Penalized Maximal F Test for Detecting Undocumented Mean Shift without Trend Change

Xiaolan L. Wang Climate Research Division, ASTD, STB, Environment Canada, Toronto, Ontario, Canada

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Abstract

In this study, a penalized maximal F test (PMFT) is proposed for detecting undocumented mean shifts that are not accompanied by any sudden change in the linear trend of time series. PMFT aims to even out the uneven distribution of false alarm rate and detection power of the corresponding unpenalized maximal F test that is based on a common-trend two-phase regression model (TPR3). The performance of PMFT is compared with that of TPR3 using Monte Carlo simulations and real climate data series.

It is shown that, due to the effect of unequal sample sizes, the false alarm rate of TPR3 has a W-shaped distribution, with much higher than specified values for points near the ends of the series and lower values for points between either of the ends and the middle of the series. Consequently, for a mean shift of certain magnitude, TPR3 would detect it with a lower-than-specified level of confidence and hence more easily when it occurs near the ends of the series than somewhere between either of the ends and the middle of the series; it would mistakenly declare many more changepoints near the ends of a homogeneous series. These undesirable features of TPR3 are diminished in PMFT by using an empirical penalty function to take into account the relative position of each point being tested. As a result, PMFT has a notably higher power of detection; its false alarm rate and effective level of confidence are very close to the nominal level, basically evenly distributed across all possible candidate changepoints. The improvement in hit rate can be more than 10% for detecting small shifts (Δ ≤ σ, where σ is the noise standard deviation).

Corresponding author address: Dr. Xiaolan L. Wang, Climate Research Division, Atmospheric Science and Technology Directorate, Environment Canada, 4905 Dufferin Street, Toronto, ON M3H 5T4, Canada. Email: xiaolan.wang@ec.gc.ca

Abstract

In this study, a penalized maximal F test (PMFT) is proposed for detecting undocumented mean shifts that are not accompanied by any sudden change in the linear trend of time series. PMFT aims to even out the uneven distribution of false alarm rate and detection power of the corresponding unpenalized maximal F test that is based on a common-trend two-phase regression model (TPR3). The performance of PMFT is compared with that of TPR3 using Monte Carlo simulations and real climate data series.

It is shown that, due to the effect of unequal sample sizes, the false alarm rate of TPR3 has a W-shaped distribution, with much higher than specified values for points near the ends of the series and lower values for points between either of the ends and the middle of the series. Consequently, for a mean shift of certain magnitude, TPR3 would detect it with a lower-than-specified level of confidence and hence more easily when it occurs near the ends of the series than somewhere between either of the ends and the middle of the series; it would mistakenly declare many more changepoints near the ends of a homogeneous series. These undesirable features of TPR3 are diminished in PMFT by using an empirical penalty function to take into account the relative position of each point being tested. As a result, PMFT has a notably higher power of detection; its false alarm rate and effective level of confidence are very close to the nominal level, basically evenly distributed across all possible candidate changepoints. The improvement in hit rate can be more than 10% for detecting small shifts (Δ ≤ σ, where σ is the noise standard deviation).

Corresponding author address: Dr. Xiaolan L. Wang, Climate Research Division, Atmospheric Science and Technology Directorate, Environment Canada, 4905 Dufferin Street, Toronto, ON M3H 5T4, Canada. Email: xiaolan.wang@ec.gc.ca

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  • Alexandersson, H., 1986: A homogeneity test applied to precipitation data. Int. J. Climatol., 6 , 661675.

  • Caussinus, H., and Mestre O. , 2004: Detection and correction of artificial shifts in climate series. Appl. Stat., 53 , 405425.

  • Davis, R. A., Lee T. C. M. , and Rodriguez-Yam G. A. , 2006: Structural breaks estimation for non-stationary time series models. J. Amer. Stat. Assoc., 101 , 223239.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • DeGaetano, A. T., 2006: Attributes of several methods for detecting discontinuities in mean temperature series. J. Climate, 19 , 838853.

  • Easterling, D. R., and Peterson T. C. , 1995: A new method for detecting undocumented discontinuities in climatological time series. Int. J. Climatol., 15 , 369377.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hanesiak, J. M., and Wang X. L. , 2005: Adverse-weather trends in the Canadian Arctic. J. Climate, 18 , 31403156.

  • Lund, R., and Reeves J. , 2002: Detection of undocumented changepoints: A revision of the two-phase regression model. J. Climate, 15 , 25472554.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lund, R., Wang X. L. , Lu Q. , Reeves J. , Gallagher C. , and Feng Y. , 2007: Changepoint detection in periodic and autocorrelated time series. J. Climate, 20 , 51785190.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Menne, M. J., and Williams C. N. Jr., 2005: Detection of undocumented changepoints using multiple test statistics and composite reference series. J. Climate, 18 , 42714286.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Peterson, T. C., and Coauthors, 1998: Homogeneity adjustments of in situ atmospheric climate data: A review. Int. J. Climatol., 18 , 14931517.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Reeves, J., Chen J. , Wang X. L. , Lund R. , and Lu Q. , 2007: A review and comparison of changepoint detection techniques for climate data. J. Appl. Meteor. Climatol., 46 , 900915.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Vincent, L. A., Zhang X. , Bonsal B. R. , and Hogg W. D. , 2002: Homogenization of daily temperatures over Canada. J. Climate, 15 , 13221334.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wan, H., Wang X. L. , and Swail V. R. , 2007: A quality assurance system for Canadian hourly pressure data. J. Appl. Meteor. Climatol., 46 , 18041817.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X. L., 2003: Comments on “Detection of undocumented changepoints: A revision of the two-phase regression model”. J. Climate, 16 , 33833385.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, X. L., 2006: Climatology and trends in some adverse and fair weather conditions in Canada, 1953–2004. J. Geophys. Res., 111 .D09105, doi:10.1029/2005JD006155.

    • Search Google Scholar
    • Export Citation
  • Wang, X. L., 2008: Accounting for autocorrelation in detecting mean-shifts in climate data series using the penalized maximal t or F test. J. Appl. Meteor. Climatol., in press.

    • Search Google Scholar
    • Export Citation
  • Wang, X. L., and Feng Y. , 2007: RHtestV2 user manual. Climate Research Division, Atmospheric Science and Technology Directorate, Science and Technology Branch, Environment Canada, 19 pp. [Available online at http://ccsma.seos.uvic.ca/ETCCDMI/software.shtml.].

  • Wang, X. L., Wen Q. H. , and Wu Y. , 2007: Penalized maximal t test for detecting undocumented mean change in climate data series. J. Appl. Meteor. Climatol., 46 , 916931.

    • Crossref
    • Search Google Scholar
    • Export Citation
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