The Method of Least Squares - Elements of Scientific Computing - page 151

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

1991 1992 1993 1994 1995 1996 1997 1998 1999 2000

Fig. 5.5 Global annual mean temperature deviation measurements (relative to the 1961-1990
mean) for the period between 1991 and 2000

what do we really mean by “as accurately as possible”? Is that uniquely determined?
No, it is not, and we shall discuss some ways of estimating ˛. An intuitive way of
estimating ˛ is to compute the average:

1 X

1
10

˛ D y i D 0:312: (5.1)

i D1

But is the average a fair approximation? To answer this, we consider the function

X

10

.˛ y i / 2 :

F.˛/ D (5.2)

i D1

Note that F measures the total deviation from ˛ to the whole data set .t i ;y i / 10
i D1

.
All these deviations, each in its square, are added up and the sum is a measure of
quality. One way of determining ˛ D ˛ is to minimize F as a function of ˛ and
let ˛ denote the value that minimizes F.˛/. We observe that F is a second-order
polynomial in ˛. Therefore, we can minimize F by computing the zero of F 0 , i.e.,
finding ˛

such that F 0 .˛ / D 0.Wehave