Allow multivariate components to share latent states

[jessegrabowski]

This PR allows multivariate components to share latent states between observed states. For example, if we want two states to have a shared component and an idiosyncratic component, as in:

$$ \begin{align} a_t &= x_t + y_t \\ b_t &= x_t + z_t \\ \end{align} $$

This can be accomplished as follows:

    shared_level = st.LevelTrendComponent(
        name="shared_level",
        order=1,
        innovations_order=1,
        observed_state_names=["a", "b"],
        share_states=True,
    )
    individual_level = st.LevelTrendComponent(
        name="individual_level",
        order=1,
        innovations_order=1,
        observed_state_names=["a", "b"],
    )

    mod = (shared_level + individual_level).build(verbose=False)

This will require a refactor for each component to add the shared_states argument. Here's the TODO list:

• LevelTrendComponent
• AutoregressiveComponent
• CycleComponent
• RegressionComponent
• TimeSeasonality
• FrequencySeasonality
• MeasurementError

[AlexAndorra]

This is 🔥
We probably wanna do it for Noise too (added it as an item in the description)

[AlexAndorra]

@jessegrabowski, while playing around with that, I realize it's breaking _hidden_states_from_data in a subtle way (the component extraction logic assumes it can always split observation indices into equal groups for each endogenous variable, but when share_states=True, there's only one set of shared states instead of separate states per endogenous variable).

I fixed it, but it'd be better if we stored the share_states boolean into _component_info (it makes the logic much simpler). This is a simple implementation, that I demoed here (ignore the changes for seasonality, which are just the ones you implemented here; focus on how share_states is passed between objects, and then to Component in core.py).

If we do that, we'd just need to make sure each component passes its share_states flag from its __init__ to its super call. LMK what you think!

[jessegrabowski]

What is noise? MeasurementError?

[jessegrabowski]

@jessegrabowski, while playing around with that, I realize it's breaking _hidden_states_from_data in a subtle way (the component extraction logic assumes it can always split observation indices into equal groups for each endogenous variable, but when share_states=True, there's only one set of shared states instead of separate states per endogenous variable).

I am getting sick of this function. I'm thinking about a big refactor that would take advantage of the computational graph

[AlexAndorra]

Looks good guys! Almost ready to merge: we need to pass the share_states flag to the super call in each component. I'll add that, and fix the merge conflict

[jessegrabowski]

Looks good guys! Almost ready to merge: we need to pass the share_states flag to the super call in each component. I'll add that, and fix the merge conflict

Why?

[jessegrabowski]

Also please rebase instead of merging

[AlexAndorra]

Looks good guys! Almost ready to merge: we need to pass the share_states flag to the super call in each component. I'll add that, and fix the merge conflict

Why?

Because I need it here, in core.py @jessegrabowski -- well, at least I think I do, but LMK if I'm over-complicating this

[AlexAndorra]

All good now!