Principal component analysis is one of the key dimension reduction tool for multivariate data used in machine learning. The finite-dimensional case has been extensively studied. We will focus on a generalization of PCA to functional data and termed functional principal component analysis (FPCA). FPCA has taken off to become the most prevalent tool in functional data analysis. This is partly because FPCA facilitates the conversion of inherently infinite-dimensional functional data to a finite-dimensional vector of random scores. Under mild assumptions, the underlying stochastic process can be expressed as a countable sequence of uncorrelated random variables, the functional principal components (FPCs) for scores, which are then truncated to a finite vector. Then the tools of multivariate data analysis can be readily applied to the resulting random vector of scores, thus accomplishing the goal of dimension reduction. ...
We have recently published a paper with GMV at Elsevier and I wanted to share it here also. Link to Article: https://www.sciencedirect.com/science/article/pii/S2352467724002248 Abstract This paper presents an exploration of the application of control theory, particularly utilizing a gradient-based algorithm, to automate and optimize the operation of photovoltaic panels and refrigeration systems in warehouse environments. The study emphasizes achieving coordination between energy generation and consumption, specifically harnessing surplus solar energy for efficient refrigeration. The complex interplay between fluctuating solar irradiance, thermal dynamics of the warehouse, and refrigeration needs underscores the significance of control theory in designing algorithms to dynamically adjust PV panel output and refrigeration system operation. The paper discusses foundational control theory principles, proposes a tailored framework for warehouse operations, and highlights the potential for sustainable energy practices. This paper explores the use of data-driven approaches based on NeuralODEs vs classical ones using physics equations. ...
What do traffic congestion, supermarket lines and fluid dynamics have in common? While we are driving, we are used to think of cars as single individuals/entities. Although, every individual has its own driving tendencies and peculiarities, at a higher-scale, we behave within certain constraints and collective behavior. This quantities can be interpreted in many cases as a homogeneous dense fluid of cars. In this post I will focus on modelling traffic flow using fluid dynamics principles. ...
Several non-linear problems relevant in practical applications can be expressed as a fixed point equation. In many cases, it is crucial to investigate how the model’s behavior changes with variations in a parameter, denoted as $\lambda$. In practical applications, $\lambda$ represents a physical or empirical magnitude of interest. Bifurcation Theory is a subfield in Nonlinear Functional Analysis that tries to study the general behavior of the equations that can be written as $\mathfrak{F}(\lambda, u)=0$ where $\lambda$ is the bifurcation parameter. ...