Exploring spatial patterning in data values - Spatial Data Analysis: An Introduction for GIS Users - page 116

Table 8.3 Coordinates of observations, variable 1 and 2 values,
distance from the fi rst observation, and geographical weights

No. x coordinate y coordinate Variable 1 ( y ) Variable 2 ( z ) Distance ( d ij ) Geog. wt. ( w ij )

1 25.00 45.00 12 6 0.00 1.0000

2 25.51 44.14 34 52 1.00 0.9950

3 21.87 48.90 32 41 5.00 0.8825

4 27.60 52.57 12 25 8.00 0.7261

5 16.69 31.33 11 22 16.00 0.2780

6 42.52 35.35 14 9 20.00 0.0889

7 9.20 65.65 56 43 26.00 0.0340

8 29.23 76.72 75 67 32.00 0.0060

9 61.37 66.01 43 32 42.00 0.0002

Bandwidth = 10 units.

of geographical weights using Equation 8.1 was illustrated in Section 8.4. Note that the
number of observations is small and this example is used purely for ease of illustration.
In this example, the weight matrix is therefore:

1.0000 0 0 0 0 0 0 0 0

È ˘

Í ˙

0 0.9950 0 0 0 0 0 0 0

Í ˙

Í ˙

0 0 0.8825 0 0 0 0 0 0

Í ˙

0 0 0 0.7261 0 0 0 0 0

Í ˙

Í ˙

0 0 0 0 0.2780 0 0 0 0

Wx () = Í

i

˙

0 0 0 0 0 0.0889 0 0 0

Í ˙

Í ˙

0 0 0 0 0 0 0.0340 0 0

Í ˙

Í 0 0 0 0 0 0 0 0.0060 0 ˙

Í ˙

0 0 0 0 0 0 0 0 0.0002

Î ˚

h e regression coei cients are obtained as for the standard regression approach
detailed above except that Y T is multiplied by W ( x i ). Following this procedure (by
referring to the example for global regression in Section 3.3 and Appendix E it should
be possible to work out what is going on):

1 0.9950 0.8825 0.7261 0.2780 0.0889 0.0340 0.0060 0.0001

È ˘

YWx

T

() = Í

˙ i

12 33.8300 28.2400 8.7132 3.0580 1.2446 1.9040 0.4500 0.0086

Î ˚

È 4.0107 89.4484 ˘

T

YWx Y () = Í

˙ i

89.4484 2494.2646

Î ˚

1.2454 - 0.0447

È ˘

T - 1

( YWx Y ( ) ) = Í

˙

i

- 0.0447 0.0020

Î ˚