x - The underlying, not directly observable, thing you are trying to track. [Boat position] Number or vector of numbers (e.g., spatial coordinates or set of formant values).
z - Thing that you can observe from which you try to deduce x. Number or vector of numbers. [Tangents of angles]
H - Mapping between unobservable and observable domains. [Trig formulas] If the true state were x and observations were accurate, Hx is what you would observe. Assumed linear for a Kalman Filter.
A - Predicted ("systematic", "deterministic") change in x between observations. [Formula for rocking motion] Can be "no change".
Q -
Average error in using A to predict next x. [Rocking
motion isn't perfectly sinusoidal] Expressed
as
(co)variance.
R -
Average error in observations. [Hard to get an
accurate reading in a rocking boat] (Co)variance. P -
Accumulated uncertainty associated with estimate of x at a given moment.
Covariance. x = Ax, P = APA' + Q K = PH'/(HPH' + R), x = x
+ K(z - Hx), P = P - KHP (notation abuse) Consider the case where z is just
an
(untrustworthy)
observation of x itself. [We just want to know the angle of elevation
and we'll work out the distance later.] Then H is simply 1 and can
be left out. So K = P/(P+R), x = x + K(z -x), P
= P - KP K represents the balance of confidence
between last estimate and current observation. If P >> R, then the likely error
in the previous estimate of x is much bigger than the likely error in the
observation. The formulas make K ~ 1 and x ~ z, i.e., believe the observation. If R >> P, then the error
in the observation is likely to be
much bigger than the error in the previous estimate.
The formulas make K ~ 0 and x ~ x, i.e., stick with the last estimate. K gives a regression
relation between the observation errors and the state estimate (if
your intuition runs to that sort of thing). Prediction Phase
Update Phase
Intuition
(Meinhold and Singpurwalla, "Understanding the Kalman Filter",
The American Statistician, v37, no 2, pp. 123-127, 1983)