An Advanced Self-Similarity Measure: Average of Level-Pairwise Hurst Exponent Estimates (ALPHEE)

Many natural processes are characterized by complex patterns of self-similarity, where repetitive structures occur across different resolutions. The Hurst exponent is a key parameter used to quantify this self-similarity. While wavelet-based techniques are effective in estimating the Hurst exponent, their performance can be compromised by noise, outliers, and modeling assumptions. This study makes a dual contribution by introducing a novel method for estimating the Hurst exponent under standard modeling assumptions and applying this method to a significant study on gait data. The novel method leverages wavelet transforms (WT) to refine the traditional assessment of self-similarity, which typically depends on the regular decay of signal energies at various resolutions. Our method integrates the standard fractional Brownian motion (fBm) model with exact probability distributions of wavelet coefficients, combining estimates of the Hurst exponent from pairs of wavelet decomposition levels into a single estimate, named ALPHEE, that offers a more precise measure of self-similarity. The study investigates the use of self-similarity features in machine learning algorithms for identifying elderly adults who have had unintentional falls. By analyzing linear acceleration (LA) and angular velocity (AV) in 147 subjects (79 fallers, 68 non-fallers), the study finds higher regularity in LA and AV for fallers. The performance of classification models is compared with and without self-similarity features, suggesting these features enhance the detection of fallers versus non-fallers. The results show that integrating self-similarity features significantly improves performance, with the proposed method achieving 89.65% accuracy, compared to 82.75% using the standard method. This improvement surpasses existing studies based on the same dataset, suggesting that the proposed method more accurately captures self-similar properties, leading to better performance in gait data analysis.