What if I set R much smaller than Q?

The filter trusts measurements and hugs the noisy data, shrinking the ±2σ band but looking jittery. Large R does the opposite: a smooth, lagging track that trusts the process model more than the sensors.

What is an innovation?

The innovation is the difference between the new measurement and what the prediction expected. For a well-tuned linear-Gaussian filter, innovations should look like zero-mean white noise; structured residuals hint at model mismatch.

Why offer random-walk and constant-velocity models?

Random walk suits slowly drifting scalars; constant velocity adds a speed state so the filter can coast through noise between measurements. Using the wrong motion model biases the estimate even if Q and R look plausible.

Kalman Filter (1-D)

Recursive optimal estimation: noisy measurements, hidden truth, predict + update with Q and R; random-walk or constant-velocity model with ±2σ band and innovations.

About this model

A one-dimensional Kalman filter recursively fuses a dynamical prior with noisy scalar measurements. At each step the filter predicts the state with process noise covariance Q, then updates with measurement noise variance R using the Kalman gain that minimizes mean-square error under linear-Gaussian assumptions. This simulator shows the hidden truth, noisy observations, the estimate, a ±2σ uncertainty band from the covariance P, and innovations (residuals). You can choose a random-walk or constant-velocity process model. Idealizations include linear dynamics, white Gaussian noises, perfect model match when tuned well, and no outliers or nonlinear sensors. Vary Q and R to see lag versus jitter, and watch innovations whiten when the filter is consistent.

Who it's for: Estimation, control, robotics, signal processing, and applied probability introductions.

Key terms

Kalman filter
Process noise Q
Measurement noise R
Innovation
Covariance
Optimal estimation

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⁻

Frequently asked questions

What if I set R much smaller than Q?: The filter trusts measurements and hugs the noisy data, shrinking the ±2σ band but looking jittery. Large R does the opposite: a smooth, lagging track that trusts the process model more than the sensors.
What is an innovation?: The innovation is the difference between the new measurement and what the prediction expected. For a well-tuned linear-Gaussian filter, innovations should look like zero-mean white noise; structured residuals hint at model mismatch.
Why offer random-walk and constant-velocity models?: Random walk suits slowly drifting scalars; constant velocity adds a speed state so the filter can coast through noise between measurements. Using the wrong motion model biases the estimate even if Q and R look plausible.

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Kalman Filter (1-D)

About this model

How it works

Key equations

Frequently asked questions

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Kalman Filter (1-D)

About this model

How it works

Key equations

Frequently asked questions