Good to know! I will check out if ODE’s make sense for my application (and if I can make sense of them :P). I also encountered Linear Dynamical Systems in the book by Bishop (Pattern Recognition and Machine Learning), which actually almost seems to be a synonym for continuous HMM’s anyway.
Here is my attempt to sort of ‘formalize’ my problem using mathematical notation:
Let D be the number of dogs, let T_d be the number of timepoints for a dog d, let P be the number of features/attributes (i.e. test scores) at every timepoint.
Then for a dog d we have \{z_{d,t}\}_{t=1}^{T_d} with only z_{d,1} being observed, where 0 \leq z \leq 4 and z could potentially be discretized as well if desired (because in the original data z are multiples of 0.25 because it is calculated based on the answers to a questionnaire), but I believe it being continuous might work to our advantage anyway.
Additionally, for a dog d we have \{X_{d,t}\}_{t=1}^{T_d} where X is P-dimensional, because in actuality there are multiple test scores at each timepoint (whereas in my initial post I assumed there to be only 1 for simplicity’s sake). X is an integer with 0 \leq X \leq 100.
I thought it might also be helpful to include a diagram:
Lastly, I did an attempt at forming a bit of an equation, inspired by Arthur’s reply in another topic.
x_{d,t,p} = \beta_{0,p} + z_{d,t} \cdot \beta_{p} + \epsilon_{d,t,p} for d=1,\dots,D, and t=1,\dots,T_d, and p=1,\dots,P.
Where:
- x_{d,t,p} is test score p for dog d at time point t
- \beta_{0,p} is the intercept for test score p
- z_{d,t} is the latent behavioral score for dog d at time t
- \beta_p is the coefficient for the behavioral score z on the test score x_p
- \epsilon_{d,t,p} are the residuals
Additionally, for the time dependency:
z_{d,t} = f(z_{d, t-1}) for d=1,\dots,D, and t=2,\dots,T_d
where f is some function of time.