4 research outputs found
Heavy-tails in Kalman filtering with packet losses
Abstract in the tex
Heavy-tails in Kalman filtering with packet losses: Confidence bounds vs second moment stability
In this paper, we study the existence of a steady state
distribution and its tail behaviour for the estimation error arising
from Kalman filtering for unstable scalar systems. Although a
large body of literature has studied the problem of Kalman
filtering with packet losses in terms of analysis of the second
moment, no study has addressed the actual distribution of the
estimation error. By drawing results from Renewal Theory, in
this work we show that under the assumption that packet loss
probability is smaller than unity, and the system is on average
contractive, a stationary distribution always exists and is heavytailed, i.e. its absolute moments beyond a certain order do not
exist. We also show that under additional technical assumptions,
the steady state distribution of the Kalman prediction error has
an asymptotic power-law tail, i.e. P[|e| > s] ∼ s
−α
, as s → ∞,
where α can be explicitly computed. We further explore how to
optimally select the sampling period assuming exponential decay
of packet loss probability with respect to the sampling period. In
order to minimize the expected value of second moment or the
confidence bounds, we illustrate that in general a larger sampling
period will need to be chosen in the latter case as a result of the
heavy tail behaviour