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How can we use a Kalman filter for tracking an object in 1D for which we we don't know the parameters of the state transition model, and only have estimates?

Consider the state transition model for the Kalman Filter given as: x(k + 1) = F(k )x(k ) + G(k )u(k ) + v(k )

If one has estimates of the form of F(k) and G(k), can we still implement the kalman filter effectively?

I believe that we may be able to implement it, but will only be able to implement it effectively if we have a large sample of data to fit the parameters of the distribution. We could then judge the form of the fit of the model by eye, and try out various models and choose the one with the lowest error. For example:

  • If the data has a straight line gradient, we may posit that we could fit a random walk with drift term therefore G(t) = 1, u(t) = gradient of the line, F(t) = 1.
  • If the data has a positive curved shape, we may try to fit a G(t) = 0, F(t) = >1.
  • If the data oscillates around a fixed mean, we may try a random walk without drift which would be F(t) = 1 and G(t) = 0.
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There are effective ways of doing this, which incorporate all the suggestions in the question.

  1. As set out in the question, you can fit different models with different parameters to the models.
  2. Test the models by fitting your data to the model and running a simulation of the model.
  3. Compare your models using AIC, BIC and log-likelihoods. The models with the lowest BIC and AIC are generally the better models. The difference between BIC and log-likelihoods is that BIC introduces a penalty term for the number of parameters in the model to penalise over-fitting.

An example of this being done is set out here: https://stackoverflow.com/q/46177279/5211911

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