2021 edition

Faithful transitive permutation representations of highly symmetric structures

10h15-10h35

Claudio Alexandre Piedade [claudio.a.piedade@ua.pt]

Scientific guidance: Maria Elisa Fernandes [maria.elisa@ua.pt]

Abstract: Cayley's theorem states that any finite group can be represented as a group of permutations, i.e. a subgroup of a symmetric group of degree n. Moreover, the action of a group on the cosets of a core-free subgroup gives a faithful transitive permutation representation of the group. These permutation representations can be used to characterize the groups of symmetric structures, such as Coxeter groups. Coxeter groups are the automorphism groups of abstract regular polytopes and, more recently, of regular hypertopes. Not many families of proper regular hypertopes are known. In this presentation, we will show how we were able to build families of locally toroidal regular hypertopes from faithful transitive permutation representations of toroidal maps {4,4} and {3,6}.

Spatio-temporal models for time series of counts

11h40-12h00

Ana Martins [a.r.martins@ua.pt]

Scientific guidance: Sónia Gouveia (UA, Aveiro, Portugal), Manuel Scotto (IST, UL, Lisboa, Portugal), Christian Weiss (Helmut-Schmidt-University, Hamburg, Germany)

Abstract: Thinning-based Integer Autoregressive Moving Average (INARMA) models are a popular approach for modelling time series of counts. These models correspond to the integer-valued counterpart of the well-known conventional ARMA models where the multiplication is replaced by the so-called binomial thinning operator (BTO). Spatio-temporal ARMA extensions have led to the formulation of (conventional) STARMA models, and yet no similar developments have been done for time series of counts. This work proposes a novel class of space-time models for counts, STINARMA, which corresponds to the integer counterpart (BTO-based) of the STARMA class and, simultaneously, to the spatio-temporal extension (STARMA inspired) of the INARMA class. So far, the moving average STINMA subclass of models have been characterized in terms of e.g., its first and second-order moments as well as the space-time autocorrelation function, highlighting existing similarities with the STMA and INMA models. Currently, estimation procedures based on the Method of Moments, Conditional Least Squares and Conditional Maximum Likelihood are being developed to compare the efficiency and consistency between the different procedures. The methods are being implemented in R programming language, to ease the broad dissemination and the awareness of the usefulness of such models.

Time series periodicity detection using area biplots: an application to hydrologic data

12h20-12h40

Alberto Silva

Abstract: Singular Spectral Analysis (SSA) is a powerful technique used in time series (ts) analysis for different purposes such as exploratory inspection and forecasting. The trajectory matrix (X) constructed in the first stage of the method allows the use of multivariate techniques from its singular values decomposition. In this work, an exploratory approach based on biplots is used to visually estimate the dominant periodicities of a ts. From the decomposition of X through the NIPALS algorithm, the method starts identifying pairs of singular values close to each other, which suggests the respective principal components (PCs) are associated with the periodicity of the ts. The selected PCs will constitute the factorial axes of the biplots. Pinning a biplot vector of interest and rotating anticlockwise by 90° the others, triangles are drawn connecting the origin of the factorial axes and the endpoints of the pinned vector and each of the rotated vectors. Depending on the percentage of variability explained by the PCs involved, the area of the triangles provides visual information regarding the magnitude of the autocorrelation between the corresponding lagged vectors (columns of X). In addition, the periodicity will emerge from the appearance of groups of similar triangles because of the strong autocorrelation between groups of lagged vectors. An application to hydrologic data containing many missing values is performed to illustrate the proposed method.

Pontryagin Maximum Principle for Distributed-Order Fractional Systems

13h00-13h20

Faïçal Ndaïrou

Abstract: We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontryagin maximum principle for distributed-order fractional optimal control problems, illustrating its applicability with an example.