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arXiv 2024年

作者： Kuehn, Christian Piazzola, Chiara Ullmann, Elisabeth Department of Mathematics TUM School of Computation Information and Technology Technical University of Munich Germany Munich Data Science Institute Technical University of Munich Germany

In this work we consider the Allen–Cahn equation, a prototypical model problem in nonlinear dynamics that exhibits bifurcations corresponding to variations of a deterministic bifurcation parameter. Going beyond the state-of-the-art, we introduce a random coefficient in the linear reaction part of the equation, thereby accounting for random, spatially-heterogeneous effects. Importantly, we assume a spatially constant, deterministic mean value of the random coefficient. We show that this mean value is in fact a bifurcation parameter in the Allen–Cahn equation with random coefficients. Moreover, we show that the bifurcation points and bifurcation curves become random objects. We consider two distinct modelling situations: (i) for a spatially homogeneous coefficient we derive analytical expressions for the distribution of the bifurcation points and show that the bifurcation curves are random shifts of a fixed reference curve;(ii) for a spatially heterogeneous coefficient we employ a generalized polynomial chaos expansion to approximate the statistical properties of the random bifurcation points and bifurcation curves. We present numerical examples in 1D physical space, where we combine the popular software package Continuation Core and Toolboxes (COCO) for numerical continuation and the Sparse Grids Matlab Kit for the polynomial chaos expansion. Our exposition addresses both, dynamical systems and uncertainty quantification, highlighting how analytical and numerical tools from both areas can be combined efficiently for the challenging uncertainty quantification analysis of bifurcations in random differential equations. Copyright © 2024, The Authors. All rights reserved.

关键词： Stochastic systems

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Partial Homoscedasticity in Causal Discovery with Linear Models

arXiv

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arXiv 2023年

作者： Wu, Jun Drton, Mathias Technical University of Munich GermanyTUM School of Computation Information and Technology Department of Mathematics and Munich Data Science Institute

Recursive linear structural equation models and the associated directed acyclic graphs (DAGs) play an important role in causal discovery. The classic identifiability result for this class of models states that when only observational data is available, each DAG can be identified only up to a Markov equivalence class. In contrast, recent work has shown that the DAG can be uniquely identified if the errors in the model are homoscedastic, i.e., all have the same variance. This equal variance assumption yields methods that, if appropriate, are highly scalable and also sheds light on fundamental information-theoretic limits and optimality in causal discovery. In this paper, we fill the gap that exists between the two previously considered cases, which assume the error variances to be either arbitrary or all equal. Specifically, we formulate a framework of partial homoscedasticity, in which the variables are partitioned into blocks and each block shares the same error variance. For any such groupwise equal variances assumption, we characterize when two DAGs give rise to identical Gaussian linear structural equation models. Furthermore, we show how the resulting distributional equivalence classes may be represented using a completed partially directed acyclic graph (CPDAG), and we give an algorithm to efficiently construct this CPDAG. In a simulation study, we demonstrate that greedy search provides an effective way to learn the CPDAG and exploit partial knowledge about homoscedasticity of errors in structural equation models. © 2023, CC BY.

关键词： Errors

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Monitoring ZA Fertilizer Production using Multivariate Maximum Chart Based on Bootstrap Control Limit 3

Monitoring ZA Fertilizer Production using Multivariate Maxim...

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3rd International Conference on Statistics, Mathematics, Teaching, and Research 2019, ICSMTR 2019

作者： Kruba, R. Mashuri, M. Prastyo, D.D. Department of Statistics Faculty of Mathematics Computation and Data Science Institut Teknologi Sepuluh Nopember Surabaya60111 Indonesia

Control charts are extensively used to monitor the production process. When there is more than one variable process are considered, the multivariate control charts are typically employed to monitor the mean vector and the variability process separately. In recent years, control charts have been developed for monitoring mean process and variability process simultaneously in a chart. A Maximum multivariate control chart (Max-Mchart) is one of the simultaneous multivariate control charts relying on exact distribution control limit. The objective of this paper is to evolve Max-Mchart based on Bootstrap control limits. This paper also compares Max-Mchart over the Hotelling T2 and Generalized Variance (GV) control chart. The interpretive examples are implemented to demonstrate the applications of the ZA fertilizer production dataset in carbonation step. © Published under licence by IOP Publishing Ltd.

关键词： Control charts

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data-driven ambiguous cognitive map for complex decision-making in supply chain management

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Journal of computational Mathematics and data science 2025年 14卷

作者： Singh, Pritpal Quantum Computation and Ambiguous Set Lab (QCASL) Department of Data Science and Analytics Central University of Rajasthan Rajasthan Ajmer 305817 India

Fuzzy cognitive maps (FCMs) have the potential to model complex systems, but they face challenges in uncertainty, complexity, and dynamic conditions. This study tackles three main issues: modeling and quantifying uncertainty in relationships and weights with imprecise inputs, managing the complexity as the number of activation levels and causal relationships increases, and determining appropriate weights and thresholds in uncertain contexts. By using ambiguous set theory, the research introduces the ambiguous cognitive map (ACM) to improve the traditional FCM and address these problems. This theory allows for the representation of states with four membership values: true, false, partially true, and partially false, which provides a more refined approach to managing uncertainty. Mathematical formulas are employed by ACM to calculate weights based on these membership values instead of randomly selecting. The introduction of rank allows for the identification of the most influential state by its highest rank in priority decisions. The application of ACM in decision-making scenarios related to the supply chain system demonstrates its efficiency in systematically prioritizing and resolving complex decisions. The ACM effectively identifies key variables and provides actionable rankings to support decision-making in the supply chain system. The results demonstrate that ACM offers a systematic approach to resolving complex decisions under uncertainty. Impact Statement ACMs replace the conventional random assignment of relationship weights with a mathematical formulation based on the four membership values, enhancing the accuracy and reliability of the modeled system. The study also introduces a rank-based decision-making process, where the most influential state is determined by the highest rank derived from the membership values. The proposed ACM framework not only addresses the limitations of traditional FCMs but also opens new avenues for artificial intelligence (AI)-dri

关键词： Ambiguous cognitive map (ACM) Ambiguous set Decision-making Fuzzy cognitive maps (FCMs) Supply chain management

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Modeling of Human Development Index in Papua Province Using Spline Smoothing Estimator in Nonparametric Regression 3

Modeling of Human Development Index in Papua Province Using ...

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3rd International Conference on Statistics, Mathematics, Teaching, and Research 2019, ICSMTR 2019

作者： Rahmawati, D.P. Budiantara, I.N. Prastyo, D.D. Octavanny, M.A.D. Department of Statistics Faculty of Mathematics Computation and Data Science Institut Teknologi Sepuluh Nopember Surabaya60111 Indonesia

The development goal of a country must be focused on the quality of human life to achieve prosperity. One important indicator for measuring the success of a country's development is the Human Development Index (HDI). In 2018, Papua was the province with the lowest HDI in Indonesia. Special attention is needed to improve HDI in Papua Province, one of them is by paying attention to the variables that affect HDI such as population growth rate, percentage of poor population, and economic growth. The relationships between HDI and the predictor variables do not have a clear pattern and tend to change at certain subintervals. This case can be approached using Spline Smoothing in multivariable nonparametric regression. Spline Smoothing is a type of estimator in nonparametric regression that has an excellent ability to handle data that tend to change at certain subintervals. Therefore, the purposes of this study are to obtain the form of Spline Smoothing estimator function in multivariable nonparametric regression, estimate the function and apply it to the HDI in Papua Province. The empirical results of modeling HDI in Papua Province show that it can be adequately applied which gives GCV = 58.108, R2 = 99.77% and RMSE = 0.0505. © Published under licence by IOP Publishing Ltd.

关键词： Population statistics

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Parameter Estimation of Multivariate Adaptive Regression Spline (MARS) with Stepwise Approach to Multi Drug-Resistant Tuberculosis (MDR-TB) Modeling in Lamongan Regency 3

Parameter Estimation of Multivariate Adaptive Regression Spl...

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3rd International Conference on Statistics, Mathematics, Teaching, and Research 2019, ICSMTR 2019

作者： Yasmirullah, S.D.P. Otok, B.W. Purnomo, J.D.T. Prastyo, D.D. Department of Statistics Faculty of Mathematics Computation and Data Science Institut Teknologi Sepuluh Nopember Surabaya60111 Indonesia

Tuberculosis (TB) is an infectious disease, caused by mycobacterium tuberculosis that affects various organs, especially the lungs. If TB treatment is not done thoroughly by the patient, then it can lead to death. Tuberculosis disease could not be healed cause TB bacteria to double immunity against anti-TB drugs, called multi-drug resistant (MDR). One identification issue toward TB infection chain is a TB case distribution analysis using mathematical modeling. The method used in this study was Multivariate Adaptive Regression Spline (MARS). Multivariate Adaptive Regression Spline is one type of non-parametric regression techniques, where the model does not assume the functional relationship between response and predictor variables, and has a flexible functional structure as well. Modeling aims to determine the factors that have the most significant influence on MDR-TB cases in Lamongan regency, as well as predict the incidence of MDR-TB in each sub-regency. The results show, the best model has a combination of BF = 28, MI = 2 and MO = 3, based on the minimum GCV, which is 5.26E-06. Furthermore, the model is statistically proper according to the criteria of APER and Press's Q. © Published under licence by IOP Publishing Ltd.

关键词： Patient treatment

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Modeling of Children Ever Born in Indonesia Using Fourier Series Nonparametric Regression 3

Modeling of Children Ever Born in Indonesia Using Fourier Se...

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3rd International Conference on Statistics, Mathematics, Teaching, and Research 2019, ICSMTR 2019

作者： Octavanny, M.A.D. Budiantara, I.N. Kuswanto, H. Rahmawati, D.P. Department of Statistics Faculty of Mathematics Computation and Data Science Institut Teknologi Sepuluh Nopember Surabaya60111 Indonesia

Nonparametric regression aims to determine the relationship between response and predictor when the data does not follow a specific pattern. Fourier series is a nonparametric regression approach which has the flexibility to follow the characteristics of data. The purpose of this study is to obtain the estimator of the nonparametric regression using Fourier series and apply the model to the fertility data. The fertility rate represented by children ever born is one of the demographic factors that determine the decline in population growth rate. The data were obtained from the Indonesia Demographic and Health Survey 2017 with children ever born as a response. The predictors are the proportion of women graduating from junior high school, the proportion of women having sex before the age of 18, the proportion of women using a modern contraceptive method, and the infant mortality rate. The relationship between response and predictors tends to have a repetitive pattern with a certain trend. The best nonparametric regression model of children ever born in Indonesia is obtained by using 3 oscillation parameters for each predictor variable with GCV = 0.0534 and R-square = 80.04%. © Published under licence by IOP Publishing Ltd.

关键词： Fourier series

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Correction to: data Assimilation Predictive GAN (DA-PredGAN) Applied to a Spatio-Temporal Compartmental Model in Epidemiology

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Journal of Scientific Computing 2023年第2期95卷 1-2页

作者： Silva, Vinicius L. S. Heaney, Claire E. Li, Yaqi Pain, Christopher C. Applied Modelling and Computation Group Imperial College London London UK Department of Earth Science and Engineering Imperial College London London UK Data Assimilation Laboratory Data Science Institute Imperial College London London UK

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Regularity and positivity of solutions of the Consensus-Based Optimization equation: unconditional global convergence

arXiv

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arXiv 2025年

作者： Fornasier, Massimo Sun, Lukang Technical University of Munich School of Computation Information and Technology Department of Mathematics Munich Germany Munich Center for Machine Learning Munich Germany Munich Data Science Institute Germany

Introduced in 2017 [28], Consensus-Based Optimization (CBO) has rapidly emerged as a significant breakthrough in global optimization. This straightforward yet powerful multi-particle, zero-order optimization method draws inspiration from Simulated Annealing and Particle Swarm Optimization. Using a quantitative mean-field approximation, CBO dynamics can be described by a nonlinear Fokker-Planck equation with degenerate diffusion, which does not follow a gradient flow structure. In this paper, we demonstrate that solutions to the CBO equation remain positive and maintain full support. Building on this foundation, we establish the unconditional global convergence of CBO methods to global minimizers. Our results are derived through an analysis of solution regularity and the proof of existence for smooth, classical solutions to a broader class of drift-diffusion equations, despite the challenges posed by degenerate diffusion. © 2025, CC BY.

关键词： Particle swarm optimization (PSO)

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Universal distances for extended persistence

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Journal of Applied and computational Topology 2024年第3期8卷 475-530页

作者： Bauer, Ulrich Botnan, Magnus Bakke Fluhr, Benedikt Department of Mathematics School of Computation Information and Technology Technical University of Munich Boltzmannstraße 3 Garching 85747 Germany Munich Data Science Institute Technical University of Munich Boltzmannstraße 3 Garching 85747 Germany Department of Mathematics VU University Amsterdam De Boelelaan 1111 Amsterdam 1081 HV Netherlands Faculty of Mathematics Bielefeld University Postfach 100 131 Bielefeld 33501 Germany

The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen–Steiner, Edelsbrunner, and Harer. We address the question of universality, which asks for the largest possible stable distance on extended persistence diagrams, showing that a more discriminative variant of the bottleneck distance is universal. Our result applies more generally to settings where persistence diagrams are considered only up to a certain degree. We achieve our results by establishing a functorial construction and several characteristic properties of relative interlevel set homology, which mirror the classical Eilenberg–Steenrod axioms. Finally, we contrast the bottleneck distance with the interleaving distance of sheaves on the real line by showing that the latter is not intrinsic, let alone universal. This particular result has the further implication that the interleaving distance of Reeb graphs is not intrinsic either. © The Author(s) 2024.

关键词： 55N31 62R40 Bottleneck distance Persistence diagrams Persistent homology Reeb graphs Topological data analysis

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