The document discusses defining spatial entropy from multivariate distributions of co-occurrences. It presents: 1) Historical perspectives on measuring spatial organization using multivariate co-occurrences to calculate spatial entropy. 2) Potential ways of counting co-occurrences and applications like CAkOO, SOOk, and SelSOOk analysis. 3) Issues involving weighting co-occurrences by topology, proximity, and order to better represent spatial patterns and interactions between different variables.

1. Defining Spatial Entropy from Multivariate
Distributions of Co-Occurrences
Didier G. Leibovici
Centre for Geospatial Sciences,
University of Nottingham, UK
didier.leibovici@nottingham.ac.uk
Spatial Statistics, Multiway Data, Marked Point Process, Spatial Pattern,
Spatial Interaction, Multi-Scale Analysis
Co-Occurrences,
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

2. • measuring spatial organisation:
- historical perspectives
- multivariate co-occurrences for Spatial
Entropy
• ways of counting co-occurrences
• potential applications ( CAkOO, SOOk, SelSOOk)
• further issues
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

3. measuring uncertainty, information content, diversity, entropy
Statistical thermodynamics economy
ecology
Signal processing geography
etc ...
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

4. measuring uncertainty, information content, diversity, entropy
Uniform distribution
normalisation /standardisation
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

5. but ... Spatially !!!???
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

6. Some solutions
• similar to Moran’s index and correlation ( Karlström & Ceccato 2002) e
nc
Sc ie
n al
io
Reg
weights
• focusing on proximities ( O’Neill et al. 1988)
y
log
eco
adjacent to
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

7. Some solutions ...
• Markov Random Field ( Maitre et al. 1994, 2000)
is
l ys
a na
e
ag
Im
of topological pattern
is
l ys
• “geo-discriminant” analysis ( Claramunt, 2005 Li and Claramunt 2006)
a na
l
ica
ph
g ra
eo
G
within class i
discriminant weights
between i and other classes
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

8. airs
• • • re of p
• ctu
• • stru
• •
weighting by topology,
using topological distribution
or both
k> 2
or der
s of
nce
urre
co -occ • • •
•
• •
• •
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

9. The co-occurrences at various orders, of attributes from the same
or different processes, build multinomial distributions at the root of
spatial organisation and interactions of processes according
to: the collocation distance, and the order of collocation.
Spatial entropy
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

10. COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

11. COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

12. an example
Spatial entropy with
increasing orders
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009
12/21

13. Counting events
a Spatio-Temporal “event”
(i) a person of age i with social class level j, diagnosed for a certain disease
k and living at location s
(ii) a plant species i, on a soil class j, at location s with annual rainfall k
(iii) a crime of type i, at time slot j, in a zone of wealth class k of location s
(i) is multivariate on the persons
(ii) is a collocation of different measurements
(ii) is a mixture of both.
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

14. ways of counting co-occurrences
geometrical aspects
Attributes (univariate/ multivariate)
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

15. potential applications
a Spatio-Temporal “event”
(i) a person of age i with social class level j, diagnosed for a certain disease
k and living at location s
(ii) a plant species i, on a soil class j, at location s with annual rainfall k
(iii) a crime of type i, at time slot j, in a zone of wealth class k of location s
SOOk CaKOO
looking for associations of vI, vJ , vK variables
in their spatial co-occurrences, scale-effect
interaction of variables spatial pattern,
looking for list of variables most structuring the events SelSOOk
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

16. SOOk, SelSOOk and CAkOO
SOOk analysis proposes to use plot
the Spatial Entropy index
against distance
"hi hi hi" "ma ma ma" "wh hi hi" "wh ma ma“
3.38% 2.23% 1.70% 1.41%
"mi mi bl" "mi bl bl" "re mi bl" "wh mi bl"
0.0152% 0.0180% 0.0365% 0.0467%
d=0.15 blackoak hickory maple misc redoak whiteoak
Hsu=0.893 7% 27% 23% 7% 16% 20%
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

17. Age
1
131 3
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1
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3 5 3 5 15355 1
5 35151333 1335133 551
5 1 <45
- - + +- - ----- -- + +
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- -+ - 1 3 111331135113511 5331
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5 1 15 3 55
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- 5 111115 331131355553351
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1 1535 1 33 5
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13
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15 55 5 513 55
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somewhere in UK
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+ - 571 478 401 55 138 235
Epidemiological study
Infectious disease dataset
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

18. SOOk, SelSOOk and CAkOO
also Spatial Entropy of Self-co-occurrences
= 0.6533133 d=2000
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

19. SOOk, SelSOOk and CAkOO
SOOk map of local Spatial entropies (d=2000)
RSA.ppp
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COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

20. Liti, north of Thessaloniki
plant OPEN. plant LOW. plant HIGH.
Ecological study – semi-natural.
plant community data plant TREE. plant S.T. plant PATCH
Transect dataset
plant community
and vegetation indices
plant TEX plant SLOPE plant ASP
fuzzy clustering
on Transect data
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

21. SOOk, SelSOOk and CAkOO
SelSOOk analysis proposes to use
the Spatial Entropy in a PEGASE (CART) like Regression Tree
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

22. SOOk, SelSOOk and CAkOO
CA2OO and CAkOO methods
• based on Correspondence Analysis (order 2)
• k-way table (order>2) using PTAk (R package)
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009

23. CAkOO d=0.1 Lansing data
COSIT’09 l’Aber Wrac'h, France, 21-25 September 2009