MATH, Flexible, due to diverse backgrounds of intended audience. Basic probability (level of MATH/STATS 425), or molecular biology (level of BIOLOGY 427), or biochemistry (level of CHEM/BIOLCHEM 451), or basic programming skills desirable or permission. (3). (BS). May not be repeated for credit. This course is open to graduate students and upper-level undergraduates in applied mathematics, bioinformatics, statistics, and engineering, who are interested in learning from data. Students with other backgrounds such as life sciences are also welcome, provided they have maturity in mathematics. The mathematical content in this course will be linear algebra, multilinear algebra, dynamical systems, and information theory. This content is required to understand some common algorithms in data science. I will start with a very basic introduction to data representation as vectors, matrices, and tensors. Then I will teach geometric methods for dimension reduction, also known as manifold learning (e.g. diffusion maps, t-distributed stochastic neighbor embedding (t-SNE), etc.), and topological data reduction (introduction to computational homology groups, etc.). I will bring an application-based approach to spectral graph theory, addressing the combinatorial meaning of eigenvalues and eigenvectors of their associated graph matrices and extensions to hypergraphs via tensors. I will also provide an introduction to the application of dynamical systems theory to data including dynamic mode decomposition and the Koopman operator. Real data examples will be given where possible and I will work with you write code implementing these algorithms to solve these problems. The methods discussed in this class are shown primarily for biological data, but are useful in handling data across many fields. A course features several guest lectures from industry and government.
This work contributes to the theoretical foundation for pricing in data markets and offers practical insights for managing digital data exchanges in the era of big data. We propose a structured pricing model for data exchanges transitioning from quasi-public to market-oriented operations. To address the complex dynamics among data exchanges, suppliers, and consumers, the authors develop a three-stage Stackelberg game framework. In this model, the data exchange acts as a leader setting transaction commission rates, suppliers are intermediate leaders determining unit prices, and consumers are followers making purchasing decisions. Two pricing strategies are examined: the Independent Pricing Approach (IPA) and the novel Perfectly Competitive Pricing Approach (PCPA), which accounts for competition among data providers. Using backward induction, the study derives subgame-perfect equilibria and proves the existence and uniqueness of Stackelberg equilibria under both approaches. Extensive numerical simulations are carried out in the model, demonstrating that PCPA enhances data demander utility, encourages supplier competition, increases transaction volume, and improves the overall profitability and sustainability of data exchanges. Social welfare analysis further confirms PCPA's superiority in promoting efficient and fair data markets.
The Laplace-Beltrami operator is studied on a stratified set consisting of two punctured circles and an interval. A complete description of all well-posed boundary value problems for the Laplace-Beltrami operator on such a set is given. In the second part of the paper, a class of self-adjoint well-posed problems for the Laplace-Beltrami operator on the specified stratified set is identified. The obtained results can be considered as a generalization of known results on geometric graphs. In particular, the stratified set under consideration can be interpreted as graphs with loops. Studies on the spectral asymptotics of SturmLiouville operators on plane curves homotopic to a finite interval are also closely related to the present results paper. Since the punctured circle is diffeomorphic to a finite interval, the spectral methods applied to differential operators on a finite interval can be modified to study the spectral properties of differential operators on the punctured circle. The main results of this paper are obtained by modifications of methods that were previously used in the study of the asymptotic behavior of the eigenvalues of the Sturm-Liouville operator on a finite interval.
Preface to the Second Edition Preface to the Third Edition Introduction to Statistical Methods for Geography Descriptive Statistics Probability and Discrete Probability Distributions Continuous Probability Distributions and Probability Models Inferential Statistics: Confidence Intervals, Hypothesis Testing and Sampling Analysis of Variance Correlation Introduction to Regression Analysis More on Regression Spatial Patterns Some Spatial Aspects of Regression Analysis Data Reduction: Factor Analysis and Cluster Analysis Epilogue Appendix A: Statistical Tables Table A.1 Random Digits Table A.2 Normal Distribution Table A.3 Student's t-Distribution Table A.4 Cumulative t-Distribution Table A.5 F-Distribution Table A.6 X2 Distribution Appendix B: Mathematical Conventions and Notation B.1 Mathematical Conventions B.2 Mathematical Notation Appendix C: Review and Extension of Some Probability Theory C.1 Expected Values C.2 Variation of a Random Variable C.3 Covariance of Random Variables
We introduce a new class of continuous distributions called the Kumaraswamy transmuted-G family which extends the transmuted class defined by Shaw and Buckley (2007). Some special models of the new family are provided. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Renyi and Shannon entropies, order statistics and probability weighted moments are derived. The maximum likelihood is used for estimating the model parameters. The flexibility of the generated family is illustrated by means of two applications to real data sets.
Information geometry [Ama16, AJLS17, Ama21] aims at unravelling the geometric structures of families of probability distributions and at studying their uses in information sciences. Information sciences is an umbrella term regrouping statistics, information theory, signal processing, machine learning and AI, etc. Information geometry was born independently from econometrician H. Hotelling (1930) and statistician C. R. Rao (1945) from the mathematical curiosity of considering a parametric family of probability distributions, called the statistical model, as a Riemannian manifold equipped with the Fisher metric tensor [Nie20]. Information geometry tackles problems by using the concepts of differential geometry (like curvature) with tensor calculus. In his pioneer work, Rao considered the Riemannian geodesic distance and geodesic balls on the manifold to study classification and hypothesis testing problems in statistics. Let (X,F, μ) denote a probability space [Kee10] (with sample space X, σ-algebra F, and finite positive measure
Problems with people's welfare typically result from the government's development efforts in a region not being done properly or not being done equally. Consider grouping and defining the traits of each region's degree of welfare as a potential answer to ensure that development policies and strategies are well-targeted. This study aims to classify 17 regency/cities in Southeast Sulawesi province based on several indicators of people's welfare. The method used is hierarchical cluster analysis with several approaches, including Single Linkage, Complete Linkage, Average Linkage, and Ward's. The data used in this study is secondary data obtained from the publication of the Central Agency of Statistics (CAS) of Southeast Sulawesi Province. Based on the results of the evaluation the best method used is Ward's method which produces three clusters. The first cluster consists of 9 regencies, namely Buton, North Buton, South Buton, Central Buton, Muna, West Muna, Wakatobi, Konawe Islands, and East Kolaka, the majority of which come from the archipelago. Some of the problems that occur in these areas are the relatively high poverty rate and the low average length of schooling and life expectancy. The same thing happened to the second cluster which consisted of 6 regencies, namely Konawe, South Konawe, North Konawe, Bombana, Kolaka, and North Kolaka with problems of poverty, the average length of schooling, and relatively low sources of proper drinking water when compared to other clusters. The third cluster consists of 2 urban areas, namely Kendari City and Baubau City, the problems that occur are the relatively high unemployment rate and population density. The government ought to offer more initiatives to handle issues with poverty, education, and health in regions in clusters 1 and 2. While in cluster 3, the government ought to offer more initiatives to combat jobless issues and prepare for rising population densities.
The main idea of this paper is to generate supra and infra topologies from simple undirected graphs. For this, we have introduced two new operators namely supra and infra operators which are defined on the power set of the vertex set of a graph. Moreover, we have also proved that the supra operator satisfying Kuratowski’s closure axiom will yield a topology. Further it was extended to develop the concept of connectedness and separation axioms on G-supra and G-infra spaces.
In this paper, some translations of bipolar valued multi fuzzy subnearring of a nearing are introduced and using these translations, some theorems are stated and proved.
This essay explores the impact of automated proof construction on three key areas of mathematical cognition: on how we judge the role one piece of mathematics plays in another, on how we make mistakes in reasoning about mathematical objects, and on how we understand what our theorems are truly about. It concludes by speculating on a new form of mathematical experience that these methods could make possible: ``glitching'', a game-like search for uncanny consequences of our definitions.
Background. According to the World Health Organization, 92% of the world's population lives in places where air quality levels exceed recommended limits. Recently, Ukraine had the most deaths per every 100,000 people (out of 120 countries) attributed to atmospheric air pollution. High levels of atmospheric air pollution have been observed not only in typically industrial regions, but in Ukraine's capital, Kyiv, as well. Objectives. The aim of the present study was to establish the state of air pollution in Kyiv and perform a risk assessment of associated human health effects. Methods. Using official statistics and state monitoring data, the study aimed to identify and analyze risks to the health of Kyiv's population associated with air pollution. The following methods were used: systematic, functional and comparative analysis, risk theory, mathematical modeling, probability theory and mathematical statistics, as well as geographic information system technologies for digital map design and objective-oriented methodology for software design systems. Results. The risk values across different areas of the city varied significantly, indicating that atmospheric air quality remains unstable. Areas with the highest and lowest risk values were identified. Conclusions. The environmental state of atmospheric air in Kyiv requires greater attention and additional research to identify the causes of air pollution, along with implementation of measures to improve air quality. Competing Interests. The authors declare no competing financial interests.
We propose to use agent-based simulation models for the development of statistical methods in Official Statistics, especially in relation with the new digital data sources. We present a mobile network data simulator which is managed through the simutils R package which provides geospatial representations of the simulated data. While the synthetic data are produced by an external tool, our simutils package allows an R user to parameterize and run this external simulation tool, to build geospatial data structures from the simulation output or to compute several aggregates. The geospatial data structures were designed with the purpose of using them in a visualization package too. Useful simulation models require the incorporation of real metadata from mobile telecommunication networks driving us to the inclusion of functionalities allowing the user to specify and validate them. All metadata are specified using XML file whose structure are defined in corresponding XSD files. Our R package includes example data sets and we show here how validate the metadata, how to run a simulation and how build the geospatial data structures and how to compute different aggregates.
Laju pertumbuhan penduduk di Kota Pasuruan pada tahun 2019 sebesar 0.68% dengan jumlah penduduk 200.422 jiwa. Tingginya pertumbuhan penduduk dapat mempengaruhi kepadatan penduduk. Penelitian ini bertujuan untuk memprediksi pertumbuhan penduduk Kota Pasuruan menggunakan metode ARIMA (Autoregressive Integrated Moving Average). Metode ARIMA adalah cara prediksi data deret waktu yang memiliki tiga model, yaitu AR (Autoregressive), MA (Moving Average), ARMA (Autoregressive Moving Average). Metode ini memiliki parameter (p,d,q) dapat diketahuidari plot ACF dan PACF untuk memastikan model yang akan digunakan untuk prediksi. Dalam penelitian ini data yang digunakan merupakan data penduduk Kota Pasuruan tahun 1983 sampai tahun 2019 sejumlah 37 data. Dari data tersebut didapatkan ARIMA model (1,1,1) dengan jumlah penduduk Kota Pasuruan pada tahun 2020 adalah 203.221 jiwa, didapatkan nilai MSE 10542507.06 dan MAPE 1.52%.