transitionMatrix | Statistical analysis and visualization of state transition | Data Visualization library

The Kalman Filter is a linear filter that is optimal for linear systems with gaussian noise. Assume there is no noise and your system is perfectly linear. The transition matrix hence describes how the state changes and is not updated by the kalman filter. If the transition matrix is just the identity it means the state never changes. The transition matrix for your case should be an identity with an entry of dt in (0,4) and (1,5) (since given the last state of your rectangle the next state is the current position (x,y) + dt *(vx,vy). It depends on the application whether you have more information about the change in your state (what happen with your width and height).

The reality is influenced by noise. There are two different kinds considered in the kalman filter. Structural /process noise and measurement noise. Both influence the gain matrix. The gain matrix is not given by the user but calculated through the history, the measurement covariance and the process covariance. If the measurement is not noisy you might decrease the measurement covariance (it might be worthwhile to check at least in which range your variances are), By "increasing" the measurement matrix on decreases the influence of the observation in the current estimation. The other part is the process noise covariance. This part influences the P'(k). This is the covariance of the current state given the last state (and its covariance). If you know perfectly how your system behaves (which means you know your transition matrix), you can assume the process noise as zero. If you increase the process noise covariance you are saying, that there is an error in the propagation between two time stamps, that is not explained by the transition matrix and is random. (Maybe you know how much the width or height change on average. This variance can be used in the diagonal element, that corresponds to w and h (2 and 3)). So to make the filter more responsive you have to tell him, that you are unsure about the actual change and hence you have to increase the process noise. For parameter tuning i suggest you have a closer look at your data. So you have to increase the process noise AND decrease the measurement noise to make the filter more responsive.

EDIT: Turns out that the stability was just that I'd found a set of values that didn't show spikes in that dataset. The instability is still there. Using this initialisation pattern does match the expected pattern until instability sets in though. So it looks like a bug in the underlying Kalman filter.

So after searching more, I found a closed issue in EMGU that pointed to the updated unit tests for the Kalman filter. Using their way of initialising, I've managed to get something that seems a lot more stable.

This has already been answered here: Kalman filter always predicting origin

OpenCV Kalman filter implementation does not let you set the an initial state. You have to save your initial state and then when you call kalman.correct you have to subtract the initial state. And when you call kalman.predict you have to add your initial state.

Something like this pseudo-code:

If you estimate from data, PyEMMA will use the largest connected subset of states to do the estimation. The background is, that you can not give a unique measure for the transition matrix, when it is not connected, e.g. every state can be reached from any other state.

You can inspect which states are connected by looking at the resulting MSM object:

print(msm.active_set)

yields some integers corresponding to descrete state indices.

kalman.correct() takes a measurement, but you're passing in the KalmanFilter's own measurementMatrix back into itself that you first assigned via the kalman.set_measurementMatrix() call. (Yes, they're different!) The measurementMatrix is a (probably static) transform from state space to measurement space, whereas measurements are your actual observations that get continually update in the loop. This also means your comment "set init measurement" is wrong and likely leading to the misunderstanding. (Yes, the opencv KF naming is confusing.) You need to add an additional measurement matrix for passing the observations to correct().

The error message is telling you that the gemm() call inside the kalman.correct() method is failing because the dimensions aren't right for how it has been configured. You're passing in a 3x9 matrix where it is expecting a 3x1.

UPDATE:

I didn't catch it the first time through your code, but the measurementNoiseCov matrix dimensions also need to be changed to measSizexmeasSize instead of stateSizexstateSize in order to match the observation size.

There was an isse in the markovchain package due to numeric tolerance when creating a markovchain. I have pushed a patch on the development version hosted in github... You can download it by:

I think I have solved it, I believe it isn't the center points it returns, it the xy coords of the top left corner of the new rect - so I just addded width and height to the second bounding box x,y coords.

Even though your code looks ok, it seems that setidentity does not work like as the name may suggest. As it is now, it will only leave the matrix with 0:

I cannot see a function in markovchain exactly for that, but it can be easily done manually too. There's one caveat though: the transition matrix does not provide the probability of observing the first A, which needs to be provided by you. Let it be 0.25, as it would be if all four states were equally likely (which is true in your example).

Then the transitions in the observed chain can be obtained with