Download My Title

Goodness-of-fit tests for point process models for forecasting
earthquakes
Rick Paik Schoenberg, UCLA Statistics
1)
Pixel-based methods
2)
Numerical summaries, K-function
3)
Error diagrams
4)
Residuals: rescaling, thinning, superposition
5)
Comparative methods
A few examples of some commonly-used conditional intensity models in seismology:
•
Stationary (homogeneous) Poisson process: (t,x) = .
•
Inhomogeneous Poisson process: (t, x) = f(t, x). (deterministic)
•
ETAS (Epidemic-Type Aftershock Sequence, Ogata 1988, 1998):
(t, x) = (x) + ∑ g(t - ti, ||x - xi||, mi),
ti < t
where g(t, x, m) =
K exp{m}
(t+c)p (x2 + d)q
1. Pixel-based methods.
Compare N(Ai) with ∫A (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.
(Baddeley, Turner, Møller, Hazelton, 2005)
2. Numerical summaries.
a) Likelihood statistics (LR, AIC, BIC).
Log-likelihood = ∑ log(ti,xi) - ∫ (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)
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 = (ti , xi)-1.
(asympt. normal, under certain regularity conditions.)
Lw(h) = centered version = √[Kw(h)/π] - h, for R2
Model(x,y;) = (x,y) + (1- ).
h (km)
2. Numerical summaries.
a) Likelihood statistics (LR, AIC, BIC).
Log-likelihood = ∑ log(ti,xi) - ∫ (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)
3. 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)]
-- Difficult to see where model fits poorly.
4. Residuals: rescaling, thinning, superposing
Rescaling.
(Meyer 1971;Berman 1984; Merzbach &Nualart 1986; Ogata 1988; Nair 1990; Schoenberg 1999; Vere-
Jones and Schoenberg 2004):
Suppose N is simple. Rescale one coordinate: move each point
{ti, xi}
to {ti , ∫oxi (ti,x) dx}
[or to {∫oti (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.
Thinning.
(Westcott 1976):
Suppose N is simple, stationary, & ergodic.
Thinning: Suppose inf (ti ,xi) = b.
Keep each point (ti ,xi) with probability b / (ti ,xi) . Can repeat many
times --> many stationary Poisson processes (but not quite ind.!)
Superposition.
(Palm 1943):
Suppose N is simple & stationary.
Then Mk --> stationary Poisson.
Superposition: Suppose sup (t , x) = c.
Superpose N with a simulated Poisson process of rate c - (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 (ti ,xi) is small, will end up with very few
points.
Superposition: Low power if c = sup (ti ,xi) is large: most of the residual points
will be simulated.
5. Comparative methods.
-- Can consider difference (for 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.
Problem: pixel choice is arbitrary, and unequal # of pts per pixel…..
Conclusions:
* Point process model evaluation is still an unsolved problem.
* Pixel-based methods have problems: non-normality, high variability,
low power, arbitrariness in choice of pixel size. Comparing loglikelihoods within each pixel seems a bit more promising.
* Numerical summary statistics and error diagrams provide very limited
information.
* Rescaling, thinning, and superposition also have problems: can have
low power, especially when the intensity is volatile.