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 = M x + 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.


x	-3.0	-2.5	-2.0	-1.5	-1.0	-0.5	0.0	0.5	1.0	1.5
z	-1.0	-0.25	0.0	0.25	0.4	0.7	1.0	1.1	1.4	1.8

Table 1: Example Data

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

xfit = (AT A )−1AT z

(2)

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

A = [ 1 | x ]

This is clarified by looking at the example code in Matlab, LinearLeastSquares.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 (Matlab)

Figure 1: Linear Fit of Example Data (rlplot)

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