An introduction to smoothing time series in python. Part III: Kalman Filter

If we have a mathematical model for the system in study, we can use that information to dramatically improve the quality of our prediction. Like in the previous filtering methods, we are taking advantage of the fact that to estimate the system state at some time $t$ we can use not only the information available at that time but also the information from the past (and the future if smoothing). But model free methods tend to want to reduce large deviations from one point in time to the next, while we may actually expect that at some specific time point the system does jump drastically in value. Adding in the model allows us to take that into account.

Let’s start by looking at the Kalman Filter, which is the optimal estimator for linear and gaussian systems. Let us define such a system first in the discrete case:

$$x_{n+1} = Ax_{n}+\xi \\ y_{n+1} = Bx_{n+1}+\zeta$$

The stochastic process in $x$ is the underlying process we want to follow. Not only is the process in $x$ a brownian process (additive white noise denoted by $\xi$), we are unable to observe it directly. The observation is denoted by $y$ and is a function of $x$ corrupted by (again) additive white noise $\zeta$. The gaussian assumption is often a reasonable approximation to the problem’s noise statistics because the timescale of whichever microscopic process produces randomness is usually much smaller than the one of the actual dynamics, allowing the central limit theorem to kick in.