Sensor Fusion Calculator

Compare inverse-variance weighting, outlier gating, and Kalman-style prior updates for IoT sensor estimates

Interactive sensor fusion calculator with scenario presets, weighted estimates, uncertainty bands, Kalman-style prior updates, outlier gating, formula checks, and mobile-safe reference material.

Animation Sensor Fusion Uncertainty

Sensor Fusion Calculator

Fuse three noisy IoT sensor readings, gate an obvious outlier, and update a prior estimate. The visual keeps the raw readings, uncertainty bands, weights, formulas, and plain-language diagnosis in sync.

-- Fused measurement

-- Fused uncertainty

-- Outlier gate

Readings are not equal

Each sensor reports a value and a standard deviation. Smaller sigma means a narrower uncertainty band.

Weights follow variance

Independent Gaussian readings are combined with weights proportional to 1 / sigma^2.

Bad readings can dominate

A confident but wrong sensor can pull the estimate away unless a residual gate rejects it first.

The prior still matters

A Kalman-style update blends the fused measurement with what the system believed before this sample.

1

Raw readings

Plot every sensor on one scale.

2

Convert variance

Square sigma to get variance.

3

Weight sensors

Precise sensors receive more trust.

4

Gate outliers

Reject readings too far from the cluster.

5

Fuse estimate

Compute the weighted estimate and sigma.

6

Update prior

Blend the measurement with the prior state.

Raw readings Plot each sensor reading and its uncertainty interval before doing any math.

Raw readings

Plot each sensor reading and its uncertainty interval before doing any math.

Stage 1 of 6

What changed Three readings are visible.

Start by checking whether the uncertainty bars overlap.

Diagnosis Ready to fuse

The default readings agree closely enough to combine.

Result --

Use Step or Play to walk through the calculation.

Controls

Choose a preset, then adjust readings and uncertainty to see how the estimate changes.

Scenario

Sensor readings

Sensor A value

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Sensor A sigma

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Sensor B value

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Sensor B sigma

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Sensor C value

--

Sensor C sigma

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Prior and gate

Prior estimate

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Prior sigma

--

Gate width

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Formula Trace

The calculator uses a simple independent-Gaussian teaching model. It is accurate for the assumptions shown here, but real deployments must validate calibration, correlation, and outlier logic.

variance_i = sigma_i^2 raw_weight_i = 1 / variance_i Smaller standard deviation produces a much larger raw weight because variance is squared.

normalized_weight_i = raw_weight_i / sum(raw_weight) fused_value = sum(normalized_weight_i * reading_i) Only accepted readings are included after the outlier gate is applied.

Kalman gain: K = P / (P + R) posterior = prior + K * (fused_value - prior) The prior variance is P. The fused measurement variance is R.

Reference Material

Use these cards to check terms, design decisions, and model limits while experimenting with the animation.

Quick Reference

Reading The reported measurement from a sensor after calibration and unit conversion.

Sigma The standard deviation of measurement error. Smaller sigma means a tighter confidence band.

Variance The squared standard deviation. In this model, fusion trust is based on inverse variance.

Weight The normalized share of trust assigned to an accepted reading.

Fused estimate The weighted mean of accepted readings, expressed in the scenario unit.

Posterior The updated estimate after blending the prior and the fused measurement.

Fusion Design Checklist

Use the same unit Do not fuse meters, RSSI, and probability directly unless they have been converted into a common state model.

Align time stamps A fast-moving system needs readings from the same time window, not stale measurements mixed with current ones.

Calibrate sensors Weighting reduces random noise. It does not automatically remove bias from a poorly calibrated device.

Check independence The simple formula assumes independent errors. Shared power, temperature, or placement can make errors correlated.

Gate outliers first Reject or downweight readings that fail a residual test before allowing a confident sensor to dominate.

Handle missing data A real fusion pipeline needs a policy for dropped packets, stale sensors, and degraded operation.

Technical Accuracy Notes

Gaussian assumption Inverse-variance weighting is optimal for unbiased, independent Gaussian measurement errors.

Fused sigma The calculator uses sqrt(1 / sum(1 / sigma_i^2)) for accepted independent sensors.

Kalman-style update This page shows a one-dimensional measurement update, not a full multi-state Kalman filter.

Outlier gate The gate is a teaching residual check around the median cluster, not a universal fault-detection rule.

Correlated errors If sensors share the same error source, the fused uncertainty will be too optimistic unless covariance is modeled.

Uncertainty is not truth A small sigma means the model trusts the sensor. It does not prove the sensor is correct.

Related Animations

These pages extend the same estimation idea from a weighted measurement into dynamic filters and visual sensor fusion.

Sensor Fusion Visualizer See how multiple sensing modalities combine into a richer IoT estimate. Kalman Filter Move from one measurement update to prediction and correction over time. Particle Filter Compare Gaussian updates with a sample-based approach for non-linear belief tracking.