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 ...

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 Presentation Transcript

  • 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. • 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. measuring uncertainty, information content, diversity, entropyStatistical thermodynamics economy ecology Signal processing geography etc ... COSIT’09 l’Aber Wrach, France, 21-25 September 2009
  • 4. measuring uncertainty, information content, diversity, entropyUniform distribution normalisation /standardisation COSIT’09 l’Aber Wrach, France, 21-25 September 2009
  • 5. but ... Spatially !!!???COSIT’09 l’Aber Wrach, 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 Wrach, 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 Wrach, France, 21-25 September 2009
  • 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. 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. COSIT’09 l’Aber Wrach, France, 21-25 September 2009
  • 11. COSIT’09 l’Aber Wrach, France, 21-25 September 2009
  • 12. an example Spatial entropy with increasing orders COSIT’09 l’Aber Wrach, France, 21-25 September 200912/21
  • 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 Wrach, France, 21-25 September 2009
  • 14. ways of counting co-occurrencesgeometrical aspectsAttributes (univariate/ multivariate) COSIT’09 l’Aber Wrach, 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 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. 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. 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. 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. 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. 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. 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. 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. CAkOO d=0.1 Lansing data COSIT’09 l’Aber Wrach, France, 21-25 September 2009
  • 24. Age 1 212 4 25 2122 5 2324 2 6 3543 3 11124 2 5 5131 3 5 635312532 5 4 3 3 5532451 6 5 3522 4 1 5 3 22 4 25 3 4 43 56533 12 154154 3 2 5 6 34 1 5 4 6 34 11 352 5 5 34 54 32 3 6 4 6 155 5165126 4 1 3 6 51 4 44423 4115 1 513 5 3 1 2 2 6 5 1 2 15 6 55 1 11 6 4 5 2 4 5 512 5 1 2 4 1 4324 23415 4 2 1 234 3 1 215334 32 462 32 4 6 1 <20 - + - not resistant 5 45 245 5 12 5625 35 336 41 3 4 545 4 212 2443 4 63 6 2 3 53656 4 3 2 2 20-45 392 1 46 4 4 54 2 2 421 32 63 25 11 23 + resitant 355 236 41 2 4 2 45 64 2 1 1 5342 55 22 3 1 555 3664112 2 511532 54 254 323 4 3 45-65 5 6 5 3 12 2 13 332 4 5 4 5 3 3 350 428 MRSA 1 4 2611 2 2132223 4 1450 5 1 1 323 633 1 22 3 6 3 25 2233 52 4 6 133542 2 6 5 3 4 65-75 - 6 6 55 3435225 2 1 4153 21 3 2 533 212125312 3 5 75-85 5 5 11 32 1 6 1553 11 3233 232 5 34 2 41 1 1 4 5 3 33 14 63 223 1 3 6 - ----- - - +++ - - +- - -------- ---- +--+--+ +-- +-- - -+ + 4 2 1 33 6 34 2 24 16 3 2 165 23 52 3552 6 >85 11 4 2 3 2 3 322 1 3 3 1 2222 1 266 6 54 6 23 34 6 523 - --++ - + - - -+- -+ 112 4 32 2 3 2-- - - -++-- + + + ++ + - - + -- - + - - -- 6 45 1 3 453 4 2 6 5 4 55 2 4 6 35 2 3 11 3 3 41 5 2 -- - ++ - - - -- - - - 6 51 51425 22 2 1 33 56 3 23 3 369 --- +--- - +-+ - 5214215 3 4 2 - -- + - +- -- ++-- - + - -+ 3 33 1416664 11 4 1325314 2 2 466 45 6 2266 4 4 1 3 44 - ++- + - - - - +-- - - - - + -+ - 6 5 25 5 564 56 6 35 6 54 1 + + 3 6 615 55 - - + -+ + + - -+ - + 2 511 5 35 35 2 33 267 53 215 563 2 1 26 622 654 2 2 46 6 5 46 46 - 53 5 46 4 5 56 253 5353 53 4 6 5 56 551 2425214 626 13 152 4 64 2 4634 4 13226 + 5 53 6 5 66122245 2 13 3 1 5 3532 2 1555 6 34251 3 6 2 5 25 3 3 2 52 1 3 34 26 + ------- -- - - - + +- + -- --+-+ 5365 4513 543156 1 132 52 45 6 42 543134 55 5 6526 2 3 44 33 6 2 32 1256536623 551631342332 2242 336 45 6 1 4 52 3 5 - --- +- +--- + - --+ -+- + 13 54212 4 41 2231 25322 6 43431264 24 5 64 2 5 5 5 2 5 4 5 - + - ++- - - - 452 3 24632 1 465416 13 531513251 241264 1 2 2 1 252 2 53363 2 4 34 13 5 6 366 6 1 33 6 -- -- --+-+-- -- - ----- -+- - - 52526 526 3 24 1 2 2 5 131 52425 61 5 6 12 45 3 56 2 44 3 3 3 2 6 311 6 15632 42 35 4 1 24 5 6 3 34 63354 55344 56553 551 62 3 5 6 4 +--+ - --- - ------ --+ +- - -- - -+ - --- - - 3 5 5 45 4633 4224513525 6 54 41 6 2 5 2 222661 4 53 5132665363 31521 5 62 546 433365 44 44 2 2 54 4 5264 6 3 - -- + --- --+------ + - - - + - -- + - +- -+-- -- + - - 4 3 1 65236 2 33 1 13 1 4 32 266 6 44 2 3 53 654 6 5 1 24 225 14 1153 22 36335 23 4 23 2 1 4 3 1 5 2 5151525 55 3532 24 52 1 22 16 3 5 6 2 5 3 46 63 6 1 431452 152 1 6 5 42 1 62 4 5 2 5 5 4 6 Onset --- --- -- - + - +- 41 26 221162253253 13 25 6 455 4 2 35125131 12 3 53 5 23545 1 3 1 2 15 2 6 5 2 - + - -+ -+---+ - -- ----- +------------ - + 3 3 56651 52155241451 3524 1 36 26 2 32 26 652 5 41 66533 645 1 25 4 44 35326 222 2 53 6 3 2 2 25 4 1 4 53 5 33 3 2 6 225 513 35 3 1354 263 2331 323 1 3 12 45 3262 6 2 b - - + +- - ----- -- + + - + +- -- - - + - -+ + - -+ +- +-- -+ -+-- - - -- + - 63 5 366 3 2245532 5 233 6 36 21 4263 554 2454 31635 624 62 2 1 41 6 333 5 251 3 5 3 22 4 14 56 2 61 4 21 462 34 aaaa ba ccb a aaccccaaa a c bb b cbbab c b cabaabbbaca b a bb + + +--- + --------- - - - + + + -+-+ 324322 2 2 4 c cc a - - - -- + + - - 422156 22 343 3 5 4236 6 c cb b bcc b b c a b b cb ba a c bb aa b cb ccbcccb a a + -- - - -+ a a bc b c b c b c - - - -- bb b b c b cc c a acc baa a b ab c c a ba a c b ab a caa abcb - +- -+-- - - -- - - - --- - - +-- + -- -- - c b ca b a c b b a c a bbaa a b c c b c ac b a cb b b a bba ac b - ++- - - + - +-- b cbc a - - - b bc b ca b -- ------+++-+- -- ---- + - --------+-- +- - --- --- -- -+ + - + -- -+ - +-- - + +++ - - a a -- -- - - -- -+- b c c a babacac + - ------ ----- ca cc ab a c - - -- acc bcac aa b c ba a b bab abaa ab ---- +++-- +- ab a a aaa bcc a c b a a bcc c b cbc b a b ab cc cccc b b a c aac ab ba ba -- ++ --- ----- +-+- -- -- + +-- +--- + --- -+ - ++--- bcb a ca b b caaa acc c cacb c - --- - + a b b c b bb b a b acc b ac a bb ab cab c ab c b a c bb a c b ba c b c + +--- - +-+--+ - - - +-- - + a b c aa bb a babaa ab c +- + -- + - a ca c a a +----+- - +- - + b acacbaa c bb b cbbbb -------+ --+ - - - -+---- - ------ - - - +- +- -- +- b c a ca b b c bb a a c caa b -- - - - -+-+ --- -+- +- - --+- -- - - - -+ + - +---+----+-+------ - ---+------+ -+ - - - - + b b ca cbc b ccc a a ab a a bcaac c a b a bbc bc bccccb ccbbbab a a c b aa c bab aac b aa c ba aaaaacbb cb b c -+------------- --- -----+-++ - - -+ - -- --- + --+ -+ - -- - +- - - + - - +-+ +- +-+-- - + - + + -- -+ +---- +++-+ - -+-+- - -- -- b a c cb b a c a b b a bb ab ab ba bc b b a a aac cca c b b abb b c c c a --- +- ++-+ +- - + - -+ --- -+ a a aa a ca + -- - -+- - + + --+ +- - - - ++ - ba a cab c b ac + -- --+ +----- --- + --------+-+--- -++- --- - ba c ab a c - -- ++ - - +- --- -+- -- -+-- ++---++- + -++ -+ -- + - + +-+ - ++- --+ -- ++---- - - c bc caa b acc c bb bc a c b a cb c b a c bc a a a --- - - --- -- + --+ -+- --- - - - -+ --- --- cc ba c b +- + --- -++-++-+-+- +- + - +- - a aaaaccaaaca a c - -+ - +-- ++ - + - - - - - - - a cb cbabbaacabccc a bac b b cc caabc ab aab c ba baa ac b +-+---- ----- ---- --- --+- - ---- + -- -- - --- + ++ -++ -++-- -++ - - -+- -+ + --- - -+ a b c c ac b c ccb b aa b b cc aa -+ - --+-+- --++-- +-+- +---- - - + - - ---+- -- -+- +- - +- ----+--- ----- - -+ b c b ca a c bbb - +- - -++ ++ - + bc c bc ----+-- --- - c b aa b b a -- - --+++--+ - -- -- + - --- b b aa b c cc aaa cbaa a cbb b ab a c aaaabbbbccb b ccca cabc aba cc c + c ab c bb ca a a caaca aa ab b a b cab bba cb bc c a b b aabaabcab c b ac a a aab c ac cc a ccabc c c aacab c a b ab b aa cb cab ac ab bcc bb c cab babaaab aa c a c ba bb a cc b c a a cc cabac ba bbaa abaa bb b cbabaaaaaaaa bbcabbcc aa c bc a b c c ba c a b a ba acbbc b ba bccc aa 687 ccccbbbba aabc baaaa baa a a caca a cc cba b c b b cc c c bc b a a b b c ca ccbaaacaabccabcaaaa c b ba caa c b c ccc b b b a b ab a c b aba aa aacabbb abaab c a a ac a c b b b a ba b ac a aa a bbab c c ac ac cb ac a bbb a b c cc b aaaa a ab bb ab bbbaa cb c c c ca ba c c a c c ab c 627 b b aba aaa c ab bca cac aaba acb ba c c a b a c aba b b b aa a a baa b a cc ab b c bbaaac b b a c a c b cb a bbb a cc babcabcb caa b cc b cc caba c aaba c b baac ab c baccaaccaba bcbbc ac a ba b b b c b b c cc a b aa b ab ab c c a ac a ab cc b aa ab a ab b b a a c c abcba bbc c c c c b a ab b aa b cc c a bbb c aaaa aaababbbaba aaa cbb b a Sep04-Dec04 564 b a baa ccbb a aca ac a b a baa ccccc cb a cc a b b bb a c b bc ba ccb ab ab c a ab a cc bca b aabc ab cabaacc cccacc ba ab a cbaca c aa b acc a baca abaa bbaba cc a acc c bc c ba c b Jan05-Apr05 b c ac b c b aab aaa c a a a bca a c ccbb c a c ac b c May05-Aug05 bcb b accacbbb b c b c bcbaa c ab c b COSIT’09 l’Aber Wrach, France, 21-25 September 2009
  • 25. ++++ FCA- 3 modes++++ = 1600 m ++ collocation Table RScont.1600 1878 4 6 ++ S I J -----Total Percent Rebuilt---- 97.12973 % ++ Percent of lack of complete independence rebuilt ++ 69.74381 % I: RS and Onset selected pctoafc > 0.1 % total= 69.70363 -a -b +a +b -no- --Sing Val-- --ssX-- --Global Pct-- --FCA-- 770 680 231 197 vs111 1 1.000000 1.10480826 90.5134437 NA 1878 vs111 4 6 3 0.015307 1.00027656 0.0212090 0.22357J: Age 4 vs111 1878 6 6 0.187667 1.08013832 3.1877832 33.603171: 234 4 vs111 1878 6 7 0.135752 1.08013832 1.6680416 17.583222: 392 6 vs111 1878 4 9 0.097394 1.02160907 0.8585808 9.050503: 350 6 vs111 1878 4 10 0.085529 1.02160907 0.6621218 6.979584: 266 vs222 11 0.029673 0.00278430 0.0796975 0.840115: 369 4 vs222 1878 6 16 0.016601 0.00140077 0.0249438 0.262946: 267 4 vs222 1878 6 17 0.011369 0.00140077 0.0116993 0.12332 6 vs222 1878 4 19 0.015158 0.00126415 0.0207967 0.21922 6 vs222 1878 4 20 0.012405 0.00126415 0.0139285 0.14682 vs333 21 0.016998 0.00099283 0.0261533 0.27569 4 vs333 1878 6 26 0.013409 0.00064126 0.0162734 0.17154 4 vs333 1878 6 27 0.010380 0.00064126 0.0097516 0.10279 6 vs333 1878 4 29 0.011269 0.00041593 0.0114937 0.12116 COSIT’09 l’Aber Wrach, France, 21-25 September 2009
  • 26. 4 vs111 1878 6 FCAk 33%4 vs111 1878 6 FCAk 17.6% 4 2 +b +a -b -a 4 vs111 1878 6 local 1.71 % 1.67 % global 1 5 1 0 6 2 -1 3 -2 -2 -1 0 1 2 COSIT’09 l’Aber Wrach, 6France,3.19 % globalSeptember 2009 4 vs111 1878 local 3.26 % 21-25
  • 27. 6 vs111 1878 4 FCAk 9%6 vs111 1878 4 FCAk 7% 2 +a 6 vs111 1878 4 local 0.72 % 0.66 % global 6 5 4 3 2 1 1 +b -b 0 -a -1 -2 COSIT’09 -2l’Aber Wrach, France, 21-25 September 2009 -1 0 1 2 6 vs111 1878 4 local 0.93 % 0.86 % global
  • 28. vs222 FCAk 0.85% RScontRSA.ppp Age - - 1 +-+++---- - +-----++ 2 1 4 35 2 22 -------+--+- - 5 2324 23 6 3543 21 111233 112 - 2 +-- - - + + 635212532 5131 5 5 5233 45 54 35 6 6 55 --- - - ++-- ++ - - ---+-+ +-+- --- - - + + + -+ ++ - - - - 3522 4 1 5 25 24 4 3 43 56533 12 154154 3 34 +-----+- - - 3 22 4 6 4 5 34 11 352 5 4 -- - +-- +---+ 1 5 4 6 32 4 4 6 155234315 5516212 6 4 1 5 6 --+ - -+-+-- + - ----++ -+- -- + - - - -- - - +- - 1 513 5 3 5 43 2 5 4 6 1 1 2 5 1 11 4 6 11 6 3 4 2 1 35 54 5 512 5 1 2 + +- + - --+ + 4 1 432 5 415 2 4 + -+ 4 - 2 1 3 + 1 1 <20 + -- ------ -- ++ -- - 225334 35 4625 32 446 2 56 44562 -- - - --+----++ - -++--+ -- +--+ -- - + 1 5 4 235 6 31 3243 4 2115 2415 2 242 544 3 4 2 3 4325 6 4 3 2 34 236 11 23 66 55 2 20-45 -- - +--+ - -- --+-- ---- -- -- - + --+ - -- -+- +--+-+---- --- - -- - - - + -- 46 355 236 41 2 4 63 542 5 2 1 53 1 5 2 13 2 45 4 2 1 5 2 5455 2511132 2 636 3 123 4 5 45563512 3 254 432 4 5 3 2 11 4 136 32 3 45-65 - -- + + -- ++- - - + --------- + 5 11 2 2132223 4 3233 33 3 61 4 3 2 3 25 6 3 543 4 65-75 + -- - -- + -+-+ + - + --- ++- - - - + - 1 6 5 2 6 5 2 1 52 1 2 5 25 1 6 2 13 251 3 1 14 5 1 3 23 2 6 -- ------- ---- ---+-- + 3 3 55 4435 2112 13 2 +++---+ +-----+-+++ 3 5311 3133 33 3 3 6 1523533 21212222 5 - - + +- ---- ----- - -- + 4 4 6 53 3 2 5 75-85 - 4 1 4163 321 52 3552 35 63 34 2 24 6 4 2 + - + + +-- + -+- -- 12 3 31 14 2 6 2 6 22 1 2 3 1 3221 +- - - - - ++ +- --- -- - + + - - -+ ----- - 1 12 2 6 4 3 4 2223 5 6 23 33 35 22 6 6 >85 4 42 1 45 2 3 + ----- - - - +- - -- - - - + +- + -- - -- - - + - -- 3 453 4 2 6 5 6 52 2 555 61 13 4 1515214212 6 4 4 11 3 3 32 36 21 5 5 32 3 - -----------+- ------ ++--+ - -+-+--- + 3 3314251465113 32613624 42 456 46 4 643653 32 2426 6 5- -a - - +- + - + - ---++-++ - --+-+ - - -- 6 51 3 5 4 5615 554 6 5 554 1 65562 - ---- ---- 5 33 2 31 5 -+- 5313 5 563 212 26- -b +- ---- +-- - + --- 2 6 2 25 6 6 61 46 44 3 463 45 56 252 65 152 -- -++------- 551 24245334 64 4 56 ++-- ---++++- 3 53 4 6 5 626 13 5214 - - - - + + -- +- - + - -- - + -+ -- ------++-- 55 5 6 4634 5213226 5 66122245 2 1 3 1 5352 2 2 5 -- 5 6 3 5323 356 22122 1 2 3 45 + - +- - -- -+--- - 15 33 34255 6 23 5 552 33 4 33 6 33 62 +a ---+--+----+--+-+------++--+- -+------ - 2461 6413 51 313 1 35 53 56 5 4 3 +--+ - 1 42+ +-- - -+- - --++ --- 42 44 6 2243 4 56 5 52 2 22 ---- - -- -------+ -+--++ - + - 543533 2 5 6 16 3 4 5 6 1 5 6 54 5 452 3 246 11 465414 13 14 2 12 5 212 4 5 41 2331 253636 4343124 24 5 65 4 5 2 5 5 6 55 2 +--+------++-----+-+--- --- - + + ++ --- ++ - - +-+-+- --++ + -------+--- - +--+- -+ -- -- + + -- - - - ---- -- -- --- 55 11455 5 26666 232 2 4 3 34 63346 232663241533 551 6 5 2 5 13136343452126522 41 4 5352 53424 6 533 3 21 2 5166 23125 6 1 524 46 4 2 2 5 35 3 6 6 316 5312345 54 15 53 3 42 5 5 6 53 66 3 2+ -- -+ +--- +-++--- - -+ -- -- - - --+ -+--++ +b- --++ --- ---++ +----+---+-+ ++ -- --+ - -+-- -- --- +++ + ------+ 2 16 32 3 26 4 1 45 2632 4264534264 6 45 31 5 2 222631 4 53 5132663363 31521 6 42 32 566 54 41 3 6 3 65 4 6 353 +- + +-- + - + - ----+ - - - +-+++-+--- + --- - - - -- - - -++ + -- - +- +- - 1425 121144 23362 2 4544 2 641 6 52 1 2322 55 4 1 5 55 5 5623 225 61421525351 35335 23 62 4 5422 22 5 6 2 54 4 12 6511151 63 3532 41 46 - + 63 6 2 431 52 552 1 3 3 4 13 24 55 6 21123253 15 123 3 6 55 1 5255 1 36 +- ++- -- ---+----++----+ -- ---+-- 4 2 552 136 6 4 ++ + - - - --- +- -- 41 2 45 -- ++ -+- -- -- ++- -- --+------- -- -+ ---++ + 1 - - + -- + ++---++----------++--+- + --- --+ + - +-- -+ - +-+ --- + 3 3 56651 54225211451 3323 3 635 13 6212155 42 2 524 2 5 31 55326 323 26 151 6 3 6 5 25 642 4 2 44 26533 125 2 55 4 6 32 22 6 5 4 1 6 2 5 213 3 2 3 36 412 326 6 2 3 412 3253 2333 321 1 5 3665326 45531 5233 3 25 3 635 6 3 66 4 2 426 554 - + --++- --+--- +-+- ----+- - +++ -- - -+- +- +--- -- -----+ --+ + - 2 3 35 4454 53 32 451 1 51 6 46 5 262 3 2 22215322 4 62 23 424 4 ---+- + - ---+-++ --+-- - --+- +-- 32334231 4223261 22145 4 36 6 2 - - 4 +---- 4 vs222 1878 6 FCAk 0.27% +a 2 5 4 vs222 1878 6 local 19.67 % 0.02 % global 4 1 -a 0 6 -b 1 2 -1 3 +b -2 -2 -1 0 1 2 COSIT’09 l’Aber Wrach, France,%21-25 September 2009 vs222 local 31.62 0.08 % global
  • 29. some conclusions & issues• Spatial Entropy from order k co-occurrences + SOOk, SelSOOk and CAkOO complementary methods + CAkOO provides a map if using S x I x J ... Data + map of local Spatial Entropies• exploratory analysis: + scale analysis + comparison to a model in SOOK can be done also in CAkOO + needs correction edge effects + needs of fuzziness (CAkOO) COSIT’09 l’Aber Wrach, France, 21-25 September 2009
  • 30. split(ppp) - + all for 3 MRSA classes: 1 3 5 135 split(ppp) 1.0 oo o o o o o o o o o - - - - - - - - -Self order 3 Spatial Entropy HsSu - - 0.9 1 3 5 + ++ + ++ ++ + + + + ++ + ++ + + ++ + ++ + + + ++ ++ + ++ + split(ppp) + ++ + ++ + ++ + +++ + + + +++ ++ ++ + 0.8 + ++ + + + + + + ++ ++ +++++ + + + + + + + + + ++ + + ++ + +++ + + + + + + ++ ++ ++ + + + ++++ + ++ + + + ++ + ++ + + + ++ + ++ ++ + + +++ ++ + + ++ + ++ ++ + + +++ +++ + + + + ++ + + + + ++ + +++ +++ + +++ +++ + ++ + + + + + + + + + + + ++ ++ +++ + + + + +++ + + +++ +++ ++ + +++ + + + + + + + ++++ + + o all + ++ + + +++ + + ++ + + + ++ + + + + ++ ++ +++ + ++ + ++ ++ + ++ + ++ + + ++ 0.7 + MRSA (+) 1 - 3 5 ---- - ---- ---- -- -- ------- - SA (-) -- -- -- --------- ---- - - -- - - - -- - -- - - - -------- -- - - - ------ - - -- --- - - - - -- -- --- - -- - 0.6 - -- --------- ----- -- - - ----- - ----- - ------------ - - - ---- -- -- - ------- - - --- - -- - - - -- - -- -- ------ - ----- -- - - --- ---------- - - ----- - ---------- -- -- - -- - --- - -- - - -- - ---- ---- ---- - - ----- -- -- -- -- -- - ---- - - -- -- - - --- --- ---- --------- - - ---- - -- - - -- -- -- -------- -- --- -- - -- - - - ---- -- ---- -- -- - ----- - - ------ - - - - -- --- - --- - -- - ---- -- --------- - - -- --------- --- -------- -- -- -- ---- --- --- - -- - - -- -- -- -- - -- -- - ---------- --- - ---- -- - - - - - --------------------- ---- -- - - --- ---- --- - ----- - - -- - --------- ------- ---- -- ----- ----- - - -- - - - - - - -- -- - --- -- - - ------- -- -- - - --- ---------- --- - ------------- - -- --- ------------- ---- - -- - 0.5 ----- -- - - ---------- ---- - - ---- -- ----------- ---- - - -- -- - - - --- --- -- ---------- -- -- - -- ----- --- --- ----- - - -- - ++ + + + + + + + + + 0.4 2000 4000 6000 8000 d COSIT’09 l’Aber Wrach, France, 21-25 September 2009
  • 31. - + all for 3 MRSA classes: all 135 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.8 1.0 0.4 0.5 0.6 0.7 0.8 0.9 1.0 oo o o o o o o o o oSelf order 3 Spatial Entropy HsSu Self collocation prob -- - - - - - - - - - - 0.8 1.0 Self collocation prob 1 3 5 o all at d=2000 + - SA (-) 1 3 5 0.0 0.2 0.4 0.6 0.8 1.0 at d=2000 Self collocation prob + MRSA (+) ++ + + + + + + + + + 1 3 5 at d=2000 2000 4000 6000 8000 d COSIT’09 l’Aber Wrach, France, 21-25 September 2009