Principal Component Analysis of the Fractional Brownian Motion for 0 < H < 0.5

Abstract

Principal component analysis (PCA) has been proposed for the estimation of the self-similarity parameter H, namely the Hurst parameter of 1/f processes, and an analytical proof is provided only for H=0.5 in a recent study [1]. In our paper, we extend this study by deriving explicit expressions and presenting an analytical proof for the range of 0 < H < 0.5 (the anti-persistent part of the fractional Brownian motion). We also show via simulations that the accuracy of the estimated H values may decrease considerably as the theoretical H value increases towards the persistent part (0.5 < H < 1).