Kalman Filter Calculator

Formula: Kalman filter equations for this calculator

In the simple 1D case with a constant state model, the prediction step carries the prior estimate forward unchanged and only inflates the variance by the process noise variance $Q$.

Prediction step

State prediction (for a constant state): ${\hat{x}}^{-}=\hat{x}$

Variance prediction:

Here ${\widehat{x}}^{-}$ and $P^{-}$ denote the predicted state and predicted variance before incorporating the new measurement.

Update step

Kalman gain: $K=\frac{P^{-}}{P^{-}+R}$

Updated state estimate: $\hat{x}+=\hat{x}-+K(z-\hat{x}-)$

Updated variance: $P^{+}=(1-K)P-$

In this calculator, the inputs are $\hat{x}$, $P$, $z$, $R$, and $Q$. The calculator computes $P^{-}$, $K$, ${\hat{x}}^{+}$ and $P^{+}$.

Interpreting the results

• Predicted variance $P^{-}$ – your uncertainty just before seeing the new measurement. Larger $Q$ leads to a larger $P^{-}$, meaning you expect more drift or disturbance between updates.
• Kalman gain $K$ – a weight between 0 and 1 (for positive variances) that determines how much you move towards the new measurement. Values near 0 mean you mostly trust the prior prediction; values near 1 mean you mostly trust the measurement.
• Updated estimate ${\hat{x}}^{+}$ – the new best estimate after combining prediction and measurement.
• Updated variance $P^{+}$ – the new uncertainty. It is usually smaller than both the predicted variance and the measurement variance, reflecting the information gained by combining them.

By experimenting with $R$ and $Q$, you can see how the filter behaves in different regimes:

• Large $R$ (very noisy sensor) → small $K$, slow response to new measurements.
• Small $R$ (high-quality sensor) → $K$ closer to 1, fast response.
• Large $Q$ (rapidly changing true state) → larger $P^{-}$ and typically larger $K$, so the filter leans more on the latest measurement.
• Small $Q$ (very stable state) → smaller $P^{-}$, so the filter becomes more conservative and smooth.

Worked example

Suppose you are tracking the temperature of a chemical reactor. Yesterday's filtered estimate was $\hat{x}=80.0°C$ with variance $P=4.0$ (standard deviation 2.0 °C).

Today you take a new measurement:

• Measured temperature: $z=86.0°C$.
• Sensor variance: $R=9.0$ (standard deviation 3.0 °C).
• Process variance between readings: $Q=1.0$.

1. Prediction

State prediction (constant model): $\hat{x}{}^{-}=80.0$.

Variance prediction: $P{-}^{}=P+Q=4.0+1.0=5.0.$

2. Kalman gain

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

$K=\frac{P^{-}}{P^{-}+R}=\frac{5.0}{5.0+9.0}=\frac{5.0}{14.0}\approx 0.357.$

3. Updated estimate

Innovation (measurement minus prediction): $z-{^x}^{-}=86.0-80.0=6.0$.

Updated estimate: $\hat{x}^+=80.0+0.357\times 6.0\approx 82.1°C.$

4. Updated variance

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

$P^+=(1-K)P^-=(1-0.357)\times 5.0\approx 3.215.$

After the update, your best estimate is 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

The Kalman filter is often compared to simple moving averages or exponential smoothing. The table below summarizes key differences in this 1D context.

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

Assumptions and limitations

This calculator intentionally focuses on a minimal, didactic setup. When using it, keep in mind the following assumptions and limitations:

• One-dimensional state: the tool handles a single scalar variable. Real systems often need multi-dimensional filters (e.g., position and velocity in 2D or 3D).
• 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 $Q$ and $R$. 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 $\hat{x}$ and $P$ at each step.
• No correlation modelling: correlations between different states or between noise sources are not represented in this 1D setup.

For educational purposes and quick sanity checks, this simplification is often sufficient. For safety-critical or high-precision applications, you should implement a full multi-dimensional filter tailored to your system, and validate it with domain-specific tests and simulations.

FAQ

How should I choose the process variance Q?

$Q$ represents how much you expect the true state to change between updates due to unmodelled dynamics, disturbances, or drift. If the underlying quantity is very stable (e.g., a slowly varying room temperature), choose a small $Q$. If it can change quickly between measurements (e.g., acceleration of a moving object), $Q$ should be larger. In practice, $Q$ is often tuned experimentally by looking at how responsive or noisy the filtered output is.

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

As $R$ becomes large relative to $P^{-}$, the Kalman gain $K$ approaches 0 and the filter trusts the prediction much more than the measurement. As $R$ becomes very small, $K$ approaches 1 and the filter closely follows each new measurement. Extremely small or zero $R$ can make the filter overly sensitive to measurement glitches.

How to use: Can I use negative or zero variances?

No. Variances $P$, $Q$, and $R$ should be non-negative, and in practice strictly positive for numerically stable filtering. Zero variance would imply perfect certainty, which rarely holds in real systems and can cause numerical issues in the underlying equations.