Statistical Properties of Marsan-Lengliné Estimates of Triggering Functions for Space-time Marked Point Processes

Transcription

1 Statistical Properties of Marsan-Lengliné Estimates of Triggering Functions for Space-time Marked Point Processes Eric W. Fox, Ph.D. Department of Statistics UCLA June 15, 2015

2 Hawkes-type Point Process Models in Seismology Conditional intensity function for the occurrence rate of earthquakes at (t, x, y), given all previously occurring earthquakes: λ(t, x, y H t ) = µ(x, y) + ν(t t k, x x k, y y k ; m k ) {k:t k <t} = µ(x, y) + {k:t k <t} κ(m k )g(t t k )f (x x k, y y k ) κ is the magnitude productivity function. g is PDF for aftershock times. f is PDF for aftershock locations.

3 Epidemic Type Aftershock Sequences Model (ETAS) (Ogata, 1998) λ(t, x, y H t ) = µ + κ(m k )g(t t k )f (x x k, y y k ) {k:t k <t} κ(m) = Ae α(m mc ), m > m c g(t) = (p 1)c (p 1) (t + c) p, t > 0 q 1 (q 1)d f (x, y) = (x 2 + y 2 + d) q, q > 1 π

4 Nonparametric Method Marsan and Lengliné (2008) proposed a nonparametric method for estimating the space-time Hawkes process model. EM-type algorithm that alternates between: (1) estimating branching structure, (2) estimating stationary background rate and triggering function. Method does not rely on specific parametric assumptions about the shape of the triggering function.

5 Research Goals Modify and improve Marsan s method in the following ways: Incorporate a non-stationary background rate; Add error bars to nonparametric estimates; Assume the separability of the triggering function; Perform density estimation on g(t) and f (r).

6 Estimating Branching Structure Let P be a N N lower triangular probability matrix with entries: probability earthquake i is an aftershock of j, i > j p ij = probability earthquake i is a mainshock, i = j 0, i < j P = p p 21 p p 31 p 32 p p N1 p N2 p N3 p NN Each row of P must sum to one: N p ij = 1 j=1

7 Estimating Branching Structure Given a space-time Hawkes model of seismicity: λ(t, x, y) = µ(x, y) + ν(t t k, x x k, y y k ; m k ) {k:t k <t} Entries of matrix P can be computed as: p ii = µ(x i, y i ) λ(t i, x i, y i ) p ij = ν(t i t j, x i x j, y i y j ; m j ) λ(t i, x i, y i ) for i > j

8 Estimation Algorithm 1. Initialize P (0) and set v = Estimate non-stationary background rate µ (v) (x, y) with a probability weighted variable kernel estimator. 3. Estimate triggering components κ (v) (m), g (v) (t), and h (v) (r) with probability weighted histogram estimators. 4. Update probabilities P (v+1) with estimates from steps 2 and Set v v + 1 and repeat steps 2 4 until convergence. Note: h(r) = 2πrf (r), where r = x 2 + y 2.

9 Simulation Analysis Simulated earthquakes from ETAS model with: Triggering function parameters set to mles from (Ogata, 1998) 1 : A α p c d q Non-stationary background rate µ(x, y): 1 Table 2, row 8.

10 Example: Simulated Realization of ETAS Space-time window: [0, 25000] [0, 4] [0, 6]. Aftershocks allowed to occur outside boundary. N = 5932 simulated earthquakes. y y x t

11 Simulation Analysis: Background Rate Simulated and re-estimated ETAS model 200 times.

12 Simulation Analysis: Triggering Function κ(m) g(t) h(r) m t r Figure: Estimates of the triggering components from 200 ETAS simulations (light grey lines) with 95% coverage error bars (cyan boxes). The black curves are the true values governing the simulation.

13 Application: Japan Dataset 6075 earthquakes of magnitude 4.0 or greater. Jan 2005 Dec E longitude and N latitude. Off the east coast of the Tohoku District, Japan. Latitude Longitude Latitude Time (days)

14 Application: Japan Dataset Estimated 809 mainshocks (13.3% of total).

15 Application: Japan Dataset κ^(m) g^(t) h^(r) m t (days) r (degrees) Figure: Estimate of triggering components from the Japan dataset. The grey error bars cover ±2 standard errors. The solid black curves are the parametric estimates from (Ogata, 1998) in the same region.

16 Conclusions Nonparametric method recovered the true form of a non-stationary background rate from simulated earthquake catalogues. Error bars added to the histogram estimates of the triggering function contain the true values governing the simulation. Amazingly, the nonparametric estimate from the Japan data closely agrees with a previously fit parametric ETAS model for this region.

17 Future Work Further analyze statistical properties of nonparametric estimators: asymptotic unbiasedness consistency optimal bin widths Earthquake forecasting using nonparametric methods: Collaboratory for the Study of Earthquake Predictability (CSEP) Incorporate directionality (fault information): Estimate anisotropic triggering density f (r, θ) nonparametrically θ is angle to the mainshock s fault plane

18 Citations Fox, E.W., Schoenberg, F.P., Gordon, J.S. A note on nonparametric estimates of space-time Hawkes point process models for earthquake occurrences. Annals of Applied Statistics, submitted 5/15. Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1): Marsan, D. and Lengliné, O. (2008). Extending earthquakes reach through cascading. Science, 319(5866): Ogata, Y. (1998). Space-time point-process models for earthquake occurrences. Annals of the Institute of Statistical Mathematics, 50(2):

19 Acknowledgements Joshua S. Gordon (UCLA, Statistics) Professor Frederic P. Schoenberg (UCLA, Statistics)