# 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 | -3.0 | -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

$${x}_{fit}={({A}^{T}A)}^{-1}{A}^{T}z$$ | (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

$$A=[\mathrm{\hspace{0.17em}\hspace{0.17em}1}|x]$$ |

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 |
---|---|

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 |