Why innovation equation in Extended Kalman filter is called innovation?

Question

In the Extended Kalman filter for SLAM, why is the innovation equation called so? Is there a reason for using the specific word "innovation" for the difference between the observed information and the predicted value?

Answer 1

From the Wikipedia entry:

In time series analysis (or forecasting) — as conducted in statistics, signal processing, and many other fields — the innovation is the difference between the observed value of a variable at time t and the optimal forecast of that value based on information available prior to time t. If the forecasting method is working correctly, successive innovations are uncorrelated with each other, i.e., constitute a white noise time series. Thus it can be said that the innovation time series is obtained from the measurement time series by a process of 'whitening', or removing the predictable component. The use of the term innovation in the sense described here is due to Hendrik Bode and Claude Shannon (1950) in their discussion of the Wiener filter problem, although the notion was already implicit in the work of Kolmogorov.

The especially noteworthy bit in that text is that, "if the forecasting method is working correctly, successive innovations ... constitute a white noise time series."

This means that you should be able to apply a feedback based on the innovations and, in the long-term, that feedback won't do anything if your predictions are correct.

To put it another way, feedback based on innovations will only change your filter parameters if your filter isn't making accurate predictions. This is the heart of how and why the Kalman filter works as well as it does - it tunes itself until it is making good predictions and then stops tuning.