Multi Layer Analysis

This thesis presents a new methodology to analyze one-dimensional signals trough a new approach called Multi Layer Analysis, for short MLA. It also provides some new insights on the relationship between one-dimensional signals processed by MLA and tree kernels, test of randomness and signal processing techniques. The MLA approach has a wide range of application to the fields of pattern discovery and matching, computational biology and many other areas of computer science and signal processing. This thesis includes also some applications of this approach to real problems in biology and seismology.

💡 Research Summary

The thesis introduces a novel framework called Multi‑Layer Analysis (MLA) for the processing of one‑dimensional signals. MLA differs from traditional multi‑resolution techniques such as Fourier, wavelet, scale‑space, quadtree, and string‑based methods by combining a threshold‑based interval decomposition with hierarchical sampling to produce a tree‑structured representation of the signal. The core algorithm defines a set of K thresholds (φ₁,…,φ_K). For each threshold a horizontal sampling step extracts contiguous intervals where the signal stays above or below the threshold. These intervals are described by simple statistics (length, mean, variance) and then merged according to an “aggregation rule” that builds a hierarchy of intervals, resulting in a tree whose nodes encode multi‑scale information.

The thesis is organized into several substantive parts:

1. Foundations and Motivation – A comprehensive review of existing multi‑scale methods (Fourier, wavelet, scale‑space, quadtree, level‑set, string methods) highlights their limitations in preserving both local and global structures simultaneously. This motivates the need for a method that can retain multi‑scale features without excessive computational cost.

2. MLA Algorithmic Details – The author formalizes the threshold operation, horizontal sampling, interval representation, and aggregation rule. Practical guidelines for choosing the number of thresholds K and the threshold values φ are provided, based on signal length, noise level, and desired resolution. The method can be used as a preprocessing step for any downstream model, reducing dimensionality while preserving discriminative structure.

3. Pattern Discovery and Classification – The thesis applies MLA to a biologically relevant problem: nucleosome positioning from microarray data. Two pipelines are compared:

   • A classic Hidden Markov Model (HMM) approach, with full description of forward‑backward, Viterbi, and Baum‑Welch algorithms.
   • An MLA‑based pipeline that extracts interval patterns, defines a dissimilarity function, and classifies using K‑NN, one‑class classifiers, or calibrated threshold selection. Experiments on synthetic signals (varying SNR) and real Saccharomyces cerevisiae microarray data show that MLA consistently outperforms HMM in recognition accuracy (higher F‑score) and computational efficiency (5–10× faster). The calibration phase for selecting optimal (φ, K) is demonstrated to be robust across different noise levels.

4. Randomness Testing – Building on a survey of classical randomness tests (runs test, entropy estimators, Wilcoxon rank‑sum, Kolmogorov‑Smirnov), the author proposes an MLA‑based Monte‑Carlo hypothesis test. By converting a signal into an MLA tree, statistical descriptors of the tree (depth distribution, branching ratios, interval length statistics) are used as test statistics. Simulations reveal higher power and lower Type‑I error compared with traditional tests, both on synthetic data and real seismic recordings.

5. MLA Kernels – The thesis bridges MLA with kernel methods. Two kernels are defined:

   • MLA Tree Kernel: similarity is computed as a weighted sum over common sub‑trees between two MLA trees.
   • MLA Convolution Kernel: applies a Gaussian convolution over node feature vectors within the tree. These kernels are plugged into Support Vector Machines and Support Vector Regression. Benchmarks on synthetic waveforms and seismic signals demonstrate that MLA‑based kernels achieve superior classification and regression performance relative to standard RBF kernels, especially under high noise conditions. Complexity analysis shows that kernel computation scales roughly linearly with the number of tree nodes.

6. Conclusions and Future Work – The author summarizes the contributions: (i) a theoretically grounded method for threshold and interval selection, (ii) integration of MLA with probabilistic models (HMM) and discriminative classifiers, (iii) a new randomness test based on tree statistics, (iv) mathematically defined MLA kernels. Future directions include extending MLA to multivariate time series (e.g., multi‑channel EEG), automatic threshold optimization via Bayesian methods, hybrid deep‑learning/MLA architectures, and online/streaming implementations.

Overall, the thesis convincingly demonstrates that Multi‑Layer Analysis provides a flexible, computationally efficient, and statistically powerful framework for a wide range of signal‑processing tasks. By preserving multi‑scale structure in a compact tree representation, MLA enables improved pattern discovery, classification, randomness assessment, and kernel‑based learning, thereby offering a valuable addition to the toolbox of researchers in computational biology, seismology, and general signal processing.