My thinking is a little confused on this so you may have to bear with me...

In a mathematical model, we often distinguish between state variables corresponding to measurable quantities (that may change), and parameters corresponding to quantities that are fixed. We may further distinguish between observable variables (like the reading on the boundary of a volume containing a field) and latent (hidden) variables (the state of the internal field) which are not directly accessible but that we can hope to infer from the observable variables with modeling.

Let me give a quick example: Assume that we can measure the horizontal and vertical sum of a grid of obstructions by observing the boundary. The grid is parameterized as an array: $$ A = \left[\begin{matrix} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ a_{31}&a_{32}&a_{33} \end{matrix} \right]\,,\hspace{2em} a_{ij}\in\mathbb{R}\,. $$ We model the boundary observation by $$ x_i = a_{1i} + a_{2i} + a_{3i}\,,\hspace{2em} y_j = a_{j1} + a_{j2} + a_{j3}\,. $$ The observed (state) variables $x_i$ and $y_i$ are determined by the latent variables $a_{ij}$, and we can try to rediscover them via methods for under-determined systems. In this case, our "model" doesn't return a vector, it actually returns a space of possible solutions. This space is in some sense (over) parameterized by the variables $a_{ij}$.

Now, lets consider a simple ODE model for prediation like $$ P' = rP - PQ\,, \\ Q' = PQ - dQ\,. \,. $$ Here, $r$ and $d$ are parameters fixed by the population and $P$ and $Q$ (and $t$?) are state variables, they're observable, you can measure them. There are no latent variables in this model. Measuring $P_0 = P(t_0)$ gives a solution to this equation. However, the solution more than a single number - it is in fact a path through state space parameterized by $t$.

In both these case, our "model" doesn't return a fixed vector it returns a space parameterized by some continuous variables. In the first case that is a defect, we cannot fully determine $a_{ij}$ from the observations. In the second case it is a feature, not only do we get the populations at single future time, we get it at a whole line of future times.

It seems like in the second case $t$ is more than a standard state variable: the object the model returns is parameterized by it, and so it is in some sense on the same footing as the $a_{ij}$. It seems like in addition to variables (which we observe) and parameters (which are fixed) we should have a class of variables which parametrize the object returned by the model. I want to say these are, I don't know, state determining variables? Or control variables? Both of these are incorrect, state and control variables are well defined already.

Does anyone have any idea how we talk about these kinds of variables? Is there a unified way?



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