I am trying to understand an example in my stats course notes, the example relates to calculating the best value for the next experiment.
The function of the line is very simple:
$$ln(Y_i) = ln(\theta^*_1 + \theta^*_2x_i) + \epsilon_i$$
The example works on a previous example where point estimates have already been obtained for $\theta^*_1$ and $\theta^*_2$ These are:
$$\theta = \begin{pmatrix} 3.9963\\ 2.3792 \end{pmatrix}$$
The values given already are:
i y x
1 4.11 0
2 6.32 1
3 8.21 2
4 10.43 3
5 14.29 4
6 16.78 5
So the question is, how to get the optimal value for $x_7$.
The equation that needs to be used is given, but doesnt make a lot of sense to me, particularly the last line, where some seemingly random numbers appear from nowhere.
Here is the notes provided: let the px1 vector $$r_{n+1} = \frac{\delta f(x_{n+1}. \theta)}{\delta\theta}$$
$$ C_{n+1} = X_{n+1}'X_{n+1} = \begin{pmatrix}X_n\\ r_{n+1}' \end{pmatrix}' \begin{pmatrix}X_n\\ r_{n+1}' \end{pmatrix} $$
$$ \phi = |C_{n+1}| = |\begin{pmatrix}X_n\\ r_{n+1}' \end{pmatrix}' \begin{pmatrix}X_n\\ r_{n+1}' \end{pmatrix}| = |X_n'X_n+r_{n+1}r_{n+1}'| $$ $$ \therefore \phi = |C_{n+1} + r_{n+1}r_{n+1}'| = |C_n|(1 + r_{n+1}'C_n^{-1}r_{n+1}) $$
Here comes the example "at which value of x, 0 <= x <= 2 should the next experiment be carried out"
$$ln(Y_i) = ln(\theta^*_1 + \theta^*_2x_i) + \epsilon_i$$
$$\theta = \begin{pmatrix} 3.9963\\ 2.3792 \end{pmatrix}$$
$$ X'X = \begin{bmatrix}0.11777 & 0.11660\\ 0.11660 & 0.33500 \end{bmatrix} $$
$$ r_7' = \begin{bmatrix} \frac{1}{\theta_1 + \theta_2 x_7} & \frac{x_7}{\theta_1 + \theta_2 x_7}\end{bmatrix} $$
Now, while I dont exactly understand all this - I can work out what its saying. What gets me, is the next bit.
$$ \phi = \frac{12.9358 - 8.9781x_7 + 4.534x_7^2}{(3.9963 + 2.3792x_7)^2} $$
Where did 12.9358, 8.9781, 4.534 come from?
Also, where did the devision part of this come from, above $\phi$ was calculated without any devision at all.
Finally, where did the $^2$ come from on the devisor?
Any help here would be greatly appreciated, I get the feeling its going to be very simple - I'm just missing something.
Thanks