Statistical Signal Processing in Multiscale Domains

In recent years, horseshoe priors have gained prominence in Bayesian statistics for their remarkable ability to induce sparsity in high-dimensional data. This talk explores the application of traditional and new horseshoe priors for shrinkage in wavelet domains, a powerful framework for function and image denoising. We begin by reviewing the theoretical underpinnings of horseshoe priors and their advantages over traditional shrinkage methods, particularly in terms of adaptivity and simplicity in estimation in the presence of noise.
Next, we delve into the integration of horseshoe priors with wavelet transforms, illustrating how this combination enhances sparsity and leads to robust denoising and compression techniques. Horseshoe priors also lead to "second posterior mode wavelet shrinkage (SPMWS)," an efficient thresholding technique. Through several simulations and real-world examples, we demonstrate the effectiveness of these techniques in recovering signals from their sparse representations and highlight their potential in various applications. 

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 84.09% accuracy, compared to 79.55% 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. 

The objective of this research focuses on the development of a statistical methodology that is able to answer the question of whether variation in the intake of sulfur amino acids (SAA) affects the metabolic process. Traditional approaches, which evaluate specific biomarkers after a series of preprocessing procedures, have been criticized as not being fully informative, as well as inappropriate for translation of methodology. Rather than focusing on particular biomarkers, our proposed methodology involves the multifractal analysis that measures the inhomogeneity of regularity of the proton nuclear magnetic resonance (1H-NMR) spectrum by wavelet-based multifractal spectrum. With two different statistical models (Model-I and Model-II), three different geometric features of the multifractal spectrum of each 1H-NMR spectrum (spectral mode, left slope, and broadness) are employed to evaluate the effect of SAA and discriminate 1H-NMR spectra associated with different treatments. The investigated effects of SAA include group effect (high and low doses of SAA), depletion/repletion effect, and time over data effect.

The 1H-NMR spectra analysis outcomes show that the group effect is significant for both models. The hourly variation in time and depletion/repletion effects does not show noticeable differences for the three features in Model-I. However, these two effects are significant for the spectral mode feature in Model-II. The 1H-NMR spectra of the SAA low groups exhibit highly regular patterns with more variability than that of the SAA high groups for both models. Moreover, the discriminatory analysis conducted using the support vector machine and the principal components analysis shows that the 1H-NMR spectra of SAA high and low groups can be easily discriminatory for both models, while the spectra of depletion and repletion within these groups are discriminatory for Model-I and Model-II. Therefore, the study outcomes imply that the amount of SAA is important, and that SAA intake affects mostly the hourly variation of the metabolic process and the difference between depletion and repletion each day. In conclusion, the proposed multifractal analysis of 1H-NMR spectra provides a novel tool to investigate metabolic processes.