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Designing Dual Modeling Task Sequences to Build Functional Analysis Learning Trajectories for engineering and Mathematical Sciences Education

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Designing Dual Modeling Task Sequences to Build Functional A...

Frontiers in Education (FIE) Conference

作者： Celil Ekici Pablo Rangel Jose Baca Devanayagam Palaniappan Mehrube Mehrubeoglu Mallikarjunaiah Muddamallappa Department of Mathematics & Statistics Texas A&M University-Corpus Christi Corpus Christi Texas Department of Engineering Texas A&M University-Corpus Christi Corpus Christi Texas

ISBN: (数字)9798350351507

ISBN: (纸本)9798350363067

This research-to-practice paper investigated the development of Functional Analysis Learning Trajectories (FALT) within the undergraduate courses for engineers including Calculus, Differential Equations, Communications Theory, Control Systems and Electromagnetism. Focusing on the local instructional practices, manifestations of functional analysis ideas are tracked along these courses. Corresponding concept maps are collaboratively within and across courses to portray the relevant connections for engineering mathematics education. These trajectories help engineering and mathematics instructors to collectively design, implement, refine task sequences to improve students' preparedness for upper-level math-heavy courses in engineering, connecting mathematics to their disciplines. Here we utilized collaborative concept mapping as a research heuristics to build curricular innovations with functional analysis learning trajectories across courses, hierarchically arranging, integrating, and conceptually connecting instructional tasks. Through sequences of dual stance learning tasks, students are given opportunities to take multiple stances in a learning task from the perspectives of engineers and mathematical scientists. A higher stance on mathematics was supported to be developed by comparing and connecting alternative disciplinary perspectives and practices with the dual modeling tasks and reflecting on the cross-cutting ideas along the learning trajectory. Here we present the collaborative design and analysis of concept maps along functional analysis learning trajectories for undergraduates. This research builds an interdisciplinary scholarship of teaching/learning mathematics across disciplines among engineering and math faculty. This work helps foster reflection and collaboration on their teaching practices to design and implement instructional tasks to build coherent mathematical perspectives across disciplines. It exemplifies how to design a research-based practices

关键词： Technological innovation Analytical models Scholarships Education Collaboration mathematics Functional analysis Mathematical models Reflection Trajectory

Phase Selection and Analysis for Multi-frequency Multi-user RIS Systems Employing Subsurfaces in Correlated Ricean and Rayleigh Environments

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arXiv 2024年

作者： Inwood, Amy S. Smith, Peter J. Martin, Philippa A. Woodward, Graeme K. Department of Electrical and Computer Engineering Wireless Research Centre University of Canterbury Christchurch New Zealand School of Mathematics and Statistics Victoria University of Wellington Wellington New Zealand

We analyse the performance of a reconfigurable intelligent surface (RIS) aided system where the RIS is divided into subsurfaces. Each subsurface is designed specifically for one user, who is served on their own frequency band. The other subsurfaces (those not designed for this user) provide additional uncontrolled scattering. We derive the exact closed-form expression for the mean signal-to-noise ratio (SNR) of the subsurface design (SD) when all channels experience correlated Ricean fading. We simplify this to find the mean SNR for line-of-sight (LoS) channels and channels experiencing correlated Rayleigh fading. An iterative SD (ISD) process is proposed, where subsurfaces are designed sequentially, and the phases that are already set are used to enhance the design of the remaining subsurfaces. This is extended to a converged ISD (CISD), where the ISD process is repeated multiple times until the SNR increases by less than a specified tolerance. The ISD and CISD both provide a performance improvement over SD, which increases as the number of RIS elements (N) increases. The SD is significantly simpler than the lowest complexity multiuser (MU) method we know of, and despite each user having less bandwidth, the SD outperforms the existing method in some key scenarios. The SD is more robust to strongly LoS channels and clustered users, as it does not rely on spatial multiplexing like other MU methods. Combined with the complexity reduction, this makes the SD an attractive phase selection method. Copyright © 2024, The Authors. All rights reserved.

关键词： Signal to noise ratio

Interpreting Temporal Graph Neural Networks with Koopman Theory

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arXiv 2024年

作者： Guerra, Michele Scardapane, Simone Bianchi, Filippo Maria UiT The Arctic University of Norway Department of Mathematics and Statistics Norway Sapienza Università di Roma Department of Information Engineering Electronics and Telecommunications Italy NORCE Norwegian Research Centre AS Norway

Spatiotemporal graph neural networks (STGNNs) have shown promising results in many domains, from forecasting to epidemiology. However, understanding the dynamics learned by these models and explaining their behaviour is significantly more complex than for models dealing with static data. Inspired by Koopman theory, which allows a simpler description of intricate, nonlinear dynamical systems, we introduce an explainability approach for temporal graphs. We present two methods to interpret the STGNN’s decision process and identify the most relevant spatial and temporal patterns in the input for the task at hand. The first relies on dynamic mode decomposition (DMD), a Koopman-inspired dimensionality reduction method. The second relies on sparse identification of nonlinear dynamics (SINDy), a popular method for discovering governing equations, which we use for the first time as a general tool for explainability. We show how our methods can correctly identify interpretable features such as infection times and infected nodes in the context of dissemination processes. Copyright © 2024, The Authors. All rights reserved.

关键词： Dynamic mode decomposition

NON-PARAMETRIC MONITORING OF SPATIAL DEPENDENCE

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arXiv 2024年

作者： Adämmer, Philipp Wittenberg, Philipp Weiß, Christian H. Testik, Murat Caner Institute of Data Science University of Greifswald Greifswald Germany Department of Mathematics and Statistics Helmut Schmidt University Hamburg Germany Hacettepe University Department of Industrial Engineering Beytepe-Ankara Turkey

In process monitoring applications, measurements are often taken regularly or randomly from different spatial locations in two or three dimensions. Here, we consider streams of regular, rectangular data sets and use spatial ordinal patterns (SOPs) as a non-parametric approach to detect spatial dependencies. A key feature of our proposed SOP charts is that they are distribution-free and do not require prior Phase-I analysis. We conduct an extensive simulation study, demonstrating the superiority and effectiveness of the proposed charts compared to traditional parametric approaches. We apply the SOP-based control charts to detect heavy rainfall in Germany, war-related fires in (eastern) Ukraine, and manufacturing defects in textile production. The wide range of applications and insights illustrate the broad utility of our non-parametric approach. Copyright © 2024, The Authors. All rights reserved.

关键词： Process control

A Geometric Framework for Distributed Frequency Models

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SSRN

SSRN 2024年

作者： Arumugam, Vishnuram Ferrante, Augusto Ntogramatzidis, Lorenzo Padula, Fabrizio Department of Mathematics and Statistics University of Western Australia Perth Australia Department of Information Engineering University of Padova Italy Curtin Centre for Optimisation and Decision Science Curtin University Perth Australia

Geometric control theory, developed by Basile and Marro, and independently, by Wonham and Morse in the 1970s revolves around characterizing the properties of finite dimensional, linear and time-invariant systems using geometry. Some examples of these properties are invariance, controllability and observability. The task addressed in this paper is to develop the geometric tools for fractional systems using the diffusive representation (also known as the distributed frequency) model. The mathematics involved in this approach is different from the classical case as fractional systems are inherently infinite dimensional. Unlike the integer order derivative, the fractional derivative is a non-local operator, and so the evolution of the so-called pseudo-state depends on not just its current value but also its past history. Thus, the notion of an initial condition for a fractional system can come in different forms. This leads to different kinds of fractional derivative operators. With some of these operators, the semigroup property is lost. With the distributed frequency model, the initial condition comes in the form of an initialization function. This takes care of the infinite dimensional nature of fractional systems. Furthermore, the distributed frequency model retains the semigroup property. This is useful in developing invariance and controlled invariance for fractional systems. Moreover, these properties of fractional systems are verified numerically using a higher order finite-dimensional approximation, which retains all the geometric properties of the distributed frequency model. © 2024, The Authors. All rights reserved.

关键词： Invariance

Functional-Coefficient Quantile Regression for Panel Data with Latent Group Structure

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arXiv 2023年

作者： Yang, Xiaorong Chen, Jia Li, Degui Li, Runze School of Statistics and Mathematics Collaborative Innovation Centre of Statistical Data Engineering Technology and Application Zhejiang Gongshang University China Department of Economics and Related Studies University of York United Kingdom Department of Mathematics University of York United Kingdom Department of Statistics Pennsylvania State University United States

This paper considers estimating functional-coefficient models in panel quantile regression with individual effects, allowing the cross-sectional and temporal dependence for large panel observations. A latent group structure is imposed on the heterogenous quantile regression models so that the number of nonparametric functional coefficients to be estimated can be reduced considerably. With the preliminary local linear quantile estimates of the subject-specific functional coefficients, a classic agglomerative clustering algorithm is used to estimate the unknown group structure and an easy-to-implement ratio criterion is proposed to determine the group number. The estimated group number and structure are shown to be consistent. Furthermore, a post-grouping local linear smoothing method is introduced to estimate the group-specific functional coefficients, and the relevant asymptotic normal distribution theory is derived with a normalisation rate comparable to that in the literature. The developed methodologies and theory are verified through a simulation study and showcased with an application to house price data from UK local authority districts, which reveals different homogeneity structures at different quantile levels. © 2023, CC BY-NC-ND.

关键词： Normal distribution

Practical Quaternion Givens Transformation with Application to Color Image Watermarking

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SSRN

SSRN 2023年

作者： Chen, Yong Jia, Zhigang Peng, Yan Peng, Yaxin School of Mechatronic Engineering and Automation Shanghai University Shanghai200444 China School of Mathematics and Statistics Jiangsu Normal University Xuzhou221116 China Department of Mathematics Shanghai University Shanghai200444 China

To solve the problem that the maximal feature element in quaternion matrix decomposition for color image watermarking is difficult to be extracted, two new quaternion Givens transformations with realvalued rotation results are proposed. Then, a structure-preserving arithmetic framework nested with three Givens transformations is proposed for the quaternion matrix decomposition, which makes the computation more flexible and efficient. This paper also compares the numerical performance of the real structure-preserving algorithms in which the quaternion Givens transformations are applied to the QR decomposition of Hessenberg matrices. The results show that the real structure-preserving algorithms based on the Givens transformation are not only the most efficient but also highly accurate. Finally, a novel color image watermarking algorithm is proposed by means of the maximal feature element generated by newly proposed Givens transformation. Compared with other watermarking algorithms, the robustness of the proposed watermarking algorithm has some advantages overall. © 2023, The Authors. All rights reserved.

关键词： Linear transformations

Stacking Integration Strategy-Based Learning Method for Prediction of Performance in Exams

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Stacking Integration Strategy-Based Learning Method for Pred...

Neuromorphic Computing (ICNC), International Conference on

作者： Hua Jiang Yaqian Bao Feng Jiang English Department Software Engineering Institute of Guangzhou Guangzhou China School of Statistics and Mathematics Zhongnan University of Economics and Law Wuhan China

Academic performance serves as a direct indicator of learning effectiveness. In the paper, a stacking learning method is proposed to predict students’ performance in exam. Firstly, an in-depth correlation analysis is conducted to examine various factors, including students’ school situation and exam scores in other subjects, which have an impact on the exam scores in a specific subject. It is found that a strong correlation exists between exam scores in different subjects, and factors such as students’ lunch level and parents’ education level significantly impact these scores. Secondly, the stacking integration strategy employs Random Forest (RF) as the base learner, while the random gradient model is selected as the meta learner to construct a predictive model for learning achievement. Finally, a comparative experiment is conducted using multiple models, including Decision Trees (DT) and support vector machines (SVM), to predict students’ exam scores. The results show that the learning achievement prediction model based on the stacking integration strategy outperforms all other tested models, exhibiting an accuracy improvement ranging from 6% to 17%. Moreover, its performance in statistical evaluation indicators and error evaluation indicators are well. The proposed learning method is more efficient to predict students’ performance in exams.

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Probabilistic Richardson Extrapolation

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arXiv 2024年

作者： Oates, Chris J. Karvonen, Toni Teckentrup, Aretha L. Strocchi, Marina Niederer, Steven A. School of Mathematics Statistics and Physics Newcastle University United Kingdom Department of Mathematics and Statistics University of Helsinki Finland School of Mathematics Maxwell Institute for Mathematical Sciences University of Edinburgh United Kingdom National Heart and Lung Institute Imperial College London United Kingdom School of Biomedical Engineering and Imaging Sciences King’s College London United Kingdom The Alan Turing Institute United Kingdom

For over a century, extrapolation methods have provided a powerful tool to improve the convergence order of a numerical method. However, these tools are not well-suited to modern computer codes, where multiple continua are discretised and convergence orders are not easily analysed. To address this challenge we present a probabilistic perspective on Richardson extrapolation, a point of view that unifies classical extrapolation methods with modern multi-fidelity modelling, and handles uncertain convergence orders by allowing these to be statistically estimated. The approach is developed using Gaussian processes, leading to Gauss–Richardson Extrapolation (GRE). Conditions are established under which extrapolation using the conditional mean achieves a polynomial (or even an exponential) speed-up compared to the original numerical method. Further, the probabilistic formulation unlocks the possibility of experimental design, casting the selection of fidelities as a continuous optimisation problem which can then be (approximately) solved. A case-study involving a computational cardiac model demonstrates that practical gains in accuracy can be achieved using the GRE method. © 2024, CC BY.

关键词： Extrapolation