2026-09-30

Discussion

Initial Direction

Extend the latent-process framework from GLM to circular time series, initially focusing on the von-Mises distribution.

• The ultimate goal is forecasting. The immediate focus is on model formulation, estimation, model fit, and comparison with existing circular time-series approaches.
• A possible future forecasting issue is that the latent-process model is generally not Markovian, so using more of the observation history may improve prediction compared with conditioning only on the most recent observation.

Data and Applications

The application should be considered from the beginning rather than after the methodology is fully developed.

Possible data sources discussed:

• UCD weather / energy-related data;
• Met Eireann high-resolution weather data;
• European weather reanalysis data;
• the new weather station planned for UCD Belfield.
  Some UCD data are available at approximately 15-minute resolution.

Paula suggested considering variables like:

• temperature;
• seasonal components;
• air pressure;
  and mentioned the availability of high-resolution weather data from Met Eireann and European weather reanalysis.

Existing Models Comparison

Existing circular time-series models can initially be fitted without covariates to:

• become familiar with the methods;
• establish benchmark results;
• test available wind-direction datasets.

One potential advantage of the proposed framework is that covariates can be incorporated directly, whereas many existing circular time-series models mainly handle non-stationarity through structures such as trends or random walks.

Concern:

Note

Directly using a multiplicative latent effect ${\mu }_t{v}_t$:

• If ${\mu }_t$ wrapped, does not guarantee $\text{wrap}({\mu }_t)\times v_t\in [-\pi ,\pi ]$;
• If ${\mu }_t$ unwrapped:
  • $({\mu }_t+2\pi )\times v_t={\mu }_t{v}_t+\underset{?}{\underbrace{2\pi v_t}}$: Same direction, different results;
  • The effect of $v_t$ depends on ${\mu }_t$:
    • ${\mu }_t\to 0$, $v_t$ less effect;
    • ${\mu }_t\to \infty$, multiple circle;
  • Start point of ${\mu }_t$;

An additive / wrapped construction such as
$\text{wrap}({\mu }_t+v_t)$
may therefore be more natural.

Wagner’s point was that the latent effect does not have to be multiplicative. The important issue is whether the chosen construction preserves the marginal structure required for estimation. For an additive construction, the corresponding expectation / moment condition may be less straightforward to derive.

Possible directions to investigate include:

• transformations of an AR(1) latent process;
• multiplicative structures with suitable constraints;
• additive / wrapped constructions;
• other transformations that respect the circular parameter space.
  The exact form is still open.

General Advice

• Start reading and writing early.
  • Follow important references when necessary;
  • Understand the assumptions behind existing methods;
  • Identify limitations or weak points;
  • Look for opportunities where the methodology could be improved.
• Keep writing preliminary ideas, model definitions, derivations, and literature notes in Overleaf.
• References and bibliography files.

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Open Questions

1. How should the latent process enter the von-Mises / circular model?
2. Is a multiplicative, additive, wrapped, or another construction more appropriate?
3. What marginal moment or estimating equation needs to be preserved so that the regression parameters can still be estimated conveniently?
4. Can an appropriate transformation of an AR(1) process satisfy the required circular constraints?
5. Which covariates are physically meaningful for wind direction?

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Next steps

Hang

• Create a shared drive and invite Wagner and Paula, then share the link with them.
• Save a version of the presentation slides shown in the meeting and upload them to the shared drive for Paula to review.
• Write a living document in the shared drive after each meeting, listing the date, attendees, and action items agreed upon.
• Start reading relevant literature, taking notes, and writing initial ideas and summaries in Overleaf or similar, focusing on circular time series and related methods.
• Investigate different model formulations (e.g. multiplicative vs. additive latent processes) and identify the marginal moment / expectation condition required for estimation.

Paula

• Put links to Met Eireann high-resolution data and other relevant weather data sources into the shared document for the team to review.

Wagner

• Find and share the email about the upcoming online meeting for all Rinn AI students and supervisors.