Library / Tracking and estimation / TRK-05
Tracking and estimation
Innovation gate statistic
Normalised squared innovation for a scalar measurement: the squared gap between the measurement and its prediction divided by the innovation variance S.
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- TRK-05
- Rev
- v1
- Field
- Tracking and estimation
- Result
- d²
The innovation ν is the gap between a measurement and its prediction, and S is the variance the filter expects it to have (the prediction variance plus the measurement variance). The statistic d² = ν²/S follows a chi-square distribution with one degree of freedom when the model is right. A measurement is accepted if d² is below a chosen quantile; for one degree of freedom, 3.84 is the 95 % value and 6.63 the 99 % value.
Normalised squared innovation for a scalar measurement: the squared gap between the measurement and its prediction divided by the innovation variance S. Under the filter model it follows a chi-square distribution with one degree of freedom.
Calculator
- Normalised innovation, |ν|/√S1.5 σ
- Share of the 95 % gate value (3.84)58.59 %
- Share of the 99 % gate value (6.63)33.94 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- d² = 2.25 is 58.6 % of the 95 % gate value of 3.84, so this measurement would be accepted at 95 %. The 95 % gate is the same as 1.96 standard deviations.
- Passing the gate does not prove the measurement belongs to the target. In clutter a wrong measurement can pass, and a wider gate (99 %: 6.63, or 2.58 standard deviations) admits more of them while rejecting fewer good ones.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| ν Innovation (measurement minus prediction) | −5 to 5 | 0 | 25 | 25 | +0.2 | +28.4 % |
| S Innovation variance | 0.25 to 4 | 0.5625 | 9 | 8.438 | +10% | −9.09 % |
Sweep of Innovation (measurement minus prediction)
| Innovation (measurement minus prediction) | Gate statistic |
|---|---|
| −5 | 25 |
| −3.75 | 14.06 |
| −2.5 | 6.25 |
| −1.25 | 1.563 |
| 0 | 0 |
| 1.25 | 1.563 |
| 2.5 | 6.25 |
| 3.75 | 14.06 |
| 5 | 25 |
Sweep one variable
Variables and how each one contributes
Gate statistic
What the equation gives you. Everything else feeds it.
Innovation (measurement minus prediction)
The innovation: measurement minus prediction, in the unit of the measurement. The statistic grows with the square of the innovation, so doubling it multiplies d² by four. Raising it by 10 % (to 1.65) moves the result by +21 %, from 2.25 to 2.723.
Innovation variance
The innovation variance, the prediction variance plus the measurement variance, in the unit squared. A larger expected variance makes the same innovation look more ordinary, so d² falls. Raising it by 10 % (to 1.1) moves the result by −9.09 %, from 2.25 to 2.045.
Worked example with the values above
- Square of the innovation: ν² = 1.5² = 2.25.
- Divide by its expected variance: d² = 2.25 / 1.
- d² = 2.25, which is 1.5 standard deviations.
Assumptions and limits
- A correctly specified linear Gaussian model, so that the innovations are zero-mean with variance S.
- A scalar measurement. For a vector of m measurements d² = νᵀS⁻¹ν has m degrees of freedom.
Common mistakes
- Using the wrong degrees of freedom. The gate value depends on the number of measurements: 3.84 for one, 5.99 for two at 95 %.
- Using the measurement variance alone instead of the full innovation variance P⁻ + R. This makes the gate too tight.
Statistics and uncertainty
If the model is correct, 5 % of good measurements will still fail a 95 % gate and 1 % a 99 % gate, so some real measurements are always rejected. Over many steps the average of d² should be about 1 for one degree of freedom. A mean well above 1 means the noise settings are too small or the target is manoeuvring, and well below 1 means they are too large.
Where it is used
- Rejecting measurements that are inconsistent with the prediction in a tracker.
- Spotting a filter whose noise settings are wrong by watching the average d².
Maths you need
- Chi-square distribution: the sum of squared standard normal values
- Innovation variance: prediction variance plus measurement variance