example of linear least squares

The assumption of linear least squares is that there is a linear relationship between our measurements $z$ and the variables to be estimated $x$

$$z=Mx+b$$(1)

For this example let us assume that our measurements are given in Table 1 and you can see them plotted in Figure 1.

Table 1: Example Data

The linear least squares solution to fit the given data is given by the equation

$$x_{fit}={(A^{T}A)}^{-1}{A}^{T}z$$(2)

The only not so obvious step before using a tool like Matlab, is to form the $A$ matrix, which is a of an identity vector and $x$ as column vectors, such that

$$A=[\mathrm{\hspace{0.17em}\hspace{0.17em}1}|x]$$

This is clarified by looking at the example in Matlab, LinearLeastSquares.mLinearLeastSquares.m. A plot of fitting the measurement data with a line such that it minimizes the the mean square of the error is shown in Figure 1. The equation of the line to fit this data is then

$$z=0.543x+0.947$$

Figure 1: Linear Fit of Example Data