COSIT 2009 - Defining a spatial entropy from co-occurrence data

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Finding geographical patterns by analysing the spatial con- 􏳅guration distribution of events, objects or their attributes has a long history in geography, ecology and epidemiology. Measuring the presence of patterns, clusters, or comparing the spatial organisation for di􏳈erent attributes, symbols within the same map or for di􏳈erent maps, is often the basis of analysis. Landscape ecology has provided a long list of in- teresting indicators, e.g. summaries of patch size distribution. Looking at content information, the Shannon entropy is also a measure of a dis- tribution providing insight into the organisation of data, and has been widely used for example in economical geography. Unfortunately, using the Shannon entropy on the bare distribution of categories within the spatial domain does not describe the spatial organisation itself. Partic- ularly in ecology and geography, some authors have proposed integrat- ing some spatial aspects into the entropy: using adjacency properties or distances between and within categories. This paper goes further with adjacency, emphasising the use of co-occurences of categories at multiple orders, the adjacency being seen as a particular co-occurence of order 2 with a distance of collocation null, and proposes a spatial entropy mea- sure framework. The approach allows multivariate data with covariates to be accounted for, and provides the 􏳉exibility to design a wide range of spatial interaction models between the attributes. Generating a mul- tivariate multinomial distribution of collocations describing the spatial organisation, allows the interaction to be assessed via an entropy for- mula. This spatial entropy is dependent on the distance of collocation used, which can be seen as a scale factor in the spatial organisation to be analysed.
see also http://c3s2i.free.fr/DidierGLeibovici

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  • Horizontal and top: 6 <> (independently of + - a b) “ showing an age disparity for the over 85: less than expected in the reds and a little association in the blue area” Vertical and left: 2 3 <> 4 (5) “ the isolated class 4 (65-75) opposed to not much of them but 3s (45-65) in the right blue blob”
  • Horizontal and top : +b <>+a in blue collocating more with +b “ More co-occurrences than expected, of cases resistant to methicillin with later contagion (rather than earlier +a) in the top left corner” Vertical ad left : -a <> +a (+b) in early – “ … but also rather with earlier resitants cases (+a) rather than just sensitives with earlier contagion”
  • vs222 +b and not 6 opposed to +a top left corner “ More than expected collocations of + with late contagion in top left corner, not collocating with age 6 (over 85)” Vertical and left: “isolated points
  • COSIT 2009 - Defining a spatial entropy from co-occurrence data

    1. 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.ukSpatial Statistics, Multiway Data, Marked Point Process, Spatial Pattern, Spatial Interaction, Multi-Scale Analysis Co-Occurrences, COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    2. 2. • measuring spatial organisation: - historical perspectives - multivariate co-occurrences for SpatialEntropy• ways of counting co-occurrences• potential applications ( CAkOO, SOOk, SelSOOk)• further issues COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    3. 3. measuring uncertainty, information content, diversity, entropyStatistical thermodynamics economy ecology Signal processing geography etc ... COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    4. 4. measuring uncertainty, information content, diversity, entropyUniform distribution normalisation /standardisation COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    5. 5. but ... Spatially !!!???COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    6. 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 Wrach, France, 21-25 September 2009
    7. 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 Wrach, France, 21-25 September 2009
    8. 8. airs • • • re of p• ctu • • stru• •weighting by topology,using topological distributionor both k> 2 or der s of nce urre co -occ • • • • • • • • COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    9. 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 Wrach, France, 21-25 September 2009
    10. 10. COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    11. 11. COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    12. 12. an example Spatial entropy with increasing orders COSIT’09 l’Aber Wrach, France, 21-25 September 200912/21
    13. 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 Wrach, France, 21-25 September 2009
    14. 14. ways of counting co-occurrencesgeometrical aspectsAttributes (univariate/ multivariate) COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    15. 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 CaKOOlooking 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 Wrach, France, 21-25 September 2009
    16. 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 Wrach, France, 21-25 September 2009
    17. 17. Age 1 131 3 5111353 333113 5 1531 3111135 53 51511353 31 1 5 5 51 351 31 311 13 3 3115 15 55 5131351 3 15333 351 3 15 1 5 3 33 5 1 55 1 3 55351555 1 11311331 1 3 53 3331 3 5 3 1 5 1 51 5 3 1355 531 11111 33 5 55 551 5 31 1 1 3 13 3313 3 - + - not resistant 1 3 + resitant 1 1 113 35351 115335 51 315 5 1450 428 MRSA 31 5 3 13331133 3 13 5 13 5 1 1 355 1 13 51 3 5 35 3 51 53 5 55 33 11 3 - 31 35 15 5 3 1 5331 551 113 1 1 51 3 3 3353 33 3 11 5 15 31 1 11 3 53 55553113 3 31 1 1 133 31 5 3 151 3 5 5 1 3 1 313 1 - ----- - - +++ - - +- - -------- ---- +--+--+ +-- +-- - -+ 5 11 515 51 1 1133 53 5 315131 1 1 13315133 53 +-- - - - - --++ - + - - -+- -+ + ++ + -- 5 1 5 3 3113 5513 133 51 1 3 5 1 3 31 3 335 153 1353 5111111 3++-- + + -- - - - + -- - + - - - 3 3 3 1 53 3 3 5 315 5 1 155 133 31131 3513 3 1 33 53 1 5 1 5 3 - ++ - - - -- - - --- +--- - +-+ - 1 5 1 1 5 31 1 15 33 1 1 3 1 1 33 15131 - ++- + - - -- + - +- -- ++-- - + - -+ - 1 5 3 3 111 5 1 13 + - - + -+ + + - -+ - +-- - +- - 11 3 351 1 1 5 33 3 5 1 1 - - + -+ + - 5 55 5 51 51 1155 5 1 113 1 3 33511 1 3 3 35 35 51 3 3 5 1 13 3 3 3 1 - 33 3 5115 1 11553115 3 335313 3 3 3 133 35 5 5 5 33 3 3 5 3553 5 5 5 555 1 5 531 1 31 55 + 5 5 3 + ------- -- - - - + +- + -- --+-+ - --- +- +--- + 5351155 3 1 13155 151 1 53 11 33 53 - - --+ -+- + - + - ++- - - 35 351 53131533 551 13 5553 3 3 5 151 1 5 511335 -- -- --+-+-- -- - ----- -+- - - 3 1 5 11 51 3 353 1 1155 +--+ - --- - ------ --+ +- - -- - -+ - --- - - 55 3 1551 5 51 531 3 35 11111 1 5 35 5 3 3 33 3 3 1551 55 - -- + --- --+------ + - - - + - -- + - +- -+-- -- + - - --- --- -- - 3553531 3351 335 33 55 31 3 5 355 3 333 11 515 53 11335 5 1 1 1 13 113313513535555 11 5135 5 5 5 15 3 3333 626 + - +- 13531 51 5 3551 3 5 15311515 1355153 13 13135353 5 353 3 55 5 5 + - -+ -+---+ - -- ----- +------------ - + 3551 5 1 5 5 1 31 3 3 5511513153133113 351 5 3 5 3 5 15355 1 5 35151333 1335133 551 5 1 <45 - - + +- - ----- -- + + + - -+ +- +-- -+ -+-- - - -- + - + +- -- - - + - -+ - 1 3 111331135113511 5331 53315515555115 1 13 5 1 15 3 55 31 5 335315353 3 1 3 3 33 5 5353 5 15 5 15 13 33 3 5 3 45-75 616 - 5 111115 331131355553351 5 3 5 3 111 311513 333 5 3 15 553 3 +- - - + +--- + --------- + - - - - - + + + -+-+ - - -- + 551 5 3 3 155 1 53 5 1 1535 1 33 5 5 3 5 53 5 11333155313513 1 351133 3 51 115513151553 3335 1 3 155135 55 5 3 551 1 51351 5 5 3 5 553 1 5 5 + -- -+ - -- 5 3 35 53 5 3331 51531111 133 3 51 3 1 135351 1 13 3 5 >75 - +- - - -+-- - - -- - - - --- - - +-- + -- -- - 513 553 315 535 13 5 3311111511155331 3 13135551135315111 5 15 55 5 513 55 53 551131533353113111 353 1 1 1 133 333 53531 5551 13 3 3 31 3 31 5 55 5 1 3 51 13 5 15 3 3 1 1 636 - ++- - - - - - + - +-- 3 531 31 3 33 31 331 1 5 311 5 -- ------+++-+- -- ---- + - --------+-- +- - --- --- -- -+ + 11355 5 31531 3 3111331 3315 333 13 15 - + -- -+ - +-- - + +++ - - -- -- 33 15 - - -- -+- + - ------ ----- - - -- ---- +++-- +- -- ++ --- ----- +-+- -- -- + +-- +--- + --- -+ - ++--- - --- - + +--- - +-+--+ - - - +-- - ++ +----+- - +- - + - - - +- +- + -- + - -------+ --+ - - - -+---- - ------ +- -- +- -- - - - -+-+ --- -+- +- - --+- -- - - - -+ + - +---+----+-+------ - ---+------+ -+ - - - - + somewhere in UK -+------------- --- -----+-++ - - -+ - -- --- + --+ -+ - -- - +- - - + - - +-+ +- +-+-- - + - + + -- -+ +---- +++-+ - -+-+- - -- -- --- +- ++-+ +- - + - -+ --- -+ + -- - -+- - + + --+ +- - - - ++ - + -- --+ +----- --- + --------+-+--- -++- --- - -- ++ - - +- --- -+- -- -+-- ++---++- + -++ -+ -- + - + +-+ - - ++- --+ -- ++---- - - --- - - --- -- + --+ -+- --- - - -+ --- --- +- + --- -++-++-+-+- +- + - +- - - -+ - +-- ++ - + - - - - - - - - +-+---- ----- ---- --- --+- - ---- + -- -- - --- + ++ -++ -++-- -++ - - -+- -+ + --- - -+ -+ - --+-+- --++-- +-+- +---- - - + - - ---+- -- -+- +- - +- ----+--- ----- - -+ - +- - -++ ++ - + -1 -3 -5 +1 +3 +5 ----+-- --- - -- - --+++--+ --- - -- -- + + - 571 478 401 55 138 235 Epidemiological study Infectious disease dataset COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    18. 18. SOOk, SelSOOk and CAkOO also Spatial Entropy of Self-co-occurrences = 0.6533133 d=2000 COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    19. 19. SOOk, SelSOOk and CAkOO SOOk map of local Spatial entropies (d=2000) RSA.ppp - oo+ +o ooooo+o + oo ooooooo ooooooo o oooo o - - --ooo +oo+ oooooo oooooo +oo+ oo + o o ooo o o o o oo o o o oo ooo+ oo oo o - oooooooo oooooooo o o oo oo o oo o o - oo ooo o - - O 0.888< HSu o o o o o o - o o + 0.7 < HSu < 0.888 +ooooo o oooo + ooo o oo ++ oo oo o o o - - oo o + ++oooooo + ++ ooo + + o +o oo o + + o oo o o o+ ++ oo o o+ +++ oooo +++ ++ - o + + oooo++ +++ o +++ + + o o o +++ + - HSu<= 0.7 - o o ooo++++ o++ + oo + o + - o o ooo++ o oo o o o+ o o ooo+++ ooo+oo oooo+++ + oo ooo oooo++ + + + - o oo ooooo+++ o o ooo o+ ++ o o+ + o oo o ++ + o o+ + ++ + - o oo o++++ ooo ++ ++ o + ++ o + + ++ o ++ + +++ o o +++ ++ o - - oo + + o+ + + o o + + + ++++++o oo+ +++o + ++++++ o + ++++++ oooo + ++++o++o + +++o+ o o ++o+ o oo + +o + oo o o + oo o o o o +oo+ ooo o o o o o + ooo oooooo oo oo ooo ooooo o oo oooooooo o oo o oo + + oo ooooo oo oooo oo o ++ + ++ ++ + ++ +o + oo ooo ++oo o o ooo oo + oo o + oo + +++++ ooooooooooooo +++++ooo ooooooo o+ +++ o o + ++++oo oooooo o oo +o + oo o o o +o o+ oooo o o+ o + o o + +o o o oo++ooooooo+oooo oo++ o + o++ooooooo ooooo oooo ++ oo +oooo oo oo o o oo+oooooooooooo o oo + o o oo + oo oo o o oooo+ooooooooooo oooo o+ + ooooooooo o oooo o o oo o oooo oo + ooo + o ooooooooooo ooo +oo oo ooo++o ooo + o oo o + ++ o oo oooo+ o ooo o+ o o ooo o oo oo oooooooo++o+ ooo o+ o o oo + o o+ ooo + +o o o + ++oooo + ooo o oo o o ++ooooo+++ooooooo + +++ o+++ oo ooo oo ++oooo++o++oooooo + o + ++ o o o + o oo+o ++ ooo +oooo++ ++o oooo o o+++ ++ oo o++o + + oo o+ +o++o+ oo++ + +o++++ oo + o o oo+ COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    20. 20. Liti, north of Thessaloniki plant OPEN. plant LOW. plant HIGH.Ecological study – semi-natural.plant community data plant TREE. plant S.T. plant PATCHTransect datasetplant communityand vegetation indices plant TEX plant SLOPE plant ASPfuzzy clusteringon Transect data COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    21. 21. SOOk, SelSOOk and CAkOOSelSOOk analysis proposes to use the Spatial Entropy in a PEGASE (CART) like Regression Tree COSIT’09 l’Aber Wrach, France, 21-25 September 2009
    22. 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 Wrach, France, 21-25 September 2009
    23. 23. CAkOO d=0.1 Lansing data COSIT’09 l’Aber Wrach, France, 21-25 September 2009

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