Inference Pipeline¶

The core contribution of FPLX. Each player is modeled independently through a multi-component system that tracks discrete form states (MV-HMM), continuous point potential (Kalman Filter), and structural statistical patterns (Enriched predictor). Their outputs are blended via calibrated inverse-variance weighting.

Overview¶

graph TD
    RAW[Raw per-fixture data] --> AGG[aggregate_dgw_timeseries\none row / GW, points_norm]
    AGG --> E[Enriched Predictor]
    AGG --> HMM[MV-HMM]
    AGG --> KF[Kalman Filter]
    N[NewsSignal] -->|Transition perturbation| HMM
    N -->|Process noise shock| KF
    HMM -->|E[Y], Var[Y]| BL[Calibrated Blend]
    E -->|mu_e, var_e, DR| BL
    KF -->|x̂, P| BL
    BL -->|per-fixture E[P], Var[P], DR| SC[scale_predictions_for_dgw\n× n_fixtures]
    SC -->|full-GW E[P], Var[P]| OPT[Two-Level ILP]

Components¶

Enriched Predictor¶

The single strongest individual model. Constructs a per-player feature vector including: EWMA of xG, xA, BPS, clean sheets, cards, own goals, penalties; fixture difficulty (home/away factor, opponent strength); and rolling statistics at windows [3, 5, 10]. A Ridge regression maps these features to expected points.

The enriched predictor also computes a semi-variance (downside risk below E[P]) used directly by the risk-averse ILP objective.

Key principle: availability prediction is separated from form prediction. The enriched predictor is fitted only on played gameweeks (minutes > 0), then availability (from recent minutes history) is applied as a multiplicative factor. This prevents the "Injured" state from absorbing variance that belongs to the form distribution.

Multivariate HMM¶

Each player is represented by a discrete hidden state $S_t \in \{ ext{Injured}, ext{Slump}, ext{Average}, ext{Good}, ext{Star}\}$.

The MV-HMM uses position-specific feature vectors (not raw points scalars):

Feature	GK	DEF	MID	FWD
xPts (composite)	✓	✓	✓	✓
Minutes fraction	✓	✓	✓	✓

Emission distributions are diagonal Gaussians per state. Parameters are learned via Baum-Welch EM with MAP regularization toward position-level priors (weight prior_weight=0.85), preventing overfitting on short histories.

Scalar HMM / KF alone are worse than rolling average

In both backtested seasons, the scalar HMM and KF individually increase MSE above the rolling average baseline. Probabilistic machinery only delivers gains when paired with structural observations (xG, xA, BPS, fixture context). The MV-HMM with position features is the correct architecture.

Kalman Filter¶

Tracks a continuous latent variable $x_t$ via a random-walk model:

\[x_{t+1} = x_t + w_t, \quad w_t \sim \mathcal{N}(0, Q_t)$$ $$y_t = x_t + v_t, \quad v_t \sim \mathcal{N}(0, R_t)\]

The filter yields minimum-MSE estimate $\hat{x}_t$ and posterior variance $P_t$.

News Signal Injection¶

News is injected inside the inference process, not as a post-hoc multiplier. When news indicates an injury:

Category	HMM boost	KF $Q_t$ multiplier
Unavailable	Injured ×10	5.0×
Doubtful	Injured ×3, Slump ×2	2.0×
Rotation	Slump ×2	1.5×
Positive	Good ×2	1.0×

Anticipatory advantage: when news arrives before the gameweek (e.g. "doubtful for Saturday"), the pipeline adjusts predictions before observing 0 points. Observation-only methods must wait one gameweek.

Calibrated Blend¶

The Enriched predictor and MV-HMM are blended per player:

\[\hat\mu = lpha \hat\mu_ ext{enr} + (1-lpha) \hat\mu_ ext{mv}$$ $$\hat\sigma^2 = lpha^2 \hat\sigma^2_ ext{enr} + (1-lpha)^2 \hat\sigma^2_ ext{mv}\]

$lpha$ is calibrated per player via a rolling one-step MSE grid search over the player's historical data. Default $lpha = 0.8$ (enriched-dominant) when fewer than 8 gameweeks of history are available.

The KF output is fused with the blend via inverse-variance weighting (the guarantee that $\hat\sigma^2_ ext{fused} \leq \min(\hat\sigma^2)$ holds).

Ablation Results¶

Model	MSE 2023–24	MSE 2024–25
Rolling average (baseline)	4.269	4.727
EWMA	4.172	4.599
HMM scalar	4.574	5.045
KF scalar	4.291	4.715
Fused scalar	4.295	4.726
MV-HMM	4.509	4.875
Enriched	3.643	4.197
Enriched + MV-HMM blend	3.656	4.145
TFT q50	4.197	4.573

95% calibration coverage: 91.3% (2023–24) and 90.1% (2024–25).

Double Gameweeks¶

DGW handling is fully integrated into the data layer. aggregate_dgw_timeseries is called automatically inside VaastavLoader.build_player_objects (historical) and build_timeseries (live API), so the inference pipeline always receives one row per GW with points_norm (per-fixture normalised) as the training target. scale_predictions_for_dgw is the only DGW-aware step after inference.

See Double Gameweeks for full details.