I’ve talked a little bit before about the potential of being driven mad by maths while carrying out mathematical modeling. Now I don’t want to alarm anyone, but currently my desk is buried underneath hundreds of bits of paper (even more than usual!) with equations and attempts to work out how to program certain things.
What particular thing has driven me mad? Well, to begin with I need to talk a little bit about my model.
My model is a dynamic state variable model, which produces a decision matrix for every combination of states. In my case I have 4 different states, including timestep (the model evolves through time until it reaches a maximum number of timesteps). So this results in a four dimensional decision matrix. Attempt to picture that for a moment..
Oo, my brain feels fuzzy.
Actually, it isn’t so bad when you picture it as a series of 3 dimensional cubes.
So for every timestep, and for every combination of my 3 state variables, which include oxygen debt and whether or not a bird has encountered prey, there is either a 1 or a 0, telling a bird what the optimum decision is: dive or rest. These will vary depending on a whole bunch of initial parameters I set, like how efficient an individual bird is at foraging.
Of course, there’s not a huge amount I can immediately do with such a decision matrix. It’s hard to answer questions with these matrices, or at least to make easy to understand graphs!
So the next step is to do forward iteration. This will let me start a bird off with a certain set of conditions and then see how it evolves through time. To do this I have to create a transition matrix, which stores the probability for each combination of states, of .. the probability of having reached those states, from any other combination of states. At every time step. Which makes it a… 7 dimensional matrix.
My computer is going to love this.