Optimal Measurement Times for Observing a Brownian Motion over a Finite Period Using a Kalman Filter

Alexandre Aksenov (1) , Pierre-Olivier Amblard (2) , Olivier J.J. Michel (2) , Christian Jutten (1)

1 GIPSA-VIBS - GIPSA - Vision and Brain Signal Processing
2 GIPSA-CICS - GIPSA - Communication Information and Complex Systems

This article deals with the optimization of the schedule of measures for observing a random process in time using a Kalman filter, when the length of the process is finite and fixed, and a fixed number of measures are available. The measure timetable plays a critical role for the accuracy of this estimator. Two different criteria of optimality of a timetable (not necessarily regular) are considered: the maximal and the mean variance of the estimator. Both experimental and theoretical methods are used for the problem of minimizing the mean variance. The theoretical methods are based on studying the cost function as a rational function. An analytical formula of the optimal instant of measure is obtained in the case of one measure. Its properties are studied. An experimental solution is given for a particular case with n > 1 measures.

Mots clés: Kalman filter; Random walk; Wiener process

Domaines: Traitement du signal et de l'image [eess.SP]

Alexandre Aksenov, Pierre-Olivier Amblard, Olivier J.J. Michel, Christian Jutten. Optimal Measurement Times for Observing a Brownian Motion over a Finite Period Using a Kalman Filter. LVA/ICA 2017 - 13th International Conference on Latent Variable Analysis and Signal Separation, Olivier Michel; Nadège Thirion-Moreau, Feb 2017, Grenoble, France. pp.509 - 518, ⟨10.1007/978-3-319-53547-048⟩. ⟨hal-01479301⟩