Sunday, February 21, 2010

Assumptions for the Markov Process

I started working on fitting a Markov process for an earthquake mechanism about two weeks ago and derived the formulas relating the number of states, N, the number of earthquakes, E, the gain function, G, and the fractional rates, r and f, as shown in the image below. The first line includes an assumption that the distribution of states is unaffected by the action of the transition matrix. It is an assumption that there is a stable distribution for the states which is more likely to hold in a long term average.

The most recent fit shows that the assumption of a g function provides a useful method in doing a curve fit if no formula is known for the distribution. The failure of an equilibrium distribution makes the assumptions in the formulas for r and f less certain and casts doubt on the assumed Markov process giving a complete picture of what is actually happening. If a distribution function exists this needs to be verified over time. If it doesn't exist then the implication would be that the conditions are rather changable and current conditions would not be a good indicator of future events. Whatever the mechanism behind earthquakes, it is a hidden mechanism and the occupation numbers of the states are less certain than the actual number of earthquakes.

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