Consider the experimental observations given in the following table:

t 1 4 8 11

y 1 2 4 5

Find the least-squares linear fit to the data.

Question

Consider the experimental observations given in the following table:

t	1	4	8	11
y	1	2	4	5

Find the least-squares linear fit to the data.

Accepted Answer

For the function defined by y = mt + c, the data lead to the overdetermined system
m + c = 1
4m + c = 2
8m + c = 4
11m + c = 5.
In matrix terms, the system is Ax = b, where

[latex]A=\left[\begin{array}{ll}1 & 1 \4 & 1 \8 & 1 \11 & 1\end{array} ight], \quad x =\left[\begin{array}{c}m \c\end{array} ight], \quad ext { and } \quad b =\left[\begin{array}{c}1 \2 \4 \5\end{array} ight][/latex].

The least-squares solution, [latex]X ^{*}[/latex], is found by solving [latex]A^{T} A x =A^{T} b[/latex], where

[latex]A^{T} A=\left[\begin{array}{rr}202 & 24 \24 & 4\end{array} ight] ext { and } A^{T} b =\left[\begin{array}{l}96 \12\end{array} ight][/latex].

There is a unique solution to [latex]A^{T} A x =A^{T} b[/latex] because A has rank 2. The solution is

[latex]x ^{*}=\left[\begin{array}{l}12 / 29 \15 / 29\end{array} ight][/latex].

Thus the least-squares linear fit is defined by

[latex]y=\frac{12}{29} t+\frac{15}{29}[/latex].

The data points and the linear fit are sketched in Fig. 3.23.

Question 3.8.2: Consider the experimental observations given in the followin...

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