Spatal Statstcs and Analyss Methods (for GEOG 104 class). Provded by Dr. An L, San Dego State Unversty. 1
Ponts Types of spatal data Pont pattern analyss (PPA; such as nearest neghbor dstance, quadrat analyss) Moran s I, Gets G* Areas Lnes Area pattern analyss (such as jon-count statstc) Swtch to PPA f we use centrod of area as the pont data Network analyss Three ways to represent and thus to analyze spatal data:
Randomly dstrbuted data Spatal arrangement The assumpton n classcal statstc analyss Unformly dstrbuted data The most dspersed pattern the antthess of beng clustered Negatve spatal autocorrelaton Clustered dstrbuted data Tobler s Law all thngs are related to one another, but near thngs are more related than dstant thngs Postve spatal autocorrelaton Three basc ways n whch ponts or areas may be spatally arranged 3
Spatal Dstrbuton wth p value 0 ) 4
Nearest neghbor dstance Questons: What s the pattern of ponts n terms of ther nearest dstances from each other? Is the pattern random, dspersed, or clustered? Example Is there a pattern to the dstrbuton of toxc waste stes near the area n San Dego (see next slde)? [hypothetcal data] 5
6
Step 1: Calculate the dstance from each pont to ts nearest neghbor, by calculatng the hypotenuse of the trangle: NND AB ( xa xb ) ( ya yb Ste NN NND NND A 1.7 8.7 B.79 B 4.3 7.7 C 0.98 C 5. 7.3 B 0.98 D 6.7 9.3 C.50 E 5.0 6.0 C 1.3 F 6.5 1.7 E 4.55 NND n 13.1 6.19 ) 13.1 7
Step : Calculate the dstances under varyng condtons The average dstance f the pattern were random? NND R 1 1 1.9 Densty 0.068 Where densty = n of ponts / area=6/88=0.068 If the pattern were completely clustered (all ponts at same locaton), then: 0 NND C Whereas f the pattern were completely dspersed, then: NND D 1.07453 Densty 1.07453 0.61 4.1 (Based on a Posson dstrbuton) 8
Step 3: Let s calculate the standardzed nearest neghbor ndex (R) to know what our NND value means: Perfectly dspersed.15 R NND NND R.19 1.9 1.14 More dspersed than random = slghtly more dspersed than random Totally random 1 More clustered than random Perfectly clustered 0 9
Hosptals & Attractons n San Dego The map shows the locatons of hosptals (+) and tourst attractons ( ) n San Dego Questons: Are hosptals randomly dstrbuted Are tourst attractons clustered? 10
Spatal Data (wth, coordnates) Any set of nformaton (some varable z ) for whch we have locatonal coordnates (e.g. longtude, lattude; or x, y) Pont data are straghtforward, unless we aggregate all pont data nto an areal or other spatal unts Area data requre addtonal assumptons regardng: Boundary delneaton Modfable areal unt (states, countes, street blocks) Level of spatal aggregaton = scale 11
Area Statstcs Questons 003 forest fres n San Dego Gven the map of SD forests What s the average locaton of these forests? How spread are they? Where do you want to place a fre staton? 1
What can we do? Preparatons Fnd or buld a coordnate system Measure the coordnates of the center of each forest Use centrod of area as the pont data (0, 763) (580,700) (380,650) (480,60) (400,500) (500,350) (300,50) (550,00) (0,0) (600, 0) 13
Mean center The mean center s the average poston of the ponts Mean center of : C Mean center of : C n n y x (0, 763) (580 380 480 400 500 550 300) C 7 455.71 (700 650 60 500 350 50 00) C 7 467.14 (0,0) # (380,650) #3 (480,60) #4 (400,500) #1 (580,700) (456,467) Mean center #5 (500,350) #6 (300,50) #7(550,00) (600, 0) 14
15 Standard dstance The standard dstance measures the amount of dsperson Smlar to standard devaton Formula ) ( ) ( ) ( ) ( c c D c c D n n S n S Defnton Computaton
Standard dstance Forests #1 580 336400 700 490000 # 380 144400 650 4500 #3 480 30400 60 384400 #4 400 160000 500 50000 #5 500 50000 350 1500 #6 300 90000 50 6500 #7 550 30500 00 40000 Sum of 1513700 Sum of 1771900 455.71 467. 14 C C S D ( n 1513700 ( 7 c ) ( 455.71 n ) c ) 1771900 ( 7 467.14 ) 08.5 16
Standard dstance (0, 763) Mean center # (380,650) #4 (400,500) (456,467) #1 (580,700) #3 (480,60) S D =08.5 #5 (500,350) #6 (300,50) #7(550,00) (0,0) (600, 0) 17
18 Defnton of weghted mean center standard dstance What f the forests wth bgger area (the area of the smallest forest as unt) should have more nfluence on the mean center? wc f f wc f f ) ( ) ( ) ( ) ( wc wc WD wc wc WD f f f f S f f f S Defnton Computaton
Calculaton of weghted mean center What f the forests wth bgger area (the area of the smallest forest as unt) should have more nfluence? Forests f(area) f (Area*) f (Area*) #1 5 580 900 700 3500 # 0 380 7600 650 13000 #3 5 480 400 60 3100 #4 10 400 4000 500 5000 #5 0 500 10000 350 7000 #6 1 300 300 50 50 #7 5 550 13750 00 5000 f 86 f 40950 f 36850 f 40950 f wc 36850 476.16 wc 48. 49 f 86 f 86 19
Calculaton of weghted standard dstance What f the forests wth bgger area (the area of the smallest forest as unt) should have more nfluence? S WD Forests f (Area) f f #1 5 580 336400 168000 700 490000 450000 # 0 380 144400 888000 650 4500 8450000 #3 5 480 30400 115000 60 384400 19000 #4 10 400 160000 1600000 500 50000 500000 #5 0 500 50000 5000000 350 1500 450000 #6 1 300 90000 90000 50 6500 6500 #7 5 550 30500 756500 00 40000 1000000 f 86 f 19974500 f 18834500 ( f f 19974500 ( 86 wc ) ( 476.16 ) f f wc ) 18834500 ( 86 48.49 ) 0.33 0
Standard dstance (0, 763) # (380,650) #1 (580,700) #3 (480,60) Mean center Weghted mean center #4 (400,500) (456,467) (476,48) #6 (300,50) #5 (500,350) #7(550,00) Standard dstance =08.5 Weghted standard Dstance=0.33 (0,0) (600, 0) 1
Standard dstance (0, 763) # (380,650) #1 (580,700) #3 (480,60) Mean center Weghted mean center #4 (400,500) (456,467) (476,48) #6 (300,50) #5 (500,350) #7(550,00) Standard dstance =08.5 Weghted standard Dstance=0.33 (0,0) (600, 0)
Spatal clustered? Gven such a map, s there strong evdence that housng values are clustered n space? Lows near lows Hghs near hghs 3
More than ths one? Does household ncome show more spatal clusterng, or less? 4
Moran s I statstc Global Moran s I Characterze the overall spatal dependence among a set of areal unts Covarance 5
Summary Global Moran s I and local I have dfferent equatons, one for the entre regon and one for a locaton. But for both of them (I and I ), or the assocated scores (Z and Z ) Bg postve values postve spatal autocorrelaton Bg negatve values negatve spatal autocorrelaton Moderate values random pattern 6
Network Analyss: Shortest routes (0, 763) Eucldean dstance #1 (580,700) # (380,650) #3 (480,60) #4 (400,500) d ( j ) ( j ) Mean center (456,467) #5 (500,350) (500 400) 180.8 (350 500) #6 (300,50) #7(550,00) (0,0) (600, 0) 7
Manhattan Dstance Eucldean medan Fnd ( e, e ) such that d e ( e) ( e ) s mnmzed Need teratve algorthms Locaton of fre staton Manhattan medan d 400 300 500 50 j 350 j j (0,0) (0, 763) # (380,650) #6 (300,50) #4 (400,500) Mean center (456,467) ( e, e ) #5 (500,350) #7(550,00) (600, 0) 8
Summary What are spatal data? Mean center Weghted mean center Standard dstance Weghted standard dstance Eucldean medan Manhattan medan Calculate n GIS envronment 9
Spatal resoluton Patterns or relatonshps are scale dependent Herarchcal structures (blocks block groups census tracks ) Cell sze: # of cells vary and spatal patterns masked or overemphaszed Vegetaton types at large (left) and small cells (rght) How to decde The goal/context of your study Test dfferent szes (Weeks et al. artcle: 50, 500, and 1,000 m) % of senors at block groups (left) and census tracts (rght) 30
Admnstratve unts Default unts of study May not be the best Many events/phenomena have nothng to do wth boundares drawn by humans How to handle Include events/phenomena outsde your study ste boundary Use other methods to reallocate the events /phenomena (Weeks et al. artcle; see next page) 31
A. Locate human settlements B. Fnd ther centrods C. Impose grds. usng RS data 3
Edge effects What t s Features near the boundary (regardless of how t s defned) have fewer neghbors than those nsde The results about near-edge features are usually less relable How to handle Buffer your study area (outward or nward), and nclude more or fewer features Varyng weghts for features near boundary a. Medan ncome by census tracts b. Sgnfcant clusters (Z-scores for I ) 33
Dfferent! c. More census tracts wthn the buffer d. More areas are sgnfcant (between brown and black boxes) ncluded 34
Applyng Spatal Statstcs Vsualzng spatal data Closely related to GIS Other methods such as Hstograms Explorng spatal data Random spatal pattern or not Tests about randomness Modelng spatal data Correlaton and Regresson analyss 35