Space‐Time Modelling of Trends in Temperature Series

Classical assessments of temperature trends are based on the analysis of a small number of time series. Considering trend to be only smooth changes of the mean value of a stochastic process through time is limiting, because it does not provide a mechanism to study changes of the mean that could also occur over space. Thus, in studies of climate there is a substantial interest in being able to jointly characterize temperature trends over time and space. In this article we build wavelet-based space-time hierarchical Bayesian models that can be used to simultaneously model trend, seasonality, and error, allowing for the possibility that the error process may exhibit space-time long-range dependence. We demonstrate how these statistical models can be used to assess the significance of trend over time and space. We motivate and apply our methods to the analysis of space-time temperature trends, based on data collected in the last five decades from central Sweden.

[1]  J. Hansen,et al.  Global temperature change , 2006, Proceedings of the National Academy of Sciences.

[2]  Timothy A. Cohn,et al.  Nature's style: Naturally trendy , 2005 .

[3]  Richard F. Gunst,et al.  Identification of model components for a class of continuous spatiotemporal models , 2003 .

[4]  Michael R. Chernick,et al.  Wavelet Methods for Time Series Analysis , 2001, Technometrics.

[5]  Richard A. Davis,et al.  A class of stochastic volatility models for environmental applications , 2011 .

[6]  Julia Brodsky,et al.  Broadband Semiparametric Estimation of the Memory Parameter of a Long‐Memory Time Series Using Fractional Exponential Models , 1998 .

[7]  Jeffrey S. Morris,et al.  Wavelet-Based Functional Mixed Models to characterize Population Heterogeneity in Accelerometer Profiles: A Case Study. , 2006 .

[8]  P. Guttorp,et al.  Geostatistical Space-Time Models, Stationarity, Separability, and Full Symmetry , 2007 .

[9]  P. Guttorp,et al.  Trend assessment in a long memory dependence model using the discrete wavelet transform , 2004 .

[10]  Jeffrey S. Morris,et al.  Journal of the American Statistical Association Using Wavelet-based Functional Mixed Models to Characterize Population Heterogeneity in Accelerometer Profiles Using Wavelet-based Functional Mixed Models to Characterize Population Heterogeneity in Accelerometer Profiles: a Case Study , 2022 .

[11]  Nouna Kettaneh,et al.  Statistical Modeling by Wavelets , 1999, Technometrics.

[12]  Liudas Giraitis,et al.  A model for long memory conditional heteroscedasticity , 2000 .

[13]  P. Jones,et al.  Representing Twentieth-Century Space-Time Climate Variability. Part II: Development of 1901-96 Monthly Grids of Terrestrial Surface Climate , 2000 .

[14]  J. Hansen,et al.  Global trends of measured surface air temperature , 1987 .

[15]  A. Walden,et al.  Wavelet Methods for Time Series Analysis , 2000 .

[16]  Claude N. Williams,et al.  Homogenization of Temperature Series via Pairwise Comparisons , 2009 .

[17]  Donald B. Percival,et al.  Asymptotic decorrelation of between-Scale Wavelet coefficients , 2005, IEEE Transactions on Information Theory.

[18]  W. González-Manteiga,et al.  Semi-parametric statistical approaches for space-time process prediction , 1998, Environmental and Ecological Statistics.

[19]  Sounak Chakraborty,et al.  A Bayesian approach to estimating the long memory parameter , 2009 .

[20]  Jane L. Harvill.  Spatio‐temporal processes , 2010 .

[21]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[22]  Marina Vannucci,et al.  Wavelet-Based Nonparametric Modeling of Hierarchical Functions in Colon Carcinogenesis , 2003 .

[23]  G. J. Janacek,et al.  DETERMINING THE DEGREE OF DIFFERENCING FOR TIME SERIES VIA THE LOG SPECTRUM , 1982 .

[24]  I. Jánosi,et al.  Detrended fluctuation analysis of daily temperature records: Geographic dependence over Australia , 2004, physics/0403120.

[25]  Catherine A. Calder,et al.  Demographic and behavioral modifiers of arsenic exposure pathways: a Bayesian hierarchical analysis of NHEXAS data. , 2008, Environmental science & technology.

[26]  C. Granger,et al.  AN INTRODUCTION TO LONG‐MEMORY TIME SERIES MODELS AND FRACTIONAL DIFFERENCING , 1980 .

[27]  Peter Guttorp,et al.  Wavelet-based parameter estimation for polynomial contaminated fractionally differenced processes , 2005, IEEE Transactions on Signal Processing.

[28]  Mark F. J. Steel,et al.  Non-Gaussian Bayesian Geostatistical Modeling , 2006 .

[29]  P. Robinson.  Semiparametric Analysis of Long-Memory Time Series , 1994 .

[30]  T. Fearn,et al.  A decision theoretical approach to wavelet regression on curves with a high number of regressors , 2003 .

[31]  Hernando Ombao,et al.  SPATIO-SPECTRAL ANALYSIS OF BRAIN SIGNALS , 2008 .

[32]  T. Fearn,et al.  Bayesian Wavelet Regression on Curves With Application to a Spectroscopic Calibration Problem , 2001 .

[33]  Wim Sweldens,et al.  The lifting scheme: a construction of second generation wavelets , 1998 .

[34]  Demetris Koutsoyiannis.  Climate change, Hurst phenomenon, and hydrologic statistics , 2006 .

[35]  P. Bloomfield.  An exponential model for the spectrum of a scalar time series , 1973 .

[36]  A. Gelfand,et al.  Handbook of spatial statistics , 2010 .

[37]  R. Lund,et al.  Changepoint Detection in Periodic and Autocorrelated Time Series , 2007 .

[38]  L. Mark Berliner,et al.  Physical‐statistical modeling in geophysics , 2003 .

[39]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[40]  Jonathan R. Stroud,et al.  Dynamic models for spatiotemporal data , 2001 .

[41]  R. Baillie,et al.  Fractionally integrated generalized autoregressive conditional heteroskedasticity , 1996 .

[42]  Christopher K. Wikle,et al.  Multiresolution Wavelet Analyses in Hierarchical Bayesian Turbulence Models , 1999 .

[43]  H. Rue,et al.  An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach , 2011 .

[44]  Thomas M. Smith,et al.  Global temperature change and its uncertainties since 1861 , 2001 .

[45]  J. R. Wallis,et al.  An Approach to Statistical Spatial-Temporal Modeling of Meteorological Fields , 1994 .

[46]  W. Sweldens.  Wavelets and the lifting scheme : A 5 minute tour , 1996 .

[47]  Demetris Koutsoyiannis,et al.  Climate change, the Hurst phenomenon, and hydrological statistics , 2003 .

[48]  L. Mark Berliner,et al.  Spatiotemporal Hierarchical Bayesian Modeling Tropical Ocean Surface Winds , 2001 .

[49]  A. Raftery,et al.  Space-time modeling with long-memory dependence: assessing Ireland's wind-power resource. Technical report , 1987 .

[50]  C. Hurvich,et al.  Model Selection for Broadband Semiparametric Estimation of Long Memory in Time Series , 1999 .

[51]  David R. Brillinger,et al.  Some uses if cumulants in wavelet analysis , 1996 .

[52]  J. Møller,et al.  Handbook of Spatial Statistics , 2008 .

[53]  D. Brillinger.  Some river wavelets , 1994 .

[54]  A. Brix,et al.  Long memory in surface air temperature: detection, modeling, and application to weather derivative valuation , 2002 .

[55]  Jeffrey S. Morris,et al.  Wavelet‐based functional mixed models , 2006, Journal of the Royal Statistical Society. Series B, Statistical methodology.

[56]  S. Hussain,et al.  Fractional order estimation and testing, application to Swedish temperature data , 1999 .

[57]  Patrick Bogaert,et al.  Spatiotemporal analysis and processing of thermometric data over Belgium , 1997 .

[58]  J. Beran.  Fitting long-memory models by generalized linear regression , 1993 .