Stochastic filtering of a random Fibonacci sequence: Theory and applications

This paper conceives a stochastic filtering problem to estimate, from noisy measurements, the numbers of the random Fibonacci sequence. The dynamical system is amenable to an exact solution, being the convolution of Bernoullian and Gaussian variables, yielding a closed form for the equations of the filter. The derived optimal filter has exponential computational complexity, thus a suboptimal filter with affordable computational load is conceived. The stochastic filter performance is then evaluated with two applications: one of theoretical value and one for a more practical application. More precisely the first case study estimates the Viswanath constant from noisy measurements of the random Fibonacci sequence. It is shown how the filter performs well in estimating the Viswanath constant even if the noise is significantly increased. The second case study refers to a model of malware propagation in a computer network. In this case, it is assumed that there is a random rate for the infection, assuming that a finite time is needed before a computer is infected. A random generalized Fibonacci sequence fits well in this case. Additional applications are possible in view of the fact that several systems both in biology and economy are well represented by Fibonacci binary random trees.