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Automation of image processing through ML algorithms of GRASS GIS using embedded Scikit-Learn library of Python

Examples and Counterexamples

Pub Date : 2025-02-03 DOI: 10.1016/j.exco.2025.100180

Polina Lemenkova

Image processing using Machine Learning (ML) and Artificial Neural Network (ANN) methods was investigated by employing the algorithms of Geographic Resources Analysis Support System (GRASS) Geographic Information System GIS with embedded Scikit-Learn library of Python language. The data are obtained from the United States Geological Survey (USGS) and include the Landsat 8 Operational Land Imager/Thermal Infrared Sensor (OLI/TIRS) multispectral satellite images. The images were collectedon 2013 and 2023 to evaluate land cover categories in each of the year. The study area covers the region of Nile Delta and the Faiyum Oasis, Egypt. A series of modules for raster image processing was applied using scripting language of GRASS GIS to process the remote sensing data. The satellite images were classified into raster maps presenting the land cover types. These include ‘i.cluster’ and ‘i.maxlik’ for non-supervised classification used as training dataset of random pixel seeds, ‘r.random’, ‘r.learn.train’, ‘r.learn.predict’ and ‘r.category’ for ML part of image processing. The consequences of various ML parameters on the cartographic outputs are analysed, such as speed and accuracy, randomness of nodes, analytical determination of the output weights, and dependence distribution of pixels for each algorithm. Supervised learning models of GRASS GIS were tested and compared including the Gaussian Naive Bayes (GaussianNB), Multi-layer Perceptron classifier (MLPClassifier), Support Vector Machines (SVM) Classifier, and Random Forest Classifier (RF). Though each algorithms was developed to serve different objectives of ML applications in RS data processing, their technical implementation and practical purposes present valuable approaches to cartographic data processing and image analysis. The results shown that the most time-consuming algorithms was noted as SVM classification, while the fastest results were achieved by the GaussianNB approach to image processing and the best results are achieved by RF Classifier.

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引用次数: 0

Counterexamples for your calculus course

Examples and Counterexamples

Pub Date : 2025-02-03 DOI: 10.1016/j.exco.2025.100177

Jürgen Appell , Simon Reinwand

We present 2 theorems and 20 counterexamples illustrating the surprising behaviour of functions between metric spaces.

引用次数: 0

Solving change of basis from Bernstein to Chebyshev polynomials

Examples and Counterexamples

Pub Date : 2025-02-02 DOI: 10.1016/j.exco.2025.100178

D.A. Wolfram

We provide two closed-form solutions to the change of basis from Bernstein polynomials to shifted Chebyshev polynomials of the fourth kind and show them to be equivalent by applying Zeilberger’s algorithm. The first solution uses orthogonality properties of the Chebyshev polynomials. The second is “modular” which enables separately verified sub-problems to be composed and re-used in other basis transformations. These results have applications in change of basis of orthogonal, and non-orthogonal polynomials.

引用次数: 0

Hölder’s inequality for shifted quantum integral operator

Examples and Counterexamples

Pub Date : 2025-02-02 DOI: 10.1016/j.exco.2025.100179

Andrea Aglić Aljinović, Lana Horvat Dmitrović, Ana Žgaljić Keko

We show by two counterexamples that Hölder’s inequality for shifted quantum integral operator does not hold in general and we prove the case in which it is valid.

引用次数: 0

Asymptotic behavior of the empirical checkerboard copula process for binary data: An educational presentation

Examples and Counterexamples

Pub Date : 2025-01-31 DOI: 10.1016/j.exco.2025.100176

Christian Genest, Johanna G. Nešlehová

The empirical multilinear or checkerboard copula process is a promising tool for statistical inference in copula models for data with ties (Genest et al., 2019a). The large-sample behavior of this process was determined in Genest et al. (2014, 2017) under very broad conditions. The purpose of this note is to provide a detailed description of this asymptotic result and to derive an expression for the limit of the process in the simplest possible case in which the data form a random sample of pairs of Bernoulli random variables. Although one would never actually fit a copula model to a 2 × 2 contingency table, this case is particularly well suited for explicit calculations and didactic explanations of the intricacies of the limiting behavior of this process and make it clear why the conditions in Genest et al. (2014, 2017) are needed and cannot be simplified.

引用次数: 0

The Hadamard-PINN for PDE inverse problems: Convergence with distant initial guesses

Examples and Counterexamples

Pub Date : 2025-01-28 DOI: 10.1016/j.exco.2025.100175

Yohan Chandrasukmana, Helena Margaretha, Kie Van Ivanky Saputra

This paper presents the Hadamard-Physics-Informed Neural Network (H-PINN) for solving inverse problems in partial differential equations (PDEs), specifically the heat equation and the Korteweg–de Vries (KdV) equation. H-PINN addresses challenges in convergence and accuracy when initial parameter guesses are far from their actual values. The training process is divided into two phases: data fitting and parameter optimization. This phased approach is based on Hadamard’s conditions for well-posed problems, which emphasize that the uniqueness of a solution relies on the specified initial and boundary conditions. The model is trained using the Adam optimizer, along with a combined learning rate scheduler. To ensure reliability and consistency, we repeated each numerical experiment five times across three different initial guesses. Results showed significant improvements in parameter accuracy compared to the standard PINN, highlighting H-PINN’s effectiveness in scenarios with substantial deviations in initial guesses.

引用次数: 0

A note on an application of discrete Morse theoretic techniques on the complex of disconnected graphs

Examples and Counterexamples

Pub Date : 2025-01-15 DOI: 10.1016/j.exco.2025.100174

Anupam Mondal , Pritam Chandra Pramanik

Robin Forman’s highly influential 2002 paper A User’s Guide to Discrete Morse Theory presents an overview of the subject in a very readable manner. As a proof of concept, the author determines the topology (homotopy type) of the abstract simplicial complex of disconnected graphs of order

n

(which was previously done by Victor Vassiliev using classical topological methods) using discrete Morse theoretic techniques, which are purely combinatorial in nature. The techniques involve the construction (and verification) of a discrete gradient vector field on the complex. However, the verification part relies on a claim that does not seem to hold. In this note, we provide a couple of counterexamples against this specific claim. We also provide an alternative proof of the bigger claim that the constructed discrete vector field is indeed a gradient vector field. Our proof technique relies on a key observation which is not specific to the problem at hand, and thus is applicable while verifying a constructed discrete vector field is a gradient one in general.

引用次数: 0

Nonuniqueness of lattice Boltzmann schemes derived from finite difference methods

Examples and Counterexamples

Pub Date : 2024-12-18 DOI: 10.1016/j.exco.2024.100171

Eliane Kummer, Stephan Simonis

Recently, the construction of finite difference schemes from lattice Boltzmann schemes has been rigorously analyzed [Bellotti et al. (2022), Numer. Math. 152, pp. 1–40]. It is thus known that any lattice Boltzmann scheme can be expressed in terms of a corresponding multi-step finite difference scheme on the conserved variables. In the present work, we provide counterexamples for the conjecture that any multi-step finite difference scheme has a unique lattice Boltzmann formulation. Based on that, we indicate the existence of equivalence classes for discretized relaxation systems.

引用次数: 0

On the effect of different samplings to the solution of parametric PDE eigenvalue problems

Examples and Counterexamples

Pub Date : 2024-12-01 DOI: 10.1016/j.exco.2024.100170

Daniele Boffi , Abdul Halim , Gopal Priyadarshi

The use of sparse sampling is a consolidated technique for the reduced order modeling of parametric PDEs. In this note we investigate the choice of sampling points within the framework of reduced order techniques for the approximation of eigenvalue problems originating from parametric PDEs. We use the standard proper orthogonal decomposition technique to obtain the basis of the reduced space and Galerkin orthogonal technique to get the reduced problem. We present some numerical results and observe that, as in the case of the source problem, also for eigenvalue problems the use of sparse sampling is a good idea and that, when the number of sampling points is assigned, sparse sampling provides better results than uniform sampling.

In the spirit of the journal, we present our results in the form of examples and counterexamples.

引用次数: 0

An example for application of Lax–Milgram’s theorem and Riesz–Schauder’s theorem

Examples and Counterexamples

Pub Date : 2024-12-01 DOI: 10.1016/j.exco.2024.100169

Tujin Kim

In this note reviewing Lax–Migram’s theorem, we show an example of its application to prove the existence of a solution to an equation in complex Hilbert space arising in the field of electromagnetic heating.

引用次数: 0

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