In this paper, we offer a solution to the stochastic realization problem for a Gaussian Markov field defined on a tridimensional lattice, which is a graph with nodes regularly positioned to form a discrete parallelepiped in the euclidean space and arcs connecting “internal' nodes with five nearest neighbors along the three coordinate directions. Next we show how the stochastic realization can be used for weather forecasting via a Kalman predictor, relying on partial observations and just a purely statistic a-priori knowledge of the Markov field, similarly to a classic Hidden Markov Model (HMM). An application carried out on real climate data shows the effectiveness of the approach taken.

The Chemical Master Equation (CME) provides a fruitful approach for the stochastic description of complex biochemical processes, because it is able to cope with random fluctuations of the chemical agents and to fit the experimental behavior in a more accurate way than deterministic concentration equations. In this work, our attention is focused on modeling and simulation of multisite phosphorylation/dephosphorylation cycles, by using the quasi-steady state approximation of enzymatic kinetics. The CME dynamics is written from the coefficients of the deterministic reaction-rate equations and the stationary distribution is computed explicitly, according to a recently developed realization scheme. Simulations are included to provide a comparison with Monte Carlo methods in terms of computational complexity.