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This open access book presents the key aspects of statistics in Wasserstein spaces, i.e. statistics in the space of probability measures when endowed with the geometry of optimal transportation. Further to reviewing state-of-the-art aspects, it also provides an accessible introduction to the fundamentals of this current topic, as well as an overview that will serve as an invitation and catalyst for further research. Statistics in Wasserstein spaces represents an emerging topic in mathematical statistics, situated at the interface between functional data analysis (where the data are functions, thus lying in infinite dimensional Hilbert space) and non-Euclidean statistics (where the data satisfy nonlinear constraints, thus lying on non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to describe data collections that are best modeled as random measures on Euclidean space (e.g. images and point processes). Such random measures carry the infinite dimensional traits of functional data, but are intrinsically nonlinear due to positivity and integrability restrictions. Indeed, their dominating statistical variation arises through random deformations of an underlying template, a theme that is pursued in depth in this monograph.

Probabilities. --- Probability Theory and Stochastic Processes. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Probability Theory and Stochastic Processes --- Optimal Transportation --- Monge-Kantorovich Problem --- Barycenter --- Multimarginal Transport --- Functional Data Analysis --- Point Processes --- Random Measures --- Manifold Statistics --- Open Access --- Geometrical statistics --- Wasserstein metric --- Fréchet mean --- Procrustes analysis --- Phase variation --- Gradient descent --- Probability & statistics --- Stochastics

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Mathematical modelling in biomedicine is a rapidly developing scientific discipline at the intersection of medicine, biology, mathematics, physics, and computer science. Its progress is stimulated by fundamental scientific questions and by the applications to public health. This book represents a collection of papers devoted to mathematical modelling of various physiological problems in normal and pathological conditions. It covers a broad range of topics including cardiovascular system and diseases, heart and brain modelling, tumor growth, viral infections, and immune response. Computational models of blood circulation are used to study the influence of heart arrhythmias on coronary blood flow and on operating modes for left-ventricle-assisted devices. Wave propagation in the cardiac tissue is investigated in order to show the influence of tissue heterogeneity and fibrosis. The models of tumor growth are used to determine optimal protocols of antiangiogenic and radiotherapy. The models of viral hepatitis kinetics are considered for the parameter identification, and the evolution of viral quasi-species is investigated. The book presents the state-of-the-art in mathematical modelling in biomedicine and opens new perspectives in this passionate field of research.

virus density distribution --- genotype --- virus infection --- immune response --- resistance to treatment --- nonlocal interaction --- quasi-species diversification --- mathematical oncology --- spatially distributed modeling --- reaction-diffusion-convection equations --- computer experiment --- spiral wave --- heterogeneity --- heart modeling --- myocardium --- left ventricle --- neural field model --- integro-differential equation --- waves --- brain stimulation --- mathematical modeling --- cardiac mechanics --- multiscale simulation --- cardiomyopathies --- left ventricle remodeling --- spatially-distributed modeling --- gradient descent --- 1D haemodynamics --- systole variations --- coronary circulation --- cardiac pacing --- tachycardia --- bradycardia --- interventricular asynchrony --- long QT syndrome --- premature ventricular contraction --- rotary blood pump --- lumped heart model --- cardiac fibrosis --- excitable media --- wave break --- elongated obstacle --- lymph flow --- mathematical modelling --- lymphatic vessels --- lymph nodes --- parameter estimation --- constrained optimization --- derivative free optimization --- multiscale models --- differential equations --- viral hepatitis

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Mathematical modelling in biomedicine is a rapidly developing scientific discipline at the intersection of medicine, biology, mathematics, physics, and computer science. Its progress is stimulated by fundamental scientific questions and by the applications to public health. This book represents a collection of papers devoted to mathematical modelling of various physiological problems in normal and pathological conditions. It covers a broad range of topics including cardiovascular system and diseases, heart and brain modelling, tumor growth, viral infections, and immune response. Computational models of blood circulation are used to study the influence of heart arrhythmias on coronary blood flow and on operating modes for left-ventricle-assisted devices. Wave propagation in the cardiac tissue is investigated in order to show the influence of tissue heterogeneity and fibrosis. The models of tumor growth are used to determine optimal protocols of antiangiogenic and radiotherapy. The models of viral hepatitis kinetics are considered for the parameter identification, and the evolution of viral quasi-species is investigated. The book presents the state-of-the-art in mathematical modelling in biomedicine and opens new perspectives in this passionate field of research.

Research & information: general --- Mathematics & science --- virus density distribution --- genotype --- virus infection --- immune response --- resistance to treatment --- nonlocal interaction --- quasi-species diversification --- mathematical oncology --- spatially distributed modeling --- reaction-diffusion-convection equations --- computer experiment --- spiral wave --- heterogeneity --- heart modeling --- myocardium --- left ventricle --- neural field model --- integro-differential equation --- waves --- brain stimulation --- mathematical modeling --- cardiac mechanics --- multiscale simulation --- cardiomyopathies --- left ventricle remodeling --- spatially-distributed modeling --- gradient descent --- 1D haemodynamics --- systole variations --- coronary circulation --- cardiac pacing --- tachycardia --- bradycardia --- interventricular asynchrony --- long QT syndrome --- premature ventricular contraction --- rotary blood pump --- lumped heart model --- cardiac fibrosis --- excitable media --- wave break --- elongated obstacle --- lymph flow --- mathematical modelling --- lymphatic vessels --- lymph nodes --- parameter estimation --- constrained optimization --- derivative free optimization --- multiscale models --- differential equations --- viral hepatitis

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The modeling and processing of empirical data is one of the main subjects and goals of statistics. Nowadays, with the development of computer science, the extraction of useful and often hidden information and patterns from data sets of different volumes and complex data sets in warehouses has been added to these goals. New and powerful statistical techniques with machine learning (ML) and data mining paradigms have been developed. To one degree or another, all of these techniques and algorithms originate from a rigorous mathematical basis, including probability theory and mathematical statistics, operational research, mathematical analysis, numerical methods, etc. Popular ML methods, such as artificial neural networks (ANN), support vector machines (SVM), decision trees, random forest (RF), among others, have generated models that can be considered as straightforward applications of optimization theory and statistical estimation. The wide arsenal of classical statistical approaches combined with powerful ML techniques allows many challenging and practical problems to be solved. This Special Issue belongs to the section “Mathematics and Computer Science”. Its aim is to establish a brief collection of carefully selected papers presenting new and original methods, data analyses, case studies, comparative studies, and other research on the topic of statistical data modeling and ML as well as their applications. Particular attention is given, but is not limited, to theories and applications in diverse areas such as computer science, medicine, engineering, banking, education, sociology, economics, among others. The resulting palette of methods, algorithms, and applications for statistical modeling and ML presented in this Special Issue is expected to contribute to the further development of research in this area. We also believe that the new knowledge acquired here as well as the applied results are attractive and useful for young scientists, doctoral students, and researchers from various scientific specialties.

mathematical competency --- assessment --- machine learning --- classification and regression tree --- CART ensembles and bagging --- ensemble model --- multivariate adaptive regression splines --- cross-validation --- dam inflow prediction --- long short-term memory --- wavelet transform --- input predictor selection --- hyper-parameter optimization --- brain-computer interface --- EEG motor imagery --- CNN-LSTM architectures --- real-time motion imagery recognition --- artificial neural networks --- banking --- hedonic prices --- housing --- quantile regression --- data quality --- citizen science --- consensus models --- clustering --- Gower’s interpolation formula --- Gower’s metric --- mixed data --- multidimensional scaling --- classification --- data-adaptive kernel functions --- image data --- multi-category classifier --- predictive models --- support vector machine --- stochastic gradient descent --- damped Newton --- convexity --- METABRIC dataset --- breast cancer subtyping --- deep forest --- multi-omics data --- categorical data --- similarity --- feature selection --- kernel density estimation --- non-linear optimization --- kernel clustering --- n/a --- Gower's interpolation formula --- Gower's metric

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The modeling and processing of empirical data is one of the main subjects and goals of statistics. Nowadays, with the development of computer science, the extraction of useful and often hidden information and patterns from data sets of different volumes and complex data sets in warehouses has been added to these goals. New and powerful statistical techniques with machine learning (ML) and data mining paradigms have been developed. To one degree or another, all of these techniques and algorithms originate from a rigorous mathematical basis, including probability theory and mathematical statistics, operational research, mathematical analysis, numerical methods, etc. Popular ML methods, such as artificial neural networks (ANN), support vector machines (SVM), decision trees, random forest (RF), among others, have generated models that can be considered as straightforward applications of optimization theory and statistical estimation. The wide arsenal of classical statistical approaches combined with powerful ML techniques allows many challenging and practical problems to be solved. This Special Issue belongs to the section “Mathematics and Computer Science”. Its aim is to establish a brief collection of carefully selected papers presenting new and original methods, data analyses, case studies, comparative studies, and other research on the topic of statistical data modeling and ML as well as their applications. Particular attention is given, but is not limited, to theories and applications in diverse areas such as computer science, medicine, engineering, banking, education, sociology, economics, among others. The resulting palette of methods, algorithms, and applications for statistical modeling and ML presented in this Special Issue is expected to contribute to the further development of research in this area. We also believe that the new knowledge acquired here as well as the applied results are attractive and useful for young scientists, doctoral students, and researchers from various scientific specialties.

Information technology industries --- mathematical competency --- assessment --- machine learning --- classification and regression tree --- CART ensembles and bagging --- ensemble model --- multivariate adaptive regression splines --- cross-validation --- dam inflow prediction --- long short-term memory --- wavelet transform --- input predictor selection --- hyper-parameter optimization --- brain-computer interface --- EEG motor imagery --- CNN-LSTM architectures --- real-time motion imagery recognition --- artificial neural networks --- banking --- hedonic prices --- housing --- quantile regression --- data quality --- citizen science --- consensus models --- clustering --- Gower's interpolation formula --- Gower's metric --- mixed data --- multidimensional scaling --- classification --- data-adaptive kernel functions --- image data --- multi-category classifier --- predictive models --- support vector machine --- stochastic gradient descent --- damped Newton --- convexity --- METABRIC dataset --- breast cancer subtyping --- deep forest --- multi-omics data --- categorical data --- similarity --- feature selection --- kernel density estimation --- non-linear optimization --- kernel clustering