, λ2}].
λ =
λ1
λ2
and Λ =
λ1 ϕ
ϕ λ2
41
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Bivariate Poisson BINAR(1) and Granger causalityFor instantaneous causality, we test
H0 : ϕ = 0 against H1 : ϕ 6= 0
Proposition Let λ̂ denote the conditional maximum likelihood estimate ofλ = (λ1, λ2, ϕ) in the non-constrained MINAR(1) model, and λ⊥ denote theconditional maximum likelihood estimate of λ⊥ = (λ1, λ2, 0) in the constrainedmodel (when innovation has independent margins), then under suitableconditions,
2[logL(N , λ̂|N0)− logL(N , λ̂⊥|N0)] L→ χ2(1), as n→∞, under H0.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Bivariate Poisson BINAR(1) and Granger causalityFor 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 diagonalmatrices for lagged independence, or a set of block triangular matrices for laggedcausality.
Proposition Let P̂ denote the conditional maximum likelihood estimate of P inthe non-constrained MINAR(1) model, and P̂
cdenote the conditional maximum
likelihood estimate of P in the constrained model, then under suitable conditions,
2[logL(N , P̂ |N0)− logL(N , P̂c|N0)] L→ χ2(d2−dim(P)), as n→∞, under H0.
Example Testing (N1,t)←(N2,t) is testing whether p1,2 = 0, or not.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Autocorrelation of MINAR(1) processesProposition Consider a MINAR(1) process with representationXt = P ◦Xt−1 + εt, where (εt) is the innovation process, with λ := E(εt) andΛ := var(εt). Let µ := E(Xt) and γ(h) := cov(Xt,Xt−h). Then µ = [I− P ]−1λ
and for all h ∈ Z, γ(h) = P hγ(0) with γ(0) solution ofγ(0) = Pγ(0)P ′ + (∆ + 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 fromhttp://www.colorado.edu/geography/foote/maps/assign/hotspots/hotspots.html
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
The dataset, and stationarity issuesWe 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 ;
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate models : comparing dynamics
X1,t
X2,t
=
p1,1 p1,2
p2,1 p2,2
◦X1,t−1
X2,t−1
+
ε1,t
ε2,t
with var
ε1,t
ε2,t
=
λ1 ϕ
ϕ λ2
Complete model, with full dependence
48
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate models : comparing dynamics
X1,t
X2,t
=
p1,1 00 p2,2
◦X1,t−1
X2,t−1
+
ε1,t
ε2,t
with var
ε1,t
ε2,t
=
λ1 ϕ
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
X1,t
X2,t
=
p1,1 00 p2,2
◦X1,t−1
X2,t−1
+
ε1,t
ε2,t
with var
ε1,t
ε2,t
=
λ1 00 λ2
Two independent INAR processes
50
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate models : comparing dynamics
X1,t
X2,t
=
p1,1 00 p2,2
◦X1,t−1
X2,t−1
+
ε1,t
ε2,t
with var
ε1,t
ε2,t
=
λ1 00 λ2
Two independent INAR processes
51
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate models : comparing dynamics
X1,t
X2,t
=
0 00 0
◦X1,t−1
X2,t−1
+
ε1,t
ε2,t
with var
ε1,t
ε2,t
=
λ1 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 fitthan 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 fitthan 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 > 5earthquakes
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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
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17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 3 hours
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17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 6 hours
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
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17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 12 hours
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12
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10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 24 hours
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
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17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 36 hours
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17
16
15
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13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Granger Causality test, 48 hours
56
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate models : frequency versus magnitudeX1,t =
∑i=1
1(Ti ∈ [t, t+ 1),Mi ≤ s) and X2,t =∑i=1
1(Ti ∈ [t, t+ 1),Mi > s)
Here we work on two sets of data : medium-size earthquakes (M ∈ (5, 6)) andlarge-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 (NorthAmerican causes West Pacific, Okhotsk causes Amur) ;
– Bi-directional causality : Okhotsk and West Pacific, South American andNasca 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
(∑Tt=1 (N1,t +N2,t) ≥ n
∣∣∣F0
)for various values of T
(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 earthquakeson West Pacific (second half of March 10th, 2011) ;
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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 0.999910 0.5650 0.7207 0.8972 0.988415 0.1027 0.2270 0.4978 0.854820 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 1.000010 0.8344 0.9064 0.9712 0.997015 0.3638 0.5288 0.7548 0.947920 0.0671 0.1573 0.3616 0.7256
59
Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Some referencesAl-Osh, M.A. & A.A. Alzaid (1987) "First-order integer-valued autoregressiveprocess", 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 withimmigration 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 GARCHprocess. Journal of Time Series Analysis 27, 923-942.
Fokianos, K. (2011). Count time series models. to appear in Handbook ofTime Series Analysis.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Franke, J. & Subba Rao, T. (1993). Multivariae first-order integer-valuedautoregressions. Forschung Universität Kaiserslautern, 95.
Gourieroux, C. & J. Jasiak (2004) "Heterogeneous INAR(1) model withapplication to car insurance", Insurance : Mathematics & Economics 34, 177-192.
Gardner, J.K. & I. Knopoff (1974) "Is the sequence of earthquakes inSouthern California, with aftershocks removed, Poissonean ?" Bulletin of theSeismological Society of America 64, 1363-1367.
Joe, H. (1996) Time series models with univariate margins in theconvolution-closed infinitely divisible class. Journal of Time Series Analysis 104,117-133.
Johnson, N.L, Kotz, S. & Balakrishnan, N. (1997). Discrete MultivariateDistributions. Wiley Interscience.
Kagan, Y.Y. & D.D. Jackson (1991) Long-term earthquake clustering,Geophysical Journal International 104, 117-133.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Kocherlakota, S. & Kocherlakota K. (1992). Bivariate DiscreteDistributions. CRC Press.
Latour, A. (1997) "The multivariate GINAR(p) process", Advances in AppliedProbability 29, 228-248.
Latour, A. (1998) Existence and stochastic structure of a non-negativeinteger-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 Poissondistribution. Journal of the American Statistical Association, 62, 25 1-258.
McKenzie, E. (1985) Some simple models for discrete variate time series, WaterResources Bulletin 21, 645-650.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Orfanogiannaki, K., D. Karlis & G.A. Papadopoulos (2010) "IdentifyingSeismicity Levels via Poisson Hidden Markov Models", Pure and AppliedGeophysics 167, 919-931.
Parsons, T., & A.A. Velasco (2011) "Absence of remotely triggered largeearthquakes beyond the mainshock region", Nature Geoscience, March 27th, 2011.
Pedeli, X. & Karlis, D. (2011) "A bivariate Poisson INAR(1) model withapplication", Statistical Modelling 11, 325-349.
Pedeli, X. & Karlis, D. (2011). "On estimation for the bivariate PoissonINAR process", To appear in Communications in Statistics, Simulation andComputation.
Rathbun, S.L. (2004), “Seismological modeling”, Encyclopedia ofEnvironmetrics.
Rosenblatt, M. (1971). Markov processes, Structure and Asymptotic Behavior.Springer-Verlag.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Shearer, P.M. & Stark, P.B (2012). Global risk of big earthquakes has notrecently increased. PNAS, 109, 717-721.Steutel, F. & K. van Harn (1979) "Discrete analogues ofself-decomposability and stability", Annals of Probability 7, 893-899.Utsu, T. (1969) "Aftershocks and earthquake statistics (I) - Some parameterswhich characterize an aftershock sequence and their interrelations", Journal ofthe Faculty of Science of Hokkaido University, 121-195.Vere-Jones, D. (2010), "Foundations of Statistical Seismology", Pure andApplied Geophysics 167, 645-653.Weiβ, C.H. (2008) "Thinning operations for modeling time series of counts : asurvey", Advances in Statistical Analysis 92, 319-341.Zucchini, W. & I.L. MacDonald (2009) "Hidden Markov Models for TimeSeries : An Introduction Using R", 2nd edition, Chapman & Hall
The article can be downloaded from http ://arxiv.org/abs/1112.0929
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