I am writing my bachelor thesis or whatever it is called about modelling of city growth, using a book called Cities and Complexity by Michael Batty. It is not that mathematical, to be honest, it has lot of text and formulas sprinkled here and there.
The part I am using (the book is about 550 pages) is simple enough, but I would like help understanding what mister Batty means when he writes thus:
To demonstrate this, we define population in area $i$ at time $t$ as $P_i(t)$ /.../. (Means removed part of text) Then the change in population between any two time periods $t$ and $t+1$ is defined as $\Delta P_i(t) = P_i(t+1)-P_i(t)$, with the rate as $\Delta P_i(t)/P_i(t)$. We are now in a position to formalize the process of growth.
First we state that the rate of growth is random: $$\frac{\Delta P_i(t)}{P_i(t)}=\varepsilon_i(t),$$ where $\varepsilon_i(t)$ is the random increment of growth associated with the area $i$ from time $t$ to $t+1$. Using an appropriate form of the equation 8.8 [mentioned above] and integrating from the initial distribution $P_i(0)$ to the current $P_i(t)$ yields $$\log[P_i(t)]-\log[P_i(0)] = \sum_{\tau = 0}^t \varepsilon_i(\tau).$$
The model thus becomes $$P_i(t)=P_i(0) \prod_{\tau = 0}^t \varepsilon_i(\tau),$$ where the change from one time period to the next is $$P_i(t+1)=\varepsilon_i(t) P_i(t).$$
What baffles me are these lines:
Using an appropriate form of the equation 8.8 [$\frac{\Delta P_i(t)}{P_i(t)}=\varepsilon_i(t)$] and integrating from the initial distribution $P_i(0)$ to the current $P_i(t)$ yields $$\log[P_i(t)]-\log[P_i(0)] = \sum_{\tau = 0}^t \varepsilon_i(\tau).$$
The model thus becomes $$P_i(t)=P_i(0) \prod_{\tau = 0}^t \varepsilon_i(\tau),$$ where the change from one time period to the next is $$P_i(t+1)=\varepsilon_i(t) P_i(t).$$
What is the "appropriate form"? What he means by "integrating" in this context? How do you get to the forms he mentions? And finally I, using simple algebra, came to the form of the model $P_i(t+1)=(1+\varepsilon_i(t)) P_i(t)$, which is in conflict what the writer gets.
Can anyone explain these to me?
Thank you.
He's hacking integrals to make an estimated growth formula, which would actually be a sum of many numbers.
For a Riemann integral, you have:
$\int_0^tg(x) dx=\lim_{n \rightarrow \infty}\sum_{i=1}^n g(x^*_i) \Delta x$, where $x_i^*$ is a sample in a segment of size $\Delta x$ and $\Delta x$ becomes infinitesimal as the number of samples $n$ goes to infinity.
It LOOKS like $\frac{\Delta f}{\Delta t}\frac{\Delta t}{f}=\frac{\Delta f}{f}=\varepsilon_i \implies \lim_{n \rightarrow \infty}\sum_{i=1}^n \frac{\Delta f}{f} = \lim_{n \rightarrow \infty}\sum_{i=1}^n \varepsilon_i$, but he's not actually letting the number of segments go to infinity, nor $\Delta f$ go infinitesimal.
If he did, though, for logarithms we have:
$\int \frac{1}{x} dx = ln(x)+C$
And the fundamental theorem of calculus says:
$\int_0^tg(x) dx=(\int g(x) dx)(t) - (\int g(x) dx)(0).$
So, $\int_0^a \frac{df}{dt}\frac{dt}{f} = \int_0^a \frac{df}{f}=ln(f(a))-ln(f(0)).$
So, on the left, he's treating the time periods as continuous, while on the right he's treating them as discrete.
Then the sum on the right is converted to a product by using both sides as the exponent to the same number, $e$.