Kalman filter
-------------
N = 1000 measurements

xt+1 = a*xt + ut + vt
yt   =   xt + et

a = 0.4

vt ~ N(0, 1)
et ~ N(0, 1)
x0 ~ N(0, 1)

ut = 5 for t = 50, 51, ..., 200, 450, 451, ..., 600
ut = 0 otherwise

Maximum likelihood estimation
-----------------------------
Task:
1. Plot the (log-)likehood function for different values of a, e.g., a = 0, 0.1, ..., 0.5

2. (If time permits,) use optimization software to estimate a automatically (e.g., SciPy's optimize > minimize)

Extended Kalman filter
----------------------
N = 1000 measurements

xt+1 = a*xt*(1 - xt) + ut + vt
yt   = xt + et

a = 3.5

vt ~ N(0,   10^-4)
et ~ N(0,   10^-4)
x0 ~ N(0.5, 10^-1)

ut = 0.1 for t = 50, 51, ..., 200, 450, 451, ..., 600
ut = 0 otherwise