Spatial Analysis and Modeling (GIST 4302/5302) Guofeng Cao Department of Geosciences Texas Tech University

Transcription

1 Spatial Aalysis ad Modelig (GIST 4302/5302) Guofeg Cao Departmet of Geoscieces Texas Tech Uiversity

2 Outlie of This Week Last week, we leared: spatial poit patter aalysis (PPA) focus o locatio distributio of evets Measure the cluster (spatial autocorrelatio)i poit patter This week, we will lear: How to measure ad detect clusters/spatial autocorrelatio i areal data (regioal data)

3 Spatial Autocorrelatio Spatial autocorrelatioship is everywhere Spatial poit patter K, F, G fuctios Kerel fuctios Areal/lattice (this topic) Geostatistical data (ext topic) 3

4 Spatial Autocorrelatio of Areal Data 4

5 Spatial Autocorrelatio Tobler s first law of geography Spatial auto/cross correlatio If like values ted to cluster together, the the field exhibits high positive spatial If there is o apparet relatioship betwee attribute value ad locatio the there is zero spatial autocorrelatio If like values ted to be located away from each other, the there is egative spatial autocorrelatio 5

6 Positive spatial autocorrelatio - high values surrouded by earby high values - itermediate values surrouded by earby itermediate values - low values surrouded by earby low values 2002 populatio desity Source: Ro Briggs of UT Dallas 6

7 Negative spatial autocorrelatio - high values surrouded by earby low values - itermediate values surrouded by earby itermediate values - low values surrouded by earby high values competitio for space Grocery store desity Source: Ro Briggs of UT Dallas 7

8 Measurig Spatial Autocorrelatio: the problem of measurig earess To measure spatial autocorrelatio, we must kow the earess of our observatios as we did for poit patter case Which poits or polygos are ear or ext to other poits or polygos? Which states are ear Texas? How to measure this? Seems simple ad obvious, but it is ot! 8

9 Spatial Weight Matrix Core cocept i statistical aalysis of areal data Two steps ivolved: defie which relatioships betwee observatios are to be give a ozero weight, i.e., defie spatial eighbors assig weights to the eighbors 9

10 Spatial Neighbors Cotiguity-based eighbors Zoe i ad j are eighbors if zoe i is cotiguity or adjacet to zoe j But what costitutes cotiguity? Distace-based eighbors Zoe i ad j are eighbors if the distace betwee them are less tha the threshold distace But what distace do we use? 10

11 Cotiguity-based Spatial Neighbors Sharig a border or boudary Rook: sharig a border Quee: sharig a border or a poit rook quee Hexagos Irregular Which use? 11

12 Problem Situatios for Irregular Polygos Close but o commo border Legth of border Is Arizoa as close to Califoria as to Utah? Base closeess o proportio of shared border, ot just oe (1) or zero (0) w ij = border legth ij /border legth j ) X 12

13 Higher-Order Cotiguity 1 st order Nearest eighbor rook hexago quee 2 d order Next earest eighbor 13

14 Distace-based Neighbors How to measure distace betwee polygos? Distace metrics 2D Cartesia distace (projected data) 3D spherical distace/great-circle distace (lat/log data) Haversie formula 14

15 Distace-based Neighbors k-earest eighbors Source: Bivad ad Pebesma ad Gomez-Rubio 15

16 Distace-based Neighbors thresh-hold distace (buffer) Source: Bivad ad Pebesma ad Gomez-Rubio 16

17 Neighbor/Coectivity Histogram Source: Bivad ad Pebesma ad Gomez-Rubio 17

18 Spatial Weight Matrix Spatial weights ca be see as a list of weights idexed by a list of eighbors If zoe j is ot a eighbor of zoe i, weights Wij will set to zero The weight matrix ca be illustrated as a image Sparse matrix 18

19 A Simple Example for Rook case Matrix cotais a: 1 if share a border 0 if do ot share a border 4 areal uits 4x4 matrix A C B D Commo border W = A B C D A B C D

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21 Sparse Cotiguity Matrix for US States -- obtaied from Aseli's web site (see powerpoit for lik) Name Fips Ncout N1 N2 N3 N4 N5 N6 N7 N8 Alabama Arizoa Arkasas Califoria Colorado Coecticut Delaware District of Columbia Florida Georgia Idaho Illiois Idiaa Iowa Kasas Ketucky Louisiaa Maie Marylad Massachusetts Michiga Miesota Mississippi Missouri Motaa Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolia North Dakota Ohio Oklahoma Orego Pesylvaia Rhode Islad South Carolia South Dakota Teessee Texas Utah Vermot Virgiia Washigto West Virgiia Wiscosi Wyomig

22 Style of Spatial Weight Matrix Row a weight of uity for each eighbor relatioship Row stadardizatio Symmetry ot guarateed ca be iterpreted as allowig the calculatio of average values across eighbors Geeral spatial weights based o distaces 22

23 Row vs. Row stadardizatio A B C D E F Divide each umber by the row sum Total umber of eighbors --some have more tha others A B C D E F Row Sum A B C D E F Row stadardized --usually use this A B C D E F Row Sum A B C D E F

24 Geeral Spatial Weights Based o Distace Decay fuctios of distace Most commo choice is the iverse (reciprocal) of the distace betwee locatios i ad j (w ij = 1/d ij ) Other fuctios also used iverse of squared distace (w ij =1/d ij2 ), or egative expoetial (w ij = e -d or w ij = e -d2 ) 24

25 Measure of Spatial Autocorrelatio 25

26 Global Measures ad Local Measures Global Measures A sigle value which applies to the etire data set The same patter or process occurs over the etire geographic area A average for the etire area Local Measures A value calculated for each observatio uit Differet patters or processes may occur i differet parts of the regio A uique umber for each locatio Global measures usually ca be decomposed ito a combiatio of local measures 26

27 Global Measures ad Local Measures Global Measures Joi Cout Mora s I (ad Getis-Ord s G) Local Measures Local Mora s I (ad Getis-Ord s G) 27

28 Joi (or Joit or Jois) Cout Statistic 60 for Rook Case 110 for Quee Case 28

29 Joi Cout: Test Statistic Test Statistic give by: Z= Observed - Expected Expected give by: SD of Expected Expected = radom patter geerated by tossig a coi i each cell. Stadard Deviatio of Expected (stadard error) give by: Where: k is the total umber of jois (eighbors) p B is the expected proportio Black, if radom p W is the expected proportio White m is calculated from k accordig to: 29

30 Gore/Bush Presidetial Electio 2000 Actual Jbb 60 Jgg 21 Jbg 28 Total

31 Joi Cout Statistic for Gore/Bush 2000 by State cadidates probability Bush Gore Actual Expected Sta Dev Z-score Jbb Jgg Jbg Total The expected umber of jois is calculated based o the proportio of votes each received i the electio (for Bush = 109*.499*.499=27.125) There are far more Bush/Bush jois (actual = 60) tha would be expected (27) Positive autocorrelatio There are far fewer Bush/Gore jois (actual = 28) tha would be expected (54) Positive autocorrelatio No strog clusterig evidece for Gore (actual = 21 slightly less tha ) 31

32 Mora s I The most commo measure of Spatial Autocorrelatio Use for poits or polygos Joi Cout statistic oly for polygos Use for a cotiuous variable (ay value) Joi Cout statistic oly for biary variable (1,0) Patrick Alfred Pierce Mora ( ) 32

33 Formula for Mora s I I N åå ij i i= 1 j= 1 = ( åå i= 1 j= 1 w w ij (x ) å i= 1 - x)(x (x i - j - x) x) 2 Where: N is the umber of observatios (poits or polygos) x is the mea of the variable X i is the variable value at a particular locatio X j is the variable value at aother locatio is a weight idexig locatio of i relative to j W ij 33

34 Mora s I ad Correlatio Coefficiet Correlatio Coefficiet [-1, 1] Relatioship betwee two differet variables Mora s I [-1, 1] Spatial autocorrelatio ad ofte ivolves oe (spatially idexed) variable oly Correlatio betwee observatios of a spatial variable at locatio X ad spatial lag of X formed by averagig all the observatio at eighbors of X

35 å i= 1 å i= 1 (y 1(y i - i - y) 2 y)(x i å i= 1 - x)/ (x i - x) 2 Correlatio Coefficiet Note the similarity of the umerator (top) to the measures of spatial associatio discussed earlier if we view Yi as beig the Xi for the eighborig polygo (see ext slide) N åå i= 1 j= 1 ( åå i= 1 j= 1 w w ij ij (x ) i å i= 1 - x)(x (x Spatial auto-correlatio i j - x) - x) 2 = åå w i= 1 (x (x i - x)(x x) 2 i= 1 x)/ ij i j i= 1 j= 1 i= 1 j= 1 å Source: Ro Briggs of UT Dallas - - å (x i åå - x) 2 w 35 ij

36 å i= 1 å i= 1 (y 1(y i - i - y) Yi is the Xi for the eighborig polygo N åå i= 1 j= 1 ( åå i= 1 j= 1 w w ij ij (x ) i å i= 1 - x)(x (x i j 2 - x) y)(x - x) i å i= 1 Mora s I 2 = - (x x)/ i - åå x) 2 w i= 1 (x (x i x) x)(x 2 i= 1 x)/ ij i j i= 1 j= 1 i= 1 j= 1 å Source: Ro Briggs of UT Dallas Correlatio Coefficiet å Spatial weights (x i åå - x) 2 w 36 ij

37 Mora Scatter Plots We ca draw a scatter diagram betwee these two variables (i stadardized form): X ad lag-x (or W_X) The slope of this regressio lie is Mora s I 37

38 Mora Scatter Plots Low/High egative SA High/High positive SA Low/Low positive SA High/Low egative SA 38

39 Mora Scatterplot: Example 39

40 Statistical Sigificace Tests for Mora s I Based o the ormal frequecy distributio with Z I - E( I) = Serror( I ) Where: I is the calculated value for Mora s I from the sample E(I) is the expected value if radom S is the stadard error Statistical sigificace test Mote Carlo test, as we did for spatial patter aalysis Permutatio test No-parametric Data-drive, o assumptio of the data Implemeted i GeoDa 40

41 Test Statistic for Normal Frequecy Distributio *techically 1/(-1) 2.5% Reject ull /(-1) % 1% 2.54 Null Hypothesis: o spatial autocorrelatio *Mora s I = 0 Alterative Hypothesis: spatial autocorrelatio exists *Mora s I > 0 Reject Null Hypothesis if Z test statistic > 1.96 (or < -1.96) ---less tha a 5% chace that, i the populatio, there is o spatial autocorrelatio ---95% cofidet that spatial auto correlatio exits Reject ull at 5% Reject ull at 1% 41

42 Null Hypothesis: o spatial autocorrelatio *Mora s I = 0 Alterative Hypothesis: spatial autocorrelatio exists *Mora s I > 0 Reject Null Hypothesis if Z test statistic > 1.96 (or < -1.96) ---less tha a 5% chace that, i the populatio, there is o spatial autocorrelatio ---95% cofidet that spatial auto correlatio exits 42

43 Local Measures of Spatial Autocorrelatio 43

44 Local Idicators of Spatial Associatio (LISA) Local versios of Mora s I, ad the Getis-Ord G statistic Mora s I is most commoly used, ad the local versio is ofte called Aseli s LISA, or just LISA See: Luc Aseli 1995 Local Idicators of Spatial Associatio-LISA Geographical Aalysis 27:

45 Local Idicators of Spatial Associatio (LISA) The statistic is calculated for each areal uit i the data For each polygo, the idex is calculated based o eighborig polygos with which it shares a border A measure is available for each polygo, these ca be mapped to idicate how spatial autocorrelatio varies over the study regio Each idex has a associated test statistic, we ca also map which of the polygos has a statistically sigificat relatioship with its eighbors, ad show type of relatioship 45

46 Example: 46

47 Bivariate LISA Mora Scatter Plot for GDI vs AL Mora s I is the correlatio betwee X ad Lag-X--the same variable but i earby areas Uivariate Mora s I Bivariate Mora s I is a correlatio betwee X ad a differet variable i earby areas. Mora Sigificace Map for GDI vs. AL 47

48 Bivariate LISA ad the Correlatio Coefficiet Correlatio Coefficiet is the relatioship betwee two differet variables i the same area Bivariate LISA is a correlatio betwee two differet variables i a area ad i earby areas. 48

49 Bivariate Mora Scatter Plot Low/High egative SA High/High positive SA Low/Low positive SA High/Low egative SA 49

50 Summary Spatial autocorrelatio of areal data Spatial weight matrix Measures of spatial autocorrelatio Global Measure Mora s I Local LISA: Mora s I Bivariate LISA Sigificace test 50

51 Cosequeces of Igorig Spatial Autocorrelatio correlatio coefficiets ad coefficiets of determiatio appear bigger tha they really are You thik the relatioship is stroger tha it really is the variables i earby areas affect each other Stadard errors appear smaller tha they really are exaggerated precisio You thik your predictios are better tha they really are sice stadard errors measure predictive accuracy More likely to coclude relatioship is statistically sigificat. 51

52 Diagostic of Spatial Depedece For correlatio calculate Mora s I for each variable ad test its statistical sigificace If Mora s I is sigificat, you may have a problem! For regressio calculate the residuals map the residuals: do you see ay spatial patters? Calculate Mora s I for the residuals: is it statistically sigificat? 52

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54 Whe (spatial) correlatio happes Try to thik of omitted variables ad iclude them i a multiple regressio. Missig (omitted) variables may cause spatial autocorrelatio Regressio assumes all relevat variables ifluecig the depedet variable are icluded If relevat variables are missig, model is misspecified 54

55 Spatial Regressio Methods Spatial Ecoometrics Approaches (available i GeoDa) Lag model Error model Spatial Statistics Approaches Simultaeous Autoregressive Models (SAR) A more geeral case of Spatial Ecoometrics Coditioal Autoregressive Models (CAR) Other methods: Bayesia spatiotemporal methods 55

56 Ed of this topic 56