# 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.

 x -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.25 0 0.25 0.4 0.7 1 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

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

 $A=[\,\,1\,\,|\,\,x\,\,]$

This is clarified by looking at the example in Matlab, \PMlinktofileLinearLeastSquares.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

Title example of linear least squares ExampleOfLinearLeastSquares 2013-03-22 16:51:19 2013-03-22 16:51:19 bloftin (6104) bloftin (6104) 6 bloftin (6104) Example msc 15-00