第八讲 空间自相关分析

7
Significance
The significance test involves calculating the standard normal deviate from the calculated value of I, the expected value I, and its standard deviation.
Substituting values into the formula we get:
The equation for the standard deviation of this value is:
I
62 *9 3*92 6*58 92 62 1
219 0.278 2835
The previously calculated value of I can now be converted into a standard normal deviate using the following equation:
8
Normality
The equation for the expected value of I under the null hypothesis of
normality is:
EI
1 n 1
The equation for the standard deviation of this value is:
The observed arrangement of values is not significantly different from random (randomly sampling from a normal distribution).
It could have easily occurred under the null hypothesis of random sampling from a normally distributed population.
2.4
0.7
0.49 0.2401
0.3
-1.4
1.96 3.8416
3.8
2.1
4.41 19.4481
0.6
-1.1
1.21 1.4641
10.2
10.32 27.7236
1.7
2
XX
10.32 1.72 1.31149
n
6
4
XX
kurtosis n 4
27.7236 1.56185 6* 2.9584
Significance Level (one-tailed)
0.1
0.05
0.01
z 1.282 1.645 2.326
-z -1.282 -1.645 -2.326
0.005 2.576 -2.576
0.001 3.09 -3.09
Significance Level (two-tailed)
0.1
2.4
0.7
0.3 -1.4
3.8
2.1
0.6 -1.1
n=6
58 10.2
1.7
(X X )2
0.81 1.44 0.49 1.96 4.41 1.21
10.32
5
Calculations for Moran’s Spatial Autocorrelation Coefficient I
Join Number
1.56185 9
62 6
6*92
2*6*58
92 6 16 26 3
The previously calculated value of I can now be converted into a standard normal deviate using the following equation:
One such measure has been devised by Moran (1950) and can be applied to area patterns and to point patterns.
For areal data the equation for Moran’s coefficient is:
Randomization: The question asked is “given a particular set of values X, what is the possibility that they could have been arranged in the observed way by chance? The null hypothesis is that the spatial distribution is random.
Hale Waihona Puke 13Calculation of Kurtosis for Randomization Significance Test of I
Area
A B C D E F
Sum Mean
X (Xi X ) (X X )2 (X X )4
2.6
0.9
0.81 0.6561
0.5
-1.2
1.44 2.0736
1 2 3 4 5 6 7 8 9
Sum
Xi ( Xi X )
0.5 -1.2
2.6
0.9
0.5 -1.2
0.5 -1.2
0.3 -1.4
2.4
0.7
0.3 -1.4
3.8
2.1
0.3 -1.4
J=9
Xj ( X j X ) Xi X ( X j X )
2.6
0.9
2.4
0.7
2.4
0.7
n
I
c Xi X X j X
2
J X X
Where I = Moran’s spatial autocorrelation coefficient n = the number of areas in the study region J = the number of joins X = a value for an area (ordinal or interval) Xi, Xj = are two contiguous areas (on either side of a join) c = a pair of contiguous areas
2
J XX
Calculated:
6 * 2.83
I
0.183
9 *10.32
Moran’s coefficient (I) is -0.183, although this value on its own is not very much use in describing the degree of spatial autocorrelation in a variable.
0.05
0.01
0.005 0.001
z 1.645
1.96 2.576
2.813 3.291
-z -1.645
-1.96 -2.576 -2.813 -3.291
11
Normality
Adopting the 0.05 significance level, the two-tailed critical value for a positive standard normal deviate is 1.96.
3
Hypothetical Study Region
4
Calculations for Moran’s Spatial Autocorrelation Coefficient I
Area
A B C D E F
Sum Mean
L L^2
2
4
3
9
4
16
4
16
3
9
2
4
X (Xi X)
2.6
0.9
0.5 -1.2
10
Normality
Note that the expected value of I for a random arrangement is small and negative (-0.2)
A smaller value, one further from zero in the negative direction implies dispersion.
n2J 3J 2 n L2
I
J 2 n2 1
Where n= the number of areas in the study region J= the number of joins L=the number of areas to which an area is joined
9
Normality
There are two possible forms of the null hypothesis: normality and randomization
Normality: The null hypothesis is that the observed values of the variable are the result of a random sample from a normally distributed population of values.
Positive values imply clustering.
After converting the observed I to a standard normal deviate, its significance can be assessed by reference to a table of critical values.