Slides binar-hec

Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management

Bivariate Counting Processesfor Risk Management

Mathieu Boudreault & Arthur Charpentier

Université du Québec à Montréal

[email protected]

http ://freakonometrics.blog.free.fr/

IFM2, Mathematical Finance Days, May 2012

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http://freakonometrics.blog.free.fr/


Figure : Mexican catastrophe bond, 2006-2009, via Cabrera (2006)

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Motivation : Mexican (earthquake) catastrophe bond

Figure : Mexican catastrophe bond, 2006-2009, via Cabrera (2006)

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Motivation

Figure : Time and distance distribution (to 6,000 km) of large (5<M<7)aftershocks from 205 M≥7 mainshocks (in sec. and h.). Parsons & Velasco(2011)

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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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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

7


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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

8


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of e

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es (

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2) p

er 1

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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

9


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

10


11


Agenda• Motivation : earthquake risk and 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

12


(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

5010

015

020

025

030

035

0

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

13


(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

14


(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.

15


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


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

17


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

18


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.

19


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)

20


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

21


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

22


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.

23


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 ))

24


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)

25


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,

26


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).

27


(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.

28


Xt+1 =Xt∑i=1

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

29


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.

30


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

.

31


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 ))

32


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).

33


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).

34


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→∞.

35


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

36


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

37


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

38


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

39


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

40


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

41


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

42


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.

43


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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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).

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Multivariate models ?

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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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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

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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

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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

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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

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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


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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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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56


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


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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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


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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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http://arxiv.org/abs/1112.0929

Economy & Finance

Slides binar-hec