Analysis of spatial correlations in marked point processes

Transcription

1 Analysis of spatial coelations in maked point pocesses with application to micogeogaphic economical data Joint wok with W. Bachat-Schwaz, F. Fleische, P. Gabanik, V. Schmidt and W. Walla Stefanie Eckel Institute of Stochastics DFG Gaduietenkolleg 1100 Ulm Univesity. 1

2 Oveview Motivation Mathematical basics Point pocesses in Ê 2 Distance dependent chaacteistics fo spatial coelations Test on independent labelling Application to eal data: puchasing powe in Baden-Wüttembeg. 2

3 Motivation Puchasing powe Sum of the funds available fo fee consumption Impotant fo site selection, egional planning, picing policy, maket eseach Data desciption 1111 townships in BW (Statistisches Landesamt Baden Wüttembeg) Relative puchasing powe fo 1987, 1993, 1998 and 2004 Pepocessing: mean elative puchasing powe is constant in all yeas Diffeences fo thee diffeent time intevals => (Change of) spatial coelations. 3

4 Motivation Puchasing powe Changes of elative puchasing powe of townships in BW fo diffeent time intevals (decease = empty cicle, incease = full cicle). 4

5 Mathematical basics 2D point pocesses Spatial point pocesses X = {X 1,X 2,...} a sequence of andom vectos with values in Ê 2 X(B) = #{n : X n B} numbe of points X n located in B B(Ê 2 ) If X(B) < fo each bounded B B(Ê 2 ), then X is called a andom point pocess. A point pocess X is called stationay (homogeneous) if the distibution of X is invaiant w..t tanslations of the oigin isotopic if the distibution of X is invaiant w..t otations aound the oigin. 5

6 Mathematical basics 2D point pocesses The intensity measue Λ is defined as In the stationay case Λ(B) = EX(B) Λ(B) = λ B λ = 0.01 λ =

7 Mathematical basics 2D point pocesses Stationay Poisson pocess Poisson distibution of point counts: X(B) Poi(λ B ) fo any bounded B B(Ê 2 ) and some λ > 0 Independent scatteing of points: the point counts X(B 1 ),...,X(B n ) ae independent andom vaiables fo any paiwise disjoint B 1,...,B n B(Ê 2 ). 7

8 Mathematical basics Distance dependent chaacteistics Ripley s K-function λk() = E X n B X(b(X n,)) 1 λ B b(x,) cicle with adius > 0 aound x Ê 2 λk() is the expected numbe of points in a cicle with adius aound a andomly selected point of X Poisson case K Poi () = π 2 K() > π 2 (K() < π 2 ) indicates clusteing (epulsion). 8

9 Mathematical basics Distance dependent chaacteistics Pai coelation function g() g() = ρ(2) () λ 2, ρ (2) () second poduct density function Let C 1 and C 2 be two discs with infinitesimal aeas dv 1 and dv 2 and midpoints x 1 and x 2 with x 1 x 2, espectively (distance = x 1 x 2 ) The pobability fo having in each disc at least one point of X is appoximately equal to ρ (2) ()dv 1 dv 2 Poisson case g Poi () = 1 g() > 1 (g() < 1) indicates clusteing (epulsion). 9

10 Mathematical basics Maked point pocesses Conside sequence Y = {Y 1,Y 2,...} with Y i = {X i,m i } X i Ê 2 and {X 1,X 2,...} (stationay and isotopic) point pocess M i Å with Å mak space Y is called a maked point pocess Independently labelled if {M 1,M 2,...} iid and independent of {X 1,X 2,...} M d = M i. 10

11 Mathematical basics Distance dependent chaacteistics Let Å be discete (e.g. Å = {1,...,m}) Bivaiate K-function K ij () = IE X (i) n B X (j) (b(x (i) n,) \ {X (i) n }) λ i λ j B λ j K ij () is the expected numbe of points of X (j) within a cicle with adius aound a andomly chosen point of X (i) Bivaiate pai coelation function g ij () = (2) ij () λ i λ j. 11

12 Mathematical basics Distance dependent chaacteistics Simpson Index D D = 1 m i=1 Independent labelling: pobability to select a point pai at andom belonging to diffeent components Measue of divesity Distance independent => Genealisation to distance dependent Simpson indices λ 2 i λ 2. 12

13 Mathematical basics Distance dependent chaacteistics α() = 1 n i=1 λ 2 ik ii () λ 2 K() Pobability to select a point pai at andom belonging to diffeent components conditional to the event that it has distance less than Independent labelling => α() = D Range of mak coelation => α() = D, α() < D (α() > D) smalle (lage) divesity than in case of independent labelling. 13

14 Mathematical basics Distance dependent chaacteistics β() = 1 n i=1 λ 2 i g ii () λ 2 g() Pobability to select a point pai at andom belonging to diffeent components conditional to the event that it has distance Independent labelling => β() = D Range of mak coelation => β() = D, β() < D (β() > D) smalle (lage) divesity than in case of independent labelling. 14

15 Mathematical basics Distance dependent chaacteistics Let Å = Ê Mak coelation function κ() M(x) mak w..t. location x Ê 2 conditional to the event that thee is a point in x κ() = IE o,x (M(o)M(x)) Expected poduct of maks M(o) and M(x), x = unde the condition of having points of X in o and in x In the following IE o M(o) = IE x M(x) = 0. 15

16 Mathematical basics Distance dependent chaacteistics Two Point Palm Mak distibution function F (s) = lim δ 0 IE#{{i,j} : X i,x j B, d(x i,x j ) δ,m i M j s} IE#{{i,j} : X i,x j B, d(x i,x j ) δ} d(x i,x j ) Euclidean distance between X i and X j κ() = Ê sdf (s) Independent labelling => κ() = 0 κ() > 0 (κ() < 0) => positive (negative) coelation w..t point pai distance. 16

17 Mathematical basics Test on independent labelling Let Å = {1,...,m} be discete Independent labelling: K 11 () = K 22 () =... = K mm () m = 3 s T = ω 12 ( l )( K 11 ( l ) K 22 ( l )) 2 +ω 13 ( l )( K 11 ( l ) K 33 ( l )) 2 l=1 +ω 23 ( l )( K 22 ( l ) K 33 ( l )) 2, ω ij ( l ) = (V a( K ii ( l ) K jj ( l ))) 1 fo l = 1,..s, i = 1,..., 3 and i < j 3 Monte Calo ank test. 17

18 Mathematical basics Test on independent labelling Simulate k ealisations based on independent labelling Given the ealisation of locations and the obseved values of the maks assign the maks completely andomly to the locations Estimate the K ii -functions at distances { 1,..., s } Estimate the vaiance ω 1 ij ( l) of the diffeence K ii ( l ) K jj ( l ) fom the k simulations (l = 1,...,s) Compute t,t 1,...,t k Aange the values t,t 1,...,t k in ascending ode Detemine ank i of t Reject null-hypothesis if i R α, e.g. fo k = 999 and α = 0.05 R α = [951, 1000]. 18

19 Application to eal data Data desciption Townships categoized in inceasing deceasing constant elative puchasing powe Locations of townships in BW. 19

20 Application to eal data Simpson index all inceasing deceasing constant D Numbe of points and distance independent Simpson index fo diffeent time intevals. 20

21 Application to eal data Distance dependent Simpson index α() 1 0,8 Alpha() 0,6 0,4 0, Distance dependent index α() and D fo diffeent time intevals ( , , ). 21

22 Application to eal data Distance dependent Simpson index β() 0,6 0,4 Beta() 0, ,2 Distance dependent index β() and D fo diffeent time intevals ( , , ). 22

23 Application to eal data Mak coelation function 6E-6 5E-6 4E-6 Kappa() 3E-6 2E-6 1E-6 0E Estimated mak coelation function κ() fo diffeent time intevals ( , , ). 23

24 Application to eal data Test on independent labelling K() K() K() K ii functions fo diffeent time intevals ( K11, K22, K33 ). 24

25 Application to eal data Test on independent labelling K() K() K() K ii functions fo diffeent time intevals ( K11, K22, K33 ) Rank Results of the Monte Calo ank test fo diffeent time intevals. 25

26 Application to eal data Mak coelation function fo Obeschwaben 4E-6 Kappa() 2E-6 0E E-6 Estimated mak coelation function κ() fo diffeent time intevals ( , , ) fo the egion of Obeschwaben. 26

27 Application to eal data Mak coelation function fo Stuttgat 4E-6 3E-6 Kappa() 2E-6 1E-6 0E Estimated mak coelation function κ() fo diffeent time intevals ( , , ) fo the egion of Stuttgat. 27

28 Application to eal data Mak coelation function fo Obeschwaben 0, E-6 0, , Kappa() 0-0, E-6 Kappa() 0E0-2E E-6 0E0 Kappa() -1E , E-6-2E wise confidence intevals fo the mak coelation function of diffeent time intevals fo the egion of Obeschwaben. 28

29 Application to eal data Mak coelation function fo Stuttgat 2E-6 3E-6 4E-6 1E-6 Kappa() 0E0-1E E-6 1E-6 Kappa() 0E0-1E E-6 2E-6 Kappa() 1E-6 0E0-1E E-6-2E-6-2E wise confidence intevals fo the mak coelation function of diffeent time intevals fo the egion of Stuttgat. 29

30 Application to eal data Summay Positive coelations in the change of elative puchasing powe Fo small point pai distances Deceasing (vanishing) fo late time intevals Fo both types of chaacteistics Mak coelation function κ() Simpson indices α() and β() Tests on independent labelling confim esults Diffeence ual/uban egions. 30

31 Application to eal data Conclusions Reasons fo positive spatial coelations Wokplaces in neighboing townships Simila entepises in neighboing townships (metal, textile,...), vetical pocess chain (automobile industy,...) Reasons fo vanishing of positive coelations Inceased mobility of population Lage divesity in type and location of the wok Inceased divesification of entepises in townships Diffeence uban/ual egions Rual: lage divesity in type/location of wok only in the last yeas Uban: aleady; aveaging out of effects. 31

32 Application to eal data Outlook Consequences Puchasing powe itself less spatially coelated Vey detailed infomation necessay w..t. location, e.g. distibution of subsidies Futhe investigations Modelling by andom fields Coss coelation to othe chaacteistics (e.g. unemployment, motality) Application to othe data (e.g. measuement points of ecological data). 32

33 Refeences W. Bachat Schwaz, S. Eckel, F. Fleische, P. Gabanik, V. Schmidt, W. Walla (2007) An investigation on the spatial coelations fo elative puchasing powe in Baden Wüttembeg. Pepint. P. J. Diggle (2003) Statistical Analysis of Spatial Point Pattens. Anold, London. D. Stoyan, H. Stoyan (1994) Factals, Random Shapes and Point Fields. John Wiley & Sons, Chiceste. Fo futhe infomation see 33