The Method of Least Squares - Elements of Scientific Computing - page 161

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

1850 1900 1950 2000

Fig. 5.11 Global annual mean temperature deviation measurements (relative to the 1961-1990
mean); from 1856 to 2000

5.1.4 Large Data Sets

Above, we derived methods for generating models of discrete data using either
constant, linear, or quadratic functions. We used the method of least squares to deter-
mine the coefficients of the approximating functions. All the calculations were done
for the data given in Table 5.1 . Here we will first extend the methods to the general
case of a data set containing n points,

.t i ;y i /i D 1;:::;n; (5.47)

and then apply our new formulas to the large data set presented in Fig. 5.11 .

Approximation by a Constant

We assume that the data ( 5.47 ) are given and that we want to determine a constant
˛ such that

p 0 .t / D ˛ (5.48)

is the best constant approximation of these data in the sense of least squares. To this
end, we define