Description of earthquake aftershock sequences using prototype point patterns

Abstract

We introduce the use of prototype point patterns to characterize the behavior of a typical aftershock sequence from the global Harvard earthquake catalog. These prototypes are used not only for data description and summary but also to identify outliers and to classify sequences into groups exhibiting similar aftershock behavior. We find that a typical shallow earthquake of magnitude between 7.5 and 8.0 has five aftershocks of magnitude at least 5.5, and these aftershocks are roughly evenly distributed in logtime between 0.113 days and 2.0 years after the mainshock. The relative magnitudes and distances from the mainshock for the typical aftershock sequence are characterized as well.

Improved Similarity Measures for Small Sets of Spike Trains

“…Such ideas may sound deceptively similar to the central spike train discussed in the literature of spike train metrics. A central spike train, or prototype spike train, minimizes the sum of the squared distances to all the spike trains of the ensemble (Schoenberg & Tranbarger, 2008). The quantityν X is not a prototype spike train; it is actually not a spike train at all but rather the rescaled sum of multiple spike trains.…”

Intra-day response of foreign exchange markets after the Tohoku-Oki earthquake

“…The core factor with respect to these methods is to use a distance for marked point processes [10,12]. Previously, we showed a possibility that a series of United States dollar/Japanese yen market is of deterministic chaos [11].…”

The application of prototype point processes for the summary and description of California wildfires

“…For purely temporal point processes, as described by Victor and Purpura (1997), moving a point t 1 to a new time t 2 would be associated with a penalty p m | t 2 − t 1 |, and this moving penalty can be extended in obvious ways to the space‐time case using p m , a vector of penalties each corresponding to one coordinate of the domain (see e.g. Schoenberg and Tranbarger, 2008). Figure 1a shows an illustration of the spike‐time distance metric.…”

“…Given a collection { N 1 , …, N m } of point patterns and cost parameters p a , p d and p m , one may define the prototype , P , of the collection as the point pattern minimizing as in Schoenberg and Tranbarger (2008). Thus, the prototype is a natural measure of central tendency for collections of point patterns, analogous in many ways to the median of a collection of real numbers.…”

“…California wildfires naturally present themselves as ideal candidates for multi‐dimensional prototyping, because of the lack of agreement on parametric models for their occurrence and due to the high variability of wildfire activity from year to year and season to season, rendering conventional averages of wildfire activity poor descriptions of typical wildfire behaviour. For properties of point process prototypes, including analogies between the prototype and the median for real‐valued data, see Schoenberg and Tranbarger (2008) and Diez et al. (2011).…”