One of the most important applications of this theory is in the area of curve fitting (often called linear or nonlinear regression).

This topic is covered in detail in Part XII of this book; here we consider only a simple example to give a flavour of how the method works, and how it involves finding the stationary points of multidimensional functions.

An example is shown in Fig. 20.6, which plots the incidence of melanoma in the non-Maori population of New Zealand as a function of latitude, measured at four different places. Let L be the latitude, and let M be the incidence of melanoma. In order to determine the background trend, we want to find some straight line, M = aL + b, for some unknown constants a and

b, that goes as close as possible to the four data points. We know this line won’t go exactly through all four data points (we can tell that just by looking at the graph) but we’d still like some approximate idea of how steeply the points go down.
Label the four data points as (Li , Mi) for i = 1, · · · , 4.

a. Show that the vertical distance from a data point to the straight line is |aLi + b − Mi |.

b. Let S

(a,

b) be the sum of the square of each of these distances, i.e., S

(a,

b) = Õ
4 i=1 (aLi + b − Mi)
2 .
Show that, in order to find the values for a and b that make S as small as possible, you need to solve the two equations Õ
4 i=1 (aLi + b − Mi) = 0, Õ
4 i=1 Li(aLi + b − Mi) = 0.

c. Show that these two equations can be written as the linear system Í
Li 4 Í
L 2 i Í
Li  a b 
=  Í
Mi Í
LiMi 
which you can solve to find a and b.