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$

$\displaystyle z = Mx + b$ (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 -3.0 -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

$\displaystyle x_{fit} = (A^TA)^{-1}A^Tz$ (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

$\displaystyle A = [    1    \vert   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

$\displaystyle z = 0.543 x + 0.947$

\includegraphics[scale=.6]{LinearLeastSquares2.eps}

Figure 1: Linear Fit of Example Data (Matlab)

\includegraphics[scale=.8]{least_squares.eps}

Figure 1: Linear Fit of Example Data (rlplot)



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