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DÜZGÜNLEŞTİRİLMİŞ FONKSİYONEL ANA BİLEŞENLER ANALİZİ İLE İMKB VERİLERİNİN İNCELENMESİ

Year 2008, Issue: 8, 1 - 32, 05.09.2011

Abstract

Fonksiyonel Veri Analizi Yöntemleri reel bir aralığın sonlu bir alt setinde değerlendirilen eğrilerden veya
gözlenen fonksiyonlardan oluşan verileri analiz etmek üzere geliştirilmiştir. Fonksiyonel Veri Analizindeki
teknikler, xi(t) (i = 1, 2, … , N) şeklinde belirtilen fonksiyonlardan veya onların türevlerinden oluşan rastgele
örneklerdeki değişimin (varyasyonun) incelenmesi ve araştırılması amacıyla kullanılabilir. Pratikte bu
fonksiyonlar sıklıkla ayrık noktalarda gözlenen verilere uygulanan düzgünleştirme (smoothing) süreçlerinin bir
sonucu olarak ortaya çıkarlar. Bu çalışmada da Splayn Düzgünleştirme Yöntemleri bu amaçla ele alınmıştır.
Bu araştırmanın amacı, ayrık noktalarda gözlenen verileri öncelikle B-Splayn Baz Fonksiyonlar ve Pürüzlü
Ceza Yaklaşımı kullanarak bir diğer deyişle bu iki yaklaşımın birlikte kullanılması olarak adlandırılan Splayn
Düzgünleştirme Yöntemi ile sürekli ve türevlenebilir fonksiyonlar haline dönüştürülmesinin incelenmesidir.
Daha sonra da veriler arasındaki yani ilgilenilen zaman aralığında hisse senetlerinin bireysel fonksiyonları
arasındaki değişkenlik yapısını ortaya koymak üzere Düzgünleştirilmiş Fonksiyonel Ana Bileşenler
Analizinden faydalanılmıştır. Burada ilgilenilen birey sayısı değişken sayısından az olduğundan dolayı klasik
yöntemler zaten bu amaç doğrultusunda yetersiz kalmaktadır.
Bu çalışmada, IMKB-100 endeksinde yer alan şirketlerin haftalık hisse senedi kapanış fiyatlarından oluşan bir
örnek üzerinde yapılan uygulamaya yer verilmektedir. Düzgünleştirilmiş Fonksiyonel Ana Bileşenler Analizi
ile incelenen 13 şirket için özellikle 2000 yılının başlarında ve 360 ıncı günden bir diğer deyişle 2005 yılından
itibaren fiyatların değişkenliğinde bir artış olduğu ve zaman noktalarının ardışık olarak birbirleriyle pozitif bir
korelasyona sahip olduğu ulaşılan önemli sonuçlardan bir tanesidir.
Geleneksel Ana Bileşenler yönteminin uygulanmasının mümkün olmadığı durumlarda bile uygulanabilen ve
sistemdeki gürültü (noise) etkisini de kaldıran bir yöntem olan Düzgünleştirilmiş Ana Bileşenler Analizi
sonucu elde edilen harmonikler sayesinde hem verilerin kovaryans yüzeyiyle açıklanamayan değişkenlik
yapısı ortaya konulmuş ve hem de genel anlamda Fonksiyonel Veri Analizi ile örneğin türev fonksiyonlarının
da incelenebilmesi gibi görsel olarak da kuvvetli bulgular sunulmuştur.

References

  • Barra V. (2004) Analysis of Gene Expression Data Using Functional Principal Components, Computer methods and programs in biomedicine, 75(11).
  • Benko M. (2004). Functional Principal Components analysis, Implementation and Applications. A Master Thesis. Humboldt University Center of Applied Statistics and Economics, Berlin.
  • Benko M., Hardle W., Kneip A.(2006). Common Functional Principal Components, SFB 649,DiscussionPaper,Erişim:10.11.2006, http://ideas.repec.org/p/hum/wpaper/sfb649dp2006-010.html
  • Besse P., Ramsay J.O. (1986). Principal Components Analysis Of Sampled Functions, Psychometrica, 51(2) Boor C . (1978). A Practical Guide to Splines. Springer-Verlag: New-York
  • Castro P. E, Lawton W. H., Sylvestre E. A. (1986). Principal Modes Of Variation for Processes with Continuous Sample Curves, Technometrics, 28(4).
  • Costanzo G.D. (2005). Functional Principal Component Analysis of Financial Time Series, Cnam-Paris.
  • Dauxois J., Pousse A., Romain Y. (1982). Asymptotic theory for the principal component analysis of a vector random function : some applications to statistical inference, J. Multivariate Analysis, 12
  • Dierckx P. (1993). Curve and Surface Fitting with Splines. Oxford University Press:New York
  • Eubank R.L. (1999). Nonparametric Regression and Spline Smoothing. Marcel Dekker: USA.
  • Green.P.J., & Silverman B.W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman & Hall:London.
  • Hall P. , Nasab H. M. (2006). On Properties Of Functional Principal Components Analysis. Journal of the Royal Statistical Society: Series B, 68(1).
  • Hyde V., Moore E., & Hodge A. (2006). Functional Pca For Exploring Bidding Acivity Times for Online Auctions. Erişim:06.07.2006,http://www.rhsmith.umd.edu/ceme/statistics/functionalpca.pdf
  • James G. M., Hastie T.J., Sugar C.A. (2000) Principal Components Models For Sparse Functional Data, Biometrica, 87(3).
  • Jank W., & Shmueli G. (2006). Functional Data Analysis in Electronic Commerce Research. Statistical Science, 21(2).
  • Jones M. C., Rice J. A. (1992). Displaying The Important Features Of Large Collections Of Similar Curves. The American Statistician, 46(2).
  • Laukaitis A., Rackauskas A. (2002). Functional Data Analysis of Payment Systems. Nonlinear Analysis: Modelling and Control, 7(2).
  • Lee H.J. (2004). Functional Data Analysis: Classification and Regression. Doctor of Philosophy, Texas A&M University.
  • Leurgans S.E., Moyeed R.A, Silverman B.W. (1993). Canonical Correlation Analysis when the Data are Curves. Journal of the Royal Statistical Society: Series B., 55(3).
  • Lillestol J., & Ollmar F. (2003). Functional Data Analysis: Introduction and Applications to Financial Electricity Contracts. Erişim: 01.03.2005, http://www.nhh.no/for/dp/2003/0603.pdf
  • Lober E.M., & Villa C. (2004). Functional Principal Component Analysis of the Yield Curve, Erişim: 05.05.2005 , http://www.u-cergy.fr/AFFI_2004/IMG/pdf/MATZNER.pdf
  • Lyche, T., Morken, K. (2002). Spline Methods Draft, Erişim: 10.07.2005, http://www.ifi.uio.no/in329/nchap1.pdf
  • Musayev B., & Alp M. (2000). Fonksiyonel Analiz. Balcı Yayınları: Kütahya
  • Müler, H. G. (2005). Functional Modelling and Classification of Longitudinal Data, Scandavian Journal of Statistics, 32(2).
  • Nürnberger G. (1989). Approximation by Spline Functions. Springer – Verlag: Berlin.
  • Ramsay, J. O., Dalzell C. (1991). Some Tools For Functonal Data Analysis, Journal of the Royal Statistical Society: Series B.,53 (3)
  • Ramsay J.O., Silverman B.W. (1997). Functional Data Analysis. Springer – Verlag: New York.
  • Ramsay J. O. , Li X. (1998). Curve Registration, Journal of the Royal Statistical Society: Series B, 60(2).
  • Ramsay J.O., (2000). Basis Functions, Erişim:11.04.2005, ftp://ego.psych.mcgill.ca
  • Ramsay J.O, Silverman B.W. (2002). Applied Functional Data Analysis: Methods and Case Studies. Springer – Verlag: New York.
  • Ramsay J.O., Silverman B.W. (2005). Functional Data Analysis. Second Edition. Springer : USA
  • Rao, C. R. (1958). Some Statistical Methods for Comparison of Growth Curves. Biometrics, 14(1).
  • Reinsch C.H. (1967). Smoothing By Spline Functions. Numerische Mathematik, 10.
  • Schumaker L.L. (1993). Spline Functions: Basic Theory. Krieger Publishing Company: Florida
  • Silverman B.W. (1985). Some Aspects Of The Spline Smoothing Approach To Non-Parametric Regression Curve Fitting. Journal of the Royal Statistical Society: Series B., 47(1).
  • Silverman B. W. (1995). Incorporating Parametric Effects into Functional Principal Components Analysis Journal of the Royal Statistical Society: Series B.,57(4).
  • Silverman B.W.(1996). Smoothed Functional Principal Component Analysis By Choice Of Norm. The Annals of Statistics, 24(1)
  • Simonoff J.S. (1996). Smoothing Methods in Statistics, Springer – Verlag: New- York.
  • Ulbricht J.(2004). Representing Functional Data as Smooth Functions. A Master Thesis, Humboldt University Institute of Statistics and Econometrics, Berlin.
  • Wittaker E.T. (1923). On a New Method of Graduation, Proc. Edinburg Mathematics Society, 41.
  • Yamanishi Y., & Tanaka Y. (2005). Sensitivity Analysis in Functional Principal Component Analysis, Computational Statistics, 20(2).
  • Yamanishi Y., (2004). Statistical Case Studies: Biostatistics and Geostatistics, Erişim: 01.07.2005, http://www.quantlet.com/mdstat/scripts/xcs/pdf/xcspdf.pdf
  • Yao F., Lee T.C.M. (2006). Penalized Spline Models For Functional Principal Component Analysis, Journal of the Royal Statistical Society: Series B., 68(1).
  • Zhang, J. T. (1999). Smoothed Functional Data Analysis, Doctor of Philosophy, University of North Carolina, 1999.
Year 2008, Issue: 8, 1 - 32, 05.09.2011

Abstract

References

  • Barra V. (2004) Analysis of Gene Expression Data Using Functional Principal Components, Computer methods and programs in biomedicine, 75(11).
  • Benko M. (2004). Functional Principal Components analysis, Implementation and Applications. A Master Thesis. Humboldt University Center of Applied Statistics and Economics, Berlin.
  • Benko M., Hardle W., Kneip A.(2006). Common Functional Principal Components, SFB 649,DiscussionPaper,Erişim:10.11.2006, http://ideas.repec.org/p/hum/wpaper/sfb649dp2006-010.html
  • Besse P., Ramsay J.O. (1986). Principal Components Analysis Of Sampled Functions, Psychometrica, 51(2) Boor C . (1978). A Practical Guide to Splines. Springer-Verlag: New-York
  • Castro P. E, Lawton W. H., Sylvestre E. A. (1986). Principal Modes Of Variation for Processes with Continuous Sample Curves, Technometrics, 28(4).
  • Costanzo G.D. (2005). Functional Principal Component Analysis of Financial Time Series, Cnam-Paris.
  • Dauxois J., Pousse A., Romain Y. (1982). Asymptotic theory for the principal component analysis of a vector random function : some applications to statistical inference, J. Multivariate Analysis, 12
  • Dierckx P. (1993). Curve and Surface Fitting with Splines. Oxford University Press:New York
  • Eubank R.L. (1999). Nonparametric Regression and Spline Smoothing. Marcel Dekker: USA.
  • Green.P.J., & Silverman B.W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman & Hall:London.
  • Hall P. , Nasab H. M. (2006). On Properties Of Functional Principal Components Analysis. Journal of the Royal Statistical Society: Series B, 68(1).
  • Hyde V., Moore E., & Hodge A. (2006). Functional Pca For Exploring Bidding Acivity Times for Online Auctions. Erişim:06.07.2006,http://www.rhsmith.umd.edu/ceme/statistics/functionalpca.pdf
  • James G. M., Hastie T.J., Sugar C.A. (2000) Principal Components Models For Sparse Functional Data, Biometrica, 87(3).
  • Jank W., & Shmueli G. (2006). Functional Data Analysis in Electronic Commerce Research. Statistical Science, 21(2).
  • Jones M. C., Rice J. A. (1992). Displaying The Important Features Of Large Collections Of Similar Curves. The American Statistician, 46(2).
  • Laukaitis A., Rackauskas A. (2002). Functional Data Analysis of Payment Systems. Nonlinear Analysis: Modelling and Control, 7(2).
  • Lee H.J. (2004). Functional Data Analysis: Classification and Regression. Doctor of Philosophy, Texas A&M University.
  • Leurgans S.E., Moyeed R.A, Silverman B.W. (1993). Canonical Correlation Analysis when the Data are Curves. Journal of the Royal Statistical Society: Series B., 55(3).
  • Lillestol J., & Ollmar F. (2003). Functional Data Analysis: Introduction and Applications to Financial Electricity Contracts. Erişim: 01.03.2005, http://www.nhh.no/for/dp/2003/0603.pdf
  • Lober E.M., & Villa C. (2004). Functional Principal Component Analysis of the Yield Curve, Erişim: 05.05.2005 , http://www.u-cergy.fr/AFFI_2004/IMG/pdf/MATZNER.pdf
  • Lyche, T., Morken, K. (2002). Spline Methods Draft, Erişim: 10.07.2005, http://www.ifi.uio.no/in329/nchap1.pdf
  • Musayev B., & Alp M. (2000). Fonksiyonel Analiz. Balcı Yayınları: Kütahya
  • Müler, H. G. (2005). Functional Modelling and Classification of Longitudinal Data, Scandavian Journal of Statistics, 32(2).
  • Nürnberger G. (1989). Approximation by Spline Functions. Springer – Verlag: Berlin.
  • Ramsay, J. O., Dalzell C. (1991). Some Tools For Functonal Data Analysis, Journal of the Royal Statistical Society: Series B.,53 (3)
  • Ramsay J.O., Silverman B.W. (1997). Functional Data Analysis. Springer – Verlag: New York.
  • Ramsay J. O. , Li X. (1998). Curve Registration, Journal of the Royal Statistical Society: Series B, 60(2).
  • Ramsay J.O., (2000). Basis Functions, Erişim:11.04.2005, ftp://ego.psych.mcgill.ca
  • Ramsay J.O, Silverman B.W. (2002). Applied Functional Data Analysis: Methods and Case Studies. Springer – Verlag: New York.
  • Ramsay J.O., Silverman B.W. (2005). Functional Data Analysis. Second Edition. Springer : USA
  • Rao, C. R. (1958). Some Statistical Methods for Comparison of Growth Curves. Biometrics, 14(1).
  • Reinsch C.H. (1967). Smoothing By Spline Functions. Numerische Mathematik, 10.
  • Schumaker L.L. (1993). Spline Functions: Basic Theory. Krieger Publishing Company: Florida
  • Silverman B.W. (1985). Some Aspects Of The Spline Smoothing Approach To Non-Parametric Regression Curve Fitting. Journal of the Royal Statistical Society: Series B., 47(1).
  • Silverman B. W. (1995). Incorporating Parametric Effects into Functional Principal Components Analysis Journal of the Royal Statistical Society: Series B.,57(4).
  • Silverman B.W.(1996). Smoothed Functional Principal Component Analysis By Choice Of Norm. The Annals of Statistics, 24(1)
  • Simonoff J.S. (1996). Smoothing Methods in Statistics, Springer – Verlag: New- York.
  • Ulbricht J.(2004). Representing Functional Data as Smooth Functions. A Master Thesis, Humboldt University Institute of Statistics and Econometrics, Berlin.
  • Wittaker E.T. (1923). On a New Method of Graduation, Proc. Edinburg Mathematics Society, 41.
  • Yamanishi Y., & Tanaka Y. (2005). Sensitivity Analysis in Functional Principal Component Analysis, Computational Statistics, 20(2).
  • Yamanishi Y., (2004). Statistical Case Studies: Biostatistics and Geostatistics, Erişim: 01.07.2005, http://www.quantlet.com/mdstat/scripts/xcs/pdf/xcspdf.pdf
  • Yao F., Lee T.C.M. (2006). Penalized Spline Models For Functional Principal Component Analysis, Journal of the Royal Statistical Society: Series B., 68(1).
  • Zhang, J. T. (1999). Smoothed Functional Data Analysis, Doctor of Philosophy, University of North Carolina, 1999.
There are 43 citations in total.

Details

Primary Language Turkish
Journal Section Makaleler
Authors

Doç. Dr. Kadir Ertaş This is me

Araş. Gör.dr. İstem Köymen Keser This is me

Publication Date September 5, 2011
Published in Issue Year 2008 Issue: 8

Cite

APA Ertaş, D. D. K., & Keser, A. G. İ. K. (2011). DÜZGÜNLEŞTİRİLMİŞ FONKSİYONEL ANA BİLEŞENLER ANALİZİ İLE İMKB VERİLERİNİN İNCELENMESİ. Istanbul University Econometrics and Statistics E-Journal(8), 1-32.
AMA Ertaş DDK, Keser AGİK. DÜZGÜNLEŞTİRİLMİŞ FONKSİYONEL ANA BİLEŞENLER ANALİZİ İLE İMKB VERİLERİNİN İNCELENMESİ. Istanbul University Econometrics and Statistics e-Journal. September 2011;(8):1-32.
Chicago Ertaş, Doç. Dr. Kadir, and Araş. Gör.dr. İstem Köymen Keser. “DÜZGÜNLEŞTİRİLMİŞ FONKSİYONEL ANA BİLEŞENLER ANALİZİ İLE İMKB VERİLERİNİN İNCELENMESİ”. Istanbul University Econometrics and Statistics E-Journal, no. 8 (September 2011): 1-32.
EndNote Ertaş DDK, Keser AGİK (September 1, 2011) DÜZGÜNLEŞTİRİLMİŞ FONKSİYONEL ANA BİLEŞENLER ANALİZİ İLE İMKB VERİLERİNİN İNCELENMESİ. Istanbul University Econometrics and Statistics e-Journal 8 1–32.
IEEE D. D. K. Ertaş and A. G. İ. K. Keser, “DÜZGÜNLEŞTİRİLMİŞ FONKSİYONEL ANA BİLEŞENLER ANALİZİ İLE İMKB VERİLERİNİN İNCELENMESİ”, Istanbul University Econometrics and Statistics e-Journal, no. 8, pp. 1–32, September 2011.
ISNAD Ertaş, Doç. Dr. Kadir - Keser, Araş. Gör.dr. İstem Köymen. “DÜZGÜNLEŞTİRİLMİŞ FONKSİYONEL ANA BİLEŞENLER ANALİZİ İLE İMKB VERİLERİNİN İNCELENMESİ”. Istanbul University Econometrics and Statistics e-Journal 8 (September 2011), 1-32.
JAMA Ertaş DDK, Keser AGİK. DÜZGÜNLEŞTİRİLMİŞ FONKSİYONEL ANA BİLEŞENLER ANALİZİ İLE İMKB VERİLERİNİN İNCELENMESİ. Istanbul University Econometrics and Statistics e-Journal. 2011;:1–32.
MLA Ertaş, Doç. Dr. Kadir and Araş. Gör.dr. İstem Köymen Keser. “DÜZGÜNLEŞTİRİLMİŞ FONKSİYONEL ANA BİLEŞENLER ANALİZİ İLE İMKB VERİLERİNİN İNCELENMESİ”. Istanbul University Econometrics and Statistics E-Journal, no. 8, 2011, pp. 1-32.
Vancouver Ertaş DDK, Keser AGİK. DÜZGÜNLEŞTİRİLMİŞ FONKSİYONEL ANA BİLEŞENLER ANALİZİ İLE İMKB VERİLERİNİN İNCELENMESİ. Istanbul University Econometrics and Statistics e-Journal. 2011(8):1-32.