How to perform multivariate linear regression on this example?

Question

So I have a table with points in time $t_i$ and measurements $s_i$ give by

\begin{bmatrix}t_i&0&1&2&3\\s_i&4.30&1.48&0.56&0.24\end{bmatrix}

As a regression function I'm given

$\hspace{6cm}s(t)=a_1 2^{-2t}+a_2 2^{-t}$

Now I'm supposed to find the parameters $a_1$ and $a_2$ with the least square method, which means I have to minimize

$\hspace{6cm} \sum\limits_{i=0}^{3}|s(t_i)-s_i|^2$.

I think I understood linear regression quite well and would be able to perform it on this example but the multiple variables of the function really throws me off.

Answer 1

$$s(t)=a \:2^{-2t}+b\: 2^{-t}$$ Compute the variable : $\quad x_i=2^{-t_i}$ $$s(x)=a \:x^2+b\: x$$ In order to find the approximates of $a,b$ one have to make a linear regression.

This means linear with respect to the sought parameters $a,b$, of course not with respect to the functions $x^2,x$.

I suppose that all is explain in your textbook on a more general manner.

If you are trouble by the function $x^2$ you can do this (not recommended, only for information) :

$$\frac{s(t)}{2^{-t}}=a \:2^{-t}+b$$ Compute $\quad y_i=\frac{s_i}{2^{-t_i}}$ $$y(x)=a \:x+b$$ You are certainely more familiar with this simpler form.

The result is slightly different from above. This is not surprising because the fitting is with respect to $y$ instead of to $s$.

Answer 2

Do you know how to do multivariate linear regression? Let one of your variables be $x_1 = 2^{-2t}$ and the other be $x_2 = 2^{-t}$, calculate those on your whole dataset and now your problem is simply optimizing $y(x) = a_1x_1 + a_2x_2$.