On the Fibonacci sequence and the Linear Time Invariant systems

The Fibonacci sequence (FS) possesses exceptional mathematical properties that have captivated mathematicians, scientists, and artists across centuries. Its intriguing nature lies in its profound connection to the golden ratio, as well as its prevalence in the natural world, exhibited through phenomena such as spiral galaxies, plant seeds, the arrangement of petals, and branching structures. This report delves into the fundamental characteristics of the FS, explores its relationship with the golden ratio using Linear Time Invariant (LTI) systems, and investigates its diverse applications in various fields. Approaching the topic from the standpoint of a digital signal processing instructor in a grade course, we depict the FS as the consequential outcome of an LTI system when subjected to the unit impulse function. This LTI system can be regarded as the original source from which one of the most renowned formulas in mathematics emerges, and its parametric definition, along with the associated systems, is intricately tied to the golden ratio, symbolized by the irrational number Phi. This perspective naturally elucidates the well-established intricate relationship between the FS and Phi. Furthermore, building upon this perspective, we showcase other LTI systems that exhibit the same magnitude in the frequency domain. These systems are characterized by either an impulse response or a difference equation, resulting in a comparable or equivalent FS in terms of absolute value. By exploring these connections, we shed light on the remarkable similarities and variations that arise within the FS under different LTI systems.

A hypothesis-driven method based on machine learning for neuroimaging data analysis

There remains an open question about the usefulness and the interpretation of Machine learning (MLE) approaches for discrimination of spatial patterns of brain images between samples or activation states. In the last few decades, these approaches have limited their operation to feature extraction and linear classification tasks for between-group inference. In this context, statistical inference is assessed by randomly permuting image labels or by the use of random effect models that consider between-subject variability. These multivariate MLE-based statistical pipelines, whilst potentially more effective for detecting activations than hypotheses-driven methods, have lost their mathematical elegance, ease of interpretation, and spatial localization of the ubiquitous General linear Model (GLM). Recently, the estimation of the conventional GLM has been demonstrated to be connected to an univariate classification task when the design matrix is expressed as a binary indicator matrix. In this paper we explore the complete connection between the univariate GLM and MLE \emph{regressions}. To this purpose we derive a refined statistical test with the GLM based on the parameters obtained by a linear Support Vector Regression (SVR) in the \emph{inverse} problem (SVR-iGLM). Subsequently, random field theory (RFT) is employed for assessing statistical significance following a conventional GLM benchmark. Experimental results demonstrate how parameter estimations derived from each model (mainly GLM and SVR) result in different experimental design estimates that are significantly related to the predefined functional task. Moreover, using real data from a multisite initiative the proposed MLE-based inference demonstrates statistical power and the control of false positives, outperforming the regular GLM.