What is the Mathematics Behind Nearest Neighbour Analysis?

[antevante]

Hi!
Does anyone know how the mathematics behind the Nearest Neighbour Analysis/Index work?
It is used in biology and geography and shows the dispersion of for example plants or shoe-shops.
/Andreas

 

[TenaliRaman]

i don't know but this sounds vaguely familiar with clustering ...
are u talking abt clustering ??

-- AI

 

[antevante]

Yes it is about clustering...
You have an area, A, in which you have a number of points, n. For every point you measure the distance to its nearest neighbour. Then you calculate the mean nearest neighbour distance, d.
Then you use the formula NNI=2d*square-root(n/A)
Values for NNI close to 0 means clustered distribution, around 1 random distribution and close to the maximum value 2,15 uniform distribution.
I do not understand why 2.15 is the largest value you can get and why a value of 1 indicates a random distribution.
Anyone that knows? I would bwe thakful if you helped me...
/Andreas

 

[TenaliRaman]

I have not done much of NNI ...
But as i see ur formula for NNI , i thought of doing a bit of reverse engineering ...

case > clustered points
If we set n and A as constant, then it may be shown that d = radius of a cluster
So it gets pretty intuitive, as to why if NNI -> 0 , would mean high clustering since NNI>0 means radii of cluster is reducing thereby increased clustering ...

Doing a bit more of this,
we may come to a conclusion that
NNI for cluster < NNI for random < NNI for uniform

However the values of 1 and 2.15 don't seem to come up anywhere throughout ...
So i feel they are statistical limits and not theoretical ones ...
I may be wrong , but i just thought if i am wrong , it may generate counter arguments ...

-- AI

 

[sloppyintuit]

The upper limit comes from an observation that in the plane hexagonal spacing (each point has six equidistant neighbours) maximizes the distance between neighbours for a given density. You can read about it in this reference:

"Distance to Nearest Neighbour as a Measure of Spatial Relationships in Populations"

Clark and Evans, Ecology, Vol 35. No 4, 1954.

They also refer to earlier work by Hertz in 1909. The appendix gives a derivation of this measure.