Marco Giannitrapani 1

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Detecting Discontinuities
Marco Giannitrapani 1 Marian Scott 1 Adrian
Bowman 1 Ron Smith 2 ( 1University of Glasgow,
2 CEH of Edinburgh )
Valencia, 9-11 April 2003
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What is a discontinuity?
A discontinuity is an abrupt-change in mean
level.
Temporary change
Permanent change
Why are we interested in detecting
discontinuities?

• It is necessary to identify any discontinuities
  to understand the cause of the change
• Emission changes.
• Change in laboratory.
• Weather effects.
• Something else.
• If possible, it is so necessary to correct the
  series before doing any trend analysis, if not,
  to analyse each sub-trend separately.

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DATA ANALYSED

• Weekly means of the natural logarithm of the
  daily data for
• SO2
• SO4 in air
• SO4 in precipitation (corrected and not for the
  sea-salt)
• Across Europe

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Austria (4) Croatia (2) Czech Republic
(2) Denmark (6) Finland (7) France (12) Germany
(19) Latvia (2) Lithuania (2) Netherlands
(7) Norway (12) Poland (5) Slovakia (4) Sweden
(8) Switzerland (5) United Kingdom (10)
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PRELIMINARY ANALYSIS
Daily data have shown the presence of two kind of
seasonal cycles
Day within the week
Week within the year
Difference between week and weekend days.
Difference between seasons of the year.
de-seasonalised data
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Theory of the Discontinuity Test
The test used was proposed by Bowman and Pope
(1997) At each data point x1, x2,, xn we
observe the data y1, y2,, yn where
yi g(xi)ei for i 1, ,n.
(where g is a smooth function) The test
statistic is based on the difference between the
left (gl(xi)) and the right (gr(xi)) smooths,
where
Left Smoother Right Smoother
where kh (the kernel smoother) is the normal
density function in z with mean xj and standard
deviation h I is the indicator function.
Criteria for detection is given by gl(xi) -
gr(xi) gt 3 standard errors.
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Plot of SO2
First Example of Discontinuity
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First Example of Discontinuity
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Plot of SO2
First Example of Discontinuity
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First Type of Discontinuity
First Example of Discontinuity
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First Example of Discontinuity
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First Example of Discontinuity
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First Example of Discontinuity
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First Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Second Type of Discontinuity
Second Example of Discontinuity
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Second Type of Discontinuity
Second Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Second Type of Discontinuity
Second Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Second Type of Discontinuity
Second Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Second Type of Discontinuity
Second Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Second Type of Discontinuity
Second Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Second Type of Discontinuity
Second Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Second Type of Discontinuity
Second Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Second Type of Discontinuity
Second Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Third Type of Discontinuity
Third Example of Discontinuity
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Third Type of Discontinuity
Third Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Third Type of Discontinuity
Third Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Third Type of Discontinuity
Third Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Third Type of Discontinuity
Third Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Third Type of Discontinuity
Third Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Third Type of Discontinuity
Third Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Third Type of Discontinuity
Third Example of Discontinuity
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Plot of SO4 in precipitation not corrected
Third Type of Discontinuity
Third Example of Discontinuity
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Final Remarks

• The test can be used to identify whether changes
  in trend have occurred.
• Once a discontinuity is detected, the series
  should be corrected before interpreting the
  trend, or doing any analysis. If it is not
  possible to do any corrections, then it becomes
  necessary, to treat each sub-trend separately.
• This version of the test requires estimation of
  the correlation and of the variance of the data.
  These have been computed on the basis of the
  weekly means after removing any trend and
  seasonality.
• Because of the edge bias in smoothers, fifty
  "testing points" at the start and fifty at the
  end of the series have been excluded.

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Conclusions

• A slight steadily decreasing trend can be noted,
  which seems more pronounced for SO2.
• However, the rate of decrease is not constant
  over the entire time period, and discontinuities
  represent a common feature.
• In particular, several temporary
  discontinuities have been identified.
• There are strong arguments for not fitting a
  single (and simple) trend function.
• Some data present still peculiar features that
  need to be revisited.