A. Elmoataz et al. (Eds.): ICISP 2012, LNCS 7340, pp. 326–332, 2012. © Springer-Verlag Berlin Heidelberg 2012 Maximum Likelihood Estimation, Interpolation and Prediction for Fractional Brownian Motion Rachid Harba 1 , Hassan Douzi 2 , and Mohamed El Hajji 2 1 Laboratoire PRISME, Polytech’Orléans, Université d'Orléans 45067 Orléans, France Rachid.Harba@univ-orleans.fr 2 Laboratoire IRF-SIC, Faculté des sciences, Université Ibn Zohr, BP8106 Agadir, Maroc {douzi_h,hajjimohmed}@yahoo.fr Abstract. The maximum likelihood (ML) estimation approach for fractional Brownian motion (fBm) is explored in this communication. First, a ML based estimation of the H parameter is implemented on the signal itself. This approach on the signal itself can easily be applied on non-uniformly sampled data or directly useful in the case of incomplete data. Secondly, the method is extended to provide a ML prediction and a ML interpolation for fBm which could be of interest in many domains. Results also help to explain errors in other interpolating methods such as the midpoint displacement algorithm used to synthesize fBm data. 1 Introduction Fractional Brownian motion (fBm) of H parameter in the range ]0 ; 1[ is defined as an extension of Brownian motion [1]. One of the main issues when dealing with such data is to estimate the H parameter [2-3]. Among the numerous methods to achieve such a goal, the maximum likelihood (ML) approach proposed by Lundahl et al. [4] is often used due to its asymptotical efficiency [5]. It is also efficient in noisy environments [6]. But the ML based estimation of the H parameter is performed on the fBm increments which may be a limiting factor in some cases. Here, we propose an ML estimate of the H parameter processed on the fBm itself. This allows direct extension of the method to include cases where there may be irregular sampling or incomplete data. Moreover, an ML based prediction and interpolation technique for a fBm signal easily result. This communication is organized as follow. In the next section, fBm is defined and its main properties are derived. Then, the ML based estimation of the H parameter is achieved and is tested on exact fBm data. Finally, ML interpolation and prediction are presented and a real data example illustrates the methods. 2 FBm Properties Continuous fBm of H parameter in ]0 ; 1[, denoted B H (t), is defined as an extension of Brownian motion B(t) [1]: