Daptive Functional Data Analysis
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Adaptive Functional Data Analysis
Statistical methods that adapt to individual observations or unknown population structures are attractive due to both numerical and theoretical advantages over their non-adaptive counterparts. In this thesis, we contribute to adaptive modeling of functional data, focusing on the fundamental aspects of representation and regression, where challenges arise from the infinite-dimensionality of their underlying spaces. For adaptive representation, the notion of mixture inner product spaces (MIPS) is developed, featuring an infinite-dimensional mixture of finite-dimensional subspaces. We show that MIPS provides a new perspective for representing functional data, in which each realization of the underlying process falls into a realization-specific component subspace whose dimension is larger for less smooth while smaller for smoother realizations. Moreover, MIPS also offers an alternative solution to the longstanding issue of nonexistent probability density for functional data. For adaptive functional regression, we are interested in the scenario that functional predictor process lies in a potentially nonlinear manifold that is intrinsically finite-dimensional but embedded in an infinite-dimensional function space. We propose a nonparametric estimator built upon local linear manifold smoothing that achieves a polynomial convergence rate and adapts to the intrinsic data geometry even when functional data are observed intermittently and contaminated by noise, in contrast to the logarithm rate in nonparametric functional regression literature. We demonstrate that both proposals enjoy favorable finite sample performance relative to commonly used methods via simulated and real data examples.
Geostatistical Functional Data Analysis
Geostatistical Functional Data Analysis Explore the intersection between geostatistics and functional data analysis with this insightful new reference Geostatistical Functional Data Analysis presents a unified approach to modelling functional data when spatial and spatio-temporal correlations are present. The Editors link together the wide research areas of geostatistics and functional data analysis to provide the reader with a new area called geostatistical functional data analysis that will bring new insights and new open questions to researchers coming from both scientific fields. This book provides a complete and up-to-date account to deal with functional data that is spatially correlated, but also includes the most innovative developments in different open avenues in this field. Containing contributions from leading experts in the field, this practical guide provides readers with the necessary tools to employ and adapt classic statistical techniques to handle spatial regression. The book also includes: A thorough introduction to the spatial kriging methodology when working with functions A detailed exposition of more classical statistical techniques adapted to the functional case and extended to handle spatial correlations Practical discussions of ANOVA, regression, and clustering methods to explore spatial correlation in a collection of curves sampled in a region In-depth explorations of the similarities and differences between spatio-temporal data analysis and functional data analysis Aimed at mathematicians, statisticians, postgraduate students, and researchers involved in the analysis of functional and spatial data, Geostatistical Functional Data Analysis will also prove to be a powerful addition to the libraries of geoscientists, environmental scientists, and economists seeking insightful new knowledge and questions at the interface of geostatistics and functional data analysis.
Multimodal and Tensor Data Analytics for Industrial Systems Improvement
This volume covers the latest methodologies for using multimodal data fusion and analytics across several applications. The curated content presents recent developments and challenges in multimodal data analytics and shines a light on a pathway toward new research developments. Chapters are composed by eminent researchers and practitioners who present their research results and ideas based on their expertise. As data collection instruments have improved in quality and quantity for many applications, there has been an unprecedented increase in the availability of data from multiple sources, known as modalities. Modalities express a large degree of heterogeneity in their form, scale, resolution, and accuracy. Determining how to optimally combine the data for prediction and characterization is becoming increasingly important. Several research studies have investigated integrating multimodality data and discussed the challenges and limitations of multimodal data fusion. This volume provides a topical overview of various methods in multimodal data fusion for industrial engineering and operations research applications, such as manufacturing and healthcare. Advancements in sensing technologies and the shift toward the Internet of Things (IoT) has transformed and will continue to transform data analytics by producing new requirements and more complex forms of data. The abundance of data creates an unprecedented opportunity to design more efficient systems and make near-optimal operational decisions. On the other hand, the structural complexity and heterogeneity of the generated data pose a significant challenge to extracting useful features and patterns for making use of the data and facilitating decision-making. Therefore, continual research is needed to develop new statistical and analytical methodologies that overcome these data challenges and turn them into opportunities.