ExampleOfLinearLeastSquares 2007-03-24 10:42:50 2007-03-24 10:42:50 Topic % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The assumption of linear least squares is that there is a linear relationship between our measurements $z$ and the variables to be estimated $x$ \begin{equation} z = Mx + b \end{equation} For this example let us assume that our measurements are given in Table 1 and you can see them plotted in Figure 1. \begin{center} \begin{tabular}{ccccccccccc} \hline \\ 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 \\ [2ex] \hline \end{tabular} \end{center} \begin{center} {\bf Table 1:} Example Data \end{center} The linear least squares solution to fit the given data is given by the equation \begin{equation} x_{fit} = (A^TA)^{-1}A^Tz \end{equation} 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 a column vectors, such that $$ A = [ \,\, 1 \,\, | \,\,x \,\,] $$ This is clarified by looking at the example code in Matlab, {LinearLeastSquares.m}{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 $$z = 0.543 x + 0.947$$ \begin{center} \includegraphics[scale=.6]{LinearLeastSquares.eps} {\bf Figure 1:} Linear Fit of Example Data \end{center}