# Linear Regression: linear or reciprocal function?

The problem is given below:

Simultaneous values of time $t$ and output $y$ from a specific sensor has been measured and is tabulated below $$\begin{array}{cc} t & y \\ \hline 1 & 17 \\ 2 & 15 \\ 3 & 11 \\ 4 & 10\\ 5&8\\ 6&7\\ 7&7 \end{array}$$ Determine whether the model $y_1(t) = \beta_0 + \beta_1 t$ or $y_2(t) = \gamma_0 + \gamma_1t^{-1}$ gives the best least squares fit to the data.

by having the design matrix and observation vector

In the first model I get following beta: $\beta_0=17.71$ and $\beta_1= -1.75$

by the formula:

$\beta = (X^TX)^{-1}X^Ty$.

My question is then, how to make the design matrix for the second model. the $t^{-1}$ confuse me!

Simply take the second column of your X1 matrix and invert the values, that is it should be $X1(:,2) = [1,1/2,1/3,\ldots,1/7]^T$

Then determine which line gives you a better fit between the two.

It should be noted, this type of regression is sometimes called "Linear Regression With Basis Functions." This concept can add a tremendous amount of flexibility to linear regression and you can get all kinds of neat fits. If you have interest, a simple search of this will give you many interesting articles/presentations/lecture notes on the topic.

Linear function

$$y_{1}(t) = \beta_{0} + \beta_{1} t$$ Linear system: \begin{align} \mathbf{A} \beta &= y \\ % \left[ \begin{array}{cc} 1 & 1 \\ 1 & 2 \\ 1 & 3 \\ 1 & 4 \\ 1 & 5 \\ 1 & 6 \\ 1 & 7 \\ \end{array} \right] % \left[ \begin{array}{c} \beta_{0} \\ \beta_{1} \end{array} \right] % &= % \left[ \begin{array}{c} 17 \\ 15 \\ 11 \\ 10 \\ 8 \\ 7 \\ 7 \end{array} \right] % \end{align} Solution: $$\beta_{LS} = % \left[ \begin{array}{c} \beta_{0} \\ \beta_{1} \end{array} \right] = % \frac{1}{28} \left[ \begin{array}{r} 496 \\ -49 \end{array} \right] % = % \left[ \begin{array}{c} 17.7143 \\ -1.753 \end{array} \right] %$$ Error: $$r = \mathbf{A} \beta - y \qquad \Rightarrow \qquad r \cdot r \approx 7.67857$$

Reciprocal function

$$y_{2}(t) = \gamma_{0} + \frac{\gamma_{1}} {t}$$ Linear system: \begin{align} \mathbf{A} \beta &= \frac{1}{y} \\ % \left[ \begin{array}{cc} 1 & 1 \\ 1 & 2 \\ 1 & 3 \\ 1 & 4 \\ 1 & 5 \\ 1 & 6 \\ 1 & 7 \\ \end{array} \right] % \left[ \begin{array}{c} \gamma_{0} \\ \gamma_{1} \end{array} \right] % &= % \left[ \begin{array}{c} \frac{1}{17} \\ \frac{1}{15} \\ \frac{1}{11} \\ \frac{1}{10} \\ \frac{1}{8} \\ \frac{1}{7} \\ \frac{1}{7} \end{array} \right] % \end{align} Solution: $$\gamma_{LS} = % \left[ \begin{array}{c} \gamma_{0} \\ \gamma_{1} \end{array} \right] = % \frac{1}{4398240} \left[ \begin{array}{c} 181296 \\ 68891 \end{array} \right] % = % \left[ \begin{array}{c} 0.0412201 \\ 0.0156633 \end{array} \right] %$$ Error: $$r = \mathbf{A} \gamma - \frac{1}{y} \qquad \Rightarrow \qquad r \cdot r \approx 0.000213205$$