Speed estimation by finite differences (3): optimal sampling period, optimal 2-samples estimator

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 22:00

*Enlace a Spanish version

Summary:

This video continues the statistical analysis of finite difference velocity prediction in a stochastic process which was detailed in the video [fdest2EN].

Here. the calculations from the previous video are run for a range of sampling period values and it is observed that the accuracy is poor at very small periods. and at very high periods, but that there is a range of sampling periods with reasonable accuracy (understanding reasonable as having a variance of the prediction error below the prior variance).

The second part of the video argues that finite differences $(p(T_s)-p(0))∕T_s$ are NOT the optimal predictor in the statistical sense. Indeed, such an optimal predictor must be formulated from the variance-covariance matrix of the signals that one wishes to work with. With this, the optimal predictor (of 2-position samples, ‘finite memory’) is built and compared with the ‘naive’ finite-difference predictor. Obviously, the statistical optimum has a lower prediction error variance.

If the number of past positions used to calculate the speed prediction tends to infinity, then the optimal predictor is the Kalman filter, whose details and comparison are addressed in the video [fdest4EN]; this will close the case study.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.