I am using a Kalman filter to fuse gyro and inclinometer data. The prediction step is given by:

$\hat{\alpha_{i}}^- = \hat{\alpha}^+_{i-1} + \omega_{i}$

Where $\hat{\alpha_{i}}^-$ is the prediction for the angle, $\hat{\alpha}^+_{i-1}$ is the previous a-posteriori state estimate and $\omega_i$ is the change in angle measured by the gyro. The innovation of the filter is given by:

$\alpha_i - \hat{\alpha_{i}}^- $

where $\alpha_i$ is the current measurement of the inclinometer. As far as my understanding of the Kalman filter goes, the innovation is expected to have a zero mean. However, the following Figure shows the histogramm for some 25000 datapoints of the innovation: Histogramm of the innovation

The mean is obviously non-zero. Would it be correct to claim that this means that some error source is present, for example an uncorrected gyro offset? Or is the non-zero mean more an "academic" property of the filter, can't be expected to be true in a real-world implementation, and thus a non-zero mean should not be overrated?

  • $\begingroup$ Yes, an uncorrected gyro offset (usually in the $z$ axis) would cause that. For example, not taking account of the earth's gravity. $\endgroup$
    – Peter K.
    Commented May 16, 2017 at 11:50

1 Answer 1


Mean of innovation sequence is zero if the Kalman filter is optimal. A non zero mean implies that there are certain errors present in your assumptions of covariance of noise or even your system matrices(if you are using a state space model).


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