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Home/Math Visualization/Kalman Filter (1-D)

Kalman Filter (1-D)

This interactive simulator explores Kalman Filter (1-D) in Math Visualization. Recursive optimal estimation: noisy measurements, hidden truth, predict + update with Q and R; random-walk or constant-velocity model with ±2σ band and innovations. Use the controls to change the scenario; watch the visualization and any graphs or readouts to connect the model with lectures, labs, and homework.

Who it's for: For learners comfortable with heavier math or second-level detail. Typical context: Math Visualization.

Key terms

  • kalman
  • filter
  • kalman 1d
  • math
  • visualization

Kalman filter

Hidden truth signal

0
-1
5s

Shortcuts

  • •Drag across plots to move the time cursor

Measured values

RMSE estimate vs truth2.313
RMSE measurement vs truth0.315
noise reduction0.14×
steady-state K0.049
σ̂ at cursor0.070

How it works

1-D Kalman filter: optimally combines a noisy measurement z of a hidden true signal with a dynamic model. Each step does predict (propagate state and covariance through the model F, growing P by process noise Q) and update (shrink toward the new measurement by Kalman gain K = P_p H^T / (H P_p H^T + R)). Crank up R and the filter trusts the model; crank up Q and it tracks the measurements aggressively. The 2-state constant-velocity mode lets the filter estimate velocity from position-only measurements — try the ramp preset to see velocity locking onto the true slope.

Key equations

predict: x̂⁻ = F x̂, P⁻ = F P Fᵀ + Q
update: K = P⁻ Hᵀ (H P⁻ Hᵀ + R)⁻¹
x̂ = x̂⁻ + K (z − H x̂⁻), P = (I − K H) P⁻