1
Continuity and Contrast: parameters in neutral point-pattern landscapes.
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Spectral density has often been used for time series and lattice data (for relevant references, see
3
Mugglestone & Renshaw 1996). Here, we use a spectral representation to calculate two measures
4
that capture essential characteristics in point-pattern landscapes. The basis of the spectral method
5
follows from the Fourier theorem, which states that continuous time series, or surfaces and higher
6
dimensions, can be represented by a combination of sine and cosine terms with different
7
frequencies and amplitudes. Thereby, analysis of the time series, or surfaces, can be performed
8
on the sine and cosine terms. From physics, we use the notation power spectral density function
9
(PSDF) for this representation of a time serie or surface by the amplitudes of sine and cosine
10
terms of different frequency.
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Continuity
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Large values of Continuity (γ) in point patterns reflect that nearby areas have similar density, and
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hence γ is a measure of autocorrelation. It is defined from the spectral representation of the
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landscape such that if the spectral density is dominated by the amplitudes of low frequencies,
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then there is a higher degree of autocorrelation. This is a point-pattern measure, akin to a two-
16
dimensional measure of 1/f noise (Halley et al. 2004, Keitt 2000). Thus a higher value of
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Continuity is analogous to a redder 1/f noise. As is done when determining the colour of 1/f
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noise, we analyse the autocorrelation as a linear regression fitted to the log(frequency) vs.
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log(amplitude) and thereby also infer self-similarity. The slope of the regression line is the
20
parameter Continuity (γ). In our analysis, we assume isotropy, which implies identical patterns in
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all directions. Hence, we can collapse the spectral analysis into a one-dimensional assessment
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(i.e. a regression line instead of a surface) after the Fourier transform of the two-dimensional
23
point pattern. To calculate the PSDF of the point pattern distribution, denoted PSDFpp, we use the
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efficient method that was developed by Mugglestone and Renshaw (1996):
2
25

 

PSDFPP    cos(C p xi )     sin( Cq yi ) 
 i
  i

2
(A.1)
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where (μp,μq) are frequencies for Fourier transform of grid data, (xi,yi) are coordinates in the point
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pattern, and C is a constant that determines the number of frequencies. Mugglestone and
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Renshaw stated that the number of frequencies used should not exceed the number of points in
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the point pattern, Np, in order to keep them independent. Therefore, given a quadratic landscape
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we chose C to be
31
number of components in the two dimensional spectral representation.
N p 2 rounded down to the nearest integer, and hence nF =2C×2C is the
32
33
Contrast
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As a point-pattern equivalent of coefficient of variation, we introduce Contrast,  , to measure
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the difference in density between sparse and dense areas. To formulate an equation for that
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measure, we first have to relate it to how variance in the continuous case can be measured using
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spectral representation. Note that when time series and surfaces are represented by sine and
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cosine terms after the Fourier transform, it is the amplitudes of the frequencies that determine the
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variance. If M is the number of components in the PSDF, the variance,  2 , is calculated as
2 
40
1
mean( PSDF  PSDF (origin ))
M
(A.2)
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The mean is represented by the amplitude in the origin1 of the spectral representation,
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PSDF(origin), and hence the coefficient of variation (standard deviation divided by the mean) is
CV 
43
1
1
mean( PSDF  PSDF (origin ))
PSDF (origin ) M
44
Substituting PSDF with PSDFpp from equation A.1 where nF is the number of spectral
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components we end up with an equation of the point-pattern measure Contrast (  ):
CVPP 
46
47
1
1
mean( PSDFPP  PSDFpp (origin ))
PSDFPP (origin ) nF
(A.3)
(A.4)
Generating neutral point-pattern landscapes
1
Note that if implemented in Matlab using built-in functions of Fourier transform, for example
the fft function, the amplitude in the PSDF(origin), represents the sum instead of the mean.
48
To obtain neutral point-pattern landscapes (NPPLs) with given characteristics, we generated
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lattice landscapes of size m×m. The density defines the probability of a point in the landscapes.
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We first generated two-dimensional 1/f-noise (denoted LG) using a method similar to that
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presented by Halley et al. (2004), and hence this is still a representation of a lattice landscape and
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not a point pattern. The values in LG are normally distributed and thus not suitable for describing
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probabilities, because they may include negative values. Although it might be possible to solve
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this by truncating, we found that that would not allow for generating sufficiently high values of δ.
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Therefore, we transformed LG using spectral mimicry. This method was introduced by Cohen et
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al. (1999) and has been employed to construct stationary time series with different Fourier
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spectra. Cohen et al. mainly used their technique to achieve transformation to normally
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distributed values with specific mean and variance. We instead transformed LG to LΓ using a
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gamma distribution (which contains no values <0) with mean=1/m2 and a coefficient of variation
60
δL. Point locations were distributed according to the probabilities given by LΓ.
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The γ and δ of the spectral point pattern are determined by γL and δL, but they are altered by the
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distribution of points as this adds some randomness. Accordingly, we measured these quantities
63
in the generated point pattern landscape (see the method described above). The relationship
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between the spectral point-pattern values of γ and δ used in our study and the γL and δL required
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to generate them was found iteratively. Furthermore, we observed that the linear relationship in
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the power spectra was maintained better for large grids (values of m), and hence we used
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m=2000. The autocorrelation parameter Continuity generates a general aggregation pattern,
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whereas the variance within the system is reflected by the Contrast parameter. A high Contrast
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parameter will impose more isolated clusters of aggregated points on the aggregation structure
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defined by the Continuity parameter.
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References
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73
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Cohen J. E., Newman C. M., Cohen A. E., Petchey O. L. & Gonzalez A. 1999 Spectral mimicry:
A method of synthesizing matching time series with different Fourier spectra. Circ. Syst. Signal
Process. 18, 431-442.
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Halley, J. M., Hartley, S., Kallimanis, A. S., Kunin, W. E., Lennon, J. J. & Sgardelis, S. P. 2004
Uses and abuses of fractal methodology in ecology. Ecol. Lett. 7, 254-271.
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Keitt, T. H. 2000 Spectral representation of neutral landscapes. Landscape Ecol. 15, 479-493.
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79
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Mugglestone, M. A. & Renshaw, E. 1996 A practical guide to the spectral analysis of spatial
point processes. Comput. Stat. Data An. 21, 43-65.