Is the Kalman filter expected to improve accuracy if there is no measurement of the final desired state variable?
Take the example of a self-driving car with accelerometer data and speedometer data, but no GPS system. Without a filter, we could integrate this data to derive location, but the noise in measurements will cause location to drift away from its true value.
Alternatively, a Kalman filter can be used to generate more accurate estimates of acceleration and speed from the noisy data -- then the estimates can be integrated to derive location.
Mathematically and practically, I don't see how the location estimate using Kalman estimates should be more accurate (this question shows an example). The estimates are closer on average to their true values, but after integration, symmetric errors tend to cancel each other regardless of noise scale. The Kalman filter's slight delay in tracking true changes in process variables can only result in greater integrated error than the previous no-filter integration.
Of the Kalman filter examples I've seen online, each falls into one of these groups:
- Direct measurement of the desired variable is available.
- Direct measurement unavailable, and integration to desired variable isn't shown.
- Direct measurement unavailable, integration to desired variable is shown, but comparison to a pre-Kalman-filter integration isn't shown (i.e., the supposed improvement to accuracy isn't demonstrated).