Kalman Filter Structural
Time Series Models

Structural time series (STSM) models decompose a time series into various components like trend, seasonality, and noise, allowing for a more nuanced understanding of the data’s underlying patterns. These models are particularly useful for forecasting and causal inference. Structural Time Series Models, or Unobserved Components models, were initially developed by Harvey (1985), Watson (1986), and Koopman et al. (1995).

In ARBOTHAI we follow the general model:

y_t = μ_t + ψ_t+ ε_t,t + g_t

where y_t is a nx1 vector and μ_t, ψ_t, ε_t and g_t represent trend, cycle, seasonal, and irregular components.

            A STSM decomposes a time series into a trend (level plus slope), a cycle, a seasonal and an irregular component. The model is estimated by the Kalman filter, and we can interpret each component directly. The computation process builds unobserved component models separately for each of the components. In STSM, we obtain each components’ forecasts separately, and they are indeed directly interpretable.

            Furthermore, cycle shifts for individual time series are incorporated within the model and estimated simultaneously with the remaining parameters. This feature permits the use of leading, coincident and lagging variables to approach the dengue cycle coincident indicator without prior analysis of their lead-lag relationship.

            An initial decomposition breaks down a time series into interpretable components, such as:

• Trend: Denotes the long-term behavior of the series (increasing, decreasing, or stable).
• Seasonality: Indicates whether recurring or periodic patterns at fixed intervals exist (e.g., weekly, monthly, yearly).
• Cyclical: Captures fluctuations with no pre-fixed period, often longer than seasonal variations.
• Irregular/Random: Accounts for unexplained or random variations.

            As a result, STSMs are very flexible as they adapt to the structure of the data and these models can efficiently handle various types of time series data, including those with complex dependencies and non-linear relationships. They can be used to better understand and estimate the causal impact of events or interventions on a time series while improving the understanding of the underlying factors driving a time series. Once the analysis facilitates the understanding of components governing dengue variability, they can be used to make more accurate predictions about the evolution of future or the ongoing epidemic.

            As key features to highlight among the statistical models is that by modeling the individual components, it can lead to better predictions and STSM is seen as an accurate tool for causal analysis. Within ARBOTHAI, whenever convenient we use a Bayesian approach that allows for flexible model specification and uncertainty quantification. In summary, structural time series models offer a powerful framework for analyzing and forecasting time series data by decomposing it into meaningful components, leading to a deeper understanding of the underlying processes and improved predictions

The multivariate unobserved components model includes a stationary higher order cycle and incorporates temperature and rainfall as explanatory covariates. As for the other modelling frameworks within ARBOTHAI, forecasts are generated at 3,6 and 12 months lead time without readjusting for observations. That is, in December 2024 we generated the forecasts for 3,6 months as well as for the entire year. As obviously, at that time we did not have either any data for temperature nor rainfall in 2025, we predicted those values for the months in 2025 from past climate data by extrapolating the trend in those variables. 95% confidence intervals for predictions are also displayed.