PlanetPhysics/Example of Linear Least Squares

The assumption of linear least [[../PiecewiseLinear/|squares]] is that there is a linear relationship between our measurements $z$ and the variables to be estimated $x$

$z=Mx+b$

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

{\mathbf Table 1:} Example Data

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

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

The only not so obvious step before using a tool like Matlab, is to form the $A$ [[../Matrix/|matrix]], which is a combination of an [[../Cod/|identity]] [[../Vectors/|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

$z=0.543x+0.947$

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

{\mathbf Figure 1:} Linear Fit of Example Data (Matlab)

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

{\mathbf Figure 1:} Linear Fit of Example Data (rlplot)