Final_1.pdf

Y 1 ,Y 2 ,Y 3 ,...,Yn, the measurements, and Z 1 ,Z 2 ,Z 3 ,...,Zn, the predicted val-
ues from the state equation, then

measurement cost =

prediction cost =

These cost functions may be considered the measures of effectiveness of
the measurement and the state transition models. A high prediction cost in-
dicates a poor prediction model. Similarly, a high observation cost indicates
a poor observation model. If the two models must be equally effective, it
makes sense to require that the values of the two cost functions be more or
less the same.
Additionally, we also know that the Kalman estimate for the state is a
convex combination of the measured and predicted states; that is, given kito
be the Kalman gain at the ith time step, we have

Xi= (1 – ki)Zi+kiXi (12.9)

XZii

i

n

()−

=

∑

2

1

XYii

i

n

()−

=

∑

2

1

Spread Inversion 195

The discussion on model selection in the case of smoothing functions

can be viewed as a situation where we attempt to separate the signal

from noise given a sum of both. On the one hand, we can be very con-

servative and treat every kink as meaningful and separate out very lit-

tle as noise. On the other hand, we can throw the baby out with the

bathwater and oversmooth the given signal, discarding part of the sig-

nal along with the noise. The model selection criterion above provides

us with a methodology to achieve a balance between both extremes.