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File S2: Wavelet analysis of total abundance and species richness for fish communities
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associated with different shoreline types.
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We performed wavelet analysis to determine how temporal fluctuations in total abundance and
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species richness varied in fish communities associated with different types of shorelines. We
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now provide a summary of these methods and direct the reader to previously published guides
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for greater details (e.g., Torrence and Compo 1998, Cazelles et al. 2008, Iles et al. 2012).
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Wavelets are flexible functions that are resolved in the time and frequency domains, and thus
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ideal for quantifying changes in the contribution of each period (or frequency) over time to the
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overall variance (or power) of a signal (e.g., a time series). Because wavelets can be scaled (i.e.,
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contracted or dilated), they can efficiently accommodate both high and low frequency structures
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in non-stationary signals whose statistical characteristics vary over time. For our analyses, we
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used the Morlet wavelet, which is defined as:
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y 0 (t) = p -1/4eiw t e-t
0
2
/2
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where i is the imaginary number, t represents nondimensional time, and w 0 = 6 is the
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nondimensional frequency (Torrence and Compo 1998). The continuous wavelet transform of a
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discrete time series x(t) with equal spacing d t and length T is defined as the convolution of
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x(t) with a normalized Morlet wavelet (Torrence and Compo 1998, Grinsted et al. 2004):
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Wx ( s, t ) =
d t T -1
å x(t)y
s
t=0
0
æ (t - t )d t ö
*ç
÷ø
è
s
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where * indicates the complex conjugate. By varying the wavelet scale s (i.e., dilating and
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contracting the wavelet) and translating along localized time position t , one can calculate the
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wavelet coefficients Wx (s, t ) , which describe the contribution of the scales s to the time series
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x(t) at different time positions
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is a parameter used to normalize the Morlet wavelet function to unit variance in order to allow
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direct comparisons of the wavelet coefficients Wx ( s, t ) across the different scales s and time
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positions t (Torrence and Compo 1998, Grinsted et al. 2004). Contour plots can be used to
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visualize how the local wavelet power spectrum (i.e., the contribution of each frequency or
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period in the time series) varies in time (Grinsted et al. 2004, Cazelles et al. 2008).
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t (Torrence and Compo 1998, Cazelles et al. 2008). Here,
Additionally, one can compute the bias-corrected global wavelet spectrum Wx2 (s) as the
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time-average (i.e., over all time locations t ) of all local wavelet spectra for each scale s
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(Torrence and Compo 1998, Liu et al. 2007, Cazelles et al. 2008):
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æ s 2 T -1
2ö
Wx2 (s) = 2 s ç x å Wx ( s, t ) ÷
è T t =0
ø
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dt
s
where s x2 represents the variance of the time series and 2 s is the bias correction factor (Liu et
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al. 2007). The global wavelet spectrum thus quantifies the average power (or variance) of the
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time series at each scale. The scale s of the Morlet wavelet is related to Fourier frequency f
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(Maraun and Kurths 2004, Cazelles et al. 2008):
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is approximately equal to the reciprocal of the Fourier frequency f (Maraun and Kurths 2004,
1
4p s
. When w 0 = 6 , the scale s
=
f w 0 + 2 + w 02
1
. Hence, in all equations, the scale s can be converted to Fourier
f
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Cazelles et al. 2008): s »
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frequency f »
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Zero-padding and the cone of influence
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In practice, the continuous wavelet transform is computed by using discrete Fourier transforms to
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calculate all T convolutions simultaneously (Torrence and Compo 1998). However, since the
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Fourier transform assumes that the data is periodic, errors in the estimation of the local wavelet
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power spectrum will occur at the beginning and at the end of any finite-length time series
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(Torrence and Compo 1998, Cazelles et al. 2008). In order to limit these edge effects, the end of
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a time series is padded with zeros prior to taking the wavelet transform and the zeroes are then
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removed (Torrence and Compo 1998, Cazelles et al. 2008). Typically, enough zeros are added in
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order for the total length T of the time series to reach the next-higher power of two. This both
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limits edge effects and improves the speed of the Fourier transform (Torrence and Compo 1998).
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1
1
or period p = » s .
f
s
Although padding with zeros limits errors due to edge effects, it introduces artificial
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discontinuities at the endpoints of the data (Torrence and Compo 1998, Cazelles et al. 2008). As
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one gets closer to the end of the data, more zeros are included in the estimation of the local
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wavelet spectrum, thus reducing its reliability (Torrence and Compo 1998, Cazelles et al. 2008).
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The region where zero padding affects the estimation of the wavelet spectrum is called the cone
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of influence (COI), and is defined as the region in which the wavelet power for a discontinuity at
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the edge drops by a factor of e -2 (Torrence and Compo 1998). Hence, any region falling below
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the COI is susceptible to edge effects.
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Statistical significance testing
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In order to determine the statistical significance of the wavelet spectrum obtained from a time
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series, one must first formulate an appropriate null hypothesis. Here, the null hypothesis is that
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the observed time series is generated by a stationary process with a given background power
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spectrum P(k) (Torrence and Compo 1998, Grinsted et al. 2004). Since many ecological and
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environmental time series exhibit strong temporal autocorrelation (i.e. high power associated
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with low frequencies; e.g. Beninca et al. 2009, see Ruokolainen et al. 2009 for review), we used
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a first order autoregressive model [AR(1)] to generate a temporally autocorrelated time series or
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red noise, which served as our null hypothesis. Specifically, the power spectrum P(k) of our red
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noise process was calculated with (Gilman et al. 1963):
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1- a 2
P(k) =
1+ a 2 - 2a cos ( 2p k / N )
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where the autocorrelation coefficient a at time lag 1 is estimated from the observed time
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series and k = 0,..., N / 2 represents the frequency index. The observed wavelet spectrum can be
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compared to the wavelet spectrum of the red noise process by means of a chi-square test. The
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distribution of the local wavelet power spectrum of a red noise process is (Torrence and Compo
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1998):
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Wx (s, t )
s
2
2
1
Þ P(k) c 22
2
where k represents the frequency index, s 2 represents the variance of the time series, “ Þ ”
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means “is distributed as”, and c 22 represents the chi-square distribution with 2 degrees of
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freedom. The value of P(k) is the mean wavelet power spectrum at frequency k that
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corresponds to the wavelet scale s (Torrence and Compo 1998). Using this equation, one can
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construct 95% confidence contour lines at each scale using the 95th percentile of the chi-square
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distribution c 22 (Torrence and Compo 1998).
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Literature Cited
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oscillations in a chaotic food web. Ecology Letters 12:1–12.
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Gilman, D. L., F. J. Fuglister, and J. M. Mitchell. 1963. On the Power Spectrum of “Red Noise”.
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Spectrum. Journal of Atmospheric and Oceanic Technology 24:2093–2102.
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Nonlin. Processes Geophys. 11:505–514.
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Figure legends
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Figure A: Wavelet analysis of monthly time series of total fish abundance in communities
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associated with shorelines characterized as (a) natural, (b) rubble with riprap, (c) vertical wall,
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and (d) vertical wall with riprap. Regions of high (low) power or variability are represented in
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warm (cold) colors. Black contours represent regions of statistically significant variability at the
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=0.05 level. Regions within the white dashed lines (the cone of influence) are not affected by
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edge effects.
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Figure B: Wavelet analysis of monthly time series of species richness in communities associated
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with shorelines characterized as (a) natural, (b) rubble with riprap, (c) vertical wall, and (d)
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vertical wall with riprap. Regions of high (low) power or variability are represented in warm
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(cold) colors. Black contours represent regions of statistically significant variability at the
125
=0.05 level. Regions within the white dashed lines (the cone of influence) are not affected by
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edge effects.
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Figure A
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Figure B