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.