4.1: Fitting a Straight Line
- Page ID
- 96051
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Suppose the fitting curve is a line. We write for the fitting curve
\[y(x)=\alpha x+\beta . \nonumber \]
The distance \(r_{i}\) from the data point \(\left(x_{i}, y_{i}\right)\) and the fitting curve is given by
\[\begin{aligned} r_{i} &=y_{i}-y\left(x_{i}\right) \\ &=y_{i}-\left(\alpha x_{i}+\beta\right) . \end{aligned} \nonumber \]
A least-squares fit minimizes the sum of the squares of the \(r_{i}\) ’s. This minimum can be shown to result in the most probable values of \(\alpha\) and \(\beta\).
We define
\[\begin{aligned} \rho &=\sum_{i=1}^{n} r_{i}^{2} \\ &=\sum_{i=1}^{n}\left(y_{i}-\left(\alpha x_{i}+\beta\right)\right)^{2} \end{aligned} \nonumber \]
To minimize \(\rho\) with respect to \(\alpha\) and \(\beta\), we solve
\[\frac{\partial \rho}{\partial \alpha}=0, \quad \frac{\partial \rho}{\partial \beta}=0 \nonumber \]
Taking the partial derivatives, we have
\[\begin{aligned} &\frac{\partial \rho}{\partial \alpha}=\sum_{i=1}^{n} 2\left(-x_{i}\right)\left(y_{i}-\left(\alpha x_{i}+\beta\right)\right)=0 \\ &\frac{\partial \rho}{\partial \beta}=\sum_{i=1}^{n} 2(-1)\left(y_{i}-\left(\alpha x_{i}+\beta\right)\right)=0 \end{aligned} \nonumber \]
These equations form a system of two linear equations in the two unknowns \(\alpha\) and \(\beta\), which is evident when rewritten in the form
\[\begin{aligned} \alpha \sum_{i=1}^{n} x_{i}^{2}+\beta \sum_{i=1}^{n} x_{i} &=\sum_{i=1}^{n} x_{i} y_{i} \\ \alpha \sum_{i=1}^{n} x_{i}+\beta n &=\sum_{i=1}^{n} y_{i} \end{aligned} \nonumber \]
These equations can be solved either analytically, or numerically in MATLAB, where the matrix form is
\[\left(\begin{array}{ccc} \sum_{i=1}^{n} & x_{i}^{2} & \sum_{i=1}^{n} \\ \sum_{i=1}^{n} & x_{i} & n \end{array}\right)\left(\begin{array}{c} \alpha \\ \beta \end{array}\right)=\left(\begin{array}{c} \sum_{i=1}^{n} x_{i} y_{i} \\ \sum_{i=1}^{n} y_{i} \end{array}\right) . \nonumber \]
A proper statistical treatment of this problem should also consider an estimate of the errors in \(\alpha\) and \(\beta\) as well as an estimate of the goodness-of-fit of the data to the model. We leave these further considerations to a statistics class.