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Detecting trends that are nonlinear and asymmetric on diurnal and seasonal time scales

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Abstract

Trends in climate time series are often nonlinear and temporally-asymmetric, i.e. the trend is different for different seasons and/or hours of the day. Here a method is developed that allows the nonlinearity and temporal asymmetry of a trend to be investigated simultaneously. First, nonlinear trend components are extracted from a univariate time series, by adapting a nonparametric dimension-reduction method. Then, the nonlinear trend components are substituted into a regression model in which the periodic mean component and the periodic variation in the amplitude of the nonlinear trend are modeled using harmonic functions of the seasonal and diurnal periods. Third, trend patterns in the positive and negative anomalies are investigated, by extending the nonlinear trend model using indicator variables. Fourth, a non-local inferential test is developed to test the statistical significance of the trend patterns. The nonlinear trend model is applied to a simulated time series, as well as to long-term high-resolution temperature records from five Southern Hemisphere sites: Lucas Heights, Sydney Airport, Cape Grim, Macquarie Island and Law Dome. Our method should be generally useful for identifying the effect of both climate-related factors and observation/site-related factors on seasonal and diurnal trends in meteorological data series.

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Acknowledgments

The authors wish to thank Dr Blair Trewin and Prof. John Dodson for helpful comments on a draft manuscript.

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Authors and Affiliations

  1. Institute for Environmental Research, Australian Nuclear Science and Technology Organisation, Menai, NSW, 2234, Australia

    Matt J. Fischer & Adrian W. Paterson

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  1. Matt J. Fischer
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Correspondence to Matt J. Fischer.

Electronic supplementary material

Online Resource 1: One electronic file containing a Supplementary Discussion section and 4 Supplementary Figures is provided. These figures are labeled Fig. S1, S2, etc. both in the manuscript text and Supplementary Material.

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Appendix

Appendix

This appendix outlines a method for matrix imputation (i.e. filling an incomplete matrix) based on Beckers and Rixen (2003) and Fuentes et al. (2006). X is the incomplete matrix. The algorithm proceeds as follows:

  1. 1.

    Determine an intial component, u 1. One method is to calculate the first eigenvector (v 1) of the correlation matrix of X (using pairwise complete observations), and calculate u 1 as the average over all the columns of X (after adjusting X by Xdiag(sign(v 1))). Set k = 1.

  2. 2.

    Remove a set of known values from X (e.g. 30 known values).

  3. 3.

    Replace all the missing values in X by regression of X on u 1.

  4. 4.

    Compute the rank-k singular value decomposition (SVD) of X.

  5. 5.

    Update the missing values in X by the elements of the rank-k SVD approximation.

  6. 6.

    Return to step 4 and iterate to convergence.

  7. 7.

    Calculate the RMSE between the known values (step 2), and their predicted values. If the RMSE is smaller than previous values, then increase k and return to step 4.

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Fischer, M.J., Paterson, A.W. Detecting trends that are nonlinear and asymmetric on diurnal and seasonal time scales. Clim Dyn 43, 361–374 (2014). https://doi.org/10.1007/s00382-014-2086-8

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