Kalman Filter (1D)

Kalman Filter (1D)

The Kalman filter is an optimal recursive estimator for linear systems with Gaussian noise. In 1D, it tracks a hidden state from noisy observations.

State space model

State:       x[t] = x[t-1] + noise_q    (process noise variance q)
Observation: z[t] = x[t] + noise_r      (observation noise variance r)

One-step update

Given current estimate x_est with uncertainty p_est:

Predict:

x_pred = x_est
p_pred = p_est + q

Update (using observation z):

K = p_pred / (p_pred + r)          (Kalman gain)
x_new = x_pred + K * (z - x_pred)  (updated estimate)
p_new = (1 - K) * p_pred           (updated uncertainty)

Task

Implement:

• kalman_update(x_est, p_est, z, q, r_noise) → returns (x_new, p_new)
• kalman_filter(observations, q, r_noise, x0, p0) → list of filtered state estimates