37.1: Least Squares Fit
- Page ID
- 70388
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Review Chapters Chapter 13 pg 225-239 of the Boyd textbook.
In this first part of this course, we try to solve the system of linear equations \(Ax=b\) with an \(m \times n\) matrix \(A\) and a column vector \(b\).
There are three possible outcomes: an unique solution, no solution, and infinite many solutions. (Review the material on this part if you are no familiar with when the three types of outcomes happen.)
When \(m<n\), we call the matrix \(A\) underdeterminated, because we can not have an unique solution for it. When \(m>n\), we call the matrix \(A\) overdeterminated, becasue we may not have a solution with high probability.
However, if we still need to find a best \(x\), even when there is no solution or infinite many solutions we use a technique called least squares fit (LSF). Least squares fit find \(x\) such that \(\|Ax-b\|\) is the smallest (i.e. we try to minimize the estimation error).
- When there is no solution, we want to find \(x\) such that \(Ax−b\) is small (here, we want \(\|Ax-b\|\) to be small).
- If the null space of \(A\) is just \(\{0\}\), we can find an unique \(x\) to obtain the smallest \(\|Ax-b\|\).
- If there is a unique solution \(x^*\) for \(Ax=b\), then \(x^*\) is the optimal \(x\) to obtain the smallest \(\|Ax-b\|\), which is 0.
- Because the null space of \(A\) is just \(\{0\}\), you can not have infinite many solutions for \(Ax=b\).
- If the null space of \(A\) is not just \(\{0\}\), we know that we can always add a nonzero point \(x_0\) in the null space of \(A\) to a best \(x^*\), and \(\|A(x^*+x_0)-b\|=\|Ax^*-b\|\). Therefore, when we have multiple best solutions, we choose to find the \(x\) in the rowspace of \(A\), and this is unique.
Let \(A=\begin{bmatrix}1\\2\end{bmatrix},\quad b=\begin{bmatrix}1.5 \\ 2\end{bmatrix}\). Find the best \(x\) such that \(\|Ax-b\|\) has the smallest value.
Compute \((A^\top A)^{-1}A^\top b\).