The Filling of Gaps in Geophysical Time Series by Artificial Neural Networks

Dergachev, V. A.; Gorban, Alexander N.; Rossiev, A. A.; Karimova, L. M.; Kuandykov, E B; Makarenko, N. G.; Steier, Peter

doi:10.1017/s0033822200038224

Cited by 15 publications

(22 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This property (zero asymptotics) must be taken into account when using the formula (10.16). When constructing invariant manifolds F (W ), it is natural to use (10.16) not for the immersion F (y), but for the deviation of F (y) from some analytical ansatz F 0 (y) [277][278][279][280].…”

Section: Carleman's Formula In the Analytical Invariant Manifolds Appmentioning

confidence: 99%

Invariant Manifolds for Physical and Chemical Kinetics

Gorban¹,

Karlin²

2005

Lecture Notes in Physics

210

418

View full text Add to dashboard Cite

Of course, it is physics. Model reduction in kinetics requires physical concepts and structures; it is impossible to make an expedient reduction of a kinetic model without thermodynamics, for example. The entropy, the Legendre transformation generated by the entropy, and the Riemann structure defined by the second differential of the entropy provide the elementary geometrical basis for the first approximation. The physical sense of the models gives many hints for their further processing. So, it is not mathematics; we care about the physical sense more than about rigorous proofs. We should deal with equations even in the absence of theorems about existence and uniqueness of solutions. Mathematics assimilates the physical notions with a considerable delay in time, but any such an assimilation leads to further insights. 1But, without doubt, it is mathematics. The story about invariant manifolds for differential equations began inside mathematics. The first significant steps were taken by two great mathematicians, A.M. Lyapunov and H. Poincaré, at the end of the XIXth century. Then N.M. Krylov and N.N. Bogolyubov, A.N. Kolmogorov, V.I. Arnold and J. Moser, J.E. Marsden, M.I. Vishik, R. Temam, and many other mathematicians developed this field of science, and many elegant theorems and useful methods were created. This is not only pure mathematics, the wide field of applications was developed too, from hydrodynamics to process engineering and control theory and methods. This is pure and applied dynamics. The language of model reduction, the basic notions that we use, the theorems and methods, all this either came from 1 The closest example: after mathematicians discovered how the entropy functional may be important for the theory of the Boltzmann equation, then they proved the existence theorem (P.L. Lions and R. DiPerna, this work was awarded the Fields medal in 1994). VIIIPreface pure and applied dynamics directly, or bears the visible imprint of its ideas and methods. Maybe the book presents a specific chapter on this subject? But, of course, the problems came from physics, from engineering. Maybe it is both physics and mathematics? Or perhaps it is something different, but what can it be? It is not so easy to answer the question, what is the subject of our book, even for the authors. But we can say what we want it to be. We want it to be a special "meeting point" of pure and applied dynamics, of physics, and of engineering sciences. This meeting point has a sufficient number of specific problems, methods and results to deserve a special name. We propose the name Model Engineering. As long as it is engineering, it is synthetic subject: if it is possible to prove something exactly, this is great, and we should follow this possibility, but if the physical sense gives us a seminal hint, well, we should use it even if the rigorous foundations are far from complete. The result is the model that works. In this enormous field of intellectual activity our book tends to be in the theoretical corner; we focus our study on c...

show abstract

Section: Carleman's Formula In the Analytical Invariant Manifolds Appmentioning

confidence: 99%

Invariant Manifolds for Physical and Chemical Kinetics

Gorban¹,

Karlin²

2005

Lecture Notes in Physics

210

418

View full text Add to dashboard Cite

show abstract

“…It is equivalent to construction of piece-wise linear manifold. Other interpolation and extrapolation methods are also applicable, for example the use of Carleman's formula [1,14,15,18,19,6] or using Lagrange polynomials [47], although they are more computationally expensive.…”

Section: Piecewise Linear Manifolds and Data Projectorsmentioning

confidence: 99%

“…5 we present two examples of 2D-datasets provided by Kégl 6 . The first dataset called spiral is one of the standard ways in the principal curve literature to show that one's approach has better performance than the initial algorithm provided by Hastie and Stuetzle.…”

Section: Test Examplesmentioning

confidence: 99%

Elastic Maps and Nets for Approximating Principal Manifolds and Their Application to Microarray Data Visualization

Gorban

Zinovyev

2008

Lecture Notes in Computational Science and Enginee

View full text Add to dashboard Cite

Summary. Principal manifolds are defined as lines or surfaces passing through "the middle" of data distribution. Linear principal manifolds (Principal Components Analysis) are routinely used for dimension reduction, noise filtering and data visualization. Recently, methods for constructing non-linear principal manifolds were proposed, including our elastic maps approach which is based on a physical analogy with elastic membranes. We have developed a general geometric framework for constructing "principal objects" of various dimensions and topologies with the simplest quadratic form of the smoothness penalty which allows very effective parallel implementations. Our approach is implemented in three programming languages (C++, Java and Delphi) with two graphical user interfaces (VidaExpert and ViMiDa applications). In this paper we overview the method of elastic maps and present in detail one of its major applications: the visualization of microarray data in bioinformatics. We show that the method of elastic maps outperforms linear PCA in terms of data approximation, representation of between-point distance structure, preservation of local point neighborhood and representing point classes in low-dimensional spaces.

show abstract

“…To avoid this effect, we introduced in the elmap package the possibility to make a linear extrapolation of the bounded rectangular manifold (extending it by continuity in different directions). Other, more complicated extrapolations can be performed as well, like using Carleman's formulas (see [1], [5], [10], [11], [14], [15]). …”

Section: Projectingmentioning

confidence: 99%

Elastic Principal Graphs and Manifolds and their Practical Applications

Gorban

Zinovyev

2005

Computing

View full text Add to dashboard Cite

Principal manifolds serve as useful tool for many practical applications. These manifolds are defined as lines or surfaces passing through "the middle" of data distribution. We propose an algorithm for fast construction of grid approximations of principal manifolds with given topology. It is based on analogy of principal manifold and elastic membrane. First advantage of this method is a form of the functional to be minimized which becomes quadratic at the step of the vertices position refinement. This makes the algorithm very effective, especially for parallel implementations. Another advantage is that the same algorithmic kernel is applied to construct principal manifolds of different dimensions and topologies. We demonstrate how flexibility of the approach allows numerous adaptive strategies like principal graph constructing, etc. The algorithm is implemented as a C++ package elmap and as a part of stand-alone data visualization tool VidaExpert, available on the web. We describe the approach and provide several examples of its application with speed performance characteristics.

show abstract

The Filling of Gaps in Geophysical Time Series by Artificial Neural Networks

Cited by 15 publications

References 16 publications

Invariant Manifolds for Physical and Chemical Kinetics

Invariant Manifolds for Physical and Chemical Kinetics

Elastic Maps and Nets for Approximating Principal Manifolds and Their Application to Microarray Data Visualization

Elastic Principal Graphs and Manifolds and their Practical Applications

Contact Info

Product

Resources

About