1. Multivariate integer-valued autoregressive models
applied to earthquake counts
(Early version of paper available on arxiv.org)
Mathieu Boudreault, Arthur Charpentier
UQAM
July 2012
Mathieu Boudreault (UQAM) MINAR - Earthquakes July 2012 1 / 14

2. Motivation
In various situations, we need to model counts of various types of
events;
I Insurance: number of claims, car accidents, etc.;
I Finance: number of trades, defaults, etc.
I Earth sciences: number of storms, hurricanes, earthquakes, etc.
I Sports: number of homeruns, goals scored, games played, etc.
Many or most of these phenomena show signs of
over/underdispersion, clustering, etc. so (pure) Poisson distribution is
not adequate;
May also want to simultaneously study the joint evolution of various
types of counts;
I Insurance: car accidents and thefts over time;
I Earth sciences: earthquake/hurricane/storm counts at various locations
Also need to account for under/overdispersion, in addition to
"space"-time clustering;
Mathieu Boudreault (UQAM) MINAR - Earthquakes July 2012 2 / 14

3. Objectives
When observations are continuous, one can use Vector Autoregression
(VAR) models to investigate the basic joint dynamics of a set of
phenomena.
I Autocorrelation (creates time clustering);
I Cross autocorrelation (creates "spatial" clustering);
I Granger causality;
When observations are discrete and positive (counts), one can use
multivariate integer-valued autoregressive models (MINAR) models;
I Not much has been done on the topic;
Objectives of the paper:
I Further investigate the properties of the MINAR model of Franke
Subba & Rao (1993);
I Generalize the bivariate INAR model of Pedeli & Karlis (2011);
I Apply a bivariate Poisson INAR model to study earthquake counts;
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4. Univariate integer-valued autoregression (INAR)
Two ways to obtain autocorrelation with discrete (and positive)
variables;
I Directly apply autoregression in the counts using a thinning operator:
models known as INAR;
I Autoregression in the mean: dynamic generalized linear models
(DGLM) (see Jung & Tremayne (2010));
Binomial thinning operator:
p N = Y1 + + YN if N 6= 0, and 0 otherwise
I N is a random variable with values in N and p 2 [0, 1]
I Y1 , Y2 , are i.i.d. Bernoulli variables, independent of N, with
P(Yi = 1) = p.
I p N has a binomial distribution with parameters N and p;
INAR models (Al-Osh and Alzaid (1987) or McKenzie (1985), Du and
Li (1991)):
q
Nt = ∑ (pj Nt j ) + εt .
j =1
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5. Multivariate integer-valued autoregression (MINAR)
Multivariate (d dimensions) INAR of order 1:
2 (1 ) 3 2 3
" (1) # 2 p1,1 p1,2 ... p1,d 3 Nt 1
(2 )
Nt 1 ...
(d )
Nt 1 εt
(1 )
Nt (1 ) (2 ) (d ) (2 )
5 6 . 7 6 7
p 2,1 p 2,2 ... p 2,d Nt 1 Nt 1 ... Nt 1 εt
= 4 ..
(2 )
Nt .
. .. .
. 4 .
.
. . .. .
.
5+4 .
.
5
(d ) . . . . . . . .
Nt p d ,1 p d ,2 ... p d ,d (1 ) (2 ) (d ) (d )
Nt 1 Nt 1 ... Nt 1 εt
where the thinning operator is applied element-by-element;
Franke & Subba Rao (1993) (unpublished manuscript):
I Based upon the binomial thinning operator;
I Conditions for stationarity and properties of the conditional maximum
likelihood estimates (CMLE);
Latour (1997) (Advances in Applied Probability):
I Multivariate extension of its generalized INAR model (Steutel and van
Harn thinning operator);
I Conditions for stationarity, properties of the least squares estimates;
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6. MINAR and bivariate INAR
MINAR of order k can be written as a MINAR of order 1 (just like
VAR);
Practical application of MINAR model is complicated by estimation:
I Especially if elements of the vector εt are dependent;
I Common shock Poisson innovation process has 2d parameters for
1
example compared to the Gaussian innovation that has 2 d (d 1)
correlation parameters;
I No practical application with d > 2 has been found;
Bivariate INAR model:
(1 ) (1 ) (2 ) (1 )
Nt = p1,1 Nt 1 + p1,2 Nt 1 + εt
(2 ) (1 ) (2 ) (2 )
Nt = p2,1 Nt 1 + p2,2 Nt 1 + εt
Interpretation: current counts of type # i are a¤ected by counts of
both types at previous time period, plus a random noise e¤ect;
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7. Bivariate integer-valued autoregression (BINAR)
No practical application until Pedeli & Karlis (2011a,b):
I Important: diagonal model, i.e. p1,2 = p2,1 = 0;
I Thus, no cross sectional e¤ects;
I Poisson and negative binomial innovations;
I They derive: forecasting moments, autocorrelations, likelihood
functions in this model;
I Application to daytime and nighttime car accidents;
Mathieu Boudreault (UQAM) MINAR - Earthquakes July 2012 7 / 14

8. Theoretical contributions
We have derived the following in the multivariate model (of Franke
Subba & Rao (1993)):
I Unconditional mean (straightforward);
I Unconditional variance (matrix …xed-point algorithm);
I Autocorrelations;
I Forecasting moments;
We have derived the following additional results in the bivariate
model (binomial thinning):
I Granger causality relationships;
I Likelihood functions;
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9. Applications to earthquakes
Need to de…ne regions and time intervals to count earthquakes
I Tectonic plates (ArcGIS shape…les from the Dept of Geography, U
Colorado (Boulder));
I List of earthquakes: Advanced National Seismic System (ANSS)
Composite Earthquake Catalog;
F To circumvent issues with the database, we used tests of changes of
structure to determine best cuto¤ dates;
F 70000 M >5 earthquakes in the period 1965-2011;
F 3000 M >6 earthquakes in the period 1992-2011;
I Time intervals of 3, 6, 12, 24, 36 and 48 hours;
We analyze:
I Space-time contagion over all possible pairs (136) of tectonic plates;
I Impact of medium-size (5<M<6) and large-size (M>6) earthquakes
over time on each plate;
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10. Pairs of tectonic plates
For all pairs of tectonic plates, at all frequencies, autoregression in
time is important (very high statistical signi…cance);
I Long sequence of zeros, then mainshocks and aftershocks;
I Rate of aftershocks decreases exponentially over time (Omori’ law);
s
For 7-13% of pairs of tectonic plates, diagonal BINAR has signi…cant
better …t than independent INARs;
I Contribution of dependence in noise;
I Spatial contagion of order 0 (within h hours);
I Contiguous tectonic plates;
For 7-9% of pairs of tectonic plates, proposed BINAR has signi…cant
better …t than diagonal BINAR;
I Contribution of spatial contagion of order 1 (in time interval [h, 2h ]);
I Contiguous tectonic plates;
Conclusion: for approximately 90% of pairs, there is no signi…cant
spatial contagion for M>5 earthquakes;
I When there is, in most cases, plates are contiguous (small distance);
I Adds evidence to Parsons & Velasco (2011);
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11. Medium- and large-size earthquakes
For each tectonic plate, 2 sets of data: medium-size earthquakes
(5<M<6) and large-size earthquakes (M>6);
Space-time contagion: "space" is now magnitude;
Want to know if past M>6 earthquakes help explain current number
of (5<M<6) earthquakes and vice-versa;
Diagonal BINAR has signi…cant better …t over all three simpler
models;
Due to the (very largely documented) presence of foreshocks and
aftershocks, should expect proposed BINAR to be (statistically)
signi…cant;
I It is indeed the case for the very large majority of plates and
frequencies;
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12. Granger causality tests
Investigate direction of relationship (which one causes the other, or
both);
Pairs of tectonic plates (see next slide):
I Uni-directional causality: most common for contiguous plates (North
American causes West Paci…c, Okhotsk causes Amur);
I Bi-directional causality: Okhotsk and West Paci…c, South American
and Nasca for example;
Foreshocks and aftershocks:
I Aftershocks much more signi…cant than foreshocks (as expected);
I Foreshocks announce arrival of larger-size earthquakes;
I Foreshocks signi…cant for Okhotsk, West Paci…c, Indo-Australian,
Indo-Chinese, Philippine, South American;
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13. Granger causality matrix
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14. Conclusion
We have presented additional results regarding the multivariate INAR;
We have extended the diagonal BINAR of Pedeli & Karlis (2011a,b)
and derived other results;
Application to earthquakes:
I Pairs of tectonic plates: spatial contagion of order 1 is important for
contiguous plates (seems to further support Parsons & Velasco (2011))
I Foreshocks and aftershocks: cross autocorrelation of order 1 is
signi…cant (aftershocks especially)
Risk management (not previously shown):
I For periods following an active day, lack of spatial contagion may
seriously understate total number of events;
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