Dependence between two records of events may be addressed using the bivariate form of the empirical K-function (Ripley 1977), introduced for this purpose by Doss (1989). In contrast to other methods for testing for synchrony (Buonaccorsi et al. 2001), this approach is suitable for events in time (temporal point processes) and thus avoids information loss from merging events into bins or smoothing records into time series of event frequency, and it does not assume any periodic structure in the data. This test only addresses dependence between the two records and preserves the first-order properties (event frequency) of each record in each randomization (Wiegand and Moloney 2004).
The K-function gives the number of events in record B occurring within ± t yr (the “temporal window”) of each event in record A, and scaled by T/(nAnB) where T is the length of the record and nA and nB are the number of events in A and B respectively. The bivariate K-function for a single dimension is expressed as:
where Ai and Bj are times of events and I is the identity function (Doss 1989). Values of K greater than 2t suggest attraction, or synchrony, between A and B, within a window of t yr. Values of K close to 2t suggest no relationship or independence between A and B, and values less than 2t suggest repulsion, or asynchrony.
We made two modifications to the empirical K-function presented in Doss (1989). First, an edge correction was used to make the function unbiased up to windows of 0.5T
where w(Ai,Bj) is a “mirror” edge correction, set to 2 if |Ai - Bj| is greater than the distance of Ai to the nearest “edge” of the record, otherwise it is set to 1. Second, because the edge correction causes 
to differ slightly from 
(i.e., whether distances are measured from A to B or vice versa), and were averaged by weighting each to produce the unbiased estimate 
(Lotwick and Silverman 1982):
For graphing purposes, the K-function is easier to interpret if its mean and variance are stabilized over t as expressed by the L-function: 
. We constructed 95% confidence envelopes for 
from 1000 randomizations in which one record was shifted a random number of years and events wrapped from the end to the start of the record. Where is greater than, within, or less than the confidence envelope indicates statistically significant synchrony, independence, or statistically significant asynchrony, respectively, in a window of t yr.
|
| FIG. B1. Demonstration of the bivariate K-function for testing synchrony over a range of temporal windows. (a) Two records, where in the first a series of events are placed on random years and the second events are placed within 50 yr of those in the first record. (b) The L-function (transform of the K-function) for the events in (a) with 95% confidence envelope (thin lines) based on 1000 randomizations. The function exceeds the upper confidence envelope from 25 to 150 yr, indicating strong correlation of event times within windows of that scale. A lack of long-term patterns in both records results in no synchrony in larger windows. (c) Two records where the number of events decreases by one each millennium. (d) The L-function for the events in (c). The function exceeds the confidence envelope at several window sizes between 500 and 1700 yr, indicating the millennial-scale pattern in common between the two records. |
LITERATURE CITED
Buonaccorsi, J. P., J. S. Elkinton, S. R. Evans, and A. M. Liebhold. 2001. Measuring and testing for spatial synchrony. Ecology 82:1668–1679.
Doss, H. 1989. On estimating the dependence between two point processes. The Annals of Statistics 17:749–763.
Lotwick, H. W., and B. W. Silverman. 1982. Methods for analysing spatial processes of several types of points. Journal of the Royal Statistical Society B44:406–413.
Ripley, B. D. 1977. Modelling spatial patterns. Journal of the Royal Statistical Society B39:172–212.
Wiegand, T., and K. A. Moloney. 2004. Rings, circles, and null-models for point pattern analysis in ecology. Oikos 104:209–229.