Reconstruction of Fractional Brownian Motion Signals From Its Sparse Samples Based on Compressive Sampling

TL;DR

This paper introduces a novel compressive sampling-based method for reconstructing fractional Brownian motion signals from sparse samples, leveraging their spectral sparsity without needing prior knowledge of the Hurst parameter.

Contribution

It develops a CS-based reconstruction technique for fBm signals that does not require Hurst parameter information, applicable to both synthetic and real-world data.

Findings

Reconstruction quality depends on Hurst parameter H and number of samples.

Method successfully reconstructs synthetic fBm signals with various H values.

Applied to Dow Jones data, demonstrating real-world applicability.

Abstract

This paper proposes a new fBm (fractional Brownian motion) interpolation/reconstruction method from partially known samples based on CS (Compressive Sampling). Since 1/f property implies power law decay of the fBm spectrum, the fBm signals should be sparse in frequency domain. This property motivates the adoption of CS in the development of the reconstruction method. Hurst parameter H that occurs in the power law determines the sparsity level, therefore the CS reconstruction quality of an fBm signal for a given number of known subsamples will depend on H. However, the proposed method does not require the information of H to reconstruct the fBm signal from its partial samples. The method employs DFT (Discrete Fourier Transform) as the sparsity basis and a random matrix derived from known samples positions as the projection basis. Simulated fBm signals with various values of H are used to show the relationship between the Hurst parameter and the reconstruction quality. Additionally, US-DJIA (Dow Jones Industrial Average) stock index monthly values time-series are also used to show the applicability of the proposed method to reconstruct a real-world data.