example of linear least squares

The assumptionPlanetmathPlanetmath 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 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-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=(ATA)-1ATz (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 vectorsMathworldPlanetmath, 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


Figure 1: Linear Fit of Example Data

Title example of linear least squares
Canonical name ExampleOfLinearLeastSquares
Date of creation 2013-03-22 16:51:19
Last modified on 2013-03-22 16:51:19
Owner bloftin (6104)
Last modified by bloftin (6104)
Numerical id 6
Author bloftin (6104)
Entry type Example
Classification msc 15-00