• ISSN: 2148-2225 (online)

Ulaştırma ve Lojistik Kongreleri

alphanumeric journal

The Journal of Operations Research, Statistics, Econometrics and Management Information Systems

A New Descriptive Statistic for Functional Data: Functional Coefficient of Variation


İstem Köymen Keser, Ph.D.

İpek Deveci Kocakoç, Ph.D.

Ali Kemal Şehirlioğlu, Ph.D.


Abstract

In this study, we propose a new descriptive statistic, coefficient of variation function, for functional data analysis and present its utilization. We recommend coefficient of variation function, especially when we want to compare the variation of multiple curve groups and when the mean functions are different for each curve group. Besides, obtaining coefficient of variation functions in terms of cubic B-Splines enables the interpretation of the first and second derivative functions of these functions and provides a stronger inference for the original curves. The utilization and effects of the proposed statistic is reported on a well-known data set from the literature. The results show that the proposed statistic reflects the variability of the data properly and this reflection gets clearer than that of the standard deviation function especially as mean functions differ.

Keywords: Coefficient of Variation Function, Descriptive Statistics, Functional Data Analysis

Jel Classification: C46

Fonksiyonel Veri İçin Yeni Bir Tanımlayıcı İstatistik: Fonksiyonel Değişkenlik Katsayısı


Öz

Bu çalışmada fonksiyonel veriler için yeni bir tanımlayıcı istatistik olan değişkenlik katsayısı fonksiyonunu önermekteyiz. Özellikle her bir eğri grubunun ortalama fonksiyonları farklı olduğu durumda, çoklu eğri gruplarının değişkenliklerini karşılaştırmada önerilen değişkenlik katsayısı fonksiyonunun kullanılmasını tavsiye ediyoruz. Değişkenlik katsayısı fonksiyonunu elde ederken kübik B-Splaynlar kullanıldığından dolayı, bu fonksiyonların birinci ve ikinci türev fonksiyonlarının da yorumlanabiliyor olması, orijinal eğriler hakkında daha güçlü çıkarımlar yapabilmemizi sağlamaktadır. Önerilen istatistiğin kullanımı ve etkileri literatürdeki iyi bilinen bir veri seti üzerinde raporlanmıştır. Sonuçlar göstermektedir ki önerilen istatistik verinin değişkenliğini düzgün bir şekilde yansıtmaktadır ve bu durum ortalama fonksiyonları farklılaştıkça standart sapma fonksiyonundan daha üstün bir hale gelmektedir.

Anahtar Kelimeler: Değişkenlik Katsayısı Fonksiyonu, Fonksiyonel Veri Analizi, Tanımlayıcı İstatistik


Cite this article

Köymen Keser, İ., Deveci Kocakoç, İ., Şehirlioğlu, AK. (2016). A New Descriptive Statistic for Functional Data: Functional Coefficient of Variation. Alphanumeric Journal, 4(2), 1-10. http://dx.doi.org/10.17093/aj.2016.4.2.5000185408

References

  • Clarkson, D.B., Fraley, C., Gu, C., Ramsay, J.(2005). S+Functional Data Analysis User's Manual for Windows ®, Springer-Verlag, New-York.
  • Coffey, N., Hinde, J.(2011). Analyzing time-course microarray data using functional data analysis-a review. Statistical Applications in Genetics and Molecular Biology, 10(1),1-32.
  • Cox, D.D., and Lee, J.S. (2008). Pointwise testing with functional data using the Westfall-Young randomization method, Biometrika, 95(3), 621-634.
  • Keser, I.K. (2014). “Comparing two mean humidity curves using functional t-tests: Turkey case”, Electronic Journal of Applied Statistical Analysis, 7(2), 254-278.
  • Keser, I.K, Deveci Kocakoç, I. (2015), FDAPackage, software. Available at http://people.deu.edu.tr/istem.koymen/fda.html.
  • Lee, J.S. (2005). Aspects of Functional Data Inference and Its Applications. Doctor of Philosophy, Houston, Texas.
  • Levitin, D.J., Nuzzo, R.L., Vines Bradley W., Ramsay J.O., (2007). Introduction to Functional Data Analysis, Canadian Psychology, 48(3), 135-155.
  • Ramsay, J.O. (1982). When the data are functions, Psychometrika 47, 379-396.
  • Ramsay, J.O. (2015). FDAPackage, software. Available at: http://www.psych.mcgill.ca/misc/fda/downloads/FDAfuns/Matlab/.
  • Ramsay, J.O., Hooker, G., Graves, S. (2009). Functional Data Analysis with R and MATLAB, Springer-Verlag, New-York.
  • Ramsay J.O, Silverman B.W. (1997). Functional Data Analysis, Springer-Verlag, New-York.
  • Ramsay, J.O, Silverman, B.W. (2005). Functional Data Analysis, Second Edition, Springer-Verlag, New-York.
  • Rice, J. A., Silverman, B.W. (1991). Estimating the Mean and Covariance Structure When the Data are Curves, Journal of the Royal Statistical Society. Series B. 53(1), 233-243.
  • Shang, H.L. (2015). Resampling Techniques for Estimating the Distribution of Descriptive Statistics of Functional Data, Communications in Statistics - Simulation and Computation, 44:3, 614-635.
  • Sun, Y., Genton, M.G. (2011) Functional Boxplots, Journal of Computational and Graphical Statistics, 20:2, 316-334, DOI: 10.1198/jcgs.2011.09224
  • Tuddenham, R., Snyder, M. (1954). Physical growth of California boys and girls from birth to age 18. California Publications on Child Development, 1, 183-364.
  • Ullah, S., Finch, C. F. (2013). Applications of functional data analysis: a systematic review, BMC Medical Research Methodology, 13(43), 539–572.
  • Yaree, K. (2011). Functional data analysis with application to ms and cervical vertebrae data. Master of Science in Statistics, Edmonton, Alberta.
  • Zhang, J-T. (2013). Analysis of Variance for Functional Data, CRC Press.

Volume 4, Issue 2, 2016

2016.04.02.ECON.01

alphanumeric journal

Volume 4, Issue 2, 2016

Pages 1-10

Received: April 13, 2016

Accepted: Aug. 11, 2016

Published: Sept. 26, 2016

Full Text [570.4 KB]

  • Share

2016 Köymen Keser, İ., Deveci Kocakoç, İ., Şehirlioğlu, AK.

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.


Contact Us

School of Transportation and Logistics, Istanbul University
Avcilar Campus 34320 Avcilar/Istanbul/TURKEY

editor@alphanumericjournal.com
+ 90 (212) 473 70 00 - 19263

alphanumeric journal

alphanumeric journal has been publishing as "International Peer-Reviewed Journal" every six months since 2013. alphanumeric serves as a vehicle for researchers and practitioners in the field of quantitative methods, and is enabling a process of sharing in all fields related to the operations research, statistics, econometrics and management informations systems in order to enhance the quality on a globe scale.