application of f-test method on model order selection and related problems a thesis submitted to ...

164 Pages·2015·25.29 MB·English

APPLICATION OF F-TEST METHOD ON MODEL ORDER SELECTION
AND RELATED PROBLEMS
A THESIS SUBMITTED TO
THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF
MIDDLE EAST TECHNICAL UNIVERSITY
BY
ALPER YAZAR
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR
THE DEGREE OF MASTER OF SCIENCE
IN
ELECTRICAL AND ELECTRONICS ENG.
AUGUST 2015

Approval of the thesis:
APPLICATION OF F-TEST METHOD ON MODEL ORDER
SELECTION AND RELATED PROBLEMS
submitted by ALPER YAZAR in partial ful(cid:28)llment of the requirements for the
degree of Master of Science in Electrical and Electronics Eng. Depart-
ment, Middle East Technical University by,
Prof. Dr. G(cid:252)lbin Dural (cid:220)nver
Dean, Graduate School of Natural and Applied Sciences
Prof. Dr. G(cid:246)n(cid:252)l Turhan Sayan
Head of Department, Electrical and Electronics Eng.
Assoc. Prof. Dr. ˙a§atay Candan
Supervisor, Electrical and Electronics Eng. Dept., METU
Examining Committee Members:
Prof. Dr. Tolga ˙ilo§lu
Electrical and Electronics Engineering Dept., METU
Assoc. Prof. Dr. ˙a§atay Candan
Electrical and Electronics Engineering Dept., METU
Assoc. Prof. Dr. Sinan Gezici
Electrical and Electronics Engineering Dept., Bilkent Univ.
Assoc. Prof. Dr. Umut Orguner
Electrical and Electronics Engineering Dept., METU
Assist. Prof. Dr. Sevin(cid:231) Figen (cid:214)ktem
Electrical and Electronics Engineering Dept., METU
Date:
I hereby declare that all information in this document has been ob-
tained and presented in accordance with academic rules and ethical
conduct. I also declare that, as required by these rules and conduct,
I have fully cited and referenced all material and results that are not
original to this work.
Name, Last Name: ALPER YAZAR
Signature :
iv
ABSTRACT
APPLICATION OF F-TEST METHOD ON MODEL ORDER SELECTION
AND RELATED PROBLEMS
Yazar, Alper
M.S., Department of Electrical and Electronics Eng.
Supervisor : Assoc. Prof. Dr. ˙a§atay Candan
August 2015, 140 pages
Signal modeling is one of the important topics of signal processing area. The in-
putsignalshouldbemodeledwithasuitablemathematicalmodel(cid:28)rst. Instatis-
tics related disciplines, there are information theory based criteria for model
order selection topic. In this thesis work, F-test based methods are proposed
on model order selection and related problems. F-test is used in statistics re-
lated disciplines. However, it is not so widely used in signal processing related
problems. Solution approaches for signal processing related problems based on
known F-test are contributions of this thesis work. This work is focused on
signals in linear spaces.
Fundamentally, F-test is a test of signi(cid:28)cance. It is used to test whether a signal
model is su(cid:30)cient to model the signal of interest or higher order models are
needed. This test is made by using two nested models with di(cid:27)erent orders.
RSS (Residual Sum of Squares) values are calculated for each model and they
v
are compared using F-test. According to the test result, it is determined that
whether the lower order model is almost good as the higher order model or the
higher order model improves the accuracy signi(cid:28)cantly. The proposed method
is basically an iterative application of F-test. It selects the suitable model order
by applying F-test many times.
In this work, some problems related with model order selection topic are solved
using F-test based approaches. An analysis window length selection method
for zero-crossing point estimation problem using line (cid:28)t is proposed as the (cid:28)rst
example. Secondly, a method is proposed for the segmentation of multi tone
signals. Similar approach is given as the third example for segmentation of
FM signals. As the fourth example, a number of pole selection algorithm is
proposed for all-pole signal modeling using Prony’s method. Lastly, a segmen-
tation method for damped sinusoidal signals with Prony’s method is proposed.
Simulation results are provided for each (cid:28)ve problems.
Keywords: Signal Modeling, Linear Models, Parameter Estimation, Model Or-
der Selection, Model Validity, Analysis Window Length Selection, Test of Sig-
ni(cid:28)cance, F-test, Nested Models, Zero-Crossing Estimation, Segmentation.
vi
(cid:214)Z
MODEL DERECES(cid:157) SE˙(cid:157)M(cid:157) VE (cid:157)LG(cid:157)L(cid:157) PROBLEMLER (cid:157)˙(cid:157)N F-TEST(cid:157)
Y(cid:214)NTEM(cid:157)N(cid:157)N UYGULANMASI
Yazar, Alper
Y(cid:252)ksek Lisans, Elektrik ve Elektronik M(cid:252)hendisli§i B(cid:246)l(cid:252)m(cid:252)
Tez Y(cid:246)neticisi : Do(cid:231). Dr. ˙a§atay Candan
A§ustos 2015 , 140 sayfa
(cid:157)‡aret modellemesi, i‡aret i‡leme alan(cid:25)n(cid:25)n en (cid:246)nemli konular(cid:25)ndan biridir. Giri‡
i‡areti ilk olarak uygun bir matematiksel model ile g(cid:246)sterilmelidir. (cid:157)statistik ile
ilgilialanlarda,modelderecesise(cid:231)imiileilgilibilgikuram(cid:25)tabanl(cid:25)(cid:231)e‡itlikriterler
bulunmaktad(cid:25)r. Bu tez (cid:231)al(cid:25)‡mas(cid:25)nda, model derecesi se(cid:231)imi ve ilgili problemler
i(cid:231)in F-testi tabanl(cid:25) bir y(cid:246)ntem (cid:246)nerilmi‡tir. F-testi, istastik ile ilgili alanlarda
kullan(cid:25)lan bir y(cid:246)ntemdir. Bu tez (cid:231)al(cid:25)‡mas(cid:25)n(cid:25)n temel katk(cid:25)s(cid:25), (cid:231)e‡itli i‡aret i‡leme
problemleri i(cid:231)in F-testi tabanl(cid:25) (cid:231)(cid:246)z(cid:252)mler sunmas(cid:25)d(cid:25)r. Bu (cid:231)al(cid:25)‡ma do§rusal uzay-
larda bulunan i‡aretler (cid:252)zerinedir.
Temel olarak F-testi bir (cid:246)nemlilik testidir. Bu test, bir modelin ilgilenen i‡a-
reti g(cid:246)sterebilmek i(cid:231)in yeterli olup olmad(cid:25)§(cid:25)n(cid:25), daha y(cid:252)ksek dereceli modellere
ihtiya(cid:231) duyulup duyulmad(cid:25)§(cid:25)n(cid:25) anlamak i(cid:231)in yap(cid:25)l(cid:25)r. Test i(cid:231)in farkl(cid:25) derecelerde
iki adet i(cid:231)i(cid:231)e modele ihtiya(cid:231) duyulmaktad(cid:25)r. Her iki model i(cid:231)in de RSS (Resi-
dual Sum of Squares) veya AKT (Art(cid:25)k Kareler Toplam(cid:25)) de§erleri hesaplan(cid:25)r
vii
ve bu de§erler F-testi kullan(cid:25)larak kar‡(cid:25)la‡t(cid:25)r(cid:25)l(cid:25)r. Test sonucuna g(cid:246)re, d(cid:252)‡(cid:252)k de-
receli modelin neredeyse y(cid:252)ksek dereceli model kadar iyi oldu§u veya y(cid:252)ksek
dereceli modelin modelleme do§rulu§unu (cid:246)nemli bir bi(cid:231)imde iyile‡tirdi§i karar(cid:25)
verilir. (cid:214)nerilen y(cid:246)ntem temel olarak F-testi y(cid:246)nteminin tekrarlamal(cid:25) olarak kul-
lanmaktad(cid:25)r. Y(cid:246)ntem, uygun olan model derecesini birden fazla F-testi yaparak
se(cid:231)mektedir.
Bu (cid:231)al(cid:25)‡mada, model derecesi se(cid:231)imi ile ilgili baz(cid:25) probemler F-testi tabanl(cid:25) yak-
la‡(cid:25)mlarla (cid:231)(cid:246)z(cid:252)lm(cid:252)‡t(cid:252)r. (cid:157)lk olarak, do§ru oturtularak yap(cid:25)lan s(cid:25)f(cid:25)r kesim noktas(cid:25)
kestirimi problemi i(cid:231)in analiz penceresi uzunlu§u se(cid:231)imi problemine bir y(cid:246)ntem
(cid:246)nerilmi‡tir. (cid:157)kinci (cid:246)rnek problem olarak, (cid:231)ok tonlu i‡aretlerin b(cid:246)l(cid:252)mlenmesi
problemi incelenmi‡tir. Benzer bir yakla‡(cid:25)m(cid:25)n FM i‡aretler i(cid:231)in uygulanmas(cid:25)
(cid:252)(cid:231)(cid:252)nc(cid:252) (cid:246)rnek problem olarak verilmi‡tir. D(cid:246)rd(cid:252)nc(cid:252) (cid:246)rnek olarak da, bir i‡aretin
Pronyy(cid:246)ntemiilesadecekutuplus(cid:252)zge(cid:231)(cid:231)(cid:25)kt(cid:25)s(cid:25)olarakmodellenmesiprobleminde
kutup say(cid:25)s(cid:25) se(cid:231)imi i(cid:231)in bir y(cid:246)ntem (cid:246)nerilmi‡tir. Son problem ise, s(cid:246)n(cid:252)ml(cid:252) si-
n(cid:252)s i‡aretlerinden olu‡an bir i‡aretin Prony y(cid:246)ntemi kullan(cid:25)larak b(cid:246)l(cid:252)mlenmesi
(cid:252)zerinedir. (cid:214)rnek problemler i(cid:231)in benzetim sonu(cid:231)lar(cid:25) sunulmu‡tur.
Anahtar Kelimeler: (cid:157)‡aret Modellemesi, Do§rusal Modeller, Parametre Kesti-
rimi, Model Derecesi Se(cid:231)imi, Model Ge(cid:231)erlili§i, Analiz Penceresi Uzunlu§u Se-
(cid:231)imi, (cid:214)nemlilik Testi, F-testi, (cid:157)(cid:231)i(cid:231)e Modeller, S(cid:25)f(cid:25)r Kesim Noktas(cid:25) Kestirimi,
B(cid:246)l(cid:252)mleme.
viii
To my loving mother and father
and
Gonca
ix
ACKNOWLEDGMENTS
First of all, I would like to express my sincere gratitude to my thesis supervisor
Assoc. Prof. Dr. Ca§atay Candan. In addition to his valuable technical com-
ments, he always motivated me for progression. He always guided me whenever
I need. I also had the opportunity of observing academic side of electronics en-
gineering with him. It was great chance and honor to me work with Dr. Candan
during last three years. Endless help of him is always to be remembered.
Secondly, I would also like to thank jury members; Dr. Orguner, Dr. ˙ilo§lu,
Dr. (cid:214)ktem and Dr. Gezici for their valuable comments and corrections.
I would like to thank Erdal Mehmetcik next. I am deeply grateful to him for
the discussions that helped me sort out the technical details of my work. His
reviews on my MATLAB codes and text were very valuable for me. Thank you
Erdal for your endless support in my life for many many years.
Gonca, it is impossible to forget your support not only during this thesis work
but also all the time. Thank you also for your e(cid:27)orts on conference paper. It
was not possible to complete it on due date without your help.
(cid:214)mer, thank you for your comments about my grammar.
I would also like to thank my company ASELSAN Inc. for supporting this thesis
work and encouraging engineers like me to make further scienti(cid:28)c studies. Also
thank all my colleagues supporting me.
I also thank Turkish Scienti(cid:28)c and Technological Research Council (T(cid:220)B(cid:157)TAK)
for their (cid:28)nancial support (2211) during my thesis study.
I want to thank developers of all free software and plug-ins that I used to pre-
pare this thesis document: LATEX3, pdfLATEX, BIBTEX, TeXstudio, LibreO(cid:30)ce,
SumatraPDF and Notepad++, GIT and so on. Without their e(cid:27)orts on these
x