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2024 | Buch

Abstract Frames Kapitel lesen Erstes Kapitel lesen

Frame Theory in Data Science

verfasst von: Zhihua Zhang, Palle E. T. Jorgensen

Verlag: Springer International Publishing

Buchreihe : Advances in Science, Technology & Innovation

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

This book establishes brand-new frame theory and technical implementation in data science, with a special focus on spatial-scale feature extraction, network dynamics, object-oriented analysis, data-driven environmental prediction, and climate diagnosis. Given that data science is unanimously recognized as a core driver for achieving Sustainable Development Goals of the United Nations, these frame techniques bring fundamental changes to multi-channel data mining systems and support the development of digital Earth platforms. This book integrates the authors' frame research in the past twenty years and provides cutting-edge techniques and depth for scientists, professionals, and graduate students in data science, applied mathematics, environmental science, and geoscience.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Abstract Frames
Abstract
Big data acquisition, transmission, and processing systems always adopt redundant representation since redundancy can make the systems optimally robust to coefficient erasures and resilient to scenario noises. Such a demand leads to the establishment of frame theory first in various abstract spaces.
Zhihua Zhang, Palle E. T. Jorgensen
Chapter 2. Harmonic Frames
Abstract
The widely used harmonic analysis is to represent complex data as the composition of harmonically related sinusoids and cosinusoids. Unfortunately, this approach can only extract integer frequency features and global frequency features inside complex data. Through embedding non-integer frequency elements and local windows, various harmonic frames overcome these drawbacks and can adaptively mine local temporal, spatial, and spectral features of data.
Zhihua Zhang, Palle E. T. Jorgensen
Chapter 3. Polynomial Frames
Abstract
Polynomial frames are an emerging topic in the frame theory. Although traditional orthogonal polynomials can well approximate any data living in a region or the whole space, they lack the ability of localized spatial-frequency analysis and enough redundancy for robustness. Polynomial frames are generated by the combination of orthogonal polynomials and different windows. The data decomposition by polynomial frames can not only reveal localized spatial-frequency structure and evolution of data, but also conserve powerful computation ability and robustness. In this chapter, we mainly focus on Legendre-Trigonometric frames, Hermite frames, Laguerre frames and Chebyshev frames.
Zhihua Zhang, Palle E. T. Jorgensen
Chapter 4. Analytic Extension of Frames
Abstract
Frames can provide a nice representation of any function in \(L^2(\mathbb {R})\). If we extend analytic frames into complex plane, it is natural to ask whether the derived analytic frame series can be used to represent any analytic functions.
Zhihua Zhang, Palle E. T. Jorgensen
Chapter 5. Quasi-Orthogonal Frames
Abstract
The core idea of multi-channel data transmission is to orthogonally project any data into several channels in the whole spectral domain and design different transmission rates for each spectral channel.
Zhihua Zhang, Palle E. T. Jorgensen
Chapter 6. Affine Frames
Abstract
Affine frames, generated by affine transform of one or several frame generator elements, can provide redundant but sparse representations of complex data. Such redundant feature guarantees the realization of robust data transmission through sparse reconstruction and noise suppression achieved at the same time. Frame tree is a tree-like connection among affine frame generators, each of which can provide a refined partition of some spatial-scale region linked with different branches in frame trees. The depth of frame trees, which determines how fine of spatial-scale cells in frame representation, may be finite or infinite. A suitable frame can be picked up fast from frame tree to mine complex data with abrupt changes only on small spatial-scale regions.
Zhihua Zhang, Palle E. T. Jorgensen
Chapter 7. Spectral Domain of Frames
Abstract
The structure of frame multiresolution analysis (FMRA) is determined largely by the spectral domain of the associated frame scaling function. The carefully selected spectral domain of FMRA can significantly suppress narrowband noises, resulting in the derived frame representation of large-scale data that can achieve compression and noise suppression simultaneously. Therefore, in this chapter, we characterize the topological features of spectral domains of FMRAs and establish the relation between frame sets and spectral domains of frames.
Zhihua Zhang, Palle E. T. Jorgensen
Chapter 8. Periodic Frames
Abstract
Various periodic oscillations have emerged in the dynamic evolution of climate and environment system, industrial system, and socio-economic system. Periodic frames are a special kind of frames with the aim to mine these periodic features.
Zhihua Zhang, Palle E. T. Jorgensen
Chapter 9. Frame Spectral Energy Distribution
Abstract
Environmental and climatic processes always respond slowly to white noise forcing arising in the Earth system, leading to that background noises in environmental and climatic evolutions exhibit obviously stronger fluctuations in low-frequency regions than those in high-frequency regions, just like the behavior of the well-known red noise.
Zhihua Zhang, Palle E. T. Jorgensen
Chapter 10. Sparse Frame-Polynomial Coupling Representation
Abstract
Frame representation is widely used in data acquisition, processing, and transmission systems. Due to abrupt discontinuity, the corresponding frame coefficients living near data boundary always have large magnitudes which have nothing to do with the data itself. In order to mitigate such artificial boundary effects, the coupling of frames and polynomials in this chapter is to represent gridded data into the combination of low-dimensional interpolation/projection polynomials and periodic smooth data which are expanded further into periodic frame series. The frame-polynomial coupling technique provides a sparse representation of databases from digital Earth platforms.
Zhihua Zhang, Palle E. T. Jorgensen
Chapter 11. Object-Oriented Frame Analysis
Abstract
Different objects of general shape from various sources can be detected, cataloged, and mined by utilizing their geometric and shape information, but internal features of these objects are often ignored due to a lack of suitable tools.
Zhihua Zhang, Palle E. T. Jorgensen
Chapter 12. Frame Networks
Abstract
As an emerging branch of statistical and deep learning, frame networks can automatically acquire novel knowledge from observation data through a statistical learning process and then makes reliable predictions and downscaling. A frame network consists of three layers: the input layer, the hidden layer, and the output layer, where various frames are embedded into each node of the hidden layer and frame coefficients are used as the weight of the directed edges from one node to another node. Frame networks provide a unique nonlinear tool for simulation, downscaling, and prediction in the mining of big data. Different from frame networks, frames on networks deal with dynamical system living on complex networks and provide novel insights into topological and dynamical features of complex nonlinear systems over a wide range of spatial/temporal scales.
Zhihua Zhang, Palle E. T. Jorgensen
Backmatter
Metadaten
Titel
Frame Theory in Data Science
verfasst von
Zhihua Zhang
Palle E. T. Jorgensen
Copyright-Jahr
2024
Electronic ISBN
978-3-031-49483-3
Print ISBN
978-3-031-49482-6
DOI
https://doi.org/10.1007/978-3-031-49483-3

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