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algorithmkalman-filter

Scalar Kalman filter implementation

Trying to use the Kálmán filter to find the steady state of time series data.

Can see Kálmán algorithm as follows for a scalar:

Predict:

x(k|k-1) = Ax(k-1|k-1) + B U(k)
P(k|k-1) = AP(k-1|k-1) + W(k)

Update: 

y(k) = c x(k) + v(k)
K(k) = P(k|k-1)C/(CP(k|k-1) + v(k)
P(k|k) = (1-K(k)C)P(k|k-1)

I am trying to understand the difference between P(k|k) and P(k|k-1).
When this is implemented, do P(k|k) becomes the P(k|k-1) in the next iteration? If yes, does the error covariance W(k) in P(k|k-1) is not included in calculations or are they different terms?

Solution

  • Both prediction and update steps in your example belong to the iteration k.

    From the perspective of the iteration k you have following covariances:

    P(k-1|k-1) - a posteriori covariance at iteration k-1 (after the measurement processing)

    P(k|k-1) - a priori covariance at iteration k (after the prediction step, before the measurement processing)

    P(k|k) - a posteriori covariance at iteration k (after the measurement processing)

    So the difference between P(k|k) and P(k|k-1) is that P(k|k) involves the information from the last measurement and P(k|k-1) does not.

    Here is the same thing for two different iterations:

    iteration k-1:

    P(k-1|k-2) = P(k-2|k-2) + W(k-1)

    P(k-1|k-1) = (1-K(k-1))P(k-1|k-2)

    iteration k:

    P(k|k-1) = P(k-1|k-1) + W(k)

    P(k|k) = (1-K(k))P(k|k-1)