Selected Model Based Architectures and Algorithms for Learning, Signal Processing and Optimization

Paweł Ksieniewicz, Mariusz Uchroński

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ISBN/ISSN:

978-83-7837-622-4

DOI:

Wydawnictwo:

Akademicka Oficyna Wydawnicza EXIT Andrzej Lang

Rok wydania:

2021

Liczba stron:

158

XML:ISBN/ISSN:

ONIX MARC21

Bibliografia:

. Selected Model Based Architectures and Algorithms for Learning, Signal Processing and Optimization. Red. Ksieniewicz, Paweł; Uchroński, Mariusz . : Akademicka Oficyna Wydawnicza EXIT Andrzej Lang, 2021, 158 s. ISBN 978-83-7837-622-4

Bibliografia refWorks:

RT Book, Whole

SR Electronic(1)

A2 Ksieniewicz, P.

A2 Uchroński, M.

T1 Selected Model Based Architectures and Algorithms for Learning, Signal Processing and Optimization

PB Akademicka Oficyna Wydawnicza EXIT Andrzej Lang

YR 2021

SN 978-83-7837-622-4

Bibliografia BibTex:

@Book{ 259880, author = "", editor = "Ksieniewicz, Paweł and Uchroński, Mariusz", title = "Selected Model Based Architectures and Algorithms for Learning, Signal Processing and Optimization", publisher = "Akademicka Oficyna Wydawnicza EXIT Andrzej Lang", year = "2021", address = "", isbn = "978-83-7837-622-4" }

Bibliografia endNote:

TY - BOOK

AU -

ED - Ksieniewicz, Paweł

ED - Uchroński, Mariusz

TI - Selected Model Based Architectures and Algorithms for Learning, Signal Processing and Optimization

PB - Akademicka Oficyna Wydawnicza EXIT Andrzej Lang

CY -

PY - 2021

SN - 978-83-7837-622-4

ER -

Netografia (standard APA):

. Red. Ksieniewicz, Paweł; Uchroński, Mariusz. Selected Model Based Architectures and Algorithms for Learning, Signal Processing and Optimization [baza danych online] : Akademicka Oficyna Wydawnicza EXIT Andrzej Lang, 2021 [dostęp: 22 07 2024]. Dostęp w Ibuk Libra: https://libra.ibuk.pl/reader/selected-model-based-architectures-and-algorithms-for-learning-signal-pawel-ksieniewicz-mariusz-259880.

automatyka

SPIS TREŚCI

1. Schur Parametrization and Orthogonal Modeling of p-Stationary Second-Order Stochastic Processes

2. Efficient DOA algorithms using multi-dimensional high-resolution Fourier transform

3. Wireless localization syst...

Więcej

ISBN/ISSN:

978-83-7837-622-4

DOI:

Wydawnictwo:

Akademicka Oficyna Wydawnicza EXIT Andrzej Lang

Rok wydania:

2021

Liczba stron:

158

XML:ISBN/ISSN:

ONIX MARC21

Spis treści niedostępny

AgnieszkaWielgus,JanZarzycki,FelicjaLwowandBartłomiejGolenko

deﬁnitesymmetricToeplitzmatrix.Therefore,parametrizationalgorithmsarefast

andsimplewhatisimportantfrompracticalpointofview.

However,inreal-lifewearefacedinmostcaseswithnon-stationarystochas-

ticprocesseswhosecovariancematricesarepositive-deﬁniteHermitian,andthe

parametrizationproblemsolutionleadstothegeneralizedSchuralgorithmoffull

complexity,comparingtotheToeplitzcase[13],eveniftheprocessisnotnecessarily

totallynonstationary,anditscovariancematrixmaybeclosetotheToeplitzmatrix.

Therefore,considerationofstationaryversusnon-stationaryprocessesisnottoo

instructive.

ManyapproachestothenonstationarySchurparametrizationalgorithmcomplex-

ityreductionhavebeenproposed,tomentiontheMatrixExtensionProblem,and

theresultingstaircaseeliminationalgorithm,proposedin[12]andgeneralizedto

thenonlinearSchurparametrizationproblemofhigher-orderstochasticprocesses

[3,4,5,14].

Inanothervein,theapproachbasedonandfollowingfromtheconceptofU-

stationarity[2],resultsinahierarchicalclassiﬁcationofnonstationaryprocessesin

termsoftheirdistancefromstationarity.Thisconceptallowsalsoforthealgorithm

complexityreductionforU-stationaryprocessesforwhichU=2(n+1)(where

n=03...3#–1)forthecovariancematrixofdimensions(#+1)⇥(#+1).

Inthispaperweshow,forasubclassofsecond-orderstochasticprocesses,which

wecallP-stationaryprocesses,whosecovariancematricesareblock-Toeplitz(with

blocksofdimension(P+1)⇥(P+1)(P=03...3#))[1],howtheirU-stationarity

isreﬂectedinthestructuresofthecorrespondingSchurparametrizationschemas.

ThepresentedapproachallowsustoreducethecomplexityoftheSchuralgorithm

forsecond-orderprocessesinauniformway.

Thispaperisorganizedasfollows:ﬁrstly,werecallthenotionofU-stationary

stochasticprocesses.Secondly,weintroduceaclassofP-stationaryprocessesand

showtheisomorphismbetweenthespaceofrandomvariablesandofcoemcient-

vectors.Next,weproposearecursiveGram-Schmidtorthogonalizationprocedureof

thebasesresultinginthegeneralizedLevinsonandSchuralgorithms,herederived

geometrically.Finally,weconsiderP-stationaryparametrizationandmodelingalgo-

rithms,usingtheKolmogorovisomorphism,andpresentstructuresoftheassociated

innovarionsandmodelingﬁltersfornear-stationarystochasticprocesses.

102"-stationarysecond-orderstochasticprocesses

Let{⌦3B3`}beaprobabilityspacewhere⌦isanabstractsetofelementsl2

⌦,B-af-algebraofBorelsubsets,and`–aprobailitymeasureonB.Via

!2{⌦3B3`}wedenoteaseparableHilbertspaceoff-measurablemappingsF:

⌦!C,satisfyingØ

⌦|F(l)|2`(dl)<∞.Weintroducetheinner-producton

!2{⌦3B3`}as(F3E)⌦

=Ø

∆

⌦F(l)¯

E(l)`(dl)=EF¯

Ewhere¯denotescomplex

conjugate,andEstandsfortheexpectationoperator.Thisinner-productinducesthe

normkFk2

⌦=Ø

⌦|F(l)|2`(dl)=E|F|2andmetricd⌦(F3E)=kF–Ek⌦.Let

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Selected Model Based Architectures and Algorithms for Learning, Signal Processing and Optimization

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