An ant farm has 1000 ants and the population of ants is observed to be 2000 after 1 year, 4200 after 2 years, and 8300 after 3 years. We want to make an ant population model.
We want to use quadratic polynomials to minimize:
$\sum_{i=0}^{3} |L_i - (\ln A + kt_i + \delta t^2_i)|^2$
So the ant model should be in the form:
$$P(t) = e^{\ln A} e^{kt+\delta t^2} = Ae^{kt+\delta t^2}$$
What we need to do is find the constants $A, k$ and $\delta$.
So, I know we need to find the best linear combination of vectors
$$[1,1,1,1]^T,[0,1,2,3]^T, [0,1,4,9]^T$$
to approximate the vector
$$L=[L_0, L_1, L_2, L_3]^T = [6.90776,7.6009,8.34284,9,02401]^T$$ The numbers above come from $\ln 1000, \ln 2000, \ln 2400 \ln 8300$
which is in $\mathbb{R}^4$. How would we use the projection formula
$ \begin{equation*} \begin{bmatrix} \ln A\\ k\\ \delta\\ \end{bmatrix} = (A^TA)^{-1}A^TL \end{equation*} $
to find the coeffients in $\ln A$, $k$, and $\delta$? I'm thinking we should use the matrix
$ \begin{equation*} A = \begin{bmatrix} 1&0&0\\ 1&1&1\\ 1&2&4\\ 1&3&9\\ \end{bmatrix} \end{equation*} $
So where do I go now?
Here $\ln P(t) = \ln A + kt+\delta t^2= [\ln A;~ k ;~\delta]^T[1;~t;~t^2]$. We have data at $t=0,1,2,3$. This gives us a matrix of form $L = Ax$ where $L =[\ln P(0);~\ln P(1);~\ln P(2);~\ln P(3) ]$ and matrix $A$ is as mentioned in question, we need $x = [\ln A;~ k ;~\delta]$.
Here $Ax=L$ is an over-constrained set of equations i.e., there are more equations than the number of unknowns. So, we generally try to find a least square solution, one which minimizes $\| Ax-L \|_2$ which leads to the equation written by you. Another option is to make the model complex (example making P(t) = exp(cubic polynomial)) and you will have four equations and four unknowns.