function x = spatialPattern(DIM,BETA)
% function x = spatialPattern(DIM, BETA),
% This function generates 1/f spatial noise, with a normal error
% distribution (the grid must be at least 10x10 for the errors to be normal).
% 1/f noise is scale invariant, there is no spatial scale for which the
% variance plateaus out, so the process is non-stationary.
% DIM is a two component vector that sets the size of the spatial pattern
% (DIM=[10,5] is a 10x5 spatial grid)
% BETA defines the spectral distribution.
% Spectral density S(f) = N f^BETA
% (f is the frequency, N is normalisation coeff).
% BETA = 0 is random white noise.
% BETA -1 is pink noise
% BETA = -2 is Brownian noise
% The fractal dimension is related to BETA by, D = (6+BETA)/2
% Note that the spatial pattern is periodic. If this is not wanted the
% grid size should be doubled and only the first quadrant used.
% Time series can be generated by setting one component of DIM to 1
% The method is briefly descirbed in Lennon, J.L. "Red-shifts and red
% herrings in geographical ecology", Ecography, Vol. 23, p101-113 (2000)
% Many natural systems look very similar to 1/f processes, so generating
% 1/f noise is a useful null model for natural systems.
% The errors are normally distributed because of the central
% limit theorem. The phases of each frequency component are randomly
% assigned with a uniform distribution from 0 to 2*pi. By summing up the
% frequency components the error distribution approaches a normal
% Written by Jon Yearsley 1 May 2004
% S_f corrected to be S_f = (u.^2 + v.^2).^(BETA/2); 2/10/05
% Generate the grid of frequencies. u is the set of frequencies along the
% first dimension
% First quadrant are positive frequencies. Zero frequency is at u(1,1).
u = [(0:floor(DIM(1)/2)) -(ceil(DIM(1)/2)-1:-1:1)]'/DIM(1);
% Reproduce these frequencies along ever row
u = repmat(u,1,DIM(2));
% v is the set of frequencies along the second dimension. For a square
% region it will be the transpose of u
v = [(0:floor(DIM(2)/2)) -(ceil(DIM(2)/2)-1:-1:1)]/DIM(2);
% Reproduce these frequencies along ever column
v = repmat(v,DIM(1),1);
% Generate the power spectrum
S_f = (u.^2 + v.^2).^(BETA/2);
% Set any infinities to zero
S_f(S_f==inf) = 0;
% Generate a grid of random phase shifts
phi = rand(DIM);
% Inverse Fourier transform to obtain the the spatial pattern
x = ifft2(S_f.^0.5 .* (cos(2*pi*phi)+i*sin(2*pi*phi)));
% Pick just the real component
x = real(x);