PRGS

๐Ÿงฎ Poisson-Reversion Goal Signal (PRGS) โ€“ Adapted Model Summary

Purpose
PRGS is a directional forecasting model that predicts whether upcoming football matches are more likely to go Over or Under 2.5 goals, based on season-level scoring trends.

โš™๏ธ Core Assumptions

  • Goals per game (GPG) in football follow a Poisson distribution.
  • Even if the average goals per game is 2.5, P(X > 2.5) < 50%, so the true Over 2.5 probability must be calculated using Poisson logic.
  • Observed scoring patterns revert toward expected distribution over time (mean reversion principle).

๐Ÿง  Model Logic

  1. Collect Season Data
    • Use all matches played so far in the current season as the sample.
    • Let:
      • Total_Games = number of games played so far
      • Total_Goals = total goals scored
      • Games_Over_2.5 = number of matches with more than 2.5 goals
  2. Calculate Metrics
    • GPG (Goals Per Game) = Total_Goals / Total_Games
    • Observed Over % = Games_Over_2.5 / Total_Games
    • Expected Over % = P(X > 2.5) using Poisson distribution with ฮป = GPG
      (i.e. 1 - P(X โ‰ค 2) from Poisson CDF)
  3. Generate Signal
    • If Observed Over % > Expected Over % โ†’ Predict Under bias ahead
    • If Observed Over % < Expected Over % โ†’ Predict Over bias ahead

๐Ÿ” Adaptive Sample Logic

  • Always use all season games played so far to maximize signal strength.
  • Early-season volatility is accepted but should be interpreted cautiously.
  • Optional: Supplement with a moving average GPG or rolling window if you want more responsiveness.

โœ… Model Strengths

  • Simple, objective, and league-agnostic.
  • Highlights inefficiencies between reality and statistical expectation.
  • Offers a probabilistic directional bias rather than deterministic outcomes.

๐Ÿ“Œ Example

  • 80 games played โ†’ 208 goals โ†’ GPG = 2.6
  • 50 games went Over 2.5 โ†’ Observed Over % = 62.5%
  • Poisson with ฮป = 2.6 โ†’ Expected Over % โ‰ˆ 52%
  • Conclusion: League overs are running hot โ†’ Predict Under trend coming via mean reversion.

โš™๏ธ PRGS โ€“ Phase 2: Intra-Batch Match Scoring

๐Ÿ“Œ Goal:

Identify which of the simultaneously played games (within the same league) are more likely Over or Under, even though they share the same Poisson-reversion environment.

๐Ÿง  Assumptions:

  1. All matches share the same expected scoring distribution (i.e., same league-wide GPG and Poisson Over %).
  2. However, small variations (like bookmaker odds or matchup characteristics) can be used to infer relative expectations.

๐Ÿ”ข Phase 2 Model Steps

Step 1: Use League-Wide GPG (from Phase 1)

  • Already calculated:
    • GPG (Goals per Game)
    • Expected Over % from Poisson
    • Signal Bias: Is the league currently Over-leaning or Under-leaning?

Step 2: Pull Game-Level Expected Goal Lines (xG or Market Odds)

  • For each game in the batch, retrieve:
    • Bookmaker line (e.g. O/U 2.5 with odds)
    • Implied total goal expectation from betting market (e.g., 2.3, 2.7, etc.)
    โš ๏ธ If no odds data is available, use a flat expectation: treat all games equally likely and randomly distribute bias (less accurate).

Step 3: Adjust Games Relative to League Mean

  • For each match:
    • Compare its implied total goals (ITG) to the league GPG.
    • Compute: iniCopyEditDeviation = ITG - League GPG
  • This ranks games as relatively higher or lower scoring vs. the season average.

Step 4: Apply Reversion Direction to Rank Matches

  • Use the signal from Phase 1 (Over-biased or Under-biased environment):
Environment Bias (Phase 1)Game Deviation from MeanPrediction
Over-biased (reversion to Under)+0.2 above meanSlight Under signal
Over-biasedโˆ’0.3 below meanStrong Under signal
Under-biased (reversion to Over)+0.4 above meanStrong Over signal
Under-biasedโˆ’0.1 below meanSlight Over signal

You could also scale this with a threshold:

  • Deviation โ‰ฅ ยฑ0.2 = Strong bias
  • Deviation between ยฑ0.1โ€“0.2 = Moderate
  • Deviation < ยฑ0.1 = Neutral

๐Ÿ” Example

Say:

  • League GPG = 2.6 (from 120 games)
  • Expected Over % = 54%
  • Observed Over % = 65%
  • โ†’ League is Over-biased โ†’ Expect reversion to Under

Now 5 upcoming games with these implied totals:

  1. Match A โ€“ 2.9
  2. Match B โ€“ 2.7
  3. Match C โ€“ 2.6
  4. Match D โ€“ 2.4
  5. Match E โ€“ 2.2

Deviations from mean (2.6):

  • A: +0.3 โ†’ Mild Under signal
  • B: +0.1 โ†’ Weak Under
  • C: 0.0 โ†’ Neutral
  • D: โˆ’0.2 โ†’ Strong Under
  • E: โˆ’0.4 โ†’ Strong Under

So Matches D & E would get the strongest “Under” signal, in line with the reversion.

๐Ÿงฉ Optional Output Format (Ranking/Labeling)

MatchImplied TotalDeviationSignal
A2.9+0.3Under
B2.7+0.1Slight Under
C2.60.0Neutral
D2.4โˆ’0.2Strong Under
E2.2โˆ’0.4Strong Under

โœ… Summary of Phase 2:

Use the Poisson-reversion signal from league-level trends (Phase 1), then rank simultaneous matches based on how far their goal expectations deviate from the season average. Games with lower implied totals get stronger “Under” tags in Over-biased environments (and vice versa).