Calculating coef. of var.
We will study our point pattern with specral methods. Spectral methods is a good tool to
study timeseries (halley vasseaur), grid data (röda boken) and can be used for point
patterns as well (Mugglestone). To make the description of our method easier to follow
we will start by describing it in 1-dimension for time series. With spectral methods we
look at the Fast Fourier transform of the signal (eq. 1) and its power spectral density
function (eq. 2). A time series with a PSDF proportional to 1/frequency^gamma is red or
aggregated. Gamma is therefore a measure of continuity, and can be estimated by plotting
log(PSDF) against log(frequency) and estimating the slope.
It is also interested to measuring the coefficient of variation of data. For time series
coefficient of variation is strait forward to calculate but it can also be expressed in
spectral functions. The reason for expressing coefficient of variation in the spectral
functions is that we want to find an analogue measure of variance for a point pattern. The
variation of a time series can be calculated from equation 3, where M is the length of the
time series. Using Parsevals formula (eq. 4) and the fact that the mean of x can be
expressed with the dc point of X (eq. 5), the variance can be expressed with spectral
functions (eq. 6). The coefficient of variation can also be expressed in the spectral world
(eq. 7).
The formulas for grid data, the 2-dimensional case, are analogue but instead of length of
the time series we need to use number of grid points of the surface (eq. 7.5). To estimate
gamma for grid data we use the r-spectrum of the PSDF, and for each frequency interval
2p / K 2p  1 / K we calculate the mean frequency and mean PSDF values. The log of
the means of the PSDF are then plotted against the log of the means of the frequency and
the slope is calculated to find gamma.
We are interested in analyzing not grid data but point pattern data. Mugglestone and
Renshaw have described a good way to calculate the PSDF (de använder inte den termen!
Extra kolla att det är samma) for point pattern data (eq 8.). Where
(  p ,  q )  (2p / K ,2q / K ) , and p ranges from 0 to pmax and q from –qmax to qmax.
Observe that x is first scaled to the unit square. We choose to let p range from –pmax to
pmax as well, this makes no large difference because the extra points are just the complex
conjugate of the calculated ones. The question is for how many frequencies NF the PSDF
should be calculated, that is what value of K should we choose. Mugglestone argue that
no more frequencies should be used than there are points in the point pattern to keep them
independent. Therefore we choose K to be sqrt(N)/2 rounded down. And the number of
frequencies used is NF =2K*2K. Equation 9 is the formula for coefficient of variation for
the point pattern. This corresponds to using a grid of size 2Kx2K for the data, but the
gain here is that we don’t have to specify a grid, instead we directly calculate the PSDF,
and therefore using the exact information on were the points are rather than in which grid
square.
We measure gammapp for the point pattern using the same method as for grid data, but
with the power spectral density function for point pattern data PSDFPP.
X (  )  DFT ( x)   x(m) e
2im
M
(1)
n
PSDF (  )  abs( X (  )) 2 (2)
 
2

1
1 
var x   E x  Ex   x(m)2  2   x(m)  (3)
M m
M  n

1
n | x(m) |2  N  | X ( ) |2 (4)
2
 x(m)
2
 DFT ( x)   0 (5)
n
var  x  
1
1
1
| X (  ) |2 
X 2 (0) 
mean( PSDF  PSDF (origo )) (6)
2 
2
M 
M
M
coef .of var . 
M
1
mean( PSDF  PSDF (origo ) (7)
PSDF (origo ) M
coef .of var . 
M 1M 2
1
mean( PSDF  PSDF (origo ) (7.5)
PSDF (origo ) M1M 2
2
2

 

PSDFPP    cos( K  p xi )     sin( K q yi )  (8)
 i
  i

NF
1
coef .of var .PP 
mean( PSDFPP  PSDFpp (origo ) (9)
PSDFPP (origo ) N F
Vet inte om det behövs fler förklaringar av variabler men iallafall:
x- tidserie, x(m) en punkt på tidsserien.
X fouriertransform av x, X(  ) fourier transformen för en frekvens 
M längden på tidsserieen.
M1xM2 storleken på grid, vid grid data.
NF antalet frekvenser vi använder
(  p ,  q ) frekvenser för fourier transform av grid data.
PSDF-power spectral density function
(xi,yi) en koordinat i point patternet. (Bör vi byta ut x:et kanske) formel 8.
K- konstant som bestämmer hur många frekvenser som är vettigt att räkna ut.