Kalman Filter Calculator

Formula: 1D Kalman filter equations used by this calculator

In this page's constant-state model, the Kalman prediction step keeps the prior estimate unchanged and adds process noise variance to the prior variance before the measurement update.

Prediction step

State prediction (for a constant state):

Variance prediction:

Here and denote the predicted state and predicted variance before the measurement is applied.

Update step

Kalman gain:

Updated state estimate:

Updated variance:

In this calculator, the inputs are , , , , and . The calculator reports , , and so you can trace the complete single-step update.

Interpreting the Kalman filter results

• Predicted variance – the uncertainty just before the new measurement is blended in. A larger makes larger, which means you expect more drift between updates.
• Kalman gain – a weight between 0 and 1 (for positive variances) that controls how far the estimate moves toward the measurement. Values near 0 mean the update stays close to the prior prediction; values near 1 mean the measurement has most of the influence.
• Updated estimate – the posterior state estimate after the innovation is scaled by .
• Updated variance – the remaining uncertainty after the Kalman update. It is usually smaller than the predicted variance and reflects the information gained from the measurement.

By changing and in this calculator, you can see the tradeoff directly:

• Large (very noisy sensor) → small , so the estimate moves only a little.
• Small (high-quality sensor) → closer to 1, so the new measurement pulls the estimate more strongly.
• Large (rapidly changing true state) → larger and usually larger , so the filter reacts more quickly to fresh data.
• Small (very stable state) → smaller , so the estimate changes more cautiously.

Worked example: a reactor temperature Kalman update

This worked example follows one Kalman filter update for a reactor temperature estimate so you can see how the prediction and measurement are combined numerically.

Suppose a process engineer is tracking the temperature of a chemical reactor. Yesterday's filtered estimate was with variance (standard deviation 2.0 °C).

Today you take a new measurement:

• Measured temperature: .
• Sensor variance: (standard deviation 3.0 °C).
• Process variance between readings: .

1. Prediction

State prediction (constant model): .

Variance prediction:

2. Kalman gain

Formula: K = P^- / (P^- + R) = 5.0 / (5.0 + 9.0) = 5.0 / 14.0 ≈ 0.357.

3. Updated estimate

Innovation (measurement minus prediction): .

Updated estimate:

4. Updated variance

Formula: P^+ = (1 − K) P^− = (1 − 0.357) × 5.0 ≈ 3.215.

After the update, the Kalman calculator gives an estimate of about 82.1 °C with variance 3.215 (standard deviation about 1.79 °C), which is both more accurate and more confident than relying purely on the prior or purely on the single noisy measurement.

Kalman filter vs. simple smoothing methods for a 1D estimate

The Kalman filter is often compared with simple moving averages or exponential smoothing, but this calculator makes the difference visible because it exposes the variances that drive the gain.

Unlike these simpler methods, the Kalman filter explicitly tracks uncertainty and automatically adjusts how much it trusts new data relative to prior estimates, based on and .

Assumptions and limitations of this 1D Kalman filter calculator

This calculator is intentionally narrow so the algebra stays easy to follow. Keep the following assumptions in mind when you use it for a scalar Kalman update:

• One-dimensional state: the calculator handles a single scalar variable. Real tracking problems often need position, velocity, or other multi-state models.
• Linear model: the state evolution and measurement model are assumed linear. Nonlinear systems require extended or unscented Kalman filters, or alternative Bayesian filters.
• Gaussian noise: both process noise and measurement noise are assumed zero-mean Gaussian with variances and . Heavy-tailed or biased noise will not be perfectly represented.
• Single-step update: the calculator performs one prediction-update cycle. A full Kalman filter implementation would iterate this over time, updating and at each step.
• No correlation modelling: correlations between different states or between noise sources are not represented in this 1D setup.

For classroom examples, quick checks, and intuition-building around this Kalman filter, the simplification is often sufficient. For safety-critical or high-precision work, use a full multi-dimensional filter tailored to your system and validate it with realistic data, domain-specific tests, and simulations.

FAQ about using this Kalman filter calculator

How should I choose the process variance Q for a Kalman filter update?

represents how much you expect the true state to change between updates because of unmodelled dynamics, disturbances, or drift. If the tracked quantity is very steady, such as a slowly drifting temperature, choose a small . If the quantity can change noticeably between measurements, such as the acceleration of a moving object, should be larger. In practice, is usually tuned by checking whether the Kalman output reacts too sluggishly or too noisily.

What happens when measurement variance R is very large or very small?

As becomes large relative to , the Kalman gain approaches 0 and the calculator weights the prediction more heavily than the measurement. As becomes very small, approaches 1 and the update follows the measurement closely. Extremely small or zero can make the estimate too sensitive to measurement glitches.

How to use: Can I use negative or zero variances in this calculator?

No. Variances , , and should be non-negative, and in practice they should usually be positive for stable Kalman filtering. A zero variance would imply perfect certainty, which rarely matches real sensor or process behavior and can create numerical issues in the update equations.