331 research outputs found
Principal manifolds and graphs in practice: from molecular biology to dynamical systems
We present several applications of non-linear data modeling, using principal
manifolds and principal graphs constructed using the metaphor of elasticity
(elastic principal graph approach). These approaches are generalizations of the
Kohonen's self-organizing maps, a class of artificial neural networks. On
several examples we show advantages of using non-linear objects for data
approximation in comparison to the linear ones. We propose four numerical
criteria for comparing linear and non-linear mappings of datasets into the
spaces of lower dimension. The examples are taken from comparative political
science, from analysis of high-throughput data in molecular biology, from
analysis of dynamical systems.Comment: 12 pages, 9 figure
The Michaelis-Menten-Stueckelberg Theorem
We study chemical reactions with complex mechanisms under two assumptions:
(i) intermediates are present in small amounts (this is the quasi-steady-state
hypothesis or QSS) and (ii) they are in equilibrium relations with substrates
(this is the quasiequilibrium hypothesis or QE). Under these assumptions, we
prove the generalized mass action law together with the basic relations between
kinetic factors, which are sufficient for the positivity of the entropy
production but hold even without microreversibility, when the detailed balance
is not applicable. Even though QE and QSS produce useful approximations by
themselves, only the combination of these assumptions can render the
possibility beyond the "rarefied gas" limit or the "molecular chaos"
hypotheses. We do not use any a priori form of the kinetic law for the chemical
reactions and describe their equilibria by thermodynamic relations. The
transformations of the intermediate compounds can be described by the Markov
kinetics because of their low density ({\em low density of elementary events}).
This combination of assumptions was introduced by Michaelis and Menten in 1913.
In 1952, Stueckelberg used the same assumptions for the gas kinetics and
produced the remarkable semi-detailed balance relations between collision rates
in the Boltzmann equation that are weaker than the detailed balance conditions
but are still sufficient for the Boltzmann -theorem to be valid. Our results
are obtained within the Michaelis-Menten-Stueckelbeg conceptual framework.Comment: 54 pages, the final version; correction of a misprint in Attachment
Generation of Explicit Knowledge from Empirical Data through Pruning of Trainable Neural Networks
This paper presents a generalized technology of extraction of explicit
knowledge from data. The main ideas are 1) maximal reduction of network
complexity (not only removal of neurons or synapses, but removal all the
unnecessary elements and signals and reduction of the complexity of elements),
2) using of adjustable and flexible pruning process (the pruning sequence
shouldn't be predetermined - the user should have a possibility to prune
network on his own way in order to achieve a desired network structure for the
purpose of extraction of rules of desired type and form), and 3) extraction of
rules not in predetermined but any desired form. Some considerations and notes
about network architecture and training process and applicability of currently
developed pruning techniques and rule extraction algorithms are discussed. This
technology, being developed by us for more than 10 years, allowed us to create
dozens of knowledge-based expert systems. In this paper we present a
generalized three-step technology of extraction of explicit knowledge from
empirical data.Comment: 9 pages, The talk was given at the IJCNN '99 (Washington DC, July
1999
Elastic principal manifolds and their practical applications
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. The 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.Comment: 26 pages, 10 figures, edited final versio
PCA Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes
Multidimensional data distributions can have complex topologies and variable
local dimensions. To approximate complex data, we propose a new type of
low-dimensional ``principal object'': a principal cubic complex. This complex
is a generalization of linear and non-linear principal manifolds and includes
them as a particular case. To construct such an object, we combine a method of
topological grammars with the minimization of an elastic energy defined for its
embedment into multidimensional data space. The whole complex is presented as a
system of nodes and springs and as a product of one-dimensional continua
(represented by graphs), and the grammars describe how these continua transform
during the process of optimal complex construction. The simplest case of a
topological grammar (``add a node'', ``bisect an edge'') is equivalent to the
construction of ``principal trees'', an object useful in many practical
applications. We demonstrate how it can be applied to the analysis of bacterial
genomes and for visualization of cDNA microarray data using the ``metro map''
representation. The preprint is supplemented by animation: ``How the
topological grammar constructs branching principal components
(AnimatedBranchingPCA.gif)''.Comment: 19 pages, 8 figure
Lyapunov-like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems
We provide Lyapunov-like characterizations of boundedness and convergence of
non-trivial solutions for a class of systems with unstable invariant sets.
Examples of systems to which the results may apply include interconnections of
stable subsystems with one-dimensional unstable dynamics or critically stable
dynamics. Systems of this type arise in problems of nonlinear output
regulation, parameter estimation and adaptive control.
In addition to providing boundedness and convergence criteria the results
allow to derive domains of initial conditions corresponding to solutions
leaving a given neighborhood of the origin at least once. In contrast to other
works addressing convergence issues in unstable systems, our results require
neither input-output characterizations for the stable part nor estimates of
convergence rates. The results are illustrated with examples, including the
analysis of phase synchronization of neural oscillators with heterogenous
coupling
- …