×

Testing the structural stability of temporally dependent functional observations and application to climate projections. (English) Zbl 1271.62097

Summary: 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 [I. Berkes, “Detecting changes in the mean of functional observations”, J. R. Stat. Soc., Ser. B, Stat. Methodol. 71, No. 5, 927–946 (2009; doi:10.1111/j.1467-9868.2009.00713.x)], 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 [the first and second author, “Testing for change points in time series”, J. Am. Stat. Assoc. 105, No. 491, 1228–1240 (2010)] 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.

MSC:

62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference

Software:

fda (R)
PDFBibTeX XMLCite
Full Text: DOI Euclid

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. · Zbl 1030.62060 · doi:10.1016/S0304-4076(03)00147-7
[2] Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation., Econometrica , 59 , 817-858. · Zbl 0732.62052 · doi:10.2307/2938229
[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. · Zbl 1241.62121
[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. · doi:10.1111/j.1467-9868.2009.00713.x
[5] Billingsley, P. (1999)., Convergence of Probability Measures; Second Edition . New York: Wiley. · Zbl 0944.60003
[6] Bosq, D. (2000)., Linear Process in Function Spaces: Theory and Applications. New York: Springer. · Zbl 0962.60004 · doi:10.1007/978-1-4612-1154-9
[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. · Zbl 1123.62063 · doi:10.1080/10629360600569394
[9] Csörgő, M. and Horváth, L. (1997)., Limit Theorems in Change-Point Analysis . New York: Wiley. · Zbl 0884.62023
[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. · Zbl 1119.62046 · doi:10.1007/0-387-36620-2
[12] Gabrys, R. and Kokoszka, R. (2007). Portmanteau test of independence for functional observations., Journal of the American Statistical Association , 102 , 1338-1348. · Zbl 1332.62322 · doi:10.1198/016214507000001111
[13] Hörmann, S. and Kokoszka, P. (2010). Weakly dependent functional data., The Annals of Statistics , 38 , 1845-1884. · Zbl 1189.62141 · doi:10.1214/09-AOS768
[14] Horváth, L., Hušková, M. and Kokoszka, P. (2010). Testing the stability of functional autoregressive process., Journal of Multivariate Analysis , 101 , 352-367. · Zbl 1178.62099 · doi:10.1016/j.jmva.2008.12.008
[15] Juhl, T. and Xiao, Z. (2009). Testing for changing mean monotonic power., Journal of Econometrics , 148 , 14-24. · Zbl 1429.62374 · doi:10.1016/j.jeconom.2008.08.020
[16] Lobato, I. N. (2001). Testing that a dependent process is uncorrelated., Journal of the American Statistical Association , 96 , 1066-1076. · Zbl 1072.62576 · doi:10.1198/016214501753208726
[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. · Zbl 1011.62002 · doi:10.1007/b98886
[20] Ramsay, J. and Silverman, B. (2005)., Functional Data Analysis . New York: Springer. · Zbl 1079.62006
[21] Riesz, F. and Sz-Nagy, B. (1955)., Functional Analysis . New York: Ungar. · Zbl 0070.10902
[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. · doi:10.1111/j.1467-9868.2009.00737.x
[23] Shao, X. and Zhang, X. (2010). Testing for change points in time series., Journal of the American Statistical Association , 105 , 1228-1240. · Zbl 1390.62184 · doi:10.1198/jasa.2010.tm10103
[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. · Zbl 0933.62092
[25] Wahba, G. (1990). Spline models for observational data., CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 59 , SIAM. · Zbl 0813.62001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.