Statistical modeling multidimensional discrete system

I have a system which its state is described by a vector $$v=(a, b, c)$$, where $$a$$, $$b$$ and $$c$$ can take any value between $$0$$ and $$100$$ and where $$a+b+c <= 100$$. I have observations of the state of the system for about 10 years (one per year, so 10 observations).

How can I model the states of the system?

Can I accurately model the state of the system using Markov chain? Or shall I use another technique?

The first approach that I used was to cluster the possible states of the system (for example a cluster contains all the observations where: $$a<33$$ & $$b<33$$ & $$c<33$$). Then I fitted a transition matrix and made the predictions (considering a homogeneous transition matrix) This works fine but the predictions are of course not accurate because of the clustering.

If $$(A+B+C)<=100$$ all three are non-negative, then you can pretend there's fourth non-negative variable $$D$$ whose value you don't include in the output, so that $$(A+B+C+D)=100$$.

Next, we can divide by 100 to rescale, so now $$(a+b+c+d) = 1$$, where $$a = A/100;b=B/100$$ etc.

This gives us something that looks a lot like a Dirichlet distribution with $$K=4$$.

Now that you can definitely fit; you can throw it into a Gibbs sampler or some variational approach at very least.

If you find a stationary distribution, all that's left is to remember to transform the 'lowercase' probabilities back into 'uppercase' values of the state by multiplying them by 100 again.

Re: comment:

Bayesian updates are asymmetric by design; if you're conditioning on time, it's time-asymmetric.

For a time-homogenous chain, by simple application of Bayes:

$$p(V_{T+dT}|V_T) = p(V_T|V_{T+dT})*p(V_{T+dT})/p(V_T)$$

where $$V_T$$ is your pick of $$(a,b,c,d)$$ at a point in time $$T$$, with $$dT$$ as a delay.

Until my time machine is fixed, past is independent of the present, so $$p(V_t|V_{T+dT}) = 1$$. This leaves us with the problem of finding the ratio $$p(V_{T+dT})/p(V_T)$$, which corresponds to a transition matrix for some time skip between two states.

For $$T_0$$, you'd use your best guess, e.g. $$0.50$$ for two variables and $$0.25$$ for four if you have no good reason to favor the odds for any one of them. Then find a transition matrix $$M \sim Dir$$ , satisfying $$V_{T_1} = V_{T_0} * M_V$$, plug it into $$V_{T_2} = V_{T_1} * M_V = V_{T_0} * {M_V}^2$$, etc.

You'll want to use whatever time step you can get based on your data. E.g. if you have daily aggregate data, this will give you a change over 1 day; however, as per the $$T_2$$ case above, you can trivially deal with missing datapoints.

The same procedure works for higher-order chains, including chains with conditional dependencies between pairs of nodes, but I'm not going to write it all up here for now.

• Thanks! Just one question, how would this method take into account the time component? Sorry but I am not that familiar with the topic – Giulia Martini Nov 23 '19 at 15:28
• Edited to explain, too long to fit a comment. – jkm Nov 23 '19 at 18:53
• -1 There's nothing dynamic about the model presented, except the process of discovery of the distribution FIXED in time. But there is a confusion of state space dimension with the number of states. – Konstantin Nov 24 '19 at 10:46
• There absolutely is something dynamic. $M$ acts as a derivative of parameters over lag $dT$. Perhaps I should have been clearer: I've described an algorithm to fit the model, where the $p(V_{T+dT}|V_T)$ is a given, based on your training dataset; the desideratum is the $M$, which you can then apply N times to any starting state to predict its evolution for lag N time. Otherwise, I don't see what you're objecting to, exactly. – jkm Nov 24 '19 at 12:02
• I would love to be proven wrong. But to me many things seemed to be off in your write up and I even slept on my comment before posting it. Maybe you could add a reference, that inspired you for this approach to the problem, some resource to help me understand? – Konstantin Nov 24 '19 at 16:54