alphanumeric journal
The Journal of Operations Research, Statistics, Econometrics and Management Information Systems
A New Descriptive Statistic for Functional Data: Functional Coefficient of Variation
bib
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
Suggested citation
(). 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.
2016.04.02.ECON.01
alphanumeric journal
Pages 1-10
Received: April 13, 2016
Accepted: Aug. 11, 2016
Published: Sept. 26, 2016
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.
scan QR code to access this article from your mobile device
Contact Us
Faculty of Transportation and Logistics, Istanbul University
Beyazit Campus 34452 Fatih/Istanbul/TURKEY
Bahadır Fatih Yıldırım, Ph.D.
editor@alphanumericjournal.com
+ 90 (212) 440 00 00 - 13219
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.