2.3 Measures based on absolute adjacency 2.3.1 Area features

1Geog. 579: GIS and Spatial Analysis - Lecture 16 OverheadsGeog. 579: GIS and Spatial Analysis - Lecture 16 Overheads

2.3 Measures based on absolute adjacency 2.3.1 Area features 2.3.1.1 Geary Index (ratio/interval) 2.3.1.2 Moran Coefficient (ratio/interval) 2.3.1.3 Joint Count Statistics (categorical) 2.3.2 Other type of features

Topics:Topics:

Lecture 16: Spatial Autocorrelation III

References:References:

Goodchild, Michael F., 1986. Spatial Autocorrelation, CATMOG 47, Geo Books, Norwich, UK, 56 pp.Griffith, Daniel A., 1987. Spatial Autocorrelation: A Primer, Resource Publications in Geography, AAG, Washington, 82 pp.Odland, John, 1987. Spatial Autocorrelation, SAGE Pulications, Inc., Beverly Hills, CA, 85 pp.


OutlinesOutlines

2.3.1 Area Features2.3.1 Area Features 2.3.1.3 Joint Count Statistics: 2.3.1.3 Joint Count Statistics: Two categories (yes or no): black and whiteTwo categories (yes or no): black and white ((The Buffalo FigureThe Buffalo Figure)) 1) Calculation:1) Calculation: Counting for number of WW, BB, BW jointsCounting for number of WW, BB, BW joints J = JJ = JWWWW+J+JBBBB+J+JBWBW

J is the total number of connectionsJ is the total number of connections 2) Interpretation:2) Interpretation: a) Under the a) Under the random arrangementrandom arrangement assumption: assumption: (1) What is random arrangement(1) What is random arrangement The population of spatial patterns for a given numberThe population of spatial patterns for a given number of events over a given number of area units is created of events over a given number of area units is created by rearranging the number of events over the area unitsby rearranging the number of events over the area units


2) Interpretation: (continued…)2) Interpretation: (continued…) a) Under the a) Under the random arrangementrandom arrangement assumption: (continued…) assumption: (continued…) (2) The expected numbers of different joints(2) The expected numbers of different joints Expected EExpected EWWWW = J N = J NWW(N(NWW-1)/N(N-1) -1)/N(N-1)

Expected EExpected EBBBB = J N = J NBB(N(NBB-1)/N(N-1) -1)/N(N-1)

Expected EExpected EBWBW = 2 J N = 2 J NBB N NWW/N(N-1)/N(N-1)

NNWW, N, NBB are the occurrence of W and B events, respectively are the occurrence of W and B events, respectively

(3) Testing:(3) Testing:

BW

BWBWBW

EJZ

For BW joints:

211

)3)(2)(1(

)1()1()]1()1([4

)1(

)]1([

BW

N

iWWBBiiWB

N

iii

BWBW ENNNN

NNNNLLJJ

NN

NNLLE


2) Interpretation: (continued…)2) Interpretation: (continued…) b) Under the b) Under the random samplingrandom sampling assumption: assumption: (1) What is a random sampling:(1) What is a random sampling: The population of spatial patterns for a given number of The population of spatial patterns for a given number of events over a given number of area units is created byevents over a given number of area units is created by sampling with replacement.sampling with replacement. (2) The expected numbers of different joints(2) The expected numbers of different joints Expected EExpected EWWWW = J (N = J (NWW))22/ N/ N22

Expected EExpected EBBBB = J (N = J (NBB))22/N/N22

Expected EExpected EBWBW = 2 J N = 2 J NBB N NWW/N/N22

NNWW, N, NBB are the occurrence of W and B events, respectively are the occurrence of W and B events, respectively

(3) Testing(3) Testing

22

12

1

][)]1([4)]1(2[N

NNLLJ

N

NNLLJ WB

N

iii

WBN

iiiBW

BW

BWBWBW

EJZ

For BW joints:


3) Example: 3) Example: (1) The Pattern (The Figure)(1) The Pattern (The Figure)

(2) The Observed Joints (BW):(2) The Observed Joints (BW): JJBWBW = 6 = 6

(3) Interpretation:(3) Interpretation: (a) Under Random Arrangement:(a) Under Random Arrangement: EEBW BW = (2x19x5x6)/11x10=10.3636= (2x19x5x6)/11x10=10.3636

∑∑LLii(L(Lii-1) = (Table7.7)=114-1) = (Table7.7)=114

σσBW BW = 1.7721= 1.7721

ZZBWBW = (6-10.3635)/1.7721= -2.462 = (6-10.3635)/1.7721= -2.462

at 95% significant level, the critical value of a at 95% significant level, the critical value of a negative standard normal deivate is -1.645, thusnegative standard normal deivate is -1.645, thus the BW joints the BW joints is very unlikelyis very unlikely to have occurred to have occurred by chance.by chance.


3) Example: (continued …)3) Example: (continued …) (3) Interpretation: (continued…)(3) Interpretation: (continued…) (b) Under Random sampling:(b) Under Random sampling: EEBW BW = (2x19x5x6)/11x11=9.42= (2x19x5x6)/11x11=9.42

∑∑LLii(L(Lii-1) = 114-1) = 114

σσBW BW = 2.2323= 2.2323

ZZBWBW = (6-9.42)/2.2323= -1.532 = (6-9.42)/2.2323= -1.532

at 95% significant level, the critical value of a at 95% significant level, the critical value of a negative standard normal deivate is -1.645, thusnegative standard normal deivate is -1.645, thus the BW joints the BW joints is likely tois likely to have occurred have occurred by chance.by chance.

2.3.2 Other Features2.3.2 Other Features

2.3.2.1 Point features (Points to Areas Conversion Figure)2.3.2.1 Point features (Points to Areas Conversion Figure)

2.3.2.2 Linear features2.3.2.2 Linear features


1. How is similarity in attribute defined in computing joint count1. How is similarity in attribute defined in computing joint count statistics and how is similarity in location defined?statistics and how is similarity in location defined?

2. What is the main difference between Joint Counts and Moran2. What is the main difference between Joint Counts and Moran Coefficient? When would you apply the Joint Counts statistics?Coefficient? When would you apply the Joint Counts statistics?

3. What is the random arrangement assumption and when should 3. What is the random arrangement assumption and when should you apply it?you apply it?

4. What is the random sampling assumption and when should you4. What is the random sampling assumption and when should you apply it? What is the difference between the two assumptions?apply it? What is the difference between the two assumptions?

5. How would you measure spatial autocorrelation using the absolute5. How would you measure spatial autocorrelation using the absolute adjacency approach for point or linear features?adjacency approach for point or linear features?

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2.3 Measures based on absolute adjacency 2.3.1 Area features

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