Session 3: Spatial
autocorrelation tests
Course on Spatial Econometrics with Applications
Profesora: Coro Chasco Yrigoyen
Universidad Autónoma de Madrid
Lugar: Universidad Politécnica de Barcelona
12-13, 18-20 de junio, 2007
©2007, Coro Chasco Yrigoyen
All Rights Reserved
Course Index
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S1: Introduction to spatial econometrics
S2: Spatial effects, spatial dependence
S3: Spatial autocorrelation tests
S4: Exploratory Spatial Data Analysis (ESDA)
S5: Specification of spatial dependence models
S6: Spatial regression models: OLS estimation and testing
PS1: GeoDa: introduction and ESDA
S7: Spatial dependence models: estimation and testing
S8: Modelling strategies in spatial regression models
PS2: SpaceStat: confirmatory spatial data analysis
S9: Specification of spatial heterogeneity models
S10: Spatial heterogeneity models: estimation and testing
PS3: Practical exercise and evaluation
@ 2007, Coro Chasco Yrigoyen
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2
. CHASCO, C. (2003), “Econometría espacial aplicada a la
predicción-extrapolación de datos microterritoriales”. Comunidad
de Madrid; pp. 62-78.
Overview and Goals
„
Global spatial autocorrelation
1.
2.
3.
4.
„
Moran’s I
Geary’s c
Mantel’s Γ
Getis and Ord’s G(d)
Local spatial autocorrelation
1. Getis and Ord’s local statistics
4. LISA tests
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3
Session 3
Session 3
3.1. Global spatial autocorrelation
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„
Used to test for the presence of general spatial trends in the
distribution of a geographical variable over a whole space.
But how can we determine the existence
of spatial autocorrelation?
3.1.1.
3.1.2.
3.1.3.
3.1.4.
Moran’s I
Geary’s c
Mantel’s Γ
Getis and Ord’s G(d)
@ 2007, Coro Chasco Yrigoyen
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Rta. disp. por hab. (1997)
(miles ptas.)
1.400 a 1.800
1.125 a 1.400
900 a 1.125
4
Session 3
. MORAN, P. (1948), “The interpretation of statistical
maps”. Journal of the Royal Statistical Society B, vol.
10; pp. 243-251.
3.1. Global spatial autocorrelation
3.1.1. Moran’s I
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Moran’I
theoretical
mean: E(I) =
W*
„
Possitive aut.
„
Negativ aut.
N: sample size
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5
. CLIFF, A. y J. ORD (1973), “Spatial
autocorrelation”. London: Pion.
. CLIFF, A. y J. ORD (1981), “Spatial processes,
models and applications”. London: Pion
Session 3
3.1. Global spatial autocorrelation
3.1.1. Moran’s I (II)
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For N → ∞, z(I) follows a standard normal
distribution: z(I) ∼ N(0,1)
Inference is typically based on a
standardized z-value,
Assumptions:
„
„
Normalisation: the variable X follows an asymptotic
normal distribution.
Randomisation by permutation: unknown
distribution function for X
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Session 3
3.1. Global spatial autocorrelation
3.1.1. Moran’s I (III)
„
Normalisation: the variable X follows a
normal distribution
1) For N → ∞, zN(I) follows a standard normal
distribution: zN(I) ∼ N(0,1)
2) Significance of zN(I): in a standard normal table
1
EN ( I ) = −
N −1
VarN ( I ) =
4 AN 2 − 8 ( A + D ) N + 12 A2
4 A2 ( N 2 − 1)
1 N
A = ∑ Li = S0
2 i =1
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1 n
D = ∑ Li (Li − 1)
2 i =1
7
Session 3
3.1. Global spatial autocorrelation
3.1.1. Moran’s I (IV)
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Permutation: randomisation with unknown
distribution function
1) A reference distribution for I is generated empirically.
2) Randomly permuting observations & computing Moran’s for a
set of n! new samples
3) E[I] & SD[I] are computed directly from the generated
distribution of Moran’s Is
4) Significance of z(I): in a standard normal table.
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3.1. Global spatial autocorrelation
3.1.1. Moran’s I (V)
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Session 3
3.1. Global spatial autocorrelation
3.1.1. Moran’s I (VI)
„
Interpretation:
Non-significant values for z(I) should be interpreted as a rejection
of H0(no spatial autocorrelation).
© Significant z(I) > 0 ⇒ positive spatial autocorrelation: it is
possible to find out similar high/low values of a variable X
spatially clustered than could be by chance.
© Significant z(I) < 0 ⇒ negative spatial autocorrelation: there is
a lack of similar high/low values of X spatially clustered than
could be by chance. This pattern is perfectly represented by a
checkerboard.
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Session 3
3.1. Global spatial
autocorrelation
3.1.1. Moran’s I (VII)
„
A negative significant z(I): spatial autocorrelation (lack of
clustering more than would be in
a random pattern)
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Session 2
. CLIFF, A. y J. ORD (1981), “Spatial processes,
models and applications”. London: Pion; chapter 5.
3.1. Global spatial autocorrelation
3.1.1. Moran’s I (VIII)
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„
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Correlogram: an analytic method that is of value
in assessing the spatial scale of a process.
Sometimes the strength of spatial interaction will
vary in a complex way with distance.
Higher-order spatial autocorrelation: spatial
correlogram
1.5
Z (I) M ORAN
1
0.5
0
1
-0.5
-1
-1.5
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2
3
4
5
6
7
8
9
Session 3
3.1. Global spatial autocorrelation
3.1.2. Geary’s c
N − 1) ∑ ( 2) wij ( xi − x j )
(
c=
2 S0
N
∑(x − x )
i =1
„
2
2
i
„
Geary’s c
theoretical
mean: E(c) = 1
Perfect possitive aut.: c = 0,
xi ≅ xj → xi – xj = 0
Geary’s c: depends on the (absolute) difference between neighboring values of
a variable. It is similar to the Durbin-Watson test. It’s a variance test.
Moran’s I: depends on the difference between each value of X variable and its
mean. It is similar to the Pearson correlation coefficient. It’s a covariance test.
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Session 3
3.1. Global spatial autocorrelation
3.1.2. Geary’s c (II)
„
„
For N → ∞, z(c) follows a standard normal
distribution: z(c) ∼ N(0,1)
Inference is typically based on a standardized z-value,
c − E (c)
z (c) =
SD ( c )
„
„
Normalisation: the variable X follows an asymptotic normal distribution.
Randomisation by permutation: unknown distribution function for X
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Session 3
3.1. Global spatial autocorrelation
3.1.2. Geary’s c (III)
„
Interpretation:
c − E (c)
z (c) =
SD ( c )
Non-significant values for z(c) should be interpreted as
a rejection of H0(no spatial autocorrelation).
© Significant z(c) < 0 ⇒ positive spatial autocorrelation: it is
possible to find out similar spatially clustered high/low values of
a variable X than it would be by chance.
© Significant z(I) > 0 ⇒ negative spatial autocorrelation: there is
a lack of clustered similar high/low values of X than it would be
by chance.
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15
Session 3
3.1. Global spatial autocorrelation
3.1.3. Mantel’s Γ
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Mantel (1967): matrix association index, which is the sum of
the cross-product of the coincident elements of matrices A, B:
Γ = ∑ ∑ a ij bij
i
j
wij
xi x j
Moran’s I
(x − x )
i
j
2
Geary’s c
Spatial association measures can be obtained, in general, expressing
similarities by means of matrices: 1) spatial similarity (e.g., the spatial
weight matrix) and 2) value similarities.
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Session 3
3.1. Global spatial autocorrelation
3.1.4. Getis and Ord G(d)
„
Spatial autocorrelation is measured as a distanced-based or spatial
clustering measure. For this test, two spatial units are neighbors if
they are located at a certain distance (d).
N
G (d ) =
N
∑∑ w ( d ) x x
i =1 j =1
N
ij
∑∑ x x
i
X>0
j
;
N
i =1 j =1
„
i
j
for j ≠ i
W = binary,
symmetric
It measures the association degree
existent between the values of X
around “i” and the association in
the value of X around “j”
j
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3.1. Global spatial autocorrelation
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Session 3
3.2. Local spatial autocorrelation
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„
Concentration -in a particular zone of the global space- of
particularly high/low values of a variable more than the expected
mean value (or mean of the variable).
This phenomenon takes place in non-stationary spatial
processes: spatial dependence changes with location.
© Sometimes there is no global spatial autocorrelation in a
variable but small spatial clusters, in which it takes a
significant concentration/lack of high values.
© Sometimes there is global spatial autocorrelation in a
variable, but each region contributes differently to it.
TESTS: 1. Getis and Ord’s local statistics
2. LISA tests.
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Session 3
3.2. Local spatial autocorrelation
3.2.1. Getis and Ord’s local statistics
„
Gi(d), Gi*(d), New Gi(d), New Gi*(d)
„
Gi(d) measures the concentration (or lack or it) of the weighted
sum of values of variable Y in a subregion of “j” locations around
“i” in the global space.
GLOBAL
N
G (d ) =
LOCAL
N
N
∑∑ w ( d ) x x
i =1 j =1
N
ij
N
∑∑ xi x j
i =1 j =1
i
j
Gi ( d ) =
∑ w (d ) x
j =1
ij
N
∑x
j =1
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j
j
⎧j≠i
⎪
; for ⎨ x j > 0
⎪
⎩W binary & symmetric
20
Session 3
3.2. Local spatial autocorrelation
3.2.1. Getis and Ord’s local statistics (II)
„
Gi*(d): Local spatial
concentration also considers
the value of variable X in “i”.
Since wii = 0, the only
difference with Gi(d) is only
in the denominator.
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N
Gi∗ ( d ) =
∑ w (d ) x
j =1
ij
j
N
∑x
j =1
j
⎧j
⎪
for ⎨ x j > 0
⎪
⎩W binary & symmetric
21
Session 3
3.2. Local spatial autocorrelation
3.2.1. Getis and Ord’s local statistics (III)
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„
„
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New Gi(d), New Gi*(d): the standardized versions of
Gi(d) and Gi*(d).
They distribute as normal variables.
Significant positive values of these tests = positive
spatial autocorrelation = concentration of high values
of the variable.
Significant negative values of these tests = positive
spatial autocorrelation = concentration of low values of
the variable.
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Session 3
Anselin, L. (1995). “Local indicators of spatial
association - LISA.” Geographical Analysis 27, 93–115.
3.2. Local spatial autocorrelation
3.2.2. LISA tests
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LISA: Local Indicators of Spatial Autocorrelation
Detect the contribution of each location to global
spatial autocorrelation
Local spatial autocorrelation statistics are useful to
identify hot spots: Spatial concentration of high/low
values or Spatial outliers
Local autocorrelation is always present in global spatial
autocorrelation, but it can also exist in the absence of it.
Local Moran’s I is the most popular.
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Session 3
3.2. Local spatial autocorrelation
Anselin, L. (1995). “Local indicators of spatial
association - LISA.” Geographical Analysis 27, 93–115.
3.2.2. LISA tests (II)
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„
Gives an indication of the extent of
significant spatial clustering of similar
values around one observation “i”.
The sum of LISAs for all observations
is proportional to the global Moran’s I.
Local Moran’s I (LISA)
zi, zj: standaridzed yi values
For a row-standardised W
„
OBS I_DIST01
1
168.0678
2
-1.155578
3
-0.88391
4
0.044727
5
-5.440304
Z_DIST01
-1.845431
0.842808
0.342822
0.291255
1.480166
P_DIST01
0.000557
0.677512
0.673284
0.950351
0.019687
The moments for Ii
statistic, under the
null hypothesis of no
spatial association,
can be derived for a
randomisation
hypothesis.
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24