Opinion Dynamics & Historical CLR Transition Models

1. Purpose of Opinion Dynamics

While the Opinion State Estimator quantifies uncertainty about current public opinion, forecasting future election-day or future polling outcomes requires modeling opinion movement over time.

This document describes the foundational empirical dynamics models evaluated against future Poll of Polls consensus trajectories:

1. point_persistence: Deterministic benchmark ($\theta_{t+h} = \text{PoP}_t$).
2. empirical_raw: Direct resampling of historical joint CLR transitions.
3. symmetric_all_history (formerly empirical_symmetric): Sign-symmetric resampling of all eligible historical CLR transitions ($\text{drift} = 0$).
4. symmetric_4y: Sign-symmetric resampling restricted to trailing 4-calendar-year window.
5. symmetric_2y: Sign-symmetric resampling restricted to trailing 2-calendar-year window.
6. symmetric_recency_weighted: Sign-symmetric resampling with 730-day exponential recency weighting ($w = \exp(-\ln(2) \cdot \text{age_days} / 730)$).

[!NOTE] Development Target Disclaimer: Poll of Polls future observations are used here as the development target for opinion-dynamics modeling. They are not assumed to be perfect observations of latent voter opinion.

2. Compositional Representation: Centered Log-Ratio (CLR)

To model multi-party transitions without reference-category asymmetries (such as those observed with small categories in ALR), dynamics are modeled in Centered Log-Ratio (CLR) space across all nine canonical categories ($D=9$):

where $g(\mathbf{p}) = \exp(\frac{1}{D}\sum_{j=1}^D \ln(p_j))$ is the geometric mean.

Mathematical Properties

• $\sum_{i=1}^D \text{clr}_i = 0$ (elements lie on the zero-sum hyperplane in $\mathbb{R}^D$).
• Coordinate symmetric: no single category acts as an asymmetric divisor.
• Inverse CLR mapping: \(p_i = 100 \cdot \frac{\exp(\text{clr}_i - m)}{\sum_{j=1}^D \exp(\text{clr}_j - m)}, \quad m = \max(\mathbf{clr})\)

3. Direct Historical Transition Construction

For each forecast horizon $h \in {7, 14, 28, 56, 84, 112}$ days:

where both $s$ and $s+h$ exist exactly in pollofpolls_timeseries.csv.

Structural Leakage Boundary

For any forecast origin $t$, a transition $(s, s+h)$ is eligible if and only if:

Transitions ending after $t$ are structurally filtered before reaching the forecasting models.

Minimum Historical Data Threshold

\(\text{MIN\_TRANSITIONS} = 30\)

If fewer than 30 eligible transitions exist for a given $(t, h)$ pair within the model’s window, the case is skipped and recorded in diagnostics.

4. Dynamics Models

A. Point Persistence (point_persistence)

\(\theta_{t+h} = \text{PoP}_t\) Uses the exact origin Poll of Polls composition as a deterministic point forecast with zero dispersion.

B. Empirical Raw Transitions (empirical_raw)

Resamples complete 9-party transition vectors uniformly with replacement from the eligible pool:

• Complete 9-party transitions are sampled jointly; party movements are never sampled independently.
• Preserves empirical historical directional drift and empirical cross-party correlation structure.

C. Symmetric Recency-Adaptive Variants

Resamples transition vectors and independently applies a random sign $S^{(i)} \in {-1, +1}$ with equal probability ($P(S=1) = 0.5$):

• Enforces zero expected historical drift ($\mathbb{E}[\mathbf{\Delta}] = \mathbf{0}$) while preserving empirical movement magnitudes and correlations.
• symmetric_all_history: Uses all eligible historical transitions ($\text{transition_end} \le t$).
• symmetric_4y: Restricts to transitions with $t - 4\text{ years} \le \text{transition_end} \le t$.
• symmetric_2y: Restricts to transitions with $t - 2\text{ years} \le \text{transition_end} \le t$.
• symmetric_recency_weighted: Samples all eligible transitions with probability proportional to $w = \exp(-\ln(2) \cdot \text{age_days} / 730)$.

5. Rolling Historical Evaluation Protocol

Because historical data through 2026 was inspected during exploratory iterations, evaluation is structured as rolling historical evaluation across annual blocks (2019, 2020, 2021, 2022, 2023, 2024, 2025, 2026 YTD):

• Avoids over-interpreting a single arbitrary split.
• Tests whether recency-adaptive transition pools perform consistently across varying political volatility regimes.

Primary Model Selection Criterion

To avoid distortion from the small, volatile derived REST category, the primary model-selection metric is:

\(\text{Equal-Weighted Annual Mean CRPS across the 8 parliamentary parties: M, L, C, KD, S, V, MP, SD}\) with central 50%, 80%, and 90% interval coverage acting as calibration guardrails.