I'm given the following model
$$\hat{y}(\theta;t)=\theta_{1}+\frac{\theta_{2}}{t+1.0}+\theta t^{3}$$
where $\theta_1,\theta_2$ and $\theta_3$ are the unknown parameters and $t$ is the independent variable. The measurement equation is
$$y_i = \hat{y}(\theta;t_i)+e_i,\ \ \ i=1,2,...,m$$
$y_i$ is the measurement at time $t_i$. It is assumed that $e_i$ is normally distributed measurement errors, i.e. $e_i \sim N(0, \sigma^{2})$ for $i=1,2,...,m$. The variance, $\sigma^{2}$ of the measurement error is not known but must be estimated. Let m = 6 in the following questions.
I'm now supposed to formulate the data fitting problem as a least squares problem
$$\underset{x\in\mathbb{R}^{n}}{\min}\ f(x)=\frac{1}{2} ||Ax-b||_{2}^{2}$$
where I have to write expressions for $A,b,x$ and $n$. I've tried for quite some time, but I really don't see how to formulate it as a least squares problem...
Any help is highly appreciated!
In general, for any linear model $$y(t)=\sum_j \theta_jf_j(t)$$ where $f_j$ are given functions and $\theta_j$ parameters to be determined, one is trying to fit $y(t_i)=y_i$. This gives a matrix equation of the form $$\pmatrix{f_1(t_1)&\cdots&f_J(t_1)\\ f_1(t_2)&\cdots&f_J(t_2)\\&\vdots\\f_1(t_N)&\cdots&f_J(t_N)}\pmatrix{\theta_1\\\vdots\\\theta_J}=\pmatrix{y_1\\\vdots\\\vdots\\y_N}$$ This overdetermined system has no solutions, so the best alternative is to find $\vec{\theta}$ which minimizes $\|A\vec{\theta}-b\|^2$.