The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Application of functional ANOVA and functional MANOVA

단변량 및 다변량 함수 데이터에 대한 분산분석의 활용

  • Kim, Mijeong (Department of Statistics, Ewha Womans University)
  • Received : 2022.07.19
  • Accepted : 2022.08.25
  • Published : 2022.10.31
Download PDF
⟨ Previous Next ⟩

Abstract

Functional data is collected in various fields. It is often necessary to test whether there are differences among groups of functional data. In this case, it is not appropriate to explain using the point-wise ANOVA method, and we should present not the point-wise result but the integrated result. Various studies on functional data analysis of variance have been proposed, and recently implemented those methods in the package fdANOVA of R. In this paper, I first explain ANOVA and multivariate ANOVA, then I will introduce various methods of analysis of variance for univariate and multivariate functional data recently proposed. I also describe how to use the R package fdANOVA. This package is used to test equality of weekly temperatures in Seoul and Busan through univariate functional data ANOVA, and to test equality of multivariate functional data corresponding to handwritten images using multivariate function data ANOVA.

함수 데이터는 다양한 분야에서 수집되고 있으며, 집단 간의 함수 데이터를 비교해야하는 경우가 종종 발생한다. 이럴 경우 점별 분산분석 방법을 이용하여 설명하기에는 무리가 있으며, 통합된 결과를 제시할 필요가 있다. 이에 대한 다양한 연구가 제안되었으며, 최근에 R 패키지 fdANOVA로 구현되었다. 이 논문에서 우선 분산분석 및 다변량 분산분석을 설명하고, 최근에 제안된 다양한 단변량 및 다변량 함수 데이터 분산분석을 설명하고자 한다. 또한 R 패키지 fdANOVA의 사용 방법을 설명하고, 이 패키지를 이용하여 서울과 부산 지역의 주별 기온을 단변량 함수 데이터 분산분석을 통해 비교하고, 손글씨 이미지를 다변량 함수 데이터로 변환하여 다변량 함수 데이터 분산분석을 이용하여 비교하고자 한다.

Keywords

Acknowledgement

이 논문은 연구재단 연구 과제 (NRF-2020R1F1A1A01074157)에 의하여 수행되었음.

References

  1. Cuevas A, Febrero M, and Fraiman R (2004). An anova test for functional data, Computational statistics & Data Analysis, 47, 111-122. https://doi.org/10.1016/j.csda.2003.10.021
  2. Cuesta-Albertos JA and Febrero-Bande M (2010). A simple multiway ANOVA for functional data, Test:Journal of the Spanish Society of Statistics and Operations Research, 19, 537-557.
  3. Don HSRA (2018). A relationship between the One-Way MANOVA test statistic and the hotelling lawley trace test statistic, International Journal of Statistics and Probability, 7, 124-131. https://doi.org/10.5539/ijsp.v7n6p124
  4. Faraway JJ (1997). Regression analysis for a functional response, Technometrics, 39, 254-261. https://doi.org/10.1080/00401706.1997.10485118
  5. Friedrich S and Pauly M (2018). MATS: Inference for potentially singular and heteroscedastic MANOVA, Journal of Multivariate Analysis, 165, 166-179. https://doi.org/10.1016/j.jmva.2017.12.008
  6. Gorecki T and Smaga L (2015). A comparison of tests for the one-way ANOVA problem for functional data, Computational Statistics, 30, 987-1010. https://doi.org/10.1007/s00180-015-0555-0
  7. Gorecki T and Smaga L (2017). Multivariate analysis of variance for functional data, Journal of Applied Statistics, 44, 2172-2189. https://doi.org/10.1080/02664763.2016.1247791
  8. Gorecki T and Smaga L (2019). fdANOVA: an R software package for analysis of variance for univariate and multivariate functional data, Computational Statistics, 34, 571-597. https://doi.org/10.1007/s00180-018-0842-7
  9. Olive DJ (2017). Robust multivariate analysis, Springer International Publishing.
  10. Ramsay JO and Silverman BW (2005). Functional Data Analysis 2nd Edition, Springer-Verlag, New York.
  11. Shen Q and Faraway J (2004). An F test for linear models with functional responses, Statistica Sinica,14, 1239-1257.
  12. Srivastava MS and Kubokawa T (2013). Tests for multivariate analysis of variance in high dimension under non-normality, Journal of Multivariate Analysis, 115, 204-216. https://doi.org/10.1016/j.jmva.2012.10.011
  13. Zhang JT (2011). Statistical inferences for linear models with functional responses, Statistica Sinica, 21, 1431-1451. https://doi.org/10.5705/ss.2009.302
  14. Zhang JT (2014). Analysis of variance for functional data, Monographs on Statistics and Applied Probability, 127, 127.
  15. Zhang JT and Chen J (2007). Statistical Inferences for functional data, The Annals of Statistics, 35, 1052-1079.
  16. Zhang JT and Liang X (2014). One-way ANOVA for functional data via globalizing the pointwise F-test, Scan-dinavian Journal of Statistics, 41, 51-71. https://doi.org/10.1111/sjos.12025
  17. Zhang JT, Cheng MY, Wu HT, and Zhou B (2019). A new test for functional one-way ANOVA with applications to ischemic heart screening, Computational Statistics & Data Analysis, 132, 3-17. https://doi.org/10.1016/j.csda.2018.05.004