I'm trying to model the response of ant populations to pheromones in this way:
The ants are simulated as Self Propelled Particles with internal energy. They undergo acceleration due to their internal energy but are slowed by a friction (due to stress, fatigue, etc.). In the absence of interaction with the environment, they undergo quasi-brownian motion.
At each time step of the simulation, the position $\textbf{r}_{i}(t)$ of the $i^{th}$ ant in the colony is determined by these differential equations:
$$ \dot{\textbf{r}}_{i} = \textbf{v}_{i} \ , \ \dot{\textbf{v}}_{i} = -\gamma v_{i}^\alpha \textbf{v}_{i} + \textbf{F}_{i}^{s} + \sqrt{2D_{v}} \bf{\xi_{i}} $$
where:
$-\gamma v_{i}^\alpha \textbf{v}_{i}$ is the friction term with $\alpha$ determining the nature of the friction force, and $\gamma$ determining the amplitude,
$\sqrt{2D_{v}} \bf{\xi_{i}}$ is the brownian motion term, with $D_{v}$ as determining the amplitude, and $\bf{\xi}_{i}$ determining the direction, chosen from a gaussian distribution.
$\textbf{F}_{i}^{s}$ is the interaction term, which incorporates two forms of interaction:
$$ \textbf{F}_{i}^{s} = \textbf{F}_{i}^{c} + \textbf{F}_{i}^{P} $$
where,
$\textbf{F}_{i}^{c}$ is the "Colony Term", which accounts for interactions with other ants, for instance a pursuit or escape response dependent on the position and velocity of other ants relative to the $i^{th}$ ant, etc. We will not worry about this term. However, it may be necessary to know that this term exists to properly answer my question.
$\textbf{F}_{i}^{P}$ is the "Pheromone Term", which is modeled as follows:
(There will also be an imposed cutoff on both the colony and pheromone terms to eliminate the possibility of acceleration or velocity values beyond the capability of real ants, but these are arbitrarily decided, and I suspect not mathematically important to the question)
Ants will be attracted to, or repelled from pheromone with the following dependence:
$$ \textbf{F}_{i}^{P} = \sum_{n = 0,1,2,...} \frac{A_{n}}{(\textbf{r}_{i} - \textbf{p})^{n}} \bf{\mathscr{r}}_{p} $$
where we are summing forces with dependence on some coefficient $A_{n}$ times one over the magnitude of the vector from the location of the pheromone $\bf{p}$ to the ant $\textbf{r}_{i}$ to some power $n$ (with $\bf{\mathscr{r}}_{p}$ being the unit length vector pointing from the pheromone to the ant). The ant is attracted to the pheromone if $A_{n} < 0$ and repulsed from it if $A_{n} > 0$.
Ants on a table top will be motion tracked to determine their position, velocity and acceleration in time steps determined by the frame rate of the camera. The information from a control test of ants on the tabletop with no pheromone present will be used to determine the values of $\gamma$, $\alpha$, and $D_{v}$, and to tune the colony term ($\textbf{F}_{i}^{c}$).
The ants will then be simulated, using successive non-scaled n-dependences. For instance, the first simulation will be run with $\textbf{F}_{i}^{P} = \frac{1}{(\textbf{r}_{i} - \textbf{p})}\bf{\mathscr{r}}_{p}$, the next with $\textbf{F}_{i}^{P} = \frac{1}{(\textbf{r}_{i} - \textbf{p})^{2}}\bf{\mathscr{r}}_{p}$, etc., where $A_{n}$ has been set to one in every case.
These forces on the ants will give rise to distinct velocity, and acceleration profiles. I will characterize these profiles using the population's mean velocity in the direction of the pheromone over time $ \langle v_{p}(t) \rangle $. I suspect that $\langle v_{p}(0) \rangle=0$ and $\langle v_{p}(T) \rangle=0$ for some characteristic time $T>0$:
$\langle v_{p}(0) \rangle=0$ because the ants will initially have no collective reason to be heading toward the pheromone yet, and
$\langle v_{p}(T) \rangle=0$ because at some point after the ants make their way to the pheromone, the imposed cutoff will eliminate drastic forces toward the pheromone and brownian motion (with all individuals congregated around, or far away from the pheromone, depending on attraction or repulsion).
I would like to take data on the $\langle v_{p}(t) \rangle$ for a wide array of n-dependencies in the pheromone term, and use these distinct cases of n-dependence as something like basis-functions that will sum to fit a real experimentally determined function for $\langle v_{p}(t) \rangle$. These basis-functions will be scalable by their coefficients $A_{n}$.
For instance, if $\langle v_{p, 1}(t) \rangle$ is the mean velocity toward the pheromone over time for $n=1$, $\langle v_{p, 3}(t) \rangle$ is the mean velocity toward the pheromone over time for $n=3$, and so on, I would like to express the experimental $\langle v_{p}(t) \rangle_{exp}$ as a sum of a small number of scaled basis-functions, for example:
$$ \langle v_{p}(t) \rangle_{exp} = A_{1} \langle v_{p,1}(t) \rangle + A_{3} \langle v_{p, 3}(t) \rangle + A_{7} \langle v_{p, 7}(t) \rangle $$
Now,
Here is my question:
Given the functions that I will be working with, is it possible to use Principle Component Analysis to find the (let's say) three basis-functions that could best be scaled and summed to fit the experimentally determined function? I understand that PCA is an orthogonal transformation that takes sets of data and converts them to sets of linearly uncorrelated data, where the 1st Principle Component contains the largest variance of the data, the second the second largest, etc. But is it possible to do this with basis-functions and an experimental function - is it possible to intelligently identify which basis-functions would be best to choose and scale to fit the experimental function with the least functions possible? I don't think, even with the constraints on $t=0$ and $t=T$, that the basis-functions will be 'orthogonal' in any sense that I know of. But, for some reason it seems like I should be able to use at least the concept behind PCA for something like this.
Any help or guidance would be much appreciated.