Principal Component Analysis for Functional Data on Riemannian Manifolds and Spheres

Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered for example as movement trajectories on the surface of the earth, are an important special case. We consider an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the time-varying Fr\'echet mean function, and then performing a classical multivariate functional principal component analysis on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained with exponential maps. The tangent-space approximation through functional principal component analysis is shown to be well-behaved in terms of controlling the residual variation if the Riemannian manifold has nonnegative curvature. Specifically, we derive a central limit theorem for the mean function, as well as root-$n$ uniform convergence rates for other model components, including the covariance function, eigenfunctions, and functional principal component scores. Our applications include a novel framework for the analysis of longitudinal compositional data, achieved by mapping longitudinal compositional data to trajectories on the sphere, illustrated with longitudinal fruit fly behavior patterns. RFPCA is shown to be superior in terms of trajectory recovery in comparison to an unrestricted functional principal component analysis in applications and simulations and is also found to produce principal component scores that are better predictors for classification compared to traditional functional functional principal component scores

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

Mathematical Methods, Modelling and Applications

This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods

Sub-Riemannian geometry and its applications to Image Processing

Master's Thesis in MathematicsMAT399MAMN-MA

Identifying Stellar Flares From Optical Lightcurves

Roihut eli flaret ovat lyhytaikaisia, korkeaenergisiä magneettisia purkauksia tähtien pinnoilla. Havainnot etenkin nuorista tähdistä ja punaisista kääpiöistä ovat osoittaneet, että purkaukset voivat olla useita kertaluokkia energisempiä kuin tähän asti havaitut aurinkomyrskyt. Koska suuri osa ihmiskunnan teknologiasta on haavoittuvainen avaruussään häiriöille, roihut ja muut tähtien magneettiset ilmiöt ovat tärkeä tutkimuskohde. Viime vuosiin asti roihuhavaintojen kerääminen tähdistä on ollut työlästä ja havaintoresursseja kuluttavaa. Tässä työssä on luotu enimmäkseen automaattinen ohjelma roihujen havaitsemiseksi ja niiden energioiden arvioimiseksi optisten valokäyrähavaintojen pohjalta. Tähtien taustasäteilyn arviointiin käytetään useista tukivektorikoneista (support vector machine) luotua mallia. Selvästi mallia kirkkaammat datapisteet merkitään todennäköisiksi roihuiksi. Näihin pisteisiin sovitetaan kirjallisuudesta haettu malli, jonka avulla arvioidaan purkauksen energisyys. Ohjelma on testattu AB Doradus ja EK Draconis -tähdistä kerätyilla valokäyrillä, joista löytyy kymmeniä roihuja. Tulokset ovat pääpiirteittäin yhteensopivia olemassa olevan kirjallisuuden kanssa, ja ohjelman voi yleistää muihin tähtiin ja muunlaisiin valokäyriin. Tässä muodossaan ohjelman luotettavuutta haastavat rajatapaukset, systemaattiset epävarmuudet ja käyttäjän tekemien valintojen vaikutus tuloksiin.Flares are short, high-energy magnetic events on stars, including the Sun. Observations of young stars and red dwarfs regularly show the occurrence of flare events multiple orders of magnitude more energetic than even the fiercest solar storms ever recorded. As our technology remains vulnerable to disruptions due to space weather, the study of flares and other stellar magnetic activity is crucial. Until recently, the detection of extrasolar flares has required much manual work and observation resources. This work presents a mostly automatic pipeline to detect and estimate the energies of extrasolar flare events from optical light curves. To model and remove the star's background radiation in spite of complex periodicity, short windows of nonlinear support vector regression are used to form a multi-model consensus. Outliers above the background are flagged as likely flare events, and a template model is fitted to the flux residual to estimate the energy. This approach is tested on light curves collected from the stars AB Doradus and EK Draconis by the Transiting Exoplanet Survey Satellite, and dozens of flare events are found. The results are consistent with recent literature, and the method is generalizable for further observations with different telescopes and different stars. Challenges remain regarding edge cases, uncertainties, and reliance on user input

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Bayesian hierarchical modelling of growth curve derivatives via sequences of quotient differences

Growth curve studies are typically conducted to evaluate differences between group or treatment-specific curves. Most analyses focus solely on the growth curves, but it has been argued that the derivative of growth curves can highlight differences between groups that may be masked when considering the raw curves only. Motivated by the desire to estimate derivative curves hierarchically, we introduce a new sequence of quotient differences (empirical derivatives) which, among other things, are well behaved near the boundaries compared with other sequences in the literature. Using the sequence of quotient differences, we develop a Bayesian method to estimate curve derivatives in a multilevel setting (a common scenario in growth studies) and show ow the method can be used to estimate individual and group derivative curves and to make comparisons. We apply the new methodology to data collected from a study conducted to explore the effect that radiation-based therapies have on growth in female children diagnosed with acute lymphoblastic leukaemia

Multi-objective optimal design of mechanical metafilters based on principal component analysis

In this paper, an advanced computational method is proposed, whose aim is to obtain an approximately optimal design of a particular class of acoustic metamaterials, by means of a novel combination of multiobjective optimization and dimensionality reduction. Metamaterials are modeled as beam lattices with internal local resonators coupled with the microstructure through a viscoelastic phase. The dynamics is governed by a set of integro-differential equations, that are transformed into the Z-Laplace space in order to derive an eigenproblem whose solution provides the dispersion relation of the free in-plane propagating Bloch waves. A multi-objective optimization problem is stated, whose aim is to achieve the largest multiplicative trade-off between the bandwidth of the first stop band and the one of the successive pass band in the metamaterial frequency spectrum. Motivated by the multi-dimensionality of the design parameters space, the goal above is achieved by integrating numerical optimization with machine learning. Specifically, the problem is solved by combining a sequential linear programming algorithm with principal component analysis, exploited as a data dimensionality reduction technique and applied to a properly sampled field of gradient directions, with the aim to perform an optimized sensitivity analysis. This represents an original way of applying principal component analysis in connection with multi-objective optimization. Successful performances of the proposed optimization method and its computational savings are demonstrated

Data-driven reduced order models using invariant foliations, manifolds and autoencoders

This paper explores how to identify a reduced order model (ROM) from a physical system. A ROM captures an invariant subset of the observed dynamics. We find that there are four ways a physical system can be related to a mathematical model: invariant foliations, invariant manifolds, autoencoders and equation-free models. Identification of invariant manifolds and equation-free models require closed-loop manipulation of the system. Invariant foliations and autoencoders can also use off-line data. Only invariant foliations and invariant manifolds can identify ROMs, the rest identify complete models. Therefore, the common case of identifying a ROM from existing data can only be achieved using invariant foliations. Finding an invariant foliation requires approximating high-dimensional functions. For function approximation, we use polynomials with compressed tensor coefficients, whose complexity increases linearly with increasing dimensions. An invariant manifold can also be found as the fixed leaf of a foliation. This only requires us to resolve the foliation in a small neighbourhood of the invariant manifold, which greatly simplifies the process. Combining an invariant foliation with the corresponding invariant manifold provides an accurate ROM. We analyse the ROM in case of a focus type equilibrium, typical in mechanical systems. The nonlinear coordinate system defined by the invariant foliation or the invariant manifold distorts instantaneous frequencies and damping ratios, which we correct. Through examples we illustrate the calculation of invariant foliations and manifolds, and at the same time show that Koopman eigenfunctions and autoencoders fail to capture accurate ROMs under the same conditions.Comment: 48 pages, 16 figures. Update: some bugs were fixed in the numerical calculation