Section: New Results

Jump point detection and parameter estimation from piecewise homogeneous Markov chains

Participants : Agustina Razetti, Xavier Descombes.

Piecewise homogeneous Markov chain processes can be applied to diverse phenomena of various nature, such as genetics, physics. Recent bibliography has focused on these systems, proposing different alternatives to detect the jump points and be able to separate between different phases of the signals. The Markov chain is usually defined by its transition matrix and the change points are modeled by a hidden Markov process. In this work, we focus on the Gaussian case with a Bernoulli distribution governing the change points. We have developped two different theoretical frameworks: one Bayesian with a Bernoulli prior, and the other one statistic-oriented, proposing a test of hypothesis based on ratio of likelihoods. For both cases we provide with robust algorithms to detect the jump points and reduce the error in the estimations of the parameters of the main model. We compare both methods and investigate their limits and advantages. We finally provided practical examples to showcase the power of the proposed approach (see Figure 16).

Figure 16. Two application examples. 10 particles of equal mass moving are shown at each case. When they collide another particle or the external limits, they follow elastic punctual collisions (shown by stars and circles). Left: particles inside a tube of diameter $D$ and length $L$; right: particles moving around a circle of radius $\rho$.