Innovation (measurement residual)

The innovation is the difference between the actual measurement and the measurement the filter expected from its predicted state: e_t = y_t − C x_pred. It is the new information the observation carries beyond what the model already anticipated, and the correction step moves the estimate only in proportion to it. Everything the filter learns at a step enters through the innovation.

The innovation sequence is also the standard tool for checking whether a filter is well-specified. For a correct linear-Gaussian model the innovations are zero-mean, white (temporally uncorrelated), and have a known covariance C P_pred C^T + Q. Autocorrelation or drift in the innovations signals a wrong dynamics matrix, mis-tuned noise covariances, or non-stationary neural activity. In BCI this is a practical health check: persistently biased innovations often precede a decoder that a user reports as 'fighting' them, and they are a trigger for recalibration.