Kalman Filter Calculator

Select data input type Sample data Raw data (comma separated)

Base value (

x

Number of steps (

n

Measurement noise σ_m:

Process noise σ_p:

Decimal Places: 2 3 5 8

Introduction

This calculator employs recursive Bayesian estimation to analyse time-series data, helping to estimate the true state of a system $x$ from noisy observations. By processing a sequence of $n$ steps, it balances prior predictions against new measurements to reduce uncertainty. It is an essential tool for students exploring stochastic processes and signal processing within a statistical framework.

What this calculator does

The tool performs a discrete-time filtering operation by accepting either raw numerical data or generating a sample set based on a base value and noise parameters. It requires inputs for measurement noise $σ_{m}$ and process noise $σ_{p}$ . The output includes filtered estimates, the Kalman gain $K$ , and performance metrics such as Root Mean Square Error and variance reduction percentages.

Formula used

The recursive process involves a prediction phase followed by an update phase. The prediction for the error covariance $P_{p}$ incorporates process noise $Q$ . The Kalman gain $K$ is then determined by the ratio of the predicted error to the sum of predicted error and measurement noise $R$ . The state update adjusts the estimate based on the innovation $y$ , which is the difference between the measurement $z$ and the prediction.

K = \frac{P_{p}}{P_{p} + R}

x_{curr} = x_{pred} + K (z - x_{pred})

How to use this calculator

1. Select the data input type: either manually entered raw data or a generated sample set.
2. Input the base value, number of steps, and the standard deviations for measurement and process noise.
3. Execute the calculation to process the recursive filtering algorithm.
4. Review the generated statistical outputs, including the performance table and the visual trend chart for analysis.

Example calculation

Scenario: An environmental scientist is analysing a series of temperature readings where the instrument has a known measurement error and the environment exhibits minor natural fluctuations.

Inputs: $x = 50$ , $σ_{m} = 5$ , $σ_{p} = 0.5$ , and a measurement $z = 55$ .

Working:

Step 1: $P_{p} = P + Q = 0.01 + 0.25 = 0.26$

Step 2: $K = \frac{0.26}{0.26 + 25}$

Step 3: $K \approx 0.01029$

Step 4: $x_{curr} = 50 + 0.01029 (55 - 50)$

Result: 50.05

Interpretation: The filtered estimate moves slightly toward the new measurement but remains heavily weighted by the previous state due to high measurement noise.

Summary: The filter successfully dampens the impact of the noisy individual measurement.

Understanding the result

The Final Filtered Estimate represents the most probable state after accounting for all noise. A low Average Kalman Gain suggests the filter relies more on predictions than noisy measurements, while a high Variance Reduction percentage indicates significant successful smoothing of the input data stream.

Assumptions and limitations

The model assumes that both process and measurement noise follow a Gaussian distribution. It also assumes the system is linear and that the noise parameters remain constant throughout the specified $n$ steps of the observation period.

Common mistakes to avoid

Users might incorrectly set the process noise to zero, which can cause the filter to ignore new data entirely after a few iterations. Another error is confusing the standard deviation with the variance; the calculator squares the standard deviation inputs to derive the noise covariance matrices $Q$ and $R$ .

Sensitivity and robustness

The output is highly sensitive to the ratio between $σ_{p}$ and $σ_{m}$ . If process noise is overestimated, the filter becomes "jittery" as it over-responds to measurements. Conversely, if measurement noise is overestimated, the filter responds too slowly to actual changes in the system state.

Troubleshooting

If the "Settling Step" shows "N/A", the filter has not yet reached a stable gain within the 1% threshold. Ensure that the data contains only valid numerical values and that the number of steps does not exceed the maximum limit of 1000 points.

Frequently asked questions

What is the innovation value?

Innovation, or the residual, is the difference between the actual observed measurement and the value predicted by the system model before the update.

How is the settling step calculated?

It is the point where the change in the Kalman gain between consecutive steps becomes less than 1% of the previous gain value, indicating stability.

Can this handle non-linear data?

This specific implementation is designed for a simple one-dimensional linear system where the state remains relatively constant or changes linearly with noise.

Where this calculation is used

This statistical method is widely utilised in academic research involving data assimilation and time-series analysis. In environmental science, it helps smooth sensor data from weather stations. In social research, it can be applied to track population trends while filtering out short-term sampling fluctuations. It is also a fundamental concept in probability theory for teaching recursive algorithms and Bayesian inference, where the posterior distribution is updated as new evidence becomes available.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.