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 Δxn,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:
Here AR1 and AR2 are autoregressive constants and Δxn,i-1 and Δxn,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. φi is the newly chosen random number, φi-1 is the random number chosen one time step before the present, and φi-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.
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.
|Significant wave height||Peak wave period||Tidal height|