SSM - Local Level Model

[paroussisc]

This issue is for tracking progress/thoughts/ideas while coding up the local level model from Chapter 2. The idea is to develop understanding and intuition that will help with studying the generalised KF in Chapter 4.

[paroussisc]

I implemented the KF recursion and got the same results as in Section 2.2.2 (commit 8c0e82614e7241bf787c1f425bebc844ae6b3129) using the river Nile data:

It would be good to note a few things at this stage:

• a_t is the state mean
• P_t is the state variance and only depends on the observation variance and state variance, which are hyperparameters (it does not depend on the observation values themselves, just on whether the observations appear - for this linear Gaussian model)
• F_t is the variance of the residuals v_t
• K_t is the Kalman gain and specifies how much to move the new state in the direction of the observed residual v_t and also how much to increase/decrease the state variance by. It is the state variance divided by the residual's variance.

In this example, F_t and P_t converge to fix values, meaning that this model has a steady state: In practice this means that we can stop computing F_t and K_t which speeds up computation. This will be investigated for this example later on.