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Kalman Filter

The Kalman Filter tracks a continuous latent variable representing the player's true point potential. It handles gradual form drift, a player slowly improving or declining across gameweeks.

State-Space Model

\[x_{t+1} = x_t + w_t, \quad w_t \sim \mathcal{N}(0, Q_t)\]
\[y_t = x_t + v_t, \quad v_t \sim \mathcal{N}(0, R_t)\]
  • \(x_t\) — true (unobserved) point potential at gameweek \(t\)
  • \(y_t\) — observed weekly points (noisy measurement)
  • \(Q_t\) — process noise (how fast true form can drift)
  • \(R_t\) — observation noise (how noisy weekly points are)

The random-walk model (\(x_{t+1} = x_t + w_t\)) means form drifts gradually. \(Q\) controls drift speed: large \(Q\) allows rapid changes, small \(Q\) assumes stable form.

Predict-Update Cycle

At each timestep:

Predict:

\[\hat{x}_{t|t-1} = \hat{x}_{t-1|t-1}\]
\[P_{t|t-1} = P_{t-1|t-1} + Q_t\]

Update:

\[K_t = \frac{P_{t|t-1}}{P_{t|t-1} + R_t} \quad \text{(Kalman gain)}\]
\[\hat{x}_{t|t} = \hat{x}_{t|t-1} + K_t \cdot (y_t - \hat{x}_{t|t-1})\]
\[P_{t|t} = (1 - K_t) \cdot P_{t|t-1}\]

Kalman Gain Interpretation

When \(Q_t\) is large → \(P_{t|t-1}\) is large → \(K_t\) is large → the filter trusts the observation more. When \(R_t\) is large → \(K_t\) is small → the filter trusts the prior more. This is exactly how news and fixtures should affect inference.

Adaptive Noise (Signal Injection)

Both \(Q_t\) and \(R_t\) can be overridden per-timestep:

Signal Parameter Multiplier Effect
Injury news \(Q_t\) 5.0 Form may have jumped; widen state uncertainty
Doubtful news \(Q_t\) 2.0 Moderate form uncertainty
Easy fixture (difficulty 1) \(R_t\) 0.8 Points more predictable, trust observation
Hard fixture (difficulty 5) \(R_t\) 1.5 Points less predictable, trust prior 
kf = KalmanFilter(Q=1.0, R=4.0, x0=4.0, P0=2.0)

# Injury at gameweek 20: process noise inflated
kf.inject_process_shock(timestep=20, multiplier=5.0)

# Hard fixture at gameweek 21: observation noise inflated
kf.inject_observation_noise(timestep=21, factor=1.5)

x_est, P_est = kf.filter(observations)

RTS Smoother

The Rauch-Tung-Striebel smoother runs a backward pass after the forward Kalman filter, producing refined estimates that use all data (past and future):

\[L_t = \frac{P_{t|t}}{P_{t+1|t}}\]
\[\hat{x}_{t|T} = \hat{x}_{t|t} + L_t \cdot (\hat{x}_{t+1|T} - \hat{x}_{t+1|t})\]

Available via KalmanFilter.smooth().

Default Parameters

Parameter Default Meaning
\(Q\) 1.0 Process noise — moderate form drift
\(R\) 4.0 Observation noise — weekly points are noisy
\(x_0\) 4.0 Initial estimate — league-average points
\(P_0\) 2.0 Initial uncertainty

Tuning

\(Q\) should be higher for young/volatile players and lower for established veterans. \(R\) can be estimated from the empirical variance of weekly points across the league.