# How can one fit a kalman filter to an object in 1D without knowledge of the parameters of the model?

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.