Finding model for time dependent functions via global model statement and constraints
I am interested in implementing a model to predict Load to Truck ratio.
The goal of the model and the training process is to find the set of functions that define the interaction between locations.
We got $N$ locations, and for a specific $t$ we have a certain amount of Loads $L(t)$ and Trucks $T(t)$. The evolution of the loads in this simplistic first stage model assumes that whenever a load disappears is due only to a truck grabbing it. So this implies that
$$ \frac{dL_i}{dt} = - \sum_{j\ne i}^Nq_{ij}(t) + s_{L_i}(t) $$
where $q_{ij}$ implies a Load to Truck assignment from location $i$ to location $j$, and this can be seen as a measure of density per unit of time of load assignments between locations (self routed locations are not taking into account because they are non existent, i.e. $q_{ii}=0$). On top of that I've added a source, new load density per unit of time, i.e. $s_{L_i}$.
On the truck side of things we got
$$ \frac{dT_i}{dt} = - \sum_{j\ne i}^Nq_{ij}(t) + \sum_{j\ne i}^Nq_{ji}(t-\Delta t_{ij}) $$
so the first terms states exactly the same in regards to load assignments. Every time one occurs both Loads and Trucks in that location take a hit. The second term of the right hand side of the equation states that trucks for which a Load Assignment was made $\Delta t_{ij}$ time units ago is arriving at location $i$.
I've got information to determine - $s_{L_i}$ - $L(t)$ - $T(t)$ - $\Delta t_{ij}$
The goal here is to find meaningful information, after a successful prediction of $T$ and $L$, on $q_{ij}$. 143
This functions represent the location-to-location interaction.
What is the most direct way of finding this interaction, time-dependent function. I have zero experience in time dependent prediction, and or models. So my naive and for sure costly approach would be,
* define a function space
* via numerical propagation find the error of the prediction
* train
The resolution of the numerical approach can manage to be on an hourly basis. Even more. Day resolution is the most desirable value. It must be said that I formulated this in a continuous way, regardless of its discrete nature, because humans are animals of habit.
Due to consistent intra-day market behavior we could state that $q_{ij}(t)$ on a low time resolution basis is periodic (4-7 day window). Also on a coarse time scale it seems to be as well a periodic function (4 - 6 months window). I guess this can be useful if I want to narrow the model space into a product of coarse and fine grained periodic functions, $f_{c}(t)g_{f}(t)$ via Fourier decomposition. But once again, maybe this is commiting to wrong assumptions.
Any comments or pointing out resources would be much appreciated. 232