1. Rick Paik Schoenberg, UCLA Statistics
A survey of goodness-of-fit tests for point process models for earthquake occurrences
Rick Paik Schoenberg, UCLA Statistics
Some point process models in seismology
Pixel-based methods
Numerical summaries
Error diagrams
Residuals: rescaling, thinning, superposition
Comparative methods, tessellation residuals
Example

2. N(A) = # of points in the set A. S = [0 , T] x X.
Some point process models in seismology.
Point process: random (s-finite) collection of points in some space, S.
N(A) = # of points in the set A S = [0 , T] x X.
Simple: No two points at the same time (with probability one).
Conditional intensity: l(t,x) = limDt,Dx -> 0 E{N([t, t+Dt) x Bx,Dx) | Ht} / [DtDx].
Ht = history of N for all times before t, Bx,Dx = ball around x of size Dx.
* A simple point process is uniquely characterized by l(t,x). (Fishman & Snyder 1976)
Poisson process: l(t,x) doesn’t depend on Ht.
N(A1), N(A2), … , N(Ak) are independent for disjoint Ai, and each Poisson.

3. Some cluster models of clustering:
Neyman-Scott process: clusters of points whose centers are formed from a stationary Poisson process. Typically each cluster consists of a fixed integer k of points which are placed uniformly and independently within a ball of radius r around each cluster’s center.
Cox-Matern process: cluster sizes are random: independent and identically distributed Poisson random variables.
Thomas process: cluster sizes are Poisson, and the points in each cluster are distributed independently and isotropically according to a Gaussian distribution.
Hawkes (self-exciting) process: parents are formed from a stationary Poisson process, and each produces a cluster of offspring points, and each of them produces a cluster of further offspring points, etc l(t, x) = m + ∑ g(t-ti, ||x-xi||).
ti < t

4. Aftershock activity typically follows the modified Omori law (Utsu 1971):
g(t) = K/(t+c)p.

5. Stationary (homogeneous) Poisson process: l(t,x) = m.
Commonly used in seismology:
Stationary (homogeneous) Poisson process: l(t,x) = m.
Inhomogeneous Poisson process: l(t, x) = f(t, x). (deterministic)
ETAS (Epidemic-Type Aftershock Sequence, Ogata 1988, 1998):
l(t, x) = m(x) + ∑ g(t - ti, ||x - xi||, mi),
ti < t
where g(t, x, m) = K exp{am}
(t+c)p (x2 + d)q

6. 2. Pixel-based methods. Compare N(Ai) with ∫A l(t, x) dt dx, on pixels Ai. (Baddeley, Turner, Møller, Hazelton, 2005) Problems: * If pixels are large, lose power. * If pixels are small, residuals are mostly ~ 0,1. * Smoothing reveals only gross features.

7. (Baddeley, Turner, Møller, Hazelton, 2005)

8. 3. Numerical summaries. a) Likelihood statistics (LR, AIC, BIC)
3. Numerical summaries. a) Likelihood statistics (LR, AIC, BIC). Log-likelihood = ∑ logl(ti,xi) - ∫ l(t,x) dt dx. b) Second-order statistics * K-function, L-function (Ripley, 1977) * Weighted K-function (Baddeley, Møller and Waagepetersen 2002, Veen and Schoenberg 2005) * Other weighted 2nd-order statistics: R/S statistic, correlation integral, fractal dimension (Adelfio and Schoenberg, 2009)

10. Weighted K-function
Usual K-function:
K(h) ~ ∑∑i≠j I(|xi - xj| ≤ h),
Weight each pair of points according to the estimated intensity at the points:
Kw(h)^~ ∑∑i≠j wi wj I(|xi - xj| ≤ h),
where wi = l(ti , xi)-1.
(asympt. normal, under certain regularity conditions.)
Lw(h) = centered version = √[Kw(h)/π] - h, for R2

11. Model: l(x,y;a) = a m(x,y) + (1- a)n.
h (km)

12. 3. Numerical summaries. a) Likelihood statistics (LR, AIC, BIC)
3. Numerical summaries. a) Likelihood statistics (LR, AIC, BIC). Log-likelihood = ∑ logl(ti,xi) - ∫ l(t,x) dt dx. b) Second-order statistics * K-function, L-function (Ripley, 1977) * Weighted K-function (Baddeley, Møller and Waagepetersen 2002, Veen and Schoenberg 2005) * Other weighted 2nd-order statistics: R/S statistic, correlation integral, fractal dimension (Adelfio and Schoenberg, 2009) c) Other test statistics (mostly vs. stationary Poisson) TTT, Khamaladze (Andersen et al. 1993) Cramèr-von Mises, K-S test (Heinrich 1991) Higher moment and spectral tests (Davies 1977) Problems: -- Overly simplistic Stationary Poisson not a good null hypothesis (Stark 1997)

13. 4. Error Diagrams Plot (normalized) number of alarms vs
4. Error Diagrams Plot (normalized) number of alarms vs. (normalized) number of false negatives (failures to predict). (Molchan 1990; Molchan 1997; Zaliapin & Molchan 2004; Kagan 2009). Similar to ROC curves (Swets 1973). Problems: Must focus near axes. [consider relative to given model (Kagan 2009)] -- Does not suggest where model fits poorly.

14. Suppose N is simple. Rescale one coordinate: move each point
5. Residuals: rescaling, thinning, superposing
Rescaling. (Meyer 1971; Merzbach & Nualart 1986; Nair 1990; Schoenberg 1999; Vere-Jones and Schoenberg 2004):
Suppose N is simple. Rescale one coordinate: move each point
{ti, xi} to {ti , ∫oxi l(ti,x) dx} [or to {∫oti l(t,xi) dt), xi }].
Then the resulting process is stationary Poisson.
Problems:
* Irregular boundary, plotting.
* Points in transformed space hard to interpret.
* For highly clustered processes: boundary effects, loss of power.

15. Thinning. (Westcott 1976): Suppose N is simple, stationary, & ergodic.

16. Thinning: Suppose inf l(ti ,xi) = b.
Keep each point (ti ,xi) with probability b / l(ti ,xi) . Can repeat many times --> many stationary Poisson processes (but not quite ind.!)

17. Then Mk --> stationary Poisson.
Superposition. (Palm 1943): Suppose N is simple & stationary.
Then Mk --> stationary Poisson.

18. Superposition: Suppose sup l(t , x) = c.
Superpose N with a simulated Poisson process of rate c - l(t , x) .
As with thinning, can repeat many times to generate many (non-independent) stationary Poisson processes.
Problems with thinning and superposition:
Thinning: Low power. If b = inf l(ti ,xi) is small, will end up with very few points.
Superposition: Low power if c = sup l(ti ,xi) is large: most of the residual points
will be simulated.

19. -- Better: consider difference between log-likelihoods, in each pixel.
6. Comparative methods, tessellation.
-- Can consider difference (between competing models) between residuals over each pixel.
Problem: Hard to interpret. If difference = 3, is this because model A overestimated by 3? Or because model B underestimated by 3? Or because model A overestimated by 1 and model B underestimated by 2?
Also, when aggregating over pixels, it is possible that a model will predict the correct number of earthquakes, but at the wrong locations and times.
-- Better: consider difference between log-likelihoods, in each pixel.
(Wong & Schoenberg 2009).
Problem: pixel choice is arbitrary, and unequal # of pts per pixel…..

20. Problem: pixel choice is arbitrary, and unequal # of pts per pixel…..
-- Alternative: use the Voronoi tessellation of the points as cells.
Cell i = {All locations closer to point (xi,yi) than to any other point (xj,yj) }.
Now 1 point per cell.
If l is locally constant, then cell area ~ Gamma (Hinde and Miles 1980)

23. 8. Example: using focal mechanisms in ETAS

24. In ETAS (Ogata 1998), l(t,x,m) = f(m)[m(x) + ∑i g(t-ti, x-xi, mi)],
where f(m) is exponential, m(x) is estimated by kernel smoothing,
and
i.e. the spatial triggering component, in polar coordinates,
has the form:
g(r, q) = (r2 + d)q .
Looking at inter-event distances in Southern California, as a
function of the direction qi of the principal axis of the prior event,
suggests: g(r, q; qi) = g1(r) g2(q - qi | r),
where g1 is the tapered Pareto distribution,
and g2 is the wrapped exponential.

25. ETAS: no use of focal mechanisms.
Summary of principal direction of motion in an earthquake, as well as resulting stress changes and tension/pressure axes.

30. tapered Pareto / wrapped exp. 
biv. normal (Ogata 1998) 
Cauchy/ ellipsoidal (Kagan 1996) 

31. Thinned residuals: Data  tapered Pareto / wrapped exp. 
Cauchy/ ellipsoidal (Kagan 1996)  biv. normal (Ogata 1998) 

32. Tapered pareto / wrapped exp. Cauchy / ellipsoidal