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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Bivariate Count Processesfor Earthquake Frequency
Mathieu Boudreault & Arthur Charpentier
UniversitΓ© du QuΓ©bec Γ MontrΓ©al
http ://freakonometrics.blog.free.fr/
SΓ©minaire GeoTop, January 2012
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
MotivationβLarge 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.β Parsons & Velasco(2011)
Figures : Number of earthquakes (magnitude exceeding 2.0, per 15 sec.)following a large earthquake (of magnitude 6.5), normalized so that the expectednumber of 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
β15 β10 β5 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
β15 β10 β5 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
4
![Page 5: Slides GEOTOP](https://reader034.vdocuments.mx/reader034/viewer/2022052621/55853632d8b42a86388b528d/html5/thumbnails/5.jpg)
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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ββββββββββββββββββββββββββββββββββββ
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βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
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βββββββββββββββββββββββββββββββββββββββ
ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
β15 β10 β5 0 5 10 15
020
040
060
080
010
00
Same techtonic plate as major oneDifferent techtonic plate as major one
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
We look at seismic risks with the eyes of actuaries and statisticians...
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
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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
9
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Part 1
Modeling dynamics of counts
10
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
(Gaussian) Auto regressive processes AR(1)Definition A time series (Xt)tβN with values in R is called an AR(1) process if
Xt = Ο0+Ο1Xtβ1 + Ξ΅t (1)
for all t, for real-valued parameters Ο0 and Ο1, and some i.i.d. random variablesΞ΅t with values in R.
It is common to assume that Ξ΅t are independent variables, with a Gaussiandistribution N (0, Ο2), with density
Ο(Ξ΅) = 1β2ΟΟ
exp(β Ξ΅2
2Ο2
), Ξ΅ β R.
Note that we assume also that Ξ΅t is independent of Xtβ1, i.e. past observationsX0, X1, Β· Β· Β· , Xtβ1. Thus, (Ξ΅t)tβN is called the innovation process.
Definition A time series (Xt)tβN is said to be (weakly) stationary ifβ’ E(Xt) is independent of t ( =: Β΅)β’ cov(Xt, Xtβh) is independent of t (=: Ξ³(h)), called autocovariance function
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Remark As a consequence, var(Xt) = E([Xt β E(Xt)]2) is independent of t(=: Ξ³(0)). Define the autocorrelation function Ο(Β·) as
Ο(h) := corr(Xt, Xtβh) = cov(Xt, Xtβh)βvar(Xt)var(Xtβh)
= Ξ³(h)Ξ³(0) , βh β N.
Proposition (Xt)tβN is a stationary AR(1) time series if and only if Ο1 β (β1, 1).
Remark If Ο1 = 1, (Xt)tβN is called a random walk.
Proposition If (Xt)tβN is a stationary AR(1) time series,
Ο(h) = Οh1 , βh β N.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Proposition If (Xt)tβN is a stationary AR(1) time series, (Xt)tβN satisfies
Xt = Ο1[Xtβ1] + Ξ΅t
= Ο1[Ο1[Xtβ2] + Ξ΅tβ1] + Ξ΅t
= Ο1[Ο1[Ο1[Xtβ2] + Ξ΅tβ2] + Ξ΅tβ1 + Ξ΅t
Β· Β· Β·
= Οt1X0 +tβ
h=0Οh1Ξ΅tβh,
which is a MA(β) representation (moving average of the innovation process).
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Vector (Gaussian) Autoregressive processes V AR(1)Definition A time series (Xt = (X1,t, Β· Β· Β· , Xd,t))tβN with values in Rd is called aVAR(1) process if
X1,t = Ο1,1X1,tβ1 + Ο1,2X2,tβ1 + Β· Β· Β·+ Ο1,dXd,tβ1 + Ξ΅1,t
X2,t = Ο2,1X1,tβ1 + Ο2,2X2,tβ1 + Β· Β· Β·+ Ο2,dXd,tβ1 + Ξ΅2,t
Β· Β· Β·Xd,t = Οd,1X1,tβ1 + Οd,2X2,tβ1 + Β· Β· Β·+ Οd,dXd,tβ1 + Ξ΅d,t
(2)
or equivalentlyX1,t
X2,t...
Xd,t
οΈΈ οΈ·οΈ· οΈΈ
Xt
=
Ο1,1 Ο1,2 Β· Β· Β· Ο1,d
Ο2,1 Ο2,2 Β· Β· Β· Ο2,d...
......
Οd,1 Οd,2 Β· Β· Β· Οd,d
οΈΈ οΈ·οΈ· οΈΈ
Ξ¦
X1,tβ1
X2,tβ1...
Xd,tβ1
οΈΈ οΈ·οΈ· οΈΈ
Xtβ1
+
Ξ΅1,t
Ξ΅2,t...Ξ΅d,t
οΈΈ οΈ·οΈ· οΈΈ
Ξ΅t
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
for all t, for some real-valued dΓ d matrix Ξ¦, and some i.i.d. random vectors Ξ΅twith values in Rd.
It is common to assume that Ξ΅t are independent variables, with a Gaussiandistribution N (0,Ξ£), with density
Ο(Ξ΅) = 1β(2Ο)d|det Ξ£|
exp(βΞ΅β²Ξ£β1Ξ΅
2
), βΞ΅ β Rd.
Thus, independent means time independent, but can be dependentcomponentwise.
Note that we assume also that Ξ΅t is independent of Xtβ1, i.e. past observationsX0,X1, Β· Β· Β· ,Xtβ1. Thus, (Ξ΅t)tβN is called the innovation process.
Definition A time series (Xt)tβN is said to be (weakly) stationary ifβ’ E(Xt) is independent of t (=: Β΅)β’ cov(Xt,Xtβh) is independent of t (=: Ξ³(h)), called autocovariance matrixRemark As a consequence, var(Xt) = E([Xt β E(Xt)]β²[Xt β E(Xt)]) is
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
independent of t (=: Ξ³(0)). Define finally the autocorrelation matrix,
Ο(h) = ββ1Ξ³(h)ββ1, where β = diag(β
Ξ³i,i(0)).
Proposition (Xt)tβN is a stationary AR(1) time series if and only if the deignvalues of Ξ¦ should have a norm lower than 1.
Proposition If (Xt)tβN is a stationary AR(1) time series,
Ο(h) = Ξ¦h, h β N.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Proposition If (Xt)tβN is a stationary AR(1) time series, (Xt)tβN satisfies
Xt = Ξ¦[Xtβ1] + Ξ΅t= Ξ¦[Ξ¦[Xtβ2] + Ξ΅tβ1] + Ξ΅t= Ξ¦[Ξ¦[Ξ¦[Xtβ2] + Ξ΅tβ2] + Ξ΅tβ1 + Ξ΅tΒ· Β· Β·
= Ξ¦tX0 +tβ
h=0Ξ¦hΞ΅tβh,
which is a VMA(β) representation.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Statistical inference for AR(1) time seriesConsider a series of observations X1, Β· Β· Β· , Xn. The likelihood is the jointdistribution of the vectors X = (X1, Β· Β· Β· , Xn), which is not the product ofmarginal distribution, since consecutive observations are not independent(cov(Xt, Xtβh) = Οh). Nevertheless
L(Ο, Ο; (X0,X )) =nβt=1
ΟΟ,Ο(Xt|Xtβ1)
where ΟΟ,Ο(Β·|Xtβ1) is a Gaussian density.
Maximum likelihood estimators are
(ΟΜ, ΟΜ) β argmax logL(Ο, Ο; (X0,X ))
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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.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Thinning operator β¦Steutel & van Harn (1979) defined a thinning operator as follows
Definition Define operator β¦ as
p β¦N = Y1 + Β· Β· Β·+ YN if N 6= 0, and 0 otherwise,
where N is a random variable with values in N, p β [0, 1], and Y1, Y2, Β· Β· Β· are i.i.d.Bernoulli variables, independent of N , with P(Yi = 1) = p = 1β P(Yi = 0). Thusp β¦N is a compound sum of i.i.d. Bernoulli variables.
Hence, given N , p β¦N has a binomial distribution B(N, p).
Note that p β¦ (q β¦N) L= [pq] β¦N for all p, q β [0, 1].
Further
E (p β¦N) = pE(N) and var (p β¦N) = p2var(N) + p(1β p)E(N).
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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)tβN with values in R is called an INAR(1) process if
Xt = p β¦Xtβ1 + Ξ΅t, (3)
where (Ξ΅t) is a sequence of i.i.d. integer valued random variables, i.e.
Xt =Xtβ1βi=1
Yi + Ξ΅t, where Y β²i s are i.i.d. B(p).
Such a process can be related to Galton-Watson processes with immigration, orphysical branching model.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Xt+1 =Xtβi=1
Yi + Ξ΅t+1, where Y β²i s are i.i.d. B(p)
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Proposition E (Xt) = E(Ξ΅t)1β p , var (Xt) = Ξ³(0) = pE(Ξ΅t) + var(Ξ΅t)
1β p2 and
Ξ³(h) = cov(Xt, Xtβh) = ph.
It is common to assume that Ξ΅t are independent variables, with a Poissondistribution P(Ξ»), with probability function
P(Ξ΅t = z) = eβλλk
k! , z β N.
Proposition If (Ξ΅t) are Poisson random variables, then (Nt) will also be asequence of Poisson random variables.
Note that we assume also that Ξ΅t is independent of Xtβ1, i.e. past observationsX0, X1, Β· Β· Β· , Xtβ1. Thus, (Ξ΅t)tβN is called the innovation process.
Proposition (Xt)tβN is a stationary INAR(1) time series if and only if p β [0, 1).
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Proposition If (Xt)tβN is a stationary INAR(1) time series, (Xt)tβN satisfies
Xt = p β¦ [Xtβ1] + Ξ΅t
= p β¦ [p β¦ [Xtβ2] + Ξ΅tβ1] + Ξ΅t
= p β¦ [p β¦ [p β¦ [Xtβ2] + Ξ΅tβ2] + Ξ΅tβ1 + Ξ΅t
Β· Β· Β·
= pt β¦X0 +tβ
h=0ph β¦ Ξ΅tβh,
which is an INMA(β) representation.
Proposition If (Xt)tβN is a stationary INAR(1) time series, (Xt)tβN is anhomogeneous Markov chain.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Ο(xt, xtβ1) = P(Xt = xt|Xtβ1 = xtβ1) =xtβk=0
P
(xtβ1βi=1
Yi = xt β k
)οΈΈ οΈ·οΈ· οΈΈ
Binomial
Β·P(Ξ΅ = k)οΈΈ οΈ·οΈ· οΈΈPoisson
.
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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, Ξ»;X0,X ) =[nβt=1
ft(Xt)]Β· Ξ»X0
(1β p)X0X0! exp(β Ξ»
1β p
)where
ft(y) = exp(βΞ»)min{Xt,Xtβ1}β
i=0
Ξ»yβi
(y β i)!
(Ytβ1
i
)pi(1β p)Ytβ1βy, for t = 1, Β· Β· Β· , n.
Maximum likelihood estimators are
(pΜ, Ξ»Μ) β argmax logL(p, Ξ»; (X0,X ))
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate Integer Autoregressive processes MINAR(1)Let Nt := (N1,t, Β· Β· Β· , Nd,t), denote a multivariate vector of counts.
Definition Let P := [pi,j ] be a dΓ d matrix with entries in [0, 1]. IfN = (N1, Β· Β· Β· , Nd) is a random vector with values in Nd, then P β¦N is ad-dimensional random vector, with i-th component
[P β¦N ]i =dβj=1
pi,j β¦Nj ,
for all i = 1, Β· Β· Β· , d, where all counting variates Y in pi,j β¦Nj βs are assumed to beindependent.
Note that P β¦ (Q β¦N) L= [PQ] β¦N .
Further, E (P β¦N) = PE(N), and
E ((P β¦N)(P β¦N)β²) = PE(NN β²)P β² + β,
with β := diag(V E(N)) where V is the dΓ d matrix with entries pi,j(1β pi,j).
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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 β¦Xtβ1 + Ξ΅t (4)
for all t, for some dΓ d matrix P with entries in [0, 1], and some i.i.d. randomvectors Ξ΅t with values in Nd.
(Xt) is a Markov chain with states in Nd with transition probabilities
Ο(xt,xtβ1) = P(Xt = xt|Xtβ1 = xtβ1) (5)
satisfying
Ο(xt,xtβ1) =xtβ
k=0P(P β¦ xtβ1 = xt β k) Β· P(Ξ΅ = k).
Proposition Let (Xt) denote a d-variate MINAR(1) process, withautoregressive matrix P with entries in [0, 1], then (Xt) admits a d-variate
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
INMA(β) representation
Xt =ββh=0
P h β¦ Ξ΅tβh.
Proposition Let P with entries in (0, 1), such that its largest eigenvalue is lessthan 1, and assume that P(Ξ΅t = 0) β (0, 1) with E(βΞ΅tββ) <β, then there existsa stationary d-variate MINAR(1) process satisfying Equation (4).
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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 ΞΈΜ of ΞΈ = (P ,Ξ)is asymptotically normal,
βn(ΞΈΜ β ΞΈ) Lβ N (0,Ξ£β1(ΞΈ)), as nββ.
Further,
2[logL(N , ΞΈΜ|N0)β logL(N ,ΞΈ|N0)] Lβ Ο2(d2 + dim(Ξ»)), as nββ.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Granger causality with BINAR(1)(X1,t) and (X2,t) are instantaneously related if Ξ΅ is a noncorrelated noise,g g gg g g g g g g g g g g
X1,t
X2,t
οΈΈ οΈ·οΈ· οΈΈ
Xt
=
p1,1 p1,2
p2,1 p2,2
οΈΈ οΈ·οΈ· οΈΈ
P
β¦
X1,tβ1
X2,tβ1
οΈΈ οΈ·οΈ· οΈΈ
Xtβ1
+
Ξ΅1,t
Ξ΅2,t
οΈΈ οΈ·οΈ· οΈΈ
Ξ΅t
, with var
Ξ΅1,t
Ξ΅2,t
=
Ξ»1 Ο
Ο Ξ»2
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Granger causality with BINAR(1)1. (X1) and (X2) are instantaneously related if Ξ΅ is a noncorrelated noise, g g g g
g g g g g g g g g g
X1,t
X2,t
οΈΈ οΈ·οΈ· οΈΈ
Xt
=
p1,1 p1,2
p2,1 p2,2
οΈΈ οΈ·οΈ· οΈΈ
P
β¦
X1,tβ1
X2,tβ1
οΈΈ οΈ·οΈ· οΈΈ
Xtβ1
+
Ξ΅1,t
Ξ΅2,t
οΈΈ οΈ·οΈ· οΈΈ
Ξ΅t
, with var
Ξ΅1,t
Ξ΅2,t
=
Ξ»1 ?
? Ξ»2
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Granger causality with BINAR(1)2. (X1) and (X2) are independent, (X1)β₯(X2) if P is diagonal, i.e.p1,2 = p2,1 = 0, and Ξ΅1 and Ξ΅2 are independent,
X1,t
X2,t
οΈΈ οΈ·οΈ· οΈΈ
Xt
=
p1,1 00 p2,2
οΈΈ οΈ·οΈ· οΈΈ
P
β¦
X1,tβ1
X2,tβ1
οΈΈ οΈ·οΈ· οΈΈ
Xtβ1
+
Ξ΅1,t
Ξ΅2,t
οΈΈ οΈ·οΈ· οΈΈ
Ξ΅t
, with var
Ξ΅1,t
Ξ΅2,t
=
Ξ»1 00 Ξ»2
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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)β(X2), if P is a lower
triangle matrix, i.e. p2,1 6= 0 while p1,2 = 0,
X1,t
X2,t
οΈΈ οΈ·οΈ· οΈΈ
Xt
=
p1,1 0? p2,2
οΈΈ οΈ·οΈ· οΈΈ
P
β¦
X1,tβ1
X2,tβ1
οΈΈ οΈ·οΈ· οΈΈ
Xtβ1
+
Ξ΅1,t
Ξ΅2,t
οΈΈ οΈ·οΈ· οΈΈ
Ξ΅t
, with var
Ξ΅1,t
Ξ΅2,t
=
Ξ»1 Ο
Ο Ξ»2
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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)β(N2,t), if P is a upper
triangle matrix, i.e. p1,2 6= 0 while p2,1 = 0,
X1,t
X2,t
οΈΈ οΈ·οΈ· οΈΈ
Xt
=
p1,1 ?
0 p2,2
οΈΈ οΈ·οΈ· οΈΈ
P
β¦
X1,tβ1
X2,tβ1
οΈΈ οΈ·οΈ· οΈΈ
Xtβ1
+
Ξ΅1,t
Ξ΅2,t
οΈΈ οΈ·οΈ· οΈΈ
Ξ΅t
, with var
Ξ΅1,t
Ξ΅2,t
=
Ξ»1 Ο
Ο Ξ»2
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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)β(N2), if P is a
full matrix, i.e. p1,2, p2,1 6= 0
X1,t
X2,t
οΈΈ οΈ·οΈ· οΈΈ
Xt
=
p1,1 ?
? p2,2
οΈΈ οΈ·οΈ· οΈΈ
P
β¦
X1,tβ1
X2,tβ1
οΈΈ οΈ·οΈ· οΈΈ
Xtβ1
+
Ξ΅1,t
Ξ΅2,t
οΈΈ οΈ·οΈ· οΈΈ
Ξ΅t
, with var
Ξ΅1,t
Ξ΅2,t
=
Ξ»1 Ο
Ο Ξ»2
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Bivariate Poisson BINAR(1)A classical distribution for Ξ΅t is the bivariate Poisson distribution, with onecommon shock, i.e. Ξ΅1,t = M1,t +M0,t
Ξ΅2,t = M2,t +M0,t
where M1,t, M2,t and M0,t are independent Poisson variates, with parametersΞ»1 β Ο, Ξ»2 β Ο and Ο, respectively. In that case, Ξ΅t = (Ξ΅1,t, Ξ΅2,t) has jointprobability function
eβ[Ξ»1+Ξ»2βΟ] (Ξ»1 β Ο)k1
k1!(Ξ»2 β Ο)k2
k2!
min{k1,k2}βi=0
(k1
i
)(k2
i
)i!(
Ο
[Ξ»1 β Ο][Ξ»2 β Ο]
)with Ξ»1, Ξ»2 > 0, Ο β [0,min{Ξ»1, Ξ»2}].
Ξ» =
Ξ»1
Ξ»2
and Ξ =
Ξ»1 Ο
Ο Ξ»2
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Bivariate Poisson BINAR(1) and Granger causalityFor instantaneous causality, we test
H0 : Ο = 0 against H1 : Ο 6= 0
Proposition Let Ξ»Μ denote the conditional maximum likelihood estimate ofΞ» = (Ξ»1, Ξ»2, Ο) in the non-constrained MINAR(1) model, and Ξ»β₯ denote theconditional maximum likelihood estimate of Ξ»β₯ = (Ξ»1, Ξ»2, 0) in the constrainedmodel (when innovation has independent margins), then under suitableconditions,
2[logL(N , Ξ»Μ|N0)β logL(N , Ξ»Μβ₯|N0)] Lβ Ο2(1), as nββ, 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 β P against H1 : P /β P,
where P is a set of constrained shaped matrix, e.g. P is the set of dΓ d diagonalmatrices for lagged independence, or a set of block triangular matrices for laggedcausality.
Proposition Let PΜ denote the conditional maximum likelihood estimate of P inthe non-constrained MINAR(1) model, and PΜ
cdenote the conditional maximum
likelihood estimate of P in the constrained model, then under suitable conditions,
2[logL(N , PΜ |N0)β logL(N , PΜc|N0)] Lβ Ο2(d2βdim(P)), as nββ, under H0.
Example Testing (N1,t)β(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 β¦Xtβ1 + Ξ΅t, where (Ξ΅t) is the innovation process, with Ξ» := E(Ξ΅t) andΞ := var(Ξ΅t). Let Β΅ := E(Xt) and Ξ³(h) := cov(Xt,Xtβh). Then Β΅ = [Iβ P ]β1Ξ»
and for all h β Z, Ξ³(h) = P hΞ³(0) with Ξ³(0) solution ofΞ³(0) = PΞ³(0)P β² + (β + Ξ).
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Part 2
Application to earthquakes
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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,β Japan is at the limit of 4 tectonic plates (Pacific, Okhotsk, Philippine and
Amur),β 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
β 1965-2011 for magnitude M > 5 earthquakes (70,000 events) ;β 1992-2011 for M > 6 earthquakes (3,000 events) ;β To count the number of earthquakes, used time ranges of 3, 6, 12, 24, 36 and
48 hours ;β 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
β¦X1,tβ1
X2,tβ1
+
Ξ΅1,t
Ξ΅2,t
with var
Ξ΅1,t
Ξ΅2,t
=
Ξ»1 Ο
Ο Ξ»2
Complete model, with full dependence
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate models : comparing dynamics
X1,t
X2,t
=
p1,1 00 p2,2
β¦X1,tβ1
X2,tβ1
+
Ξ΅1,t
Ξ΅2,t
with var
Ξ΅1,t
Ξ΅2,t
=
Ξ»1 Ο
Ο Ξ»2
Partial model, with diagonal thinning matrix, no-crossed lag correlation
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate models : comparing dynamics
X1,t
X2,t
=
p1,1 00 p2,2
β¦X1,tβ1
X2,tβ1
+
Ξ΅1,t
Ξ΅2,t
with var
Ξ΅1,t
Ξ΅2,t
=
Ξ»1 00 Ξ»2
Two independent INAR processes
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate models : comparing dynamics
X1,t
X2,t
=
p1,1 00 p2,2
β¦X1,tβ1
X2,tβ1
+
Ξ΅1,t
Ξ΅2,t
with var
Ξ΅1,t
Ξ΅2,t
=
Ξ»1 00 Ξ»2
Two independent INAR processes
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate models : comparing dynamics
X1,t
X2,t
=
0 00 0
β¦X1,tβ1
X2,tβ1
+
Ξ΅1,t
Ξ΅2,t
with var
Ξ΅1,t
Ξ΅2,t
=
Ξ»1 Ο
Ο Ξ»2
Two (possibly dependent) Poisson processes
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate models : tectonic plates interactionsβ For all pairs of tectonic plates, at all frequencies, autoregression in time is
important (very high statistical significance) ;β Long sequence of zeros, then mainshocks and aftershocks ;β Rate of aftershocks decreases exponentially over time (Omoriβs law) ;
β For 7-13% of pairs of tectonic plates, diagonal BINAR has significant better fitthan independent INARs ;β Contribution of dependence in noise ;β Spatial contagion of order 0 (within h hours) ;β Contiguous tectonic plates ;
β For 7-9% of pairs of tectonic plates, proposed BINAR has significant better fitthan diagonal BINAR ;β Contribution of spatial contagion of order 1 (in time interval [h, 2h]) ;β Contiguous tectonic plates ;
β 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 β N2 or N1 β 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
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
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
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
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
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Granger causality N1 β N2 or N1 β 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
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
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
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
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, 24 hours
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Granger causality N1 β N2 or N1 β 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
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
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
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
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, 48 hours
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Multivariate models : frequency versus magnitudeX1,t =
βi=1
1(Ti β [t, t+ 1),Mi β€ s) and X2,t =βi=1
1(Ti β [t, t+ 1),Mi > s)
Here we work on two sets of data : medium-size earthquakes (M β (5, 6)) andlarge-size earthquakes (M > 6).
β Investigate direction of relationship (which one causes the other, or both) ;β Pairs of tectonic plates :
β Uni-directional causality : most common for contiguous plates (NorthAmerican causes West Pacific, Okhotsk causes Amur) ;
β Bi-directional causality : Okhotsk and West Pacific, South American andNasca for example ;
β Foreshocks and aftershocks :β Aftershocks much more significant than foreshocks (as expected) ;β Foreshocks announce arrival of larger-size earthquakes ;β Foreshocks significant for Okhotsk, West Pacific, Indo-Australian,
Indo-Chinese, Philippine, South American ;
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
Risk management issuesβ Interested in computing P
(βTt=1 (N1,t +N2,t) β₯ n
β£β£β£F0
)for various values of T
(time horizons) and n (tail risk measure) ;β Total number of earthquakes on a set of two tectonic plates ;
β 100 000 simulated paths of diagonal and proposed BINAR models ;β Use estimated parameters of both models ;β Pair : Okhotsk and West Pacific ;
β 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
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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.
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.
Franke, J. & Subba Rao, T. (1993). Multivariae first-order integer-valuedautoregressions. Forschung UniversitΓ€t Kaiserslautern, 95.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
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.
Kocherlakota, S. & Kocherlakota K. (1992). Bivariate DiscreteDistributions. CRC Press.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
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.
LΓΌtkepohl, 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.
Orfanogiannaki, K., D. Karlis & G.A. Papadopoulos (2010) "IdentifyingSeismicity Levels via Poisson Hidden Markov Models", Pure and AppliedGeophysics 167, 919-931.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
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), βSeismological modelingβ, Encyclopedia ofEnvironmetrics.
Rosenblatt, M. (1971). Markov processes, Structure and Asymptotic Behavior.Springer-Verlag.
Steutel, F. & K. van Harn (1979) "Discrete analogues ofself-decomposability and stability", Annals of Probability 7, 893-899.
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Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate count processes for earthquake frequency
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.
WeiΞ², 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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