Mark W. Denny, Luke J. H. Hunt, Luke P. Miller, and Christopher D. G. Harley. 2009. On the prediction of extreme ecological events. Ecological Monographs 79:397–421.

Appendix C. Simulated time series.

In the absence of long-term environmental data for intertidal sites, we desire to create a long-term set of reasonably realistic hypothetical data with which we can test the accuracy of our resampling approach. Given this long-term data, we choose a 7-yr subset as an example of the type of short-term data available for the real world. We then apply our resampling method to this 7-yr subset to predict return times, which can then be compared to the return times actually “observed” in the full data series.

Second order autoregressive moving-average (ARMA) processes provide a convenient mechanism to model the random fluctuations in standardized environmental residuals. The standardized residual Δx_n,i at a particular present time i depends on both chance (in the form of a random numbers, φ) and previous values of the residual:

(C.1)

Here AR₁ and AR₂ are autoregressive constants and Δx_n,i-1 and Δx_n,i-2 are values of the residual one and two times steps before the present, respectively. MA₁ and MA₂ are moving-average constants. The random numbers φ are chosen from a standard normal distribution (mean = 0, standard deviation = 1) and each is multiplied by a constant gain, G. φ_i is the newly chosen random number, φ_i-1 is the random number chosen one time step before the present, and φ_i-2 is the random number chosen two time steps before the present. For appropriate values of the autoregressive constants, the ARMA process outlined here is asymptotically stationary; that is, after an initial interval during which the process may have a net trend, it settles into stationarity (Priestley 1981, Chapter 3).

Appropriate values of AR and MA for significant wave heights, peak wave periods, and tidal elevations were determined through analysis of the standardized residuals calculated from our empirical 7-yr data series, and are shown in Table C1. For each variable, we created a year’s worth of hourly data to allow the ARMA processes to become stationary, and then calculated 10,000 years of simulated standardized residuals. These hypothetical standardized residuals were then combined with the annual cycles of each variable measured as described in the Materials and Methods to create 10,000-year long time series of hypothetical wave height, wave period, and tidal height data. We then analyzed a random 7-yr segment of these simulated data in exactly the same fashion as for the real 7-yr time series: we determined the predicted variations (means and standard deviations) for each factor and from these predicted values we calculated the stochastic standardized residuals. Standardized residuals were resampled to create 10,000 year-long realizations of the wave environment, and these realizations were played through the model for wave stress. These 10,000 year-long records of stress provide an estimate of return time as a function of imposed stress (based on 7 yr of data), and this estimate can then be compared to the actual return times observed in the 10,000 year simulated series (Fig. 7 in the main text).

Note that we do not intend the ARMA processes used here to model the actual wave and tide data precisely. Instead, we use them solely to calculate a reasonably realistic, long-term data set with which we can assess the validity of our resampling approach.

LITERATURE CITED

Priestley, M. B. 1981. Spectral analysis and time series. Academic Press, New York, New York, USA. 890 pp.

TABLE C1. Coefficients for the autoregressive moving-average simulation.