The Kalman Filter

The Kalman filter is the optimal recursive estimator for a linear system with Gaussian noise, and the ancestor of most twin assimilation.

An optimal recursive estimator

The Kalman filter estimates the hidden state of a linear dynamical system disturbed by noise, using a stream of noisy measurements. It is recursive: it keeps only the current estimate and its uncertainty, folding in each new measurement without reprocessing history. For linear dynamics with Gaussian noise it is provably the minimum-variance estimator, which is why it anchors the field.

State and covariance

The filter carries two objects: a state vector, the best estimate of the quantities of interest, and a covariance matrix, the uncertainty in that estimate and the correlations among its components. Both are propagated in time and both are corrected at each measurement.

The two steps

Predict: advance the state through the system model and grow the covariance by the process noise. Update: compute the Kalman gain from the current covariance and the measurement noise, then correct the state toward the measurement and shrink the covariance. The gain is the formal answer to how much to trust the new data.

python

import numpy as np

def kalman_step(x, P, z, F, Q, H, R):
    # predict
    x = F @ x
    P = F @ P @ F.T + Q
    # update
    y = z - H @ x                 # innovation
    S = H @ P @ H.T + R           # innovation covariance
    K = P @ H.T @ np.linalg.inv(S)  # Kalman gain
    x = x + K @ y
    P = (np.eye(len(x)) - K @ H) @ P
    return x, P

Assumptions and their limits

The classical filter assumes linear dynamics and measurement, Gaussian noise, and known noise statistics. Real fusion systems are strongly nonlinear, so the plain filter is used mostly for near-linear subsystems such as some magnet or power-supply dynamics. Nonlinear extensions handle the rest: see the extended and unscented variants, and the ensemble filter for high-dimensional fields.

In a Kronos twin the filter's covariance is as valuable as its state, because honest uncertainty on an inferred quantity, such as an unmeasured internal temperature, is what makes a prediction trustworthy. See uncertainty quantification.