Ecological Archives M074-011-A8

Mark W. Denny, Brian Helmunth, George H. Leonard, Christopher D. G. Harley, Luke J. H. Hunt, and Elizabeth K. Nelson. 2004. Quantifying scale in ecology: lessons from a wave-swept shore. Ecological Monographs 74:513–532.

Appendix H. Making Measurements in a 1/f-Noise Environment.

The type of variation exhibited by our spatial variables (the larger the scale of measurement, the larger the associated variance) raises several practical issues, two of which we examine here.

(1) We propose in the text that the measured spectrum of a phenomenon can be used in conjunction with a defined, biological measurement scale to quantify the variance, ,encompassed by that scale (Fig. 8). This approach is particularly useful if there is an inflection point (a "sill") in this curve of vs. measurement scale (Fig. 3B). Indeed, this inflection point was the basis for defining the variance scale (Eq. 7, B.10). One might assume that 1/f-noise processes with b 1 would encounter problems with this approach. If the variance increases without limit as the spatial frequency gets smaller (as it does in this case), won't the standard deviation increase problematically fast as the measurement scale increases, jeopardizing our ability to define the variance scale?

In fact, the situation is not all that bad. Consider, for instance, the spatial pattern of primary productivity, which appears to be a 1/f-noise process with a of 1.36 (Table 3). Despite the inverse relationship between variance and spatial frequency, the standard deviation of maximum force is most sensitive to changes in scale at small scales and a discrete v-scale can be discerned (Fig. H1A).  The larger is, the more sensitive the standard deviation becomes to changes in measurement scale at large scales (Fig. H1B), and therefore, the weaker our estimate of the v-scale. But for variables we have encountered (0 < < 1.5) this behavior does not pose a serious problem for the estimate of scale.

(2) In this report we have concentrated on measurements of the variation associated with ecological processes and the physical environment. Although variation is undoubtedly important, there are times where it is not the central issue. For example, one might want to explore the effect of wave exposure on the rate at which snails prey on mussels. Two categories of sites could be chosen in which to measure predation intensity, those on exposed shores, and the others on protected, shores. It would then be traditional to characterize the exposure of each site by measuring some appropriate index (e.g., maximum wave force) at several points within the site, and calculating the mean. Significant differences between the mean exposures of the two categories of sites would validate the assumption that they do, indeed, represent different experimental treatments. However, our ability to discern differences in means is affected by the standard error of the mean (SEM). Given the type of spatial variation found in our study, how are the mean and its standard error affected?

We approach this question by making the following calculation for each of our spatial variables. Each series of data with k number of points was divided into a set of contiguous segments, each of length n (n = 2,3,...k/3). The limit of k/3 is set so that we have at least three segments, as required for the calculations described below.The average value of the data within each segment was computed. These averages were themselves then averaged, and the standard deviation of the averages (by definition, the standard error of the segment mean) was calculated. For each value of n, the extent of the measurement was assumed to be (n - 1)x, where x is the spacing between locations (or times) at which the measurements were taken. A plot of the average of averages and the standard error as a function of the extent of measurement then provides the information we seek.

In general, the extent of measurement has little practical effect on the mean itself. In 12 of 25 cases there is a statistically significant trend in the measured value of the mean as a function of extent (9 with a negative slope, 3 with a positive slope), however, the trend is slight (Table H1). In only two cases is the relative slope (the slope divided by the overall mean) greater than 1%, and on average the relative slope is only 0.3%.

As one might expect, the standard error of the mean decreases with increasing measurement scale (e.g., Fig. H2). However, the autocorrelation of data within each series results in a rate of decrease that is less than one might expect. If data are independent (for instance, if = 0), at a given scale of measurement the standard error of the mean varies as n-1/2 where n (the number of measurements) is in this case proportional to the extent of the data series. If fact, due to autocorrelation, the standard error decreases more slowly, varying on average ( ± SD) as  for the spatial variables measured here.

These results suggest that, despite the 1/f-noise pattern of variation in our spatial variables, there seems to be no unusual problem in quantifying the mean. As might be expected, the more samples one takes (in this case, implying a larger extent of measurement), the greater the confidence in the resulting average.

We emphasize, however, that the ability to accurately measure a mean and the ability to interpret that mean, may not be the same. For example, on our short transect (with a length of 44 m) the average maximum wave force is 44 N when the offshore significant wave height is 2 m. However, within this short distance the largest force (157 N) exceeds the smallest (15 N) by a factor of greater than 10. Given this large amount of variation in such a short stretch of shore, how useful is the mean maximum force as an index of wave exposure?

Table H1. Scale dependence of the mean.

 

Scale dependence of the mean

 

Units

Mean

Slope/mean

Probability

Force

       

    short

1/m

41.17

-0.0054

0.000

    medium

1/m

42.45

ns

0.048

    long

1/m

37.73

ns

0.939

         

Wave force index

       

    short

1/m

0.575

-0.0032

0.000

    medium

1/m

0.551

ns

0.061

    long

1/m

0.548

ns

0.016

         

Temperature

       

    short

1/m

0.765

ns

0.720

    medium

1/m

0.793

0.0022

0.000

         

Chlorophyll

       

    medium

1/m

0.976

ns

0.195

         

Diversity

       

    short

1/m

0.576

ns

0.101

    medium

1/m

0.5898

ns

0.049

    long

1/m

0.557

0.000187

0.000

         

Mussel density

       

    short

1/m

0.177

-0.01102

0.000

    medium

1/m

0.217

-0.00231

0.000

    long

1/m

0.149

ns

0.933

         

Mussel disturbance

       

    short

1/m

0.027

-0.01285

0.000

    medium

1/m

0.023

-0.00254

0.000

    long

1/m

0.007

-0.0011

0.000

         

Mussel recruitment

       

    medium

1/m

14.51

ns

0.027

         

Predators

       

    short

1/m

0.103

ns

0.259

    medium

1/m

0.128

-0.00258

0.000

    long

1/m

0.067

ns

0.024

         

Grazers

       

    short

1/m

3.4

-0.00406

0.000

    medium

1/m

3.02

ns

0.003

    long

1/m

2.62

0.00068

0.000

   Notes: Significance level is set at 0.002 (0.05 with a Bonferroni adjustment for multiple comparisons). ns = not significant.

 

 
   FIG. H1. A. Although its > 1, the log-log curve of standard deviation versus measurement extent for microalgal productivity exhibits a clear inflection point, thereby allowing the variance scale to be measured. B. Standard deviation as a function of measurement extent for 1/f-noise processes with various s. Unless approaches 2, it is feasible to use the approximately linear regions of these curves at small and large measurement extent to unambiguously define a variance scale.

 

 
   FIG. H2. The standard error of the mean maximum force decreases with the extent of measurement.



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