Key concepts 2 Statistics - Spatial Data Analysis: An Introduction for GIS Users - page 33

i gures that tell us something about the nature of the relationship between variables.
h e least squares method is the most common approach to i tting a line to data and
obtaining values of b 0 and b 1 . h is method minimizes the squared dif erence between
the observed value ( z i , the measurement) and the value given by the line i tted with
regression ( z ˆ):

n

Â

ˆ

2

( zz - )

(3.6)

i i

i = 1

h e following text describes how the intercept and slope are obtained through the
ordinary least squares (OLS) method. h e slope coei cient, b 1 , is obtained from:

Â

n

( yyzz - )( - )

i i

b =

i = 1

(3.7)

1

Â

n

2

( yy - )

i

i = 1

h e numerator gives the covariance between the independent and dependent val-
ues. h e covariance is a measure of the degree to which two variables vary together
and is the dif erence in one value from its mean multiplied by the dif erence in the
second value from its mean. h e covariances for each location are summed. h e
denominator is the sum of squared dif erences between the independent values and
their mean.
h e intercept, b 0 , is given by:

ÂÂ

n n

z - b y

(3.8)

i 1 i

b =

i = 1 i = 1

= z - b y

0 1

n

h e values used in Table 3.1 are used to illustrate the regression procedure. Note
that this sample is very small and in practice regression analyses should be based on
much larger samples. However, this small sample allows direct illustration of the
methods. h is topic is discussed further in Section 3.4.
h e slope is computed i rst. Initially, we compute the numerator of Equation 3.7:

n

 1 (

yyzz - )( - )

i i

i =

We take each value of y and subtract the mean value of y ; in turn we take each
value of z and subtract the mean value of z . h e dif erence between each y value and its
mean and each z value and its mean is then multiplied together as shown in Table 3.2
(in the column headed ( -¥ -). h is is done for all of the observations and

the multiplied values are added together. h e mean value of y is 32.11 and the mean
value of z is 33.
h e sum of the mul ti plied dif erences in Table 3.2 is 3092. h e denominator of
Equation 3.7,

yy zz ) ( )

i i

2

= Â , is then used. In words, we take each value of y , subtract
its mean, square this dif erence (see the column headed

n
i

(

i yy

)

1

(

i yy

- )

2

in Table 3.2), and add