Multi-Resolution Studies

Purpose

A multi-resolution study is a single study whose stages span more than one temporal resolution — most commonly a monthly head followed by a quarterly tail, or a weekly head followed by a monthly tail. All stages belong to one SDDP horizon and share one set of value-function cuts; no coupling boundary exists. The modelling challenge is that the PAR(p) inflow model is defined at a fixed seasonal grid, yet stages at different resolutions must draw noise from the same underlying stochastic process and produce statistically coherent inflow realisations. This chapter describes how Cobre handles that challenge.

Boundary with the coupled-studies chapter. A multi-resolution study is one study with mixed-resolution stages. Weekly+monthly coupled studies (Weekly+Monthly Coupled Studies) are instead two separate studies that hand off at an explicit coupling boundary. Where the same stochastic aggregation concept appears in both contexts, this chapter owns the mechanism description; the coupled-studies chapter cross-references here.

The PAR(p) model associates each stage $t$ with a season index $m(t)$ . In a purely monthly study, $m(t)$ cycles over twelve months and stages with the same calendar month share the same seasonal parameters $(\mu_m, s_m, \psi_{m,\ell})$ .

When some stages run at quarterly resolution, a quarterly stage covers the same calendar period as three monthly stages. Cobre assigns every quarterly stage the season index of its middle month so that the seasonal noise vector for that quarter is drawn from the same distribution as the corresponding monthly season. The noise draw $\varepsilon_t$ for a quarterly stage therefore represents the same seasonal signal that monthly stages in the same calendar quarter would share — the stochastic process is not re-defined at the coarser resolution; it is sampled at the resolution that the stage demands.

This alignment guarantees that forward trajectories at different stage resolutions remain on the same probability space. A quarterly stage and the three monthly stages it replaces draw noise that is centred on the same seasonal mean and scaled by a statistic derived from the same historical record after aggregation (see section 3).

2. Duration-Weighted Aggregation

Monthly inflow observations cannot be fed directly into a PAR fitting procedure intended for quarterly stages because the seasonal moments (mean, standard deviation, and autocorrelations) must refer to the resolution at which the AR model will be evaluated. Cobre aggregates the historical observations from the fine to the coarse resolution before fitting, using duration weighting.

Aggregation rule. The quarterly equivalent of a set of monthly observations within the same quarter is the duration-weighted sum of those observations:

a^{(Q)}_q = \sum_{m \in Q} d_m \cdot a^{(M)}_m

where $a^{(M)}_m$ is the monthly observation, $d_m$ is the duration weight of month $m$ within quarter $Q$ (normalised so that $\sum_{m \in Q} d_m = 1$ ), and $a^{(Q)}_q$ is the resulting quarterly observation. The seasonal mean and standard deviation of the quarterly series are then estimated from the aggregated observations $\{a^{(Q)}_q\}$ in the usual way.

The duration weights reflect the fraction of the quarter’s total duration contributed by each month. For calendar quarters with three months of unequal length (e.g., January/31, February/28–29, March/31), the weights differ from the uniform $1/3$ value that a naive average would apply.

Weekly-to-monthly aggregation follows the same rule: a monthly observation aggregates the weekly observations whose periods fall within that month, weighted by the fraction of the month’s duration each week covers.

3. PAR Fitting on Aggregated Statistics

After duration-weighted aggregation, the aggregated series has the same form as a homogeneous-resolution series at the coarser resolution. The standard five-step PAR fitting procedure described in PAR(p) Inflow Model section 5 applies without modification.

For stages that run at the fine resolution (e.g., the monthly head of a monthly-quarterly study), the seasonal statistics $(\mu_m, s_m)$ and AR coefficients $(\psi_{m,\ell})$ are derived from monthly observations in the normal way. For stages that run at the coarse resolution (e.g., the quarterly tail), the statistics and coefficients are derived from the duration-weighted quarterly aggregates. The two sets of parameters co-exist in the same parameter files; the system resolves the appropriate set for each stage when building the stage-indexed preprocessing arrays described in Scenario Generation section 1.

No separate fitting pipeline exists for multi-resolution studies: the aggregation step is a preprocessing transformation that produces a coarser-resolution observation series, after which the standard procedure runs once per resolution band that appears in the study.

4. Opening-Tree Generation Across Resolutions

The opening tree (see Scenario Generation section 2.3) is generated per stage. Each stage’s noise vectors are drawn from the PAR model at that stage’s native resolution: monthly stages use monthly seasonal parameters; quarterly stages use quarterly parameters derived from the aggregated statistics of section 2.

Because the season index assignment of section 1 aligns quarterly stages with the same seasonal calendar as the corresponding monthly stages, the opening tree is coherent across the resolution boundary. A quarterly stage in the fourth quarter and the three monthly stages of that quarter draw noise with the same seasonal conditioning — the cross-resolution noise alignment is a structural consequence of season-index assignment, not a separate sampling rule.

The backward pass iterates over the full opening tree without distinguishing stage resolution; cut aggregation proceeds as in a single-resolution study.

5. Trade-offs

Sub-resolution information loss. Duration-weighted aggregation discards the intra-quarter variation among monthly observations. A monthly series with pronounced within-quarter autocorrelation loses that signal when aggregated to quarterly. The fitted PAR model at quarterly resolution can only reproduce variation at the quarterly timescale; sub-quarterly dynamics are invisible to the value function in the quarterly tail.

Timescale interpretation. The analyst must track which resolution each reported statistic refers to. A mean inflow reported for a quarterly stage is a duration-weighted quarterly mean, not a monthly mean. Comparing statistics across the resolution boundary requires explicit re-scaling; automated cross-resolution comparisons are outside the scope of the optimizer.

Season-index choice. Assigning a quarterly stage the season index of its middle month is a convention, not a physical law. It aligns the noise distribution with the seasonal calendar in a symmetric way, but introduces a small boundary effect at the start and end of each quarter: the first and last months of a quarter contribute less to the quarterly mean than the middle month does to the monthly statistics. The analyst should inspect seasonal means and standard deviations in the aggregated statistics file to verify that the aggregated distribution matches expectations before running training.

Cross-References

PAR(p) Inflow Model — The fitting procedure (section 5) that applies to aggregated statistics; the parameter set that the aggregated statistics populate; the LP-ready form that quarterly stages use at runtime.
Scenario Generation — Opening-tree generation (section 2.3) that produces per-stage noise vectors; the PAR preprocessing pipeline (section 1) that resolves seasonal statistics into stage-indexed arrays at the native resolution.
Weekly+Monthly Coupled Studies — The boundary chapter: two studies coupled at a handoff point rather than one study with mixed stages; sub-monthly lag accumulation and terminal boundary cuts live there, not here.