Technical Appendix v3

Goal model

Two XGBoost Poisson regressors predict expected goals per side:

λ_home = f_H(elo_h, elo_a, fifa_h, fifa_a, host_h, alt, climate, travel_h, rest_h, squad_h, …)
λ_away = f_A(elo_h, elo_a, fifa_h, fifa_a, host_a, alt, climate, travel_a, rest_a, squad_a, …)

Each regressor is trained on 27,000+ post-1990 international matches (poisson objective, early-stopping on a holdout slice).

Negative Binomial parameterisation

To handle the over-dispersion in real football scorelines (variance > mean), the marginal goal distribution is Negative Binomial:

G ~ NB(λ, φ)         (mean = λ, variance = λ + λ²/φ)
P(G = k) = Γ(k+φ) / (k! · Γ(φ)) · (φ/(φ+λ))^φ · (λ/(φ+λ))^k

The dispersion parameter φ is tuned on the training set; shipped default: φ = 5.0 (where φ → ∞ reduces to Poisson — lower φ means more upset variance). The sensitivity audit varies φ across [3, 5, 8, 15].

Dixon-Coles τ correction

The independent-marginals product under-predicts low-score draws and over-predicts 1-0 / 0-1. The Dixon-Coles correction multiplies the joint pmf on the four affected cells:

P(h, a) := P(H=h) · P(A=a) · τ(h, a, λ_h, λ_a, ρ)

τ(0, 0) = 1 − λ_h λ_a ρ
τ(0, 1) = 1 + λ_h ρ
τ(1, 0) = 1 + λ_a ρ
τ(1, 1) = 1 − ρ
τ(h, a) = 1   otherwise

Calibrated on training data, current value: ρ = −0.13. The full joint matrix is then re-normalised. Sensitivity varies ρ across [−0.18, −0.08].

Per-match λ noise (Gamma multiplier)

Beyond NB dispersion, each simulation pass draws a Gamma multiplier for each side to capture "off days" / "purple patches":

α ~ Gamma(k=σ⁻², θ=σ²)        E[α] = 1, Var[α] = σ²
λ_sim = λ · α

Shipped α (Gamma shape) is 12.0, which corresponds to σ ≈ 0.289 (≈ 29% per-match relative noise via σ = 1/√α). This is what turns identical λ values into the realistic 14–18pp spread of W/D/L probabilities seen in the live match cards.

Bracket & tiebreakers

2026 is the first 48-team World Cup. FIFA published the deterministic Round-of-32 bracket in advance (which group-position fills which slot); the 8-best-third-placer slot assignment uses FIFA's Annex C lookup table — 495 rows covering every possible combination of the 8 qualifying-group identities.

Group tiebreaker cascade (per simulation)

1. Most points
2. Best goal difference
3. Most goals scored
4. Head-to-head points among tied teams
5. Head-to-head goal difference
6. Head-to-head goals scored
7. Fair-play points (approximated as zero in simulation)
8. FIFA World Ranking (new in 2026 — replaces drawing of lots)

The pre-launch validator asserts annex_c_misses == 0 across the production run.

Monte Carlo configuration

• Seeds: 5 independent RNGs (NumPy PCG64, seeded deterministically). Cross-seed variance is reported as the simulation range (5th–95th percentile across seeds) for every aggregate probability. This captures Monte-Carlo sampling noise from independent tournament rollouts — not parameter / model uncertainty (those would need bootstrap or posterior resampling, which this build does not perform).
• Sims per seed: 5,000 → 25,000 total tournaments. Per-team SE on champion probability ≈ √(p(1-p)/25000) ≤ 0.32pp at the 50% level.
• Knockout overtime: if tied at 90', play extra time with halved λ (15-min halves); if still tied, penalty shootout sampled with team-specific shootout slope (calibrated on historic finals).
• Travel + venue: per-match effective Elo = base Elo + host boost − travel penalty − altitude penalty − heat penalty + form decay + squad-value prior. All inputs are pre-computed once per tournament; only the goal sample is stochastic.
• Live mode: with --live, completed matches are locked verbatim (their realised goals enter the goal-difference / head-to-head accounting); remaining matches simulate as normal.
• Matchday intelligence: elo_eff_base picks up get_team_elo_adjustment() from scripts/live/apply_matchday_adjustments.py, summing the injury (cap ±25, extreme ±35), weather (±15), lineup (±20), and post-match stats-proxy (±8/match, ±20/group) layers. Aggregate matchday cap is ±35 per team-match; grand total with mid-tournament live-form delta is ±45. Every per-tick decision is appended to data/live/matchday_intelligence_log.jsonl for full audit.

Calibration & performance metrics

Holdout (4,839 unseen modern matches)

Walk-forward backtest

Each test World Cup is run with only training data strictly older than that tournament (no peeking). All hyperparameters tuned once on pre-2010 data.

Scope: walk-forward validates the base goal-model layer only (Elo + form + rest + neutrality + importance). The deployed simulator additionally applies host-boost, squad-value, altitude, climate, travel, and injury bonuses pre-injected as Elo deltas (see scripts/03_simulate.py:806-810). Those layers are validated separately via ablation and sensitivity — they are not in the walk-forward chain.

Feature ablation

Sensitivity audit (27 scenarios)

Every hand-coded assumption was varied across the ranges below. Each scenario is a full 5×5,000 re-run.

Top-team ranges from latest sensitivity run:

Top-4 rank ordering (Spain > Argentina > France > England) is identical across all 27 scenarios; positions 5–6 (Brazil / Colombia) swap under altitude-penalty extremes.

Reproducibility

To rerun the full pipeline from a fresh clone:

git clone <repo-url> wc26-matchday-intelligence
cd wc26-matchday-intelligence

# 1. Environment
python3 -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt

# 2. Data preparation + Elo
python scripts/01_prepare_data.py

# 3. Train goal model (≈ 45 s on M-series laptop)
python scripts/02_goal_model.py

# 4. Evaluation suite
python scripts/04_evaluate.py
python scripts/06_ablation.py
python scripts/07_walk_forward.py
python scripts/05_sensitivity.py           # slowest — ≈ 5 min for 27 scenarios

# 5. Production simulations
python scripts/03_simulate.py --no-travel --out predictions_no_travel.json --seeds 5 --sims 5000
python scripts/03_simulate.py --seeds 5 --sims 5000        # ≈ 30 s
python scripts/08_travel_impact.py

# 6. Sync artefacts to dashboard
cp data/processed/predictions.json data/processed/calibration.json data/processed/travel_impact.json dashboard/
cp models/walk_forward.json models/ablation.json models/sensitivity.json dashboard/

# 7. Pre-launch validation — must pass before publishing
python scripts/09_validate.py

# 8. Serve locally
cd dashboard && python3 -m http.server 8765

Seeds are fixed (seed_0=42, seed_1=43, …) — rerunning the simulator with identical inputs reproduces the same probabilities to within floating-point precision. The walk-forward backtest is the strictest reproducibility test: it trains on pre-WC-year data only and tests on the held-out tournament.

Data schema — predictions.json

{
  "generated_at":          ISO-8601 UTC,
  "n_simulations_total":   25000,
  "n_seeds":               5,
  "n_simulations_per_seed":5000,

  "team_predictions": [{
    "team":               str,
    "group":              str (A–L),
    "elo":                float,
    "fifa_pts":           float,
    "squad_value_eur_m":  float,
    "p_advance_groups":   [0,1],
    "p_reach_r16":        [0,1],
    "p_reach_qf":         [0,1],
    "p_reach_sf":         [0,1],
    "p_reach_final":      [0,1],
    "p_champion":         [0,1],
    "p_third_place":      [0,1],
    "p_champion_p05":     [0,1],   // 5th percentile across seeds
    "p_champion_p95":     [0,1],   // 95th percentile across seeds
    "p_finish_1st_group": [0,1],
    "p_finish_2nd_group": [0,1],
    "p_finish_3rd_group": [0,1],
    "p_finish_4th_group": [0,1]
  }, ...],

  "match_predictions": [{
    "m":                  int (1–72),
    "date":               YYYY-MM-DD,
    "time":               HH:MM (local kickoff),
    "group":              str,
    "home":               str,
    "away":               str,
    "venue":              str,
    "venue_country":      str,
    "altitude_m":         int,
    "climate":            "mild" | "hot" | "very_hot" | "high_altitude_*",
    "lam_home":           float,  // mean expected goals
    "lam_away":           float,
    "p_home_win":         [0,1],
    "p_draw":             [0,1],
    "p_away_win":         [0,1],
    "elo_home":           float,
    "elo_away":           float,
    "effective_elo_home": float,  // base Elo + boosts − penalties
    "effective_elo_away": float,
    "home_travel_km":     float,
    "away_travel_km":     float,
    "home_rest_days":     int,
    "away_rest_days":     int,
    "home_travel_penalty":float,   // Elo points
    "away_travel_penalty":float,
    "locked_score":       null | "H-A"
    // Note: when locked_score is non-null (completed match), the
    // p_home_win / p_draw / p_away fields remain the original PRE-MATCH
    // forecast — they are NOT collapsed to a one-hot of the realised
    // result. Inside each Monte Carlo rollout the scoreline is forced to
    // the locked value (scripts/03_simulate.py:388-394); the displayed
    // forecast is preserved for backtest / calibration use.
  }, ...],

  "concentration":   { top1_champion_p, top2_combined, top5_combined },
  "annex_c_misses":  0,
  "live_mode":       bool,
  "completed_matches": int
}

Raw data downloads

All artefacts are static JSON. They are the same files the dashboard reads — no server, no auth, no rate limits.

• predictions.json — team + match probabilities with 5-seed simulation ranges (p05/p95)
• predictions_live.json — live-mode rerun with locked completed matches
• calibration.json — predicted-vs-actual bins on 4,839 unseen matches
• walk_forward.json — backtest on WC 2010/14/18/22
• ablation.json — feature ablation log-loss + accuracy
• sensitivity.json — 27-scenario hyper-parameter audit
• travel_impact.json — champion-prob diff with/without travel penalty
• live_state.json — mode, completed count, last-updated timestamp
• live_delta.json — top movers between static and live predictions
• matchday_intelligence.json — consolidated injury / weather / lineup / stats-proxy adjustments per team-match, with caps and provenance

The dashboard only reads the artefacts above. The repository also keeps four upstream feeds in data/live/ for traceability (injuries_2026.json, weather_2026.json, lineups_2026.json, match_stats_2026.json) plus the append-only matchday_intelligence_log.jsonl — one record per matchday-intel tick, retained for the tournament duration.