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.
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FIG.
H1. A. Although its ![]() ![]() ![]() |
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FIG. H2. The standard error of the mean maximum force decreases with the extent of measurement. |