Climate data: Long range dependent or nonstationary?
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Climate data: Long range dependent or nonstationary? Dependence, Stability, and Extremes Workshop The Fields Institute Toronto, Ontario, Canada 6 May 2016 Mark M. Meerschaert Department of Statistics and Probability Michigan State University mcubed@stt.msu.edu http://www.stt.msu.edu/users/mcubed Partially supported by NSF FRG grant DMS-1462156, NSF INSPIRE grant EAR-1344280, and ARO MURI grant W911NF-15-1-0562. Abstract A North American Regional Climate Change Assessment Program (NARCCAP) climate model was used to generate 29 years of daily maximum temperature data at around 16,000 spatial locations in North America. Our ultimate goal is to develop accurate models for forecasting extreme weather conditions, e.g., heat waves. The goal of this talk is to develop a reasonable time series model for the temperature data at any given spatial location. Future work will also consider the spatial pattern. Since the data shows a strong seasonal variation, as well as long range correlations, it is not clear a priori whether to apply a model with long range dependence, or periodic stationarity, or both. It has been observed, e.g., in financial time series, that nonstationarity can mimic long range dependence. We will see that this is also the case for climate data. Hence a periodically stationary time series model may be indicated. Acknowledgments Paul L. Anderson, Maths and Computer Science, Albion College Metin Eroglu, Statistics and Probability, Michigan State U Joshua French, Math and Statistical Sci., U Colorado Denver Piotr Kokoszka, Statistics, Colorado State University Stilian Stoev, Statistics, University of Michigan NARCCAP data set Data is freely available at http://www.narccap.ucar.edu/ from the North American Regional Climate Change Assessment Program (NARCCAP) . Data Source: Canadian Regional Climate Model (CRCM) + Community Climate System Model (CCSM) CRCM is a detailed local climate model from the Canadian Centre for Climate Modelling and Analysis. CCSM is a global circulation climate model from the USA National Center for Atmospheric Research (NCAR) in Boulder CO. A complete data set (no missing values) of maximum daily surface air temperature (◦K) on a 140×115 grid over much of North America covers N = 29 years, 2041–2069. NARCCAP data set Xn(sk , tj ) = max temperature at location sk on day tj of year n Xn(sk , tj ) − μ̂(sk , tj ) Standardize: Yn(sk , tj ) = where σ̂(sk , tj ) N 1 X n (s k , t j ) μ̂(sk , tj ) = N n=1 σ̂(sk , tj )2 = N 2 1 Xn(sk , tj ) − μ̂(sk , tj ) . N − 1 n=1 Heat wave statistic: Z ∗ (s k , t j ) = ⎡ min k∈IK (k) ⎤ ⎢1 ⎣ t −<t ≤t j i j ⎥ Y n (s k , t j ) ⎦ K = 50 nearest neighbors, = 9 day moving average Nearest neighbor map Examples of 50 nearest neighbors G GG G GGGGGGGG G G G G G G G G G G G G G GGGGG GGGG G G G G G G G G G GG G G GGGG G GGGGGGG G G G GGG GGG G G G G G G G G G G G G G GG G G G G G G G G G GGG G G G G G G G G G G GG G G G G G GGGGGG G G GGGGGGGGGGG GGGGGG G GGGGGGGGGGG GGGGG G G G G G G G G G G G G G GG G G G G G G GGGGG G G G G G G G G G GGGG GGGG G GG GGGGG G GG G GG G G G G G G GG G G G G GGGG GG G GG G GGGG GGGGG G G G G G G G G G G G G G G G G G G G G GG G GG GGGGGGGGGG G G G G G G G G G G GGG GGGGGGG GG G G G G G G G G G G GGGGG G GG GG G G G G G GG G GG GGG GG G GGG GG GG G G GG G G G GG GG G G GG GG G GGG GGGGG GG GGG GG GG G GG GG GGG GG GGGGGGGGGGGGGGGGGGGGGGGG G GG G GG GGGGGGGGGG GG G GG GGGGG GG G G GGGG GG G G G GGGGGGG G G GG GG G G GGG GGGGGGGG GG GG G G GG GG G G G G GG G G G GG GGGGGGGGGGG GGGGG 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G G GG GG G GG G GG G GG G G G G G G G G G G GG G G G G G GG G G G G G G G G G GG G G G G G G G G G G GG G G G G GG G G G GG G GG G G G GG G G G G GG G G G G GGGG G G G GG G G G GG G G G G G G G G G G G GGG G G G GG G G G G G G G G G G G G G GGG G G G G GGG G G G G G G G G G G G GGG G G G GGG G G G GGG G GGG G GGG G G G G G G G GGG G G G GG G G G G G G GGG G G G GG G G G G G G G G G GG G G GG G G G G G G GG G G G G G G G G G G G G G GG G G G G G G G G G G G G G G GG G G G G G G G G G G GG GGGG G G G G G G G G G GG G G G G G G GGG G G G GG G G G G G G G G G G G G G G GG G G G GGG G G GGG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GGGG G G G GGGG G G GG GGG G G G G G GG G G GG GG G G G G G GGG G GG G GGG G G G G G G GG G GGGG G G GGG G G GGG GGG G G G G GGG G G G GG G G G G G G GG G G GG G GG G G G G G G G G G G GG G G G G G G G GG G G G G G G G G G G G GG G G G G G GG G G G G GG G G GG G G GG G G G G GG G G G GG G G G G G GG G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GGG G G G GGGG G G G G GGG G G G G G G GGG G GG G G G G G G G G G G GGG G G G GGG G GGG G GGG G G G G G G G GGG G G G G G G G G G G G GG G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G GG G G GG G G G G GG G G G G G G G GG G G G G G G G G G G GG G G G G G G G GG G G G G G G G G G G G G G G G GG G G G G G GGG G G G G G G GG GGG G G G G G G G G G GG G G G G GGG GGG G G G GGG G GG G GG G G G G G GG G GG G G GG G G G G GG G G G G GG G GGG G GGG G G GG G G G G GG G GGG G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GGG G G G G G G G G G G G GGG G G G G G G G G G G G G G G G G G GGG G G GGG G G G GG G GG G G G G G G GG G G G G G G GGG G G G G G G G GG G GGG G G G G G G G G G GGG G GG G G G G G G G G G GG G GG G G G G G G G G GG G G G G G G G G G G G G G G G G GG GG G G G G G G G G G G G G GG G G G G G G G GG G G G G G GG G GG G GG G G G G G G G G G G G G GG G G G G GG G G G G G G G G G G G G G GGG G G G G G G G G G G GGG G G G G G G G G G G G G G G GGG G G GGG G G G G G G G G G G G G G G G G G G G G G G G G G G GGG G G G G GGG G G G G G G G G G G GGG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G GG G G G G G G G G G G G GG G G G G GGG G G G GG G G G G G G G G G G G G G G G G GGG G G G G G G G G GGG G GGG G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G GG G GGG G G GGGG G G G G GGG G G G GG G G G G G GGG G G G G G G G G G G GG G G G G GGG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G GG G G GG G G GG G G G G G G G G GG G G G G G G G G G GG G G GG G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GGG G G G G G G GGG G G G G G G G G G G G G G G G GGG G G G G G G G G G G G G G GG G G G GGG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G GG G G G G G G G GG G G G G G G G GG G G G G G G G G G GG G G G G G G G G GG GG G G G G G G G G G GG G G G GGG G G G G G G G G GG G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G GG G G G G G G GG G G G G G G GG G G G G G G GG G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G GG G G G GG G G GG G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G GG G G G GG G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G Heat wave map Probability that Z ∗(sk , tj ) > 1 (mild heat wave) 70 z=1 60 0.8 50 40 0.4 30 0.2 20 latitude 0.6 200 220 240 260 280 300 320 Heat wave map Probability that Z ∗(sk , tj ) > 2 (strong heat wave) 70 z=2 60 0.08 50 40 0.04 30 0.02 0.00 20 latitude 0.06 200 220 240 260 280 300 320 Long range dependence? Autocorrelation at one site suggests LRD. 0.0 0.2 0.4 ACF 0.6 0.8 1.0 Series Y 0 10 20 Lag 30 40 Check spectral density −2 −4 −6 −8 log(spec) 0 2 4 Power law spectral density also suggests LRD d = 0.9. −8 −6 −4 log(freq) −2 LRD and heavy tails Hill estimator also indicates heavy tail with α ≈ 1.5. +,//6+)7 DOSKD U 3DJH Question: Is Yn(sk , tj ) stationary? Oscillation in the ACF suggests periodic behavior. 0.0 0.2 0.4 ACF 0.6 0.8 1.0 Series Y 0 100 200 Lag 300 400 Periodic ARMA model One spatial location: X̃365 (n−1)+t = Xn(sk , t) PARMAS (p, q) model: Xt − p φt(j)Xt−j = εt + j=1 q θt(j)εt−j j=1 Mean-centered data Xt = X̃t − μt Standardized errors δt = σt−1εt assumed IID Parameters φt(j), θt(j), σt all periodic with period S = 365. R package perARMA is quite convenient Periodic mean 290 285 280 275 270 seasonal mean 295 300 Plot suggests some seasonal variation. Is SARIMA enough? 0 100 200 t 300 Periodic standard deviation 7 6 5 4 3 seasonal standard deviation 8 SARIMA needs σt constant. Is standardizing enough? 0 100 200 t 300 Periodic autocorrelation function at lag 1 0.6 0.4 0.2 seasonal ACF at lag 1 0.8 Standardizing does not produce a stationary process. 0 100 200 t 300 Fit a periodic AR(1) via perARMA Residuals indicate a good fit. Nonstationary data mimics LRD! 0.0 0.2 0.4 ACF 0.6 0.8 1.0 Series resid 0 10 20 30 40 Model residuals are approximately normal PAR(1) residuals via perARMA close to a normal PDF. )UHTXHQF\ 3$5UHVLG A cautionary tale Temperature data exhibits a heavy tail with infinite mean α < 2 Standardized data exhibits LRD d = 0.9 Periodic ARMA model removes the heavy tail Periodic ARMA model removes the LRD Nonstationary data mimics LRD and heavy tail! Smoothing the PARMA parameters Discrete Fourier transform smooths PAR(1) coefficients φt(1) ϭ͘Ϯ ϭ Ϭ͘ϴ ƉŚŝ Ϭ͘ϲ ƉŚŝ^ŵŽŽƚŚ Ϭ͘ϰ Ϭ͘Ϯ Ϭ Ϭ ϭϬϬ ϮϬϬ ϯϬϬ Smoothing the seasonal standard deviation Discrete Fourier transform smooths PAR(1) coefficients σt ϲ ϱ ϰ ƐŝŐŵĂ ϯ ƐŝŐŵĂ^ŵŽŽƚŚ Ϯ ϭ Ϭ Ϭ ϭϬϬ ϮϬϬ ϯϬϬ Residuals of the smoothed PAR(1) model Smoothed model has only 4 parameters. Residuals still uncorrelated. 0.0 0.2 0.4 ACF 0.6 0.8 1.0 Series stdresid 0 10 20 Lag 30 40 Residuals of the smoothed PAR(1) model Histogram of smoothed PAR(1) residuals fits a normal PDF reasonably well, but with a bit of negative skewness. 900 800 700 Frequency 600 500 400 300 200 100 0 -4.8 -3.6 -2.4 -1.2 0.0 perARMAresid 1.2 2.4 3.6 Test on three 10 × 10 grids Try a PAR(p) model on three test grids. Find p = 1 is sufficient in SW grid, p = 3 enough for North-Central and Eastern grids. Check ACF for some randomly selected sites Residual ACF plots, at four sites in the SW grid, PAR(1) model fit using Yule-Walker method via perARMA. Series residout[, 50] 0.6 0.2 0.0 0.0 10 20 30 40 0 10 20 30 Lag Lag Series residout[, 75] Series residout[, 100] 40 0.6 0.4 0.0 0.0 0.2 0.2 0.4 ACF 0.6 0.8 0.8 1.0 1.0 0 ACF 0.4 ACF 0.6 0.4 0.2 ACF 0.8 0.8 1.0 1.0 Series residout[, 25] 0 10 20 Lag 30 40 0 10 20 Lag 30 40 Spatial cross-correlation at nearby sites Center site in SW grid and 4 nearest neighbors (in longitude) residout[, 55] & residout[, 54] 0.0 0.0 0.2 0.4 ACF 0.2 ACF 0.4 0.6 0.6 0.8 residout[, 55] & residout[, 53] −20 0 20 −20 0 20 Lag residout[, 55] & residout[, 56] residout[, 55] & residout[, 57] 0.2 0.4 ACF 0.4 0.0 0.2 0.0 ACF 0.6 0.6 0.8 0.8 Lag −20 0 Lag 20 −20 0 Lag 20 Spatial cross-correlation at nearby sites Center site in SW grid and 4 nearest neighbors (in latitude). Perhaps a spatial AR(1) will be sufficient? residout[, 55] & residout[, 45] 0.6 0.0 0.0 0.2 0.4 ACF 0.4 0.2 ACF 0.6 0.8 0.8 residout[, 55] & residout[, 35] −20 0 20 −20 0 20 Lag residout[, 55] & residout[, 65] residout[, 55] & residout[, 75] 0.2 0.4 ACF 0.4 0.2 0.0 0.0 ACF 0.6 0.6 0.8 0.8 Lag −20 0 Lag 20 −20 0 Lag 20 References 1. I.B. Aban and M.M. Meerschaert, Shifted Hill’s estimator for heavy tails. Communications in Statistics: Simulation and Computation, Vol. 30 (2001), No. 4, pp. 949–962. 2. I.B. Aban, M.M. Meerschaert, and A.K. Panorska, Parameter Estimation for the Truncated Pareto Distribution, Journal of the American Statistical Association: Theory and Methods, Volume 101 (2006), Number 473, pp. 270–277. 3. P.L. Anderson and M.M. Meerschaert, Periodic moving averages of random variables with regularly varying tails, The Annals of Statistics, Vol. 25 (1997), No. 2, pp. 771–185. 4. P.L. Anderson, M.M. Meerschaert, Y.G. Tesfaye, Fourier-PARMA Models and Their Application to Modeling of River Flows, Journal of Hydrologic Engineering, Vol. 12 (2007), No. 5, pp. 462–472. 5. Y.G. Tesfaye, P.L. Anderson, and M.M. Meerschaert, Asymptotic results for FourierPARMA time series, Journal of Time Series Analysis, Vol. 32 (2011), No. 2, pp. 157–174.
