Limitations and glossary

Interpretation limits

• Football scores are sparse, and team strength changes for reasons the state model cannot directly observe. Intervals cover modeled uncertainty, not every real-world uncertainty.
• Score matrices stop at 10 goals per side and renormalize the retained cells. Result probabilities therefore use a finite-support approximation; expected goals are integrated from the rates separately and are not truncated at ten.
• Seven-point Gauss–Hermite quadrature is the default approximation to the Gaussian uncertainty integral.
• Monte Carlo outputs contain sampling error. Small development runs are not publication precision.
• Motivation represents modeled table-objective leverage, not player psychology. It activates only through remaining-match gates supplied by the loaded policy.
• Venue and three-derby adjustments are narrow selected policies, not universal causal estimates.
• Defaults for clubs promoted from I liga and for summer transitions require independent validation because their supporting datasets and selection history are not exported.
• Discipline and fair-play tie-break data are absent, so random noise can decide an otherwise unresolved final table tie.
• Cup projections require caller-supplied lower-tier ratings, bridge state, entrants, and bracket assumptions. The export does not provide a joint live calendar.
• Market odds are excluded from forecasts by design. This is not a market-consensus model and is not wagering advice.
• A synthetic or newly fitted output from this code is not a reproduction of the private operational forecast.

Glossary

attack rating : A latent additive effect on a club's own log scoring rate. Higher is stronger.

defence rating : A latent quantity subtracted from the opponent's log scoring rate. Higher is stronger defence.

home-field advantage (HFA) : An additive home log-rate effect. It may be fixed or dynamic and can be attenuated for a supplied relocated-venue period.

joint covariance : One matrix describing uncertainty and correlation among all active attack, defence, and dynamic HFA parameters.

Dixon–Coles correction : Multipliers applied to the four lowest cells of an independent Poisson score matrix to model low-score dependence.

Gauss–Hermite quadrature : Deterministic weighted nodes used to integrate match probabilities over Gaussian log-rate uncertainty.

causal date batch : All matches on one calendar date are predicted before any result from that date updates the state.

xG (expected goals) : A continuous description of chance quality. Caller-supplied calibrated xG can be blended with goals; simulated future xG supports chronologically consistent path updates.

club remaining in Ekstraklasa : A club that competes in Ekstraklasa in consecutive seasons. It can carry an Ekstraklasa state through the summer transition.

club promoted from I liga : A club moving from I liga, Poland's second tier, into Ekstraklasa. It needs a new Ekstraklasa opening prior rather than a carried top-tier state.

club relegated to I liga : A club moving out of Ekstraklasa into I liga after the previous season.

season prior : The opening distribution for team strength. The code can incorporate previous state for clubs remaining in Ekstraklasa, caller covariates, retention, and policies for clubs promoted from I liga.

retained-minutes share : The share of prior-season player minutes belonging to players retained at a chosen cutoff; one possible input to continuity and uncertainty for clubs promoted from I liga.

settled score : A late-season probability-like measure of low table-objective leverage.

urgency score : A late-season measure based on the largest modeled win-versus-loss probability swing for title, European-place, or relegation objectives.

static posterior simulation : A season path in which one sampled latent team-strength state remains fixed through future fixtures.

chronological simulation : A season path with one persistent, drifting latent truth that generates scores and synthetic xG. A separate model estimate observes those outputs in date batches and updates its ratings.

prediction-only policy : A temporary fixture-rate adjustment that should be accounted for during observation but should not become persistent team strength.

canonical club ID : An identity stable across season-local team IDs. The derby policy uses canonical pairs; the mapping dataset is not exported.

finish distribution : The simulated probability of each team ending in each league place. Season APIs return these values as percentages.

point interval : The simulator's 10th and 90th percentiles of final points, often described as an 80% model interval.

LiveSnapshot : An immutable output dataclass with JSON serialization. This export supplies the contract but not the operational builder that populates it.

source-only reference : Algorithms and interfaces without the datasets, fitted artifacts, orchestration, and lineage required to recreate the source application's forecasts.