Hilbertsche Zerlegungen eingebetteter Prozessräume und ihre Anwendung auf die Vorhersage von Zeitreihen
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Philipps-Universität Marburg
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Abstract
The Theory of Time Series or Stochastic Processes
is partly of functional analytic character. Well-known examples
are the Reproducing Kernel Hilbert space associated with a
process, and the Karhunen-Loève expansion. More generally, the
Spectral Theory of stationary processes, and orthogonal
projection as a principle of prediction, are functional
analytic aspects of time series. This dissertation aims at
enhancing this theory by transfering modern methods of
functional analysis to the field of Stochastic Processes,
providing new resp. broadened results. The above-mentioned
topics are just a few out of Time Series Analysis but they
convey the interface between Stochastic Processes and Analysis
on which this thesis focuses. We give a more detailed
description: To begin with, it is the structure of stationary
processes that allows successful application of analytic tools.
For instance, the stationary Prediction Theory started by
Wiener and Kolmogorov is of abstract and (Fourier) analytic
nature. Generalizations leading to similar results without the
restriction of stationarity are still of interest. They might
require alternative time domain methods formalized only by
means of the process' indexset (time). Besides, Representation
Theory of Stochastic Processes seems limited to its original
setting. As initiated by Karhunen and Loève the theory depends
on elementary isometries between the process space and a space
of quadratically integrable functions - called Spectral Domain.
Continuous processes on compact intervals allow precise
deductions and reveal the refineable connection to eigenvector
bases of integral operators (Mercer's Theorem). Finally, the
Theory of Hilbert Subspaces associated with processes could be
modernized, too. Parzen published the relationship by linking a
Kernel Hilbert space (in the sense of Aronszajn) to a process.
This space of functions on the indexset gives rise to an
isometric description of the process space. The relationship is
still of relevance, even though it is of rather elementary
character due to the discreteness of the time domain. In
summary, the open problems and the way the presented thesis
approaches them are as follows: (1) Hilbert Subspaces
associated with processes are introduced until now assuming a
discrete topology on the indexset and this leads solely to
Kernel Hilbert spaces in the sense of Aronszajn. This
dissertation analyses in how far the actual topology of the
time domain could be preserved and what consequences this
brings for the construction as well as for the properties of
the process space. Time is modelled topologically in form of a
(pivotal) Hilbert space and covariance functions are understood
as generalized functions. The developed theory of embeddable
processes provides associated Hilbert subspaces within this
generalized setting. A 'Reproducing Property' of the embedded
process space is proved. (2) Issues of bases (for the process
spaces) and their constructions follow immediately. So far,
they have been formulated as representation problems for the
related process by means of a denumerable orthonormal system.
Solutions depend on elementary isometries within Hilbert
spaces. This thesis shows how modern decomposition techniques
of Hilbert Subspaces give rise to new (esp. continuous) bases
and describes two construction methods: image and spectral
decompositions. Both are independent of conditions of
denumerability, incorporate existing methods and allow a
representation of the process. (3) The known Karhunen-Loève
expansion builts on the usual isometry arguments limited to a
formalism of denumerability. However, it is the Spectral Theory
of special, positive integral operators which yields the
expansion. The dissertation clarifies the abstract formalism by
means of unbounded positive operators. The relevance of such
operators' Spectral Theory for the resulting spectral
decomposition is illustrated, abstracting the rather specific
influence of Mercer's Theorem. (4) The use of decompositions
for prediction purposes is obvious. So far, preferred Fourier
analytic approaches seem to disguise important time domain
aspects of forecasting, though. The presented thesis aims at
characterizing a general prediction decomposition using time
domain terminology. The found prediction methods show an
intrinsic 'Gram-Schmidt Principle' and indicate the
relationship with Cholesky's Factorization. Predictors in terms
of decomposition bases are given as well as their connection to
previous results.
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Supervisor:
Dates
Created: 2004Issued: 2004-03-18Updated: 2011-08-10
Faculty
Fachbereich Mathematik und Informatik
Publisher
Philipps-Universität Marburg
Language
ger
Data types
DoctoralThesis
Keywords
(Generalized) eigenfunction expansions; rigged Hilbert spacesCovariance operators , Factorization theoryGeneral second-order processesPrediction theoryKarhunen-Loève-ZerlegungPrediction theoryHilbert subspaces (= operator ranges); complementation (Aronszajn, de Branges,...)Series expansion of processesHilbert spaces with reproducing kernelsDreieckszerlegung , Selbstähnliche ProzesseKovarianzoperatorAbschließbarer Operator
DFG-subjects
Verallgemeinerter Funktionenraum , DekompositionVorhersagetheorieStochastischer ProzessPrädikHilbert-Unterraum , Kovarianzfunktion
DDC-Numbers
510
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Jäger, Ralf (128940158): Hilbertsche Zerlegungen eingebetteter Prozessräume und ihre Anwendung auf die Vorhersage von Zeitreihen. : Philipps-Universität Marburg 2004-03-18. DOI: https://doi.org/10.17192/z2004.0097.
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This item has been published with the following license: In Copyright