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

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, Horvath 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 SNbased 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.

[1]  X. Shao,et al.  A self‐normalized approach to confidence interval construction in time series , 2010, 1005.2137.

[2]  D. Andrews.  Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation , 1991 .

[3]  Pierre Perron,et al.  Dealing with Structural Breaks , 2005 .

[4]  Zhijie Xiao,et al.  Tests for changing mean with monotonic power , 2009 .

[5]  D. Parker,et al.  A new daily central England temperature series, 1772–1991 , 1992 .

[6]  Piotr Kokoszka,et al.  Testing the stability of the functional autoregressive process , 2010, J. Multivar. Anal..

[7]  X. Shao,et al.  Testing for Change Points in Time Series , 2010 .

[8]  C. Folland,et al.  A NEW DAILY CENTRAL ENGLAND TEMPERATURE SERIES , 1992 .

[9]  Piotr Kokoszka,et al.  Portmanteau Test of Independence for Functional Observations , 2007 .

[10]  P. Jones,et al.  Uncertainty estimates in regional and global observed temperature changes: A new data set from 1850 , 2006 .

[11]  Piotr Kokoszka,et al.  Detecting changes in the mean of functional observations , 2009 .

[12]  L. Horváth,et al.  Limit Theorems in Change-Point Analysis , 1997 .

[13]  James O. Ramsay,et al.  Applied Functional Data Analysis: Methods and Case Studies , 2002 .

[14]  G. Wahba.  Spline models for observational data , 1990 .

[15]  C. Crainiceanu,et al.  Nonmonotonic power for tests of a mean shift in a time series , 2001 .

[16]  Filippo Altissimo,et al.  Strong Rules for Detecting the Number of Breaks in a Time Series , 2003 .

[17]  S. Klein,et al.  GFDL's CM2 Global Coupled Climate Models. Part I: Formulation and Simulation Characteristics , 2006 .

[18]  Frédéric Ferraty,et al.  Nonparametric Functional Data Analysis: Theory and Practice (Springer Series in Statistics) , 2006 .

[19]  T. Vogelsang.  Sources of nonmonotonic power when testing for a shift in mean of a dynamic time series , 1999 .

[20]  P. Kokoszka,et al.  Weakly dependent functional data , 2010, 1010.0792.

[21]  Hans-Georg Müller,et al.  Functional Data Analysis , 2016 .

[22]  Ignacio N. Lobato.  Testing That a Dependent Process Is Uncorrelated , 2001 .

[23]  J. Aston,et al.  Detecting and estimating epidemic changes in dependent functional data , 2011 .