Electronic Journal of Statistics

Testing the structural stability of temporally dependent functional observations and application to climate projections

Xianyang Zhang, Xiaofeng Shao, Katharine Hayhoe, and Donald J. Wuebbles

Full-text: Open access

Abstract

We develop a self-normalization (SN) based test to test the structural stability of temporally dependent functional observations. Testing for a change point in the mean of functional data has been studied in Berkes, Gabrys, Horváth and Kokoszka [4], but their test was developed under the independence assumption. In many applications, functional observations are expected to be dependent, especially when the data is collected over time. Building on the SN-based change point test proposed in Shao and Zhang [23] for a univariate time series, we extend the SN-based test to the functional setup by testing the constant mean of the finite dimensional eigenvectors after performing functional principal component analysis. Asymptotic theories are derived under both the null and local alternatives. Through theory and extensive simulations, our SN-based test statistic proposed in the functional setting is shown to inherit some useful properties in the univariate setup: the test is asymptotically distribution free and its power is monotonic. Furthermore, we extend the SN-based test to identify potential change points in the dependence structure of functional observations. The method is then applied to central England temperature series to detect the warming trend and to gridded temperature fields generated by global climate models to test for changes in spatial bias structure over time.

Article information

Source
Electron. J. Statist., Volume 5 (2011), 1765-1796.

Dates
First available in Project Euclid: 13 December 2011

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1323785608

Digital Object Identifier
doi:10.1214/11-EJS655

Mathematical Reviews number (MathSciNet)
MR2870150

Zentralblatt MATH identifier
1271.62097

Subjects
Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

Keywords
Change point CUSUM functional data time series self-normalization

Citation

Zhang, Xianyang; Shao, Xiaofeng; Hayhoe, Katharine; Wuebbles, Donald J. Testing the structural stability of temporally dependent functional observations and application to climate projections. Electron. J. Statist. 5 (2011), 1765--1796. doi:10.1214/11-EJS655. https://projecteuclid.org/euclid.ejs/1323785608


Export citation

References

  • [1] Altissimo, F. and Corradi, V. (2003). Strong rules for detecting the number of breaks in a time series, Journal of Econometrics, 117, 207-244.
  • [2] Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation., Econometrica, 59, 817-858.
  • [3] Aston, J. A. D. and Kirch, C. (2011). Detecting and estimating epidemic changes in dependent functional data., CRiSM Research Report 11-07, University of Warwick.
  • [4] Berkes, I., Gabrys, R., Horváth, L. and Kokoszka, P. (2009). Detecting changes in the mean of functional observations., Journal of Royal Statistical Society, Series B, Methodology, 71, 927-946.
  • [5] Billingsley, P. (1999)., Convergence of Probability Measures; Second Edition. New York: Wiley.
  • [6] Bosq, D. (2000)., Linear Process in Function Spaces: Theory and Applications. New York: Springer.
  • [7] Brohan, P., Kennedy, J. J., Harris, I., Tett, S. F. B. and Jones, P. D. (2006). Uncertainty estimates in regional and global observed temperature changes: A new data set from 1850., Journal of Geographical Research Atmosphere, 111, D12106.
  • [8] Crainiceanu, C. M. and Vogelsang, T. J. (2007). Spectral density bandwidth choice: source of nonmonotonic power for tests of a mean shift in a time series., Journal of Statistical Computation and Simulation, 77, 457-476.
  • [9] Csörgő, M. and Horváth, L. (1997)., Limit Theorems in Change-Point Analysis. New York: Wiley.
  • [10] Delworth, T. L., Broccoli, A. J., Rosati, A., Stouffer, R. J., Balaji, V., Beesley, J. A., Cooke, W. F., Dixon, K. W. et al. (2006). GFDL’s CM2 global coupled climate models—Part 1: Formulation and simulation characteristics., Journal of Climate, 19, 643-674.
  • [11] Ferraty, F. and Vieu, P. (2006)., Nonparametric Functional Data analysis. New York: Springer.
  • [12] Gabrys, R. and Kokoszka, R. (2007). Portmanteau test of independence for functional observations., Journal of the American Statistical Association, 102, 1338-1348.
  • [13] Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data., The Annals of Statistics, 38, 1845-1884.
  • [14] Horváth, L., Hušková, M. and Kokoszka, P. (2010). Testing the stability of functional autoregressive process., Journal of Multivariate Analysis, 101, 352-367.
  • [15] Juhl, T. and Xiao, Z. (2009). Testing for changing mean monotonic power., Journal of Econometrics, 148, 14-24.
  • [16] Lobato, I. N. (2001). Testing that a dependent process is uncorrelated., Journal of the American Statistical Association, 96, 1066-1076.
  • [17] Parker, D. E., Legg, T. P. and Folland, C. K. (1992). A new daily central England temperature series, 1772-1991., International Journal of Climatology, 12, 317-342.
  • [18] Perron, P. (2006). Dealing with structural breaks. in, Palgrave Handbook of Econometrics, Vol. 1: Econometric Theory, eds. K. Patterson and T. C. Mills, Palgrave Macmillan, pp. 278-352.
  • [19] Ramsay, J. and Silverman, B. (2002)., Applied Functional Data Analysis: Methods and Case Studies. New York: Springer.
  • [20] Ramsay, J. and Silverman, B. (2005)., Functional Data Analysis. New York: Springer.
  • [21] Riesz, F. and Sz-Nagy, B. (1955)., Functional Analysis. New York: Ungar.
  • [22] Shao, X. (2010). A self-normalized approach to confidence interval construction in time series., Journal of the Royal Statistical Society, Series, B, 72, 343-366.
  • [23] Shao, X. and Zhang, X. (2010). Testing for change points in time series., Journal of the American Statistical Association, 105, 1228-1240.
  • [24] Vogelsang, T. J. (1999). Sources of nonmonotonic power when testing for a shift in mean of a dynamic time series., Journal of Econometrics, 88, 283-299.
  • [25] Wahba, G. (1990). Spline models for observational data., CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 59, SIAM.