Continuity and Contrast: parameters in neutral point pattern landscapes.
Spectral density has been used frequently for time series and lattice data (see Mugglestone &
Renshaw 1996 for relevant references). Here we present how to, by a spectral representation,
calculate two measures which capture essential characteristics in point pattern landscapes. The
basis of the spectral method follows from the Fourier theorem, where it is stated that continuous
time series, or surfaces and higher dimensions, can be represented by a combination of sine and
cosine terms with different frequencies and amplitudes. Thereby, analysis of the time series, or
surfaces, can be performed on the sine and cosine terms. From physics we use the notation Power
Spectral Density Function, PSDF, for this representation of a time series or surface by the
amplitudes of sine and cosine terms of different frequency.
Continuity
Large values of Continuity (γ) in point pattern reflects that nearby areas have similar density and
hence 𝛾 is a measure of autocorrelation. It is defined from the spectral representation of the
landscape such that if the spectral density is dominated by the amplitudes of low frequencies
there is a higher degree of autocorrelation. The measure is a point pattern measure, akin to a two
dimensional measure of 1/f noise (Halley et al. 2004, Keitt 2000). A higher value on Continuity is
then analogous to a redder 1/f noise. We analyze, as is done when determining the color of 1/f
noise, the autocorrelation as a linear regression fitted to the log(frequency) vs. log(amplitude) and
thereby also infer self similarity. The slope of the regression line is the parameter Continuity, 𝛾.
In our analysis we have assumed isotropy which implies identical patterns in all directions.
Hence, we may collapse the spectral analysis into an analysis in one dimension, a regression line
instead of a surface, after the Fourier transform of the two dimensional point pattern. To calculate
the Power Spectral Density Function of the point pattern distribution, denoted PSDFpp, we use
the work of Mugglestone & Renshaw (1996) where they present an efficient way to calculate
PSDFpp as
2
2
𝑃𝑆𝐷𝐹𝑃𝑃 = (∑ 𝑐𝑜𝑠(𝐶𝜇𝑝 𝑥𝑖 )) + (∑ 𝑠𝑖𝑛(𝐶𝜇𝑞 𝑦𝑖 ))
𝑖
(A.1)
𝑖
Here, (μp,μq) are frequencies for Fourier transform of grid data, (xi,yi) are coordinates in the point
pattern, and C is a constant that determines the number of frequencies. Mugglestone & Renshaw
(1996) state that no more frequencies should be used than there are points in the point pattern, Np,
to keep them independent. Therefore we chose C to be
and the number of frequencies used is nF =2C×2C.
Contrast
N p 2 rounded down to nearest integer
As a point pattern equivalent of coefficient of variation we introduce Contrast, 𝛿, to measure
difference of density between sparse and dense areas. To formulate an equation for that measure,
we first have to relate it to how variance in the continuous case can be measured using spectral
representation. Note that when time series and surfaces are represented by sine and cosine terns
after the Fourier transform it is the amplitudes of the frequencies that determines the variance. If
M is the number of frequencies in the PSDF the variance, 𝜎2, is calculated as
𝜎2 =
1
𝑚𝑒𝑎𝑛(𝑃𝑆𝐷𝐹 − 𝑃𝑆𝐷𝐹(𝑜𝑟𝑖𝑔𝑖𝑛))
𝑀
(A.2)
The mean is represented by the amplitude in the origin1 of the frequency plane, PSDF(origin),
and hence the coefficient of variation, standard deviation divided by the mean, is
𝐶𝑉 =
𝑀
1
√ 𝑚𝑒𝑎𝑛(𝑃𝑆𝐷𝐹 − 𝑃𝑆𝐷𝐹(𝑜𝑟𝑖𝑔𝑖𝑛))
𝑃𝑆𝐷𝐹(𝑜𝑟𝑖𝑔𝑖𝑛) 𝑀
(A.3)
Substituting PSDF with PSDFpp from equation A.1 we end up with an equation of the point
pattern measure Contrast, 𝛿:
𝛿 = 𝐶𝑉𝑃𝑃
=
𝑁𝐹
1
√ 𝑚𝑒𝑎𝑛(𝑃𝑆𝐷𝐹𝑃𝑃 − 𝑃𝑆𝐷𝐹𝑃𝑃 (𝑜𝑟𝑖𝑔𝑖𝑛))
𝑃𝑆𝐷𝐹𝑃𝑃 (𝑜𝑟𝑖𝑔𝑖𝑛) 𝑀
(A.4)
Generating neutral point pattern landscapes
To get NPPL with given characteristics we generated lattice landscapes of size m×m. The density
defines the probability of a point in the landscapes. We first generated 2-dimensional 1/f-noise
(denoted LG) using a method similar to that presented by Halley et al. (2004). Hence this is still a
representation of a lattice landscape not a point pattern. The values in LG are normally distributed
and hence not suitable for describing probabilities because it may include negative values. While
this could be solved by truncating we found that it would not allow for generating sufficiently
high values of δ. We therefore transformed LG using spectral mimicry. This method was
introduced by Cohen et al. (1999) and has been used to construct stationary time series with
different Fourier spectra. Cohen et al. presents the method for transformation to normally
distributed values with specific mean and variance. We instead transformed LG to LΓ using a
1
Note that if implemented in Matlab using built in functions of Fourier transform, for example
the fft function, the amplitude in the origin, PSDF(origin), represents the sum instead of the
mean.
gamma distribution (which contains no values <0) with mean=1/m2 and coefficient of variation
δL. Point locations were distributed according to the probabilities given by LΓ.
While γ and δ of the spectral point pattern is determined by γL and δL, they are altered by both the
gamma transformation of the grid values and the distribution of points. Hence we measured these
quantities in the generated landscape (see method given above). The relationship between
spectral point pattern values of γ and δ used in the study and the γL and δL required to generate
them was found iteratively. Furthermore, we found that the linear relationship in the power
spectra was maintained better for large grids (values of m) and hence we used m=2000. The
autocorrelation parameter Continuity generates a general aggregation pattern while the variance
within the system is reflected by the Contrast parameter. High Contrast parameter will impose
more isolated clusters of aggregated points onto the aggregation structure defined by the
Continuity parameter.