A randomness test for functional panels

Functional panels are collections of functional time series, and arise often in the study of high frequency multivariate data. We develop a portmanteau style test to determine if the cross-sections of such a panel are independent and identically distributed. Our framework allows the number of functional projections and/or the number of time series to grow with the sample size. A large sample justification is based on a new central limit theorem for random vectors of increasing dimension. With a proper normalization, the limit is standard normal, potentially making this result easily applicable in other FDA context in which projections on a subspace of increasing dimension are used. The test is shown to have correct size and excellent power using simulated panels whose random structure mimics the realistic dependence encountered in real panel data. It is expected to find application in climatology, finance, ecology, economics, and geophysics. We apply it to Southern Pacific sea surface temperature data, precipitation patterns in the South-West United States, and temperature curves in Germany.

[1]  Debashis Paul,et al.  CONSISTENCY OF RESTRICTED MAXIMUM LIKELIHOOD ESTIMATORS OF PRINCIPAL COMPONENTS , 2008, 0805.0465.

[2]  Dag Tjøstheim,et al.  Modelling panels of intercorrelated autoregressive time series , 1999 .

[3]  J. Aston,et al.  Evaluating stationarity via change-point alternatives with applications to fMRI data , 2012, 1301.2894.

[4]  Denis Bosq,et al.  Linear Processes in Function Spaces , 2000 .

[5]  P. Hall,et al.  Properties of principal component methods for functional and longitudinal data analysis , 2006, math/0608022.

[6]  M. Jirak.  Optimal eigen expansions and uniform bounds , 2015, 1501.01271.

[7]  Dominik Liebl,et al.  Modeling and forecasting electricity spot prices: A functional data perspective , 2013, 1310.1628.

[8]  Cheng Hsiao,et al.  Analysis of Panel Data , 1987 .

[9]  Wolfgang Härdle,et al.  Yield curve modeling and forecasting using semiparametric factor dynamics , 2012 .

[10]  André Mas,et al.  Testing hypotheses in the functional linear model , 2003 .

[11]  B. Li,et al.  Nonlinear sufficient dimension reduction for functional data , 2017 .

[12]  E. S. Pearson,et al.  THE USE OF CONFIDENCE OR FIDUCIAL LIMITS ILLUSTRATED IN THE CASE OF THE BINOMIAL , 1934 .

[13]  G. Nkiet,et al.  Testing for lack of dependence between functional variables , 2010 .

[14]  D. Ruppert,et al.  Multilevel Bayesian framework for modeling the production, propagation and detection of ultra-high energy cosmic rays , 2012, 1206.4569.

[15]  Hanxiang Peng,et al.  Asymptotic normality of quadratic forms with random vectors of increasing dimension , 2018, J. Multivar. Anal..

[16]  Kevin E. Trenberth,et al.  Indices of El Niño Evolution , 2001 .

[17]  Richard A. Davis,et al.  Time Series: Theory and Methods , 2013 .

[18]  T. Tony Cai,et al.  Prediction in functional linear regression , 2006 .

[19]  Katharine Hayhoe,et al.  Testing the structural stability of temporally dependent functional observations and application to climate projections , 2011 .

[20]  A. Schick,et al.  Empirical likelihood approach to goodness of fit testing , 2013, 1307.5982.

[21]  W. Fung,et al.  Testing model adequacy for dynamic panel data with intercorrelation , 2002 .

[22]  Bruce D. McCullough,et al.  Diagnostic Checks in Time Series , 2005, Technometrics.

[23]  Richard A. Davis,et al.  Introduction to time series and forecasting , 1998 .

[24]  Piotr Kokoszka,et al.  Functional data analysis with increasing number of projections , 2013, J. Multivar. Anal..

[25]  Wolfgang Härdle,et al.  Spatial Risk Premium on Weather Derivatives and Hedging Weather Exposure in Electricity , 2011 .

[26]  T. W. Anderson.  ASYMPTOTIC THEORY FOR PRINCIPAL COMPONENT ANALYSIS , 1963 .

[27]  Marie Husková,et al.  Test of independence for functional data , 2013, J. Multivar. Anal..

[28]  Piotr Kokoszka,et al.  Asymptotic normality of the principal components of functional time series , 2013 .

[29]  Alexander Graham,et al.  Kronecker Products and Matrix Calculus: With Applications , 1981 .

[30]  Leslie Godfrey,et al.  Testing the adequacy of a time series model , 1979 .

[31]  Ratnam V. Chitturi.  Distribution of Multivariate White Noise Autocorrelations , 1976 .

[32]  Victor M. Panaretos,et al.  Cramér–Karhunen–Loève representation and harmonic principal component analysis of functional time series , 2013 .

[33]  A. I. McLeod,et al.  On the Distribution of Residual Autocorrelations in Box–Jenkins Models , 1978 .

[34]  Piotr Kokoszka,et al.  Inference for Functional Data with Applications , 2012 .

[35]  M. Hallin,et al.  Dynamic functional principal components , 2015 .

[36]  Helmut Ltkepohl,et al.  New Introduction to Multiple Time Series Analysis , 2007 .

[37]  Vladimir V. Senatov.  Normal Approximation: New Results, Methods and Problems , 1998 .

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

[39]  D. Bosq.  Linear Processes in Function Spaces: Theory And Applications , 2000 .

[40]  B. M. Brown,et al.  Martingale Central Limit Theorems , 1971 .

[41]  Julio Rodríguez,et al.  A Powerful Portmanteau Test of Lack of Fit for Time Series , 2002 .

[42]  Marie Frei,et al.  Functional Data Analysis With R And Matlab , 2016 .

[43]  G. Box,et al.  On a measure of lack of fit in time series models , 1978 .

[44]  J. Dauxois,et al.  Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference , 1982 .

[45]  Nils Lid Hjort,et al.  Extending the Scope of Empirical Likelihood , 2009, 0904.2949.

[46]  Kevin E. Trenberth,et al.  The Definition of El Niño. , 1997 .

[47]  G. Box,et al.  Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models , 1970 .

[48]  David S. Matteson,et al.  A Bayesian Multivariate Functional Dynamic Linear Model , 2014, 1411.0764.

[49]  Thomas J. Fisher,et al.  New Weighted Portmanteau Statistics for Time Series Goodness of Fit Testing , 2012 .

[50]  J. R. M. Hosking,et al.  The Multivariate Portmanteau Statistic , 1980 .