I think that this was a great intro into Kalman filtering.
The one important point that I think warrants a small paragraph near the end is that the example you gave is a way of doing forecasting (estimating the future state) and nowcasting (estimating the current state), but Kalman filters can also be used retrospectively to do retrocasting (using the present data to get a better estimate of the past).
Nowcasting and retrocasting are concepts that a lot of people have trouble with. That trouble is the crux of the Kalman filter ... combining (noisy) measurements with (noisy) dead reckoning gives us (better) knowledge. For complete symmetry, it is important to point out that we can't just use old measurements to describe the past any more than we should only use current and past measurements to define our estimate of the present.
The one important point that I think warrants a small paragraph near the end is that the example you gave is a way of doing forecasting (estimating the future state) and nowcasting (estimating the current state), but Kalman filters can also be used retrospectively to do retrocasting (using the present data to get a better estimate of the past).
Nowcasting and retrocasting are concepts that a lot of people have trouble with. That trouble is the crux of the Kalman filter ... combining (noisy) measurements with (noisy) dead reckoning gives us (better) knowledge. For complete symmetry, it is important to point out that we can't just use old measurements to describe the past any more than we should only use current and past measurements to define our estimate of the present.