Slides binar-hec
2 likes1,003 views
This document discusses using bivariate counting processes to model risk from correlated catastrophic events like earthquakes. It provides motivation for why correlated risks are important from examples of large earthquakes triggering aftershocks and increased seismic activity near the main shock. Specifically, it examines using bivariate counting processes to model the number of earthquakes before and after a major earthquake, which shows a significant increase in seismic activity for a few rupture lengths around the main shock.
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
Motivation
“Large earthquakes are known to trigger earthquakes elsewhere. Damaging large
aftershocks occur close to the mainshock and microearthquakes are triggered by
passing seismic waves at significant distances from the mainshock. It is unclear,
however, whether bigger, more damaging earthquakes are routinely triggered at
distances far from the mainshock, heightening the global seismic hazard after
every large earthquake. Here we assemble a catalogue of all possible earthquakes
greater than M5 that might have been triggered by every M7 or larger mainshock
during the past 30 years. [...] We observe a significant increase in the rate of
seismic activity at distances confined to within two to three rupture lengths of the
mainshock. Thus, we conclude that the regional hazard of larger earthquakes is
increased after a mainshock, but the global hazard is not.” Parsons & Velasco
(2011)
Figure : Number of earthquakes (magnitude exceeding 2.0, per 15 sec.) following
a large earthquake (of magnitude 6.5), normalized so that the expected number
of earthquakes before and after is 100.
6](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-6-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
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
cov(Xt , Xt−h ) γ(h)
ρ(h) := corr(Xt , Xt−h ) = = , ∀h ∈ N.
var(Xt )var(Xt−h ) 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) = φh ,
1 ∀h ∈ N.
19](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-19-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
Remark As a consequence, var(X t ) = E([X t − E(X t )] [X t − E(X t )]) is
independent of t (=: γ(0)). Define finally the autocorrelation matrix,
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.
23](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-23-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
Poisson distribution - and process - for counts
k
−λ λ
N as a Poisson distribution is P(N = k) = e where k ∈ N.
k!
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](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-25-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
Thinning operator ◦
Steutel & van Harn (1979) defined a thinning operator as follows
Definition Define operator ◦ as
p ◦N = Y1 + · · · + YN if N = 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 ).
27](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-27-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
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
d
[P ◦ N ]i = 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 ) ) = P E(N N )P + ∆,
with ∆ := diag(V E(N )) where V is the d × d matrix with entries pi,j (1 − pi,j ).
33](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-33-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
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
xt
π(xt , xt−1 ) = P(P ◦ xt−1 = xt − k) · P(ε = k).
k=0
34](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-34-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
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
Rao (1993)), then the conditional maximum likelihood estimate θ of θ = (P , Λ)
is asymptotically normal,
√ L
n(θ − θ) → N (0, Σ−1 (θ)), as n → ∞.
Further,
L
2[log L(N , θ|N 0 ) − log L(N , θ|N 0 )] → χ2 (d2 + dim(λ)), as n → ∞.
35](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-35-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
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
min{k1 ,k2 }
(λ1 − ϕ)k1 (λ2 − ϕ)k2 k1 k2 ϕ
e−[λ1 +λ2 −ϕ] i!
k1 ! k2 ! i=0
i i [λ1 − ϕ][λ2 − ϕ]
with λ1 , λ2 > 0, ϕ ∈ [0, min{λ1 , λ2 }].
λ λ ϕ
λ= 1 and Λ = 1
λ2 ϕ λ2
42](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-42-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
Bivariate Poisson BIN AR(1) and Granger causality
For instantaneous causality, we test
H0 : ϕ = 0 against H1 : ϕ = 0
Proposition Let λ denote the conditional maximum likelihood estimate of
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
2[log L(N , λ|N 0 ) − log L(N , λ |N 0 )] → χ2 (1), as n → ∞, under H0 .
43](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-43-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
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.
Proposition Let P denote the conditional maximum likelihood estimate of P in
c
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
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.
44](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-44-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
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 + (∆ + Λ).
45](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-45-320.jpg)
![Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management
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](https://image.slidesharecdn.com/slides-binar-hec-120426135351-phpapp02/85/Slides-binar-hec-53-320.jpg)
Slides binar-hec
- 1. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Bivariate Counting Processes for Risk Management Mathieu Boudreault & Arthur Charpentier Université du Québec à Montréal charpentier.arthur@uqam.ca http ://freakonometrics.blog.free.fr/ IFM2, Mathematical Finance Days, May 2012 1
- 2. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Figure : Mexican catastrophe bond, 2006-2009, via Cabrera (2006) 2
- 3. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Motivation : Mexican (earthquake) catastrophe bond Figure : Mexican catastrophe bond, 2006-2009, via Cabrera (2006) 3
- 4. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 4
- 5. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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) 5
- 6. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Motivation “Large earthquakes are known to trigger earthquakes elsewhere. Damaging large aftershocks occur close to the mainshock and microearthquakes are triggered by passing seismic waves at significant distances from the mainshock. It is unclear, however, whether bigger, more damaging earthquakes are routinely triggered at distances far from the mainshock, heightening the global seismic hazard after every large earthquake. Here we assemble a catalogue of all possible earthquakes greater than M5 that might have been triggered by every M7 or larger mainshock during the past 30 years. [...] We observe a significant increase in the rate of seismic activity at distances confined to within two to three rupture lengths of the mainshock. Thus, we conclude that the regional hazard of larger earthquakes is increased after a mainshock, but the global hazard is not.” Parsons & Velasco (2011) Figure : Number of earthquakes (magnitude exceeding 2.0, per 15 sec.) following a large earthquake (of magnitude 6.5), normalized so that the expected number of earthquakes before and after is 100. 6
- 7. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Number of earthquakes (magnitude >2) per 15 sec., average before=100 q 180 q q q q 160 q q q qq q q q 140 qq qq qq qq q qq q q q q qq qqq qq qq q q qqq q q qq q q q q q q q qqq q qq q qq qq qq q q 120 qqqqq q qqq qq qq qq q qq q q q qqq q q q qq qq q q q qq q q q qqq qq q q q q qq qq q qq qq qqq q qqqq q qq q qq q q q qqqqq q q q qqq qqqq qq q qqq q q qq q q qq q q qq qq q q q q q q q qq qqq q q q qq q q q q q q q q q q qq q qqq q qqq qq q qq q q q q qq q qq q q q q q q q qq q q q q q q qq q q q q qq q qqq qq q q q q q q q q q q q qq qq qq q q q q qq qq q qqq q qq q qqqq qq q qq q q q q q qq q qq q q q qq qqqq qq q q q q q q qq q q q q q q q qq qq qq qq q q q qq qq qqqq qq q qq q qqq qq q q qq q q q qq q q qqqq qq qq q qqq q q q qq q qq qq qq qq q q qq qq q q qq q q q q q qqq q qq q qqqqq qqq qqq q q q q qq q qq q q qq q q q q qqq qq q q qq qq q qq q q q qq q q q q q q q q q qq q q q q q q q qq q q qq q qq q q q qq qq qqq q q q q qq q qq q q q q q q q q q qqq q qq qq qq q q qq q q qqq q q q q qq q q q q q q qq q q qq q q qq q qq q qq q qqq q qq q qq q q q qq q q q qqqq q q q q qq q qq q qqqqqqqqqq qqqqqqqq q qqqq q q qq q qqqq qq q qqqq q q qq qq qqq q qqqq qqq qqqqqqqqqqq q q q q q q qqqqq qq q q q q q q q qqqq q q q qqq q qq q q q q q q q qq qqq q q q q qq q q qq qqq qq q q q q qq qqqqqq q q q q qq qq qq q q q q q qq q q q qq q q q qqq q q q q q qq qq qq q q q q q qqq qqqq q q qqq qq q qqq qqq q q qq q q q q q q q qq qqq qqqqq q q q q q qq q qq q qq q q q q qq qq q q q q q qq q q q q q qq q q q q q q q q q q qq q qqq qq q q qqqq q qqqq qqqqq q qq qqqqqqqqq qq q q q qq q qq q qqq q q q q qq q q qq qqqq q qqqq q qq q q q q qqq q q q q q q q q q q q qqqqq q qqqq q qqqq qq q qq qq qq qqqqqq q qq qq qqq q q q q qq qq q q q qq q q qq q 100 qq qq qq qq q q qq q qq q q q q qq q q q qqq qq q qq q qq q q q q qq q q q q qq q qq qq q q q q q qq qqq q qqqq qqq qq qqqqqqq qqq q q q qq q qqqq qqqqq qq qqqq q q qq qq q q q q q q q qq q q q q qqq qq q q q qq qqq q q qq q q qq qqq q q qqqq qqqq q qqq q q qq q q q q q q q q q q qq qq qq qq qqqqqq q qqqqqq qqqqq q qqqq qq qqqqqqqq qq qq qqqqqq qqqq q q qq qq q q q q q qq q q q q q q q qq q q q q q q q q qqq q qqq q qqq qq qqq q q q qq qqq q q q q q q q q q q q qq q q q q q qq qq q qqqqqqq qqqq qqqqqq qqqqqq qq q qqqq qqqqq qq q qq q q q q q q q qq q q qqq q q qq q q q qq qq q q q qq qqq q q q q q q q qqq qq q qqq q q qqqq qq q q q qqqqq qqqqqq q qqq qq qqqq qq qq q qq qqq q q q qq qqq qqq qqq qq q qq q q q q qq q q q q qqq qqq q q qqq qqqqqq q qq qq qq q qqqqqqq q q qq qqq q q q qq q q qq q q q q q q q qqqq qq qq qq qqqq qq q qq qq q q qq qqqq q q q qqqq q qq qqq qqq qqqq q q q q qq qqq q q q q q q q qqq q qq q qq q q q q q qq q qq qq qqq qq qqq q q q q qqq q qqqq q qqq q qqqqqqqq q q qq qqq qq q qq q q q qq q q q q qq q qq q q qqq q q q qq q q qqq q qq qqqq qqqqqq q qq qqq q q qqqqq qq qq q q q qqq qqqqq qqqq q qq qq q qq q q q q q qq q q q q q q q qqq q q qqqq qq q qqq qq qq q q qqq qqq q q qqqq qqq q qqq qqqqqqq qq qqqqqq q q q q q q qqq q qq qq qqqq q qq q q qq q qqq qqq qqq qq qqq q q qqq qqq q qq q qq q q q qqq q q qq q q q qq q q qqqqq qqqqq qqq q q qqqqqqqq qqq q qqqqqqqqq qq qqq q qqqq q q qq qq qq qq q q q q q q qq qq q q q qqq qqqqqq q qq q q qqqq q q qq q qq qqqq qqq q qqq q qqqqq q q q q qq q q qqq q qq q qq qq q q q q q q q q q q q qq q q q q qq q qq q q q q qq qq qq q q qq q q q q qq qqq qq q q qqq qq qqqq qq qqqq qqqq qqq q q q q q q qq q q qq qq qq qqqq q q qq qq q q qq qqq qqqq qq qq q qq qqqqqqqq q qq q q qq q q q qq q qqq q q qqqq q q qqq q q q q q q q qq q qq qqq q qq qqq qqqqqq qq qqqq q qq q q qq qq q qqqqqq q q q q q qq q q qqqq qqq qq qq q qq q q qq qqq q qq qq q q q q qq q qqq q qq q q q qq q q q qq q q qq q qq q q q q q qq q qqqq q qq q q qq qq q qq qq q q q qq q qq q qq qq q qqqq qqqqqqqqqqqq qqq q qq q q qq q q q qqqqqqq q q qq qq q q q qq q qq q q q q qq q qq q q q q q q q q q q qqq q qq qq q q q q q qq q qq qqq q q q qqq q q q q q q q q qq q qqq q q qq q qq q qq q qq q qq qq q q q q q q q q q qq q q qq q q q q q q q q qqq q qq q qq q q q q q q q qq qq qqq qq q q q q q q qqq q q q qq q qq qq q q q q q qq q q q q qq qq qq qq q q q qq q q q q q q q q q qq q q q q q q qq q qq qq q q qq q q qq q q q q q q q qq q q q q q q qq q q q q q q q q q qq q q q q q q q q q q 80 −15 −10 −5 0 5 10 15 Time before and after a major eathquake (magnitude >6.5) in days 7
- 8. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Number of earthquakes (magnitude >4) per 15 sec., average before=100 Number of earthquakes before and after a major one, magnitude of the main event, small events more than 4 220 q q Main event, magnitude > 6.5 q Main event, magnitude > 6.8 200 q Main event, magnitude > 7.1 Main event, magnitude > 7.4 q q 180 q qq q qq 160 q q q q qq q qqq q q q q qq qq q qq qq q qq q q q qqq q qqq q q q q q qq qq qq qq q qqqqq 140 q q q qqqq q qq q qq q q qq q qq q q qqq q q qqq q qq qq qq q qq qqq qq q qq q q q q qq q q q q q qqq q q qq q qq q qq q q q q q q q q q q q q q q qq q 120 qq qq qq q q qq qq q q q q qq q q q q q q q q qq q qq q q q q q q q qq q qq qq q q q q qq q qq q q q qq q qq q q q q qq q q q q q q q qqq q q q q q q q qq q qq qq q qqqqqqq q qq q q qq qqqq q qq q qq q qq q q q q q q q qq qq q q q q q qq q q q qq q q q qq q q q q q qq q q qq q qq q q q q q qq q qq q q q q q q q q q q q q q q qq qq q q q q q qq qq q q q q q q q q q q qq q q q q q qq q qq qqq q q q qq q q qq qqq qqqqqqqq q q q qq q q q qqqqqq q qq q q q qq qqq qq qqqqq q q q q q q q q qq qq q q q q q qq q q qq q qq q q qq q q qq q q qq qqq q qq q qq q q q q qqq q q q q qq q q qq qq q q q q q qqq q q q q q qq q q q qq q q q q q qq q q q q q q q q qq qq qqqq q q qq q q qqq q q q q qq qqqqqqq q q q q q q qq qqq q q qq q qq qq q q q qqq qqq qq qq qq qqqq qq qq qqq qq qq q q qq qqqq q q q qq q qqq q q q q q q q qq q qqq qqq q qqqq qq q q qqqq qq qqq qqq qq qq q q q q q q q qq qqqq qqqq qqq qqqqq q qq q q qq qqq q qqqqqqqqq q q q q q qq q q q q q qq q q qq q q qq q q q qq q q qq q q q q q qq q q q q q qq q q q q q q q q q q q q q qqq q qq q qq qq q qqq q q qq qq q qq q q q qq q q qq q q q q qqq q qqq qq q q q qq qqq q q q qqqq qq qq q qq q q q q qqq q q qq q q q q q q qq q q qq q q q q q q qq q q q q qq qqqq qq q qq qq q q q q qqq qqqq qqqq qqqqq q qqq qqq qq qq q q qqqqqq q q q qq q q q qqq q q q qq qq qqqqqq qq qq q qqqqqq qq qqq qq qq qqqq qq qqqqqq qq qq q qq q q qq qq qq q q q q q q qq qq qqqq q qq qq qq qq q q q q q q q qq q q q qq q q q q qq q q qq qqqqq qqqq q qqq qq qqq q qq q qqqqqq q qq qqq qqqqq qq qq qq q q q q qq q q q q q q q qqq q qqq q 100 q q qqq q qqqqqq q q q qqq q q q q qq qq q q q q q qqqqqqq q q qq q qq qq q q qqqqq q q qq q qq q qq q q qq q q q qqqq qqqqqq q q q qqq q qqqqqqq q qqqq q q q qqq qqqqq qq q qq qqq qqqqqqqqq q qqq qq qq q q qq q q q q qq q qq q q q qq q q q q qq q q q q q q qqq q q q qq q q qqq qqq qqq qq q q qq q qqqq q qq qq qqqq q qq qqqqqqqq q q q qq q qq qq q q q q q qq q q q q qq q q q q qq q q q qqq q q qq qq q qq qq q q qqq qq q q qq q qqqq qq q q q q qqq qqq qq q q q q qq q qq q q q q q q q q q qq q q q q q q qqqqq qqqqq q q qq qq q qqqq q q qq qq q q q q q qqqqqq qq qq qqqqq qqqq qq qqqqq qqq q q q q q q qqq qqqqq qq q q q q q qq q q qq q q q qq q qqq qq q qqqq q q qqqq q q qq qqq q qq q qqqq q qq q q q q qq qq q qq q q q q q qq qqqqqqq q qq q qq qq qq q q qq q qqq q q q q qq q qq qq q qq qqqq qq q qqq q qqq qq q q qqqq q q q q qq qq q qqq qqq qq qq qqqqqqq q qqq q q q q q q qq qq q q q q q qqq q q q q q q qq q q q qq q qq qqqq q q qqq q q qqq q q q qq q q qqq q q q qq q qq q qqq qq q qqq q q qq q qqq qqqqq q qq q q q q qq q q q qq q q q qqq q q q q q q q q qq qqq q q qq q q q q q q q qq qq qqqqqq qq q q q q q qq q qq qq q q q q qq q q q q q q q q q qq q qqq qq q qq q q q q q qq q q qq qq qqqq qq qqq q q qqq q q qqq q qq q q qq q qqq qq q qqq qq q q qq qqqq qqqqqqqq q qqqq qqqq qq qq qqq q q q qqq q q qqq qq qqq qq q q qq q q q q q q q qq q q q q qq q qqqqq q q q qq q q qqq q q q q qq q q q q q q qq qq q qq q q q q qq q qqq q q q q q q q qq q q qq q qq qq q qq q q q q qqqq q qqq q qq qqq qqqqq q q qq qqq q q qqqq q q qq q q q q q qq q q qq q q q qq q q q q q q q qqqqqq q qqq qq qq q q q qqq q q qq q qqq qqq q q q qq q q q q qq q qqqqq q q q q qq q q qq q q q q qq q q q q qqqqqqq qqq q qq q q q qq q qqq q qqqq q q q q q q q qq q qq q q qq qq q qq q q q q qqq q qq q q q q q q q q q q qq qq q q q qq qq qqqq q q q q q q q qq q q q q q q qqqq qqqqqq q q q q q q q qq q qqq q q qq qq q q qqqq q qq qq q q q qq q q q q qq q qq q q q q q q qqq q q qqqq q qqq q qq q q q qq q q qq qq q q q q qq q q q qq q q q q q qqq q q q q q qq qq q q q qq q q q q qq qq q q q q q qq q q q q q q q q qq qq qq q q q qqqq q q q qq q q q qq q q qqq qq q q q qq q q qq qq q q qq q qq q qqq q q q q qqqq q q q qq q q q q qq qq q qq q q qq q q q qq q qq qq q q q q q q qq q qq qq q qqq q q q q q q qq qq q q q q qq q qq q q q q qq q q qqqq q q qq q q qq q q q q qq q q q qq q qq q q q q q q q q q qqq q q qq q q q qq q q qq q q q q q q q q q q q q q q q q 80 q q q q q q −15 −10 −5 0 5 10 15 Time before and after a major eathquake (magnitude >6.5) in days 8
- 9. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Number of earthquakes (magnitude >2) per 15 sec., average before=100 q 1000 q q Same techtonic plate as major one 800 q q Different techtonic plate as major one q q q q q 600 q q q q q q q qq qq q qq q qq q qq 400 q qq q qq q qqq qq qqq qq qq q qq qq q qq qq q q qq qq q qq q qq q qqq qq qqqq qq qq qqq qq qqqq qqq qqqq q q qq q q qq qq qq qq q qqqq q q 200 qq qq q q qq q qqq q q q qqqqqq q qqqq q q q q q qq qq q qq qqqq qq q q qqqqqq q qq q q qq qq qq q q qqqq q q q q qq q q q q qq q qqqqqqqq qq qqqqqqqqq q qq q q qqq q q qq qq qq q qqq qq qq q q qqqqq qqqqqq qq q q q qq q q qq qq qq qq q q qq q q qqqqq qq qq qq q q q q qqqq q q qqq q qq q qq q qqqqq qqqqqq q q q qqq qqq qqqqq q q qq qq qqq qqq q qqq qq qqqqqq q qqqqq qq qq q qqqqq qq qqqq qq q qqq qqqqqq q q q q q q q qq q qq qq q q qq q qq q qqq qq qqq qqq q q qq q q q qq q qq q q q q qqqqqqqqqqqqqq qqqqqqqq qq qqq qqq qq qqqqqq q q qq qqq qqq qq q qqqq qq qq qqqq q q q q qqq qq q qq q q qqq q qqqq qq q q qq qq q qq qqqq q qqq q qq q qq qq qq qq q q q qqqq q q q qq q q qq qq qq q q qq qq q q qqq q qqq qqqq q qqqqqqq q q q q qqq q q q qq q qq q q q qqqq qq qq q q qqqqq q q q q q qqqq qq q qqqq qq qq q qq q q qqqqqqq q q q q q qq q qqqq qq q qq q q q q q q qq qq q qqqqqqqqq qqqqq qqq q qqqq q qqqq qq qqqq qqq q qqqqq qq qqqqqqqqqqqq qqqqqqqqqq qq q qqq q qqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqq qqq qqqqqqq qqqqqqqq qq q qqq q qqqqqq qqqqq qqqqqqqqqq q q q qqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqq qqqqqqqq qq qqqq qq q q q q q qq q q q qq qq qqqqq q qqqq qqqqqqqqq qq q q q qqqq qqqqqqqqqqqqqqqqqqqqq qqqqqqqqqq qq qqqqqqqqqqqq qqqq q qq q q q q q q qqqq qq q qqq q q q q q q q q q q q qqqq q qq q q q q q q q q qqqqqq qqqqq qqqqqqqqq q qq qq qqqqqq qq qqqqqqq qq q q qq q q q qq qqqq q q q qq qq q qq qq q qqqqqqqqqqqqqqqqqqqq q q qqqq q q qqq qqqqq qq qqqqq qqqq qqqqqqqqqqqqqqqqqq q q qq q q q q q qq q q q q q q qqqq qqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqq qq q qqqqqqqqqq qqqqq qq qqqq q qqqqqqqqqqqqqqqq qqq q q q qq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqq q q qqqqqqqqq q qq qq qqq qq qq qqqq qqqq qqq qqqqqqqqqqqqqqqqqqqqq qqqq q qq q qqqq qqqq q qq q q q qqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqq qqqqq qqqqq qqqqqq qqq q q qq qqqq qqq q qqq qqq qqq qq qqqq qqqqq q qqq q qqq qqqq q qqq q qq q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqq q q q q qqq q q qqq qq qqqqqqq qq q qqq qqqqqqqqqq qqq q q q q qqqqq qq qq qq qq qq qq qq q qqq q q q q qqq qqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q qqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqq qq qqqqq qqqqq qq qqq qqqqqqq q q q q qqqqqq qqqqq qqqqqqqqqqqqq qqqqqqqqqq qqq q q qqqqqq q qq qq q q qq qq q qq qq qq q q qqq qqqqqq q qq qqq q qqqq qq qq q q qqq qq qq q qqqqq qq qq qq q qqqq qqqqqq qq q q q q q q q q qqqqqqqq qq qqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qqqqqqqqqqqq q q qqq q q qq q q qq qqq q qq qq q qq q q q q qq qqq qqq q qqq q qqqqqqqqqqqq qqqq qqqqqqqqqq qq q qq q qqqqqqqqqqqq q q qq q q qqqqqqqqqqqqqqqqqq qq qqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqq qq qqqqqqqqq qq qq q qq q q qqqqqqqqqq qqq q q q qqqqqqqqqqqqqqqqqqqqqqq qq qqqqqqqq qqqq qqqqqqqqq q q qqq qqqqqqqqqqq qqqqq q qqqqqqqqqqqqqqqqq qq qqqqqqq qqqq qqqqqqqqq q qqq qqqqqqqqqq qq qq q qq q qqqqqqqq qqqq qq q qq q qq qqqq q q q qqqqq qq q q qq qq q q qqq qqqq qq qqq qqqqqqqqqqqqqqq q qqqqqq q qq q q q q q qqqqqqqqqqqqqq q qqq qqqq qqqqqq q qq qqqqqqq qqqqqq qqqqqqqqqq q qq q q q q qqq q q q q qqqqq qq qqq q q qqqqqq qqqqq qqqqqqqqq q qq q qqqqq qq qq q q q qq q qq q qq q q q qqq q qqqqqq q qq qq q qq q q q q q qq q q q q q q q q q q qqq q q qqqqqqqqqqq qqqqqq q qqq q qqq qqq q q qqq q qq qqqqqqq q qqq qqqqqqqq qqqqqqqqqqq q q q q q q qq q q qq qqq qqq qqqqqqqq qqq qqqqqqqq q qq q q q q qq q q qq q qq qq qq q q q qq q q q q q q q q qqqqq qqqq qq q q q qq q q q q qqqq qq q q q q q qq qqq qq q q qq q qqq q q qq q q q q qqqq q q qq q q q q qq q q q q qq qq qqqqqq q qqqqq qqq q qqqq qq qq q qq q q qq q q q q qq qq q q q q q q q q qq q qqq q q qqq qqqqq qqqq q qqqqqq q qqq qqqqqq q q q qq q q q q qq qq q q qq q q q q q q q q q q qqq qq q q qq q q q q qq q q q q q q q qq qqq q q q qq q q q 0 −15 −10 −5 0 5 10 15 Time before and after a major eathquake (magnitude >6.5) in days 9
- 10. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Shapefiles from http://www.colorado.edu/geography/foote/maps/assign/hotspots/hotspots.html 10
- 11. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 11
- 12. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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
- 13. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management (ANSS) http://www.ncedc.org/cnss/catalog-search.html Number of earthquakes (Magnitude ≥ 5) per month, worldwide 350 q q 300 q q q q q q q 250 q q q q q q q q q q q 200 q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q q q q q qq qq q q q q q q q q q q 150 q qq q q q q q qq qq q q q qq q q q q q q q q q q qq q q q q q qqq q q q q q q q q q q q q q q q qq q q q qq q q q q q q q q q q q q qqq qq q q qq q q q q q q qq q q q q qq qq q qq q qq q q q q q q q q q q q q qq q q q q q q q q q qq q q q q q q q q qq q q qq q q q q q q q q q q q q q q q q qq qq qq q q q q qq q q q q q qq q q q q q q q qq q qq q q q qq q qq q qq q q q q q q q q q q qq q qq q q q q qq q q q q qq q q q qq q qq qq q qq q q q q q q q q q q q q q qq q qq q q q q qq qq q q q q qq qq q q q q q qq q q q q q q q q qq q q qq q q q q q q q qq q q q q q q q q q qq q q q q q q qq q q qq q q q 100 q q q q q q q q q qq q q q q qq qqq q q q q q q q q qq q q qq q q q q q q q q q qq q qq q q qq q qq q q qq q q q q qqqq q q q q q q qq qq q q q q q qq q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q q qq q q q q q q q 50 1970 1980 1990 2000 2010 Number of earthquakes (Magn.>5) worldwide per month 13
- 14. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management (ANSS) http://www.ncedc.org/cnss/catalog-search.html Number of earthquakes (Magnitude ≥ 5) per month, in western U.S. q 15 q q q 10 q q q q q q 5 q q q q q q q q q qq q q q q q q q qqq qq q q qq q q q q q q q q q q q q q q q q qq qq q q q q q q qqqq q q q q q q qq q q q q q q q qq q qqq q q qq q q qq q q q qq q qq q q q q q q qqqqqq q q qq q qq qq qq qqq qq q qq qq q q qq q q q q qqq qq qq q q q q q q q qqq q q q q q qq q q qqq qq q qq q qq q q qq q q q qqqqqq qqqqqqq qq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqq qqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqq qq qqqqqqq qqqq qqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qqqqq qqqqqqqq q q q qqqqqqq qq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqqqqqq qq qq qqqqq qqqqqq qqqqq qqqq qqqqqqqqqqqqqqq qqqqqqqqqqqqqqq qqq qq q qq q qq qqqq qqq qq qq q q qqqq qqq qqq qqqqq qqqqqq q q qqqqqq q q q qqqq q q qq q q qq q qq qq q qq q qq qqq qqq q qqqqqq qq q q q qq q q q qqq qqqqqqqq qqqq qqqqqqqqqqqqqqq qqqqqq qqq q q qq qqq q q qq q qq q q q q q q qq qq qqqq qqq qqqqq qq qqqqqqqqqqqqq qq q q q qqq qqqqqq q qqqq q qq 0 1950 1960 1970 1980 1990 2000 2010 Number of earthquakes (Magn.>5) in West−US per month 14
- 15. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management (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 1 ε2 ϕ(ε) = √ exp − 2 , ε ∈ R. 2πσ 2σ 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. 15
- 16. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Example : Xt = φ1 Xt−1 + εt with εt ∼ N (0, 1), i.i.d., and φ = 0.6 q q q q q q q q q q q q qq 2 q q q q q qq q q q q q q q q q q qq q q q q q q q q q q qq q q q qq qq q q q q q q q q q q q q qq q q q q q q q qq q q qq qq q q q q q q qq q q q q qq q q q q 0 q q q qq q q q q q q qq qq q q q q q q q qq q q q q q qq q qq q q q q qq q q q q q q q q qq q q q qq q q q q q q q q q q qq q q q q q q qq q q q q q q qq q q q q q qq qq q q q q q q q q qq q q q q q q q q q q qqq q q q q q qq q q q q q q qq −2 q q qq q q q q q q q q q q q q q q q −4 q q 0 50 100 150 200 250 300 16
- 17. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Example : Xt = φ1 Xt−1 + εt with εt ∼ N (0, 1), i.i.d., and φ = 0.6 q q q q q q q q q q q q qq 2 q q q q q qq q q q q q q q q q q qq q q q q q q q q q q qq q q q qq qq q q q q q q q q q q q q qq q q q q q q q qq q q qq qq q q q q q q qq q q q q qq q q q q 0 q q q qq q q q q q q qq qq q q q q q q q qq q q q q q qq q qq q q q q qq q q q q q q q q qq q q q qq q q q q q q q q q q qq q q q q q q qq q q q q q q qq q q q q q qq qq q q q q q q q q qq q q q q q q q q q q qqq q q q q q qq q q q q q q qq −2 q q qq q q q q q q q q q q q q q q q −4 q q 0 50 100 150 200 250 300 17
- 18. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Example : Xt = φ1 Xt−1 + εt : autocorrelation ρ(h) = corr(Xt , Xt−h ) = φh 1.0 0.8 0.6 1 ACF 0.4 0.2 0.0 0 5 10 15 20 Lag 18
- 19. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 cov(Xt , Xt−h ) γ(h) ρ(h) := corr(Xt , Xt−h ) = = , ∀h ∈ N. var(Xt )var(Xt−h ) 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) = φh , 1 ∀h ∈ N. 19
- 20. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management From univariate to multivariate models Density of the Gaussian distribution Univariate gaussian distribution N (0, σ 2 ) 1 x2 ϕ(x) = √ exp − 2 , for all x ∈ R 0.20 2πσ 2σ 0.15 0.10 0.05 0.00 Multivariate gaussian distribution N (0, Σ) 3 2 1 x Σ−1 x 1 ϕ(x) = exp − , 0 (2π) d | det Σ| 2 −1 2 3 for all x ∈ Rd . 1 −2 0 −1 −3 −2 −3 X = AZ where AA = Σ and Z ∼ N (0, I) (geometric interpretation) 20
- 21. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 2,1 1,t−1 + φ2,2 X2,t−1 + · · · + φ2,d Xd,t−1 + ε2,t X = φ X 2,t (2) ··· d,1 1,t−1 + φd,2 X2,t−1 + · · · + φd,d Xd,t−1 + εd,t X = φ X d,t or equivalently X φ φ1,2 ··· φ1,d X ε 1,t 1,1 1,t−1 1,t X2,t φ2,1 φ2,2 ··· φ2,d X2,t−1 ε2,t . = . . . . + . . . . . . . . . . . . . Xd,t φd,1 φd,2 ··· φd,d Xd,t−1 εd,t Xt Φ X t−1 εt 21
- 22. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 1 ε Σ−1 ε ϕ(ε) = exp − , ∀ε ∈ Rd . (2π) d | det Σ| 2 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 22
- 23. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Remark As a consequence, var(X t ) = E([X t − E(X t )] [X t − E(X t )]) is independent of t (=: γ(0)). Define finally the autocorrelation matrix, 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. 23
- 24. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 n L(φ, σ; (X0 , X )) = πφ,σ (Xt |Xt−1 ) t=1 where πφ,σ (·|Xt−1 ) is a Gaussian density. Maximum likelihood estimators are (φ, σ) ∈ argmax log L(φ, σ; (X0 , X )) 24
- 25. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Poisson distribution - and process - for counts k −λ λ N as a Poisson distribution is P(N = k) = e where k ∈ N. k! 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
- 26. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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, 26
- 27. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Thinning operator ◦ Steutel & van Harn (1979) defined a thinning operator as follows Definition Define operator ◦ as p ◦N = Y1 + · · · + YN if N = 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 ). 27
- 28. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management (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 = Yi + εt , where Yi 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. 28
- 29. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Xt Xt+1 = Yi + εt+1 , where Yi s are i.i.d. B(p) i=1 29
- 30. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 k −λ λ P(εt = k) = e , k ∈ N. k! 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. 30
- 31. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management xt xt−1 π(xt , xt−1 ) = P(Xt = xt |Xt−1 = xt−1 ) = P Yi = xt − k · P(ε = k) . k=0 i=1 Poisson Binomial 31
- 32. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Inference of Integer AutoRegressive processes IN AR(1) Consider a Poisson INAR(1) process, then the likelihood is n λX0 λ L(p, λ; X0 , X ) = ft (Xt ) · exp − t=1 (1 − p)X0 X0 ! 1−p where min{Xt ,Xt−1 } λy−i Yt−1 i ft (y) = exp(−λ) p (1 − p)Yt−1 −y , for t = 1, · · · , n. i=0 (y − i)! i Maximum likelihood estimators are (p, λ) ∈ argmax log L(p, λ; (X0 , X )) 32
- 33. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 d [P ◦ N ]i = 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 ) ) = P E(N N )P + ∆, with ∆ := diag(V E(N )) where V is the d × d matrix with entries pi,j (1 − pi,j ). 33
- 34. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 xt π(xt , xt−1 ) = P(P ◦ xt−1 = xt − k) · P(ε = k). k=0 34
- 35. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 Rao (1993)), then the conditional maximum likelihood estimate θ of θ = (P , Λ) is asymptotically normal, √ L n(θ − θ) → N (0, Σ−1 (θ)), as n → ∞. Further, L 2[log L(N , θ|N 0 ) − log L(N , θ|N 0 )] → χ2 (d2 + dim(λ)), as n → ∞. 35
- 36. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Granger causality with BIN AR(1) (X1,t ) and (X2,t ) are instantaneously related if ε is a noncorrelated noise,g g g ggggggggggg X1,t p1,1 p1,2 X ε ε λ ϕ = ◦ 1,t−1 + 1,t , with var 1,t = 1 X2,t p2,1 p2,2 X2,t−1 ε2,t ε2,t ϕ λ2 Xt P X t−1 εt 36
- 37. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Granger causality with BIN AR(1) 1. (X1 ) and (X2 ) are instantaneously related if ε is a noncorrelated noise, g g g g gggggggggg X1,t p1,1 p1,2 X ε ε λ = ◦ 1,t−1 + 1,t , with var 1,t = 1 X2,t p2,1 p2,2 X2,t−1 ε2,t ε2,t λ2 Xt P X t−1 εt 37
- 38. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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, X1,t p1,1 0 X ε ε λ 0 = ◦ 1,t−1 + 1,t , with var 1,t = 1 X2,t 0 p2,2 X2,t−1 ε2,t ε2,t 0 λ2 Xt P X t−1 εt 38
- 39. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 = 0 while p1,2 = 0, X1,t p1,1 0 X ε ε λ ϕ = ◦ 1,t−1 + 1,t , with var 1,t = 1 X2,t p2,2 X2,t−1 ε2,t ε2,t ϕ λ2 Xt P X t−1 εt 39
- 40. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 = 0 while p2,1 = 0, X1,t p1,1 X ε ε λ ϕ = ◦ 1,t−1 + 1,t , with var 1,t = 1 X2,t 0 p2,2 X2,t−1 ε2,t ε2,t ϕ λ2 Xt P X t−1 εt 40
- 41. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 = 0 X1,t p1,1 X ε ε λ ϕ = ◦ 1,t−1 + 1,t , with var 1,t = 1 X2,t p2,2 X2,t−1 ε2,t ε2,t ϕ λ2 Xt P X t−1 εt 41
- 42. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 min{k1 ,k2 } (λ1 − ϕ)k1 (λ2 − ϕ)k2 k1 k2 ϕ e−[λ1 +λ2 −ϕ] i! k1 ! k2 ! i=0 i i [λ1 − ϕ][λ2 − ϕ] with λ1 , λ2 > 0, ϕ ∈ [0, min{λ1 , λ2 }]. λ λ ϕ λ= 1 and Λ = 1 λ2 ϕ λ2 42
- 43. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Bivariate Poisson BIN AR(1) and Granger causality For instantaneous causality, we test H0 : ϕ = 0 against H1 : ϕ = 0 Proposition Let λ denote the conditional maximum likelihood estimate of 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 2[log L(N , λ|N 0 ) − log L(N , λ |N 0 )] → χ2 (1), as n → ∞, under H0 . 43
- 44. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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. Proposition Let P denote the conditional maximum likelihood estimate of P in c 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 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. 44
- 45. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 + (∆ + Λ). 45
- 46. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Multivariate models ? 46
- 47. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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
- 48. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Multivariate models : comparing dynamics X p p1,2 X ε ε λ ϕ 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 λ2 Complete model, with full dependence 48
- 49. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Multivariate models : comparing dynamics X p 0 X ε ε λ ϕ 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 λ2 Partial model, with diagonal thinning matrix, no-crossed lag correlation 49
- 50. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Multivariate models : comparing dynamics X p 0 X ε ε λ 0 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 λ2 Two independent INAR processes 50
- 51. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Multivariate models : comparing dynamics X p 0 X ε ε λ 0 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 λ2 Two independent INAR processes 51
- 52. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Multivariate models : comparing dynamics X 0 0 X ε ε λ ϕ 1,t = ◦ 1,t−1 + 1,t with var 1,t = 1 X2,t 0 0 X2,t−1 ε2,t ε2,t ϕ λ2 Two (possibly dependent) Poisson processes 52
- 53. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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
- 54. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 Granger Causality test, 6 hours 1 2 3 4 5 6 7 8 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 q q 1 1 q q 2 2 q q 3 3 q q 4 4 q q 5 5 q q 6 6 q q 7 7 q q 8 8 q q 9 9 q q 10 10 q q 11 11 q q 12 12 q q 13 13 q q 14 14 q q 15 15 q q 16 16 q q 17 17 q q 54
- 55. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 Granger Causality test, 24 hours 1 2 3 4 5 6 7 8 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 q q 1 1 q q 2 2 q q 3 3 q q 4 4 q q 5 5 q q 6 6 q q 7 7 q q 8 8 q q 9 9 q q 10 10 q q 11 11 q q 12 12 q q 13 13 q q 14 14 q q 15 15 q q 16 16 q q 17 17 q q 55
- 56. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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 Granger Causality test, 48 hours 1 2 3 4 5 6 7 8 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 q q 1 1 q q 2 2 q q 3 3 q q 4 4 q q 5 5 q q 6 6 q q 7 7 q q 8 8 q q 9 9 q q 10 10 q q 11 11 q q 12 12 q q 13 13 q q 14 14 q q 15 15 q q 16 16 q q 17 17 q q 56
- 57. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Multivariate models : frequency versus magnitude X1,t = 1(Ti ∈ [t, t + 1), Mi ≤ s) and X2,t = 1(Ti ∈ [t, t + 1), Mi > s) i=1 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
- 58. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management Risk management issues T – Interested in computing P t=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 earthquakes on West Pacific (second half of March 10th, 2011) ; 58
- 59. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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
- 60. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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. Cabrera, B., B. (2006). Pricing catastrophic bonds for earthquakes in Mexico. Master thesis, Humboldt-Universität zu Berlin. 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
- 61. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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
- 62. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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
- 63. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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
- 64. Arthur CHARPENTIER & Mathieu BOUDREAULT, Bivariate counting processes in risk management 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