[Math Evenings] Usage of Markov Chains and Eigenvectors in Long-term behavior estimation

Today, I learnt new techniques in the field of statistics that is extremely useful. Previously, when I was working on my HiMCM and IMMC (both modelling competitions), there were several questions that I couldn’t answer—most notably:

1. A lot of the variables are interrelated. To find how a variable would change after, say, 10 years, do I have to calculate how every value would change for every single year?

2. But what if I want to know when the variables will stabilize, or what will happen when the year approaches infinity?

My deepest concerns were entirely addressed by this new mathematical concept: Markov chains and eigenvalue.

First, a Markov chain is a method of approximating the future when one has full knowledge of the present conditions and past history. For instance, consider a SIR model, often used for disease modelling. S stands for susceptible, while I is for infected, and R is for removed. But what makes the calculations complicated in a SIR model is that all three variables are closely interrelated with each other. Markov chains express this entangled nature in a diagram (fig. 1) so that it is much easier to formulate.

fig. 1. Markov Chain in SIR model

However, after creating the linear multivariable equations that express each variable at time , a problem arises: because the value at time t is dependant on the values of every other variable at time , one has to calculate the value of every single variable until reaching time t. That is to say, because of the interlinked nature of such variables, it is hard to predict long-term behavior.

To this, what I did in my previous project was simple (and quite stupid at the same time). I chose an arbitrary value t, that was big enough to represent long-term behavior, and I bashed until I got to that value of t. Needless to say, this method is likely to have high inaccuracies.

Instead, the use of matrices can largely help us predict their long term behavior. Consider a model for variables x_t, y_t, z_t represented by this system of equations:

Then, this system can be expressed as

Note that multiplying this vector by itself produces a diagonal vector as well!

Generalizing this form, we can see that

Note that in this form, we can obtain a value for each of as tends to infinity. This would allow us to predict long-term behavior. This technique would be extremely useful especially when modelling long-term effects of government policies.