Geography Reference
In-Depth Information
80
70
-1.8
60
50
7.9
-17.6
40
-8.1
9.2
30
-7.1
20
-4.9
10.4
11.9
10
0
0
10
20
30
40
50
60
70
80
y
Figure 3.5
Scatter plot. Elevation (
y
) in metres (m) against
snowfall (
z
) in centimetres (cm) showing residuals.
under-predicts the observed value by 4.862. Figure 3.5 shows residuals (to one
decimal place) for the scatter plot in Figure 3.4.
Examining the residuals, perhaps by mapping them, may prove illuminating and
may help to highlight regions with unusual characteristics. One goal of regression is
to minimize the squared residuals while another is to minimize clustering in the
values of the mapped residuals (see Sections 4.8 and 8.2 for discussions on a related
topic).
Before discussing measures of goodness of i t, some grounding is briel y given in
more ei cient means for obtaining regression coei cients than was provided above.
h is background is necessary to enable readers to make the most of the descriptions
of local regression procedures that come later. h e following text introduces the idea
of matrices and matrix multiplication—concepts essential to the worked example.
Another key topic, inversion, is outlined in Appendix E.
Using matrix notation, the ordinary least squares (OLS) regression (the standard
method of i nding regression coei cients as outlined above) coei cients can be
obtained using:
b
=
YY
(
T
)
-
1
Yz
T
(3.9)
h e upper case bold letters indicate a set of values that can be arranged in a rectan-
gle with at least two rows and columns. h e lower case bold letter indicates the case
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