Appendix F. Estimating absolute maxima.
In our approach to predicting ecological extremes, we resample a time series of environmental factors, and each resampling provides a realization of how the environment might play out. If the time series from which we resample is finite, there must be at least one particular pattern in which we can resample the series that leads to the largest imposed stress. How can we calculate the value of this absolute maximum extreme?
It would be impractical simply to try all possible resamplings. If there are n points in the time series of each environmental variable at which we can start a block and q blocks chosen for each year-long realization, there are nq different ways in which the time series can be resampled. For example, in our analysis of hydrodynamic stress, n = 2,550 (the number of days in 7 years) and q = 13 (the number of 30-day blocks in a year), so there are 1.93 × 1044 possible resamplings, a prohibitively large number. Is there a more efficient method of determining absolute maximum hydrodynamic stress?
One’s first impulse might be to take the largest residuals in tides, wave height, and wave period and add them to the largest predicted values for these variables. Wouldn’t the stress resulting be the largest possible? There are two reasons why this approach is inappropriate. First, recall that to maintain any cross-correlation among factors when we resample standardized residuals, segments of all residuals are sampled at the same time point in the record (Fig. 3 in the main text). Thus, unless the maximum tidal residual occurs at the same time as the maximal residuals of wave-height and wave-period, these maxima can never act in concert in our resampling scheme. Similarly, unless the highest predicted tide occurs at the same time as the highest predicted wave height and longest predicted wave period, the extreme residuals in each variable can never be applied simultaneously to their respective extreme predicted values. Second, the maximum values of each variable do not necessarily combine to give the maximum output. As noted in the text, hydrodynamic stress is not a monotonic function of wave period: at certain tide heights, stress decreases as T increases above some intermediate value. As a result, if we combine maximum residuals and maximum predicted values for Hs, T, and tidal height from our HMS data, we predict an absolute maximum stress that is actually smaller than some measured values.
Direct calculation of absolute maximal body temperature is even more difficult than direct estimation of absolute maximal wave stress. Body temperature depends not only on the instantaneous values of air temperature, wind speed, solar irradiance, etc., but also on their history. Thus, even if one could line up the extreme residuals for each variable (keeping in mind that it is the minimum wave height, tide, and wind speed that lead to the highest temperatures), and in turn line these up with the extreme expected values, the resulting body temperature would not necessarily be the maximum possible. Only if the conditions preceding these extreme circumstances provided sufficient time for the substratum to heat up would a maximal body temperature be achieved. In sum, the absolute maximum cannot be calculated by simply toting up the various individual extremes.
However, absolute maximum stress can be estimated using the statistics of extremes (Gaines and Denny 1993, Denny and Gaines 2000, Coles 2001, Katz et al. 2005). Our resampling method produces an ensemble of yearlong hypothetical environmental realizations, each of which can be processed to predict the maximum value of a biological relevant parameter. We can use the distribution of these predicted annual maxima to test for the existence of an absolute largest annual maximum. In theory, the cumulative distribution of our predicted annual maxima should asymptotically approach a member of the family of generalized extreme value distributions (GEV) described by Eq. 14 in the main text. This asymptotic distribution is estimated by fitting the “observed” distribution of annual maxima (i.e., the distribution of stress values calculated using our resampling technique) to Eq. 14 using a maximum likelihood criterion. If the best estimate of coefficients α and β are both positive, their ratio is an estimate of the absolute maximal value the distribution can attain. For example, the distribution of annual maximum hydrodynamic stresses for D = 1 m is best fit with values (±95% confidence limits) of α = 4512 N m-2 (±842) and β = 0.09952 (±0.02811). The ratio of α to β (our best estimate of the absolute maximum) is 4.53 × 104 N m-2, only slightly above the maximum of 3.88 × 104 N m-2 recorded in the 10,000 years of our realizations. Given the limited statistical power available from our 1000 year-long realizations of limpet body temperature, the best-fit value of β (although nearly always positive) cannot be statistically distinguished from 0 for any substratum orientation, so a reliable estimate of absolute maximum body temperature cannot be calculated from our data.
The ability of univariate extreme-value analysis to estimate absolute maximum hydrodynamic stress or body temperature depends on the ability of our bootstrap method to provide information about the distribution of these extremes. There are cases in which bootstrap resampling is known to fail in this task. The classic example is the case in which one attempts to estimate the distribution of maximum values drawn from a uniform distribution extending from 0 to 1 (Bickell and Freeedman , Efron and Tibshirani [1993, pg 81]). A nonparametric bootstrap does not provide a good estimate of the shape of the distribution of sample maxima.
The difficulty with the bootstrap procedure in this case arises from the limited size of the empirical sample from which bootstrap estimates are drawn. If one takes 50 samples from the uniform distribution (as Efron and Tibshirani do in their example), there are likely to be relatively few samples near the upper end of the distribution. As a consequence, no matter how many times one resamples these 50 measurements, little information is available about the shape of the distribution of maxima.
However, this example differs from the situation we encounter in our analysis of hydrodynamic forces and body temperatures. Because we have only 7 years of empirical data, when we choose a bootstrap sample from the time series of standardized residuals we have limited information available by which to judge the shape of the upper end of its distribution. But it is not the extremes of the standardized residuals that govern the ecological consequences, rather it is how the residuals combine with their respective mean annual cycles and how the resulting variables interact. Thus, while we have limited data in the upper end of the distributions of individual maximum standardized residuals, we have a very large number of combinations of residuals with their annual cycles and of the resulting variables with each other. It is these combinations (rather than the extremes of individual variables) that determines the distribution of ecological maxima.
Bickel, P. J., and D. A. Freedman. 1981. Some asymptotic theory for the bootstrap. The Annals of Statistics. 9:11961217.
Coles, S. T. 2001. An introduction to statistical modeling of extreme values. Springer-Verlag. London, UK, 208 pp.
Denny, M. W., and S. Gaines. 2000. Chance in biology. Princeton University Press, Princeton, New Jersey, USA, 291 pp.
Efron, B., and R. J. Tibshirani. 1993. An introduction to the bootstrap. Chapman and Hall/CRC Boca Raton, Florida, USA, 436 pp.
Gaines, S. D., and M. W. Denny. 1993. The largest, smallest, highest, lowest, longest and shortest: extremes in ecology. Ecology 74:16771692.
Katz, R. W., G. S. Brush, and M. B. Parlange. 2005. Statistics of extremes: Modeling ecological disturbances. Ecology 86:11241134.