Local Level Trend Model

I believe I'm missing something basic about setting-up a simple local level trend state space model. I can follow the model structure in regards to the A,B,C and D matricies by which I mean that I believe I understand the implications of the elements in those matrices. However, I don't know what to do with the error distribution data. For instance, if I'm given that the covariance of the state equations is distributed W = [NaN 0; 0 NaN], how do I incorporate that information into the model? My initial thought was that it was part the the "D" matrix, and I've also tried to put it in the initial Cov0 matrix.

Thank you

Answer 1

[Amish]

Hi Jeffrey,

I see that you are trying to firgure out about incorporating the covariance of the state equations, represented by W = [NaN 0; 0 NaN], into a local level trend state space model.

The "W" matrix you're referring to is meant to represent the process noise covariance in the state space model, not to be confused with the "D" matrix or the initial state covariance "Cov0". The presence of "NaN" values indicates missing or undefined elements, which would typically need to be defined with actual numerical values for a practical application.

Therefore, once correctly defined, the "W" matrix will essentially become the "Q" matrix. This will represent the process noise covariance. The "D" matrix should be used for direct feedthrough from input to output and does not incorporate process noise.

Note that, When setting up your state space model in MATLAB, ensure that the "Q" matrix (corrected "W") is used appropriately, especially if applying estimation or control algorithms that require knowledge of the process noise characteristics.

You can refer to the following MathWorks documentation for additional information:

State Space Models: https://in.mathworks.com/help/control/ug/what-are-state-space-models.html

Kalman Filtering using MATLAB: https://in.mathworks.com/help/control/ug/kalman-filtering.html

Hope this helps!