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

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  • 1 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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