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Daily Stew Kickoff – 27. January 2011
First Steps
Data:
Climate stations of DWD with daily data (or even 7, 14, 21 h)
Although the project is focused on weather indices and extremes, we
consider initially a much easier parameter: Monthly mean temperature
Derive and test methods to:
Create reference time seriesDifferences between a station and its reference are expected be zero
Detect breaksMaximize the external variance by a minimum of breaks
Daily Stew Kickoff – 27. January 2011
DWD Climate Stations
20001900
1920
1960
1200 Stations in total,but not coexistent.
1900: 251940: 1001960: 5002000: 600
Daily Stew Kickoff – 27. January 2011
Kriging Approach
• n observations xi at the locations Pi are given.
• Perform a prediction x0 for the location P0 , where no obs is available.
• Construct the prediction by a weighted average of the observations xi.
• Take into account the observation errors xi.
• Determine the weights i.
min1
2
10
m
t
n
iiii xxx
Daily Stew Kickoff – 27. January 2011
Matrix and Input
Correlations
The spatial autocorrelationis dervided from all availabledata for each of the 12 months.
High correlations for monthly mean temperature.
Daily Stew Kickoff – 27. January 2011
Potsdam and Reference
A reference for each station is created by kriging of the surrounding 16 stations.
Normalized temperature anomaly in January for station Potsdam.
Station and Reference seems to be nearly identical.
Daily Stew Kickoff – 27. January 2011
Potsdam and Reference
A reference for each station is created by kriging of the surrounding 16 stations.
Normalized temperature anomaly in January for station Potsdam.
Station and Reference seems to be nearly identical.
However, there is a difference showing a positive trend from 1930 to 2000
Daily Stew Kickoff – 27. January 2011
Defining breaks
Breaks are defined by abrupt changes in the station-reference time series.
Internal variance
within the subperiods
External variance
between the means of different subperiods
Maximize the external variance by
a minimum number of breaks
Daily Stew Kickoff – 27. January 2011
Decomposition of Variance
m yearsN subperiodsnk members
The external variance is a weightedmeasure for the variability of thesubperiods‘ means.
The internal variance containsinformation about the error of thesubperiods‘ means.
The seeming external variance hasto be diminished by this errorto obtain the true external variance.
Daily Stew Kickoff – 27. January 2011
Break Criterion
The true external variance is used as criterion for breaks.
Daily Stew Kickoff – 27. January 2011
The first break
The difference time series increase from 1930 to 2000 (as already shown)
Between 1965 and 1985 the criterion reaches maximum values.
More than 20% of the total variance can be explained by a break in one of these years.
1970 1968 1969 1979 1967 1978 1980 1971 1972 1981
21.77 21.76 21.67 21.64 21.41 21.33 21.07 20.95 20.87 20.77
criterion
time series
Daily Stew Kickoff – 27. January 2011
Break Searching Method
Now the first break is not simply fixed where the maximum criterion occured (1970).
But combinations of two breaks are tested which contain one of the 10 best
first-break candidates (10 times 100 permutations).
The 10 best two-breaks combinations are used as seed for the search of
three-breaks combinations.
Daily Stew Kickoff – 27. January 2011
1970 0.3197 0.3176 0.3150 0.3029 0.2968 0.2941 0.2904 0.2869 0.2857 0.2824 1930 1929 1928 1927 1925 1926 1924 1923 1920 1922
1968 0.3296 0.3270 0.3240 0.3110 0.3039 0.3014 0.2969 0.2931 0.2911 0.2881
1930 1929 1928 1927 1925 1926 1924 1923 1920 1922
1969 0.3232 0.3209 0.3181 0.3056 0.2991 0.2965 0.2924 0.2888 0.2872 0.2840 1930 1929 1928 1927 1925 1926 1924 1923 1920 1922
1979 0.2821 0.2815 0.2804 0.2718 0.2686 0.2656 0.2642 0.2632 0.2621 0.2591 1930 1929 1928 1927 1925 1926 1924 1920 1923 1922
1967 0.3301 0.3273 0.3240 0.3106 0.3032 0.3007 0.2959 0.2919 0.2896 0.2868 1930 1929 1928 1927 1925 1926 1924 1923 1920 1922
1978 0.2818 0.2810 0.2799 0.2710 0.2675 0.2646 0.2630 0.2617 0.2608 0.2577 1930 1929 1928 1927 1925 1926 1924 1920 1923 1922
1980 0.2720 0.2716 0.2707 0.2624 0.2595 0.2565 0.2553 0.2547 0.2534 0.2506 1930 1929 1928 1927 1925 1926 1924 1920 1923 1922
1971 0.3041 0.3023 0.3001 0.2887 0.2831 0.2804 0.2771 0.2739 0.2732 0.2697 1930 1929 1928 1927 1925 1926 1924 1923 1920 1922
1972 0.2987 0.2971 0.2951 0.2841 0.2790 0.2761 0.2732 0.2702 0.2698 0.2662 1930 1929 1928 1927 1925 1926 1924 1923 1920 1922
1981 0.2654 0.2651 0.2643 0.2564 0.2537 0.2508 0.2499 0.2497 0.2497 0.2495 1930 1929 1928 1927 1925 1926 1967 1924 1968 1920
The second break
Daily Stew Kickoff – 27. January 2011
4 Breaks
Where to stop?
The searching method is applied to a random time series to define a stop criterion
Daily Stew Kickoff – 27. January 2011
Decreasing of internal variance
1 to 400 breakswithin 1000 years
1 to 50 breakswithin 100 years
The remaining internal varianceshrinks rather smoothly for a1000 years time series.
Actually, we are dealing with only a 100 years time series.
Similar behaviour, but lessregular.
Repeat the procedure 500times and consider the change in variance for each added break.
Daily Stew Kickoff – 27. January 2011
Many Breaks for many random time series
In average 6% of the variance is gained by the first breaks.
The 50th break gains only 0.3%
The 90 and the 95 percentile remain nearly constant at a few percent.
The first step is an exception as here only 100 possibilities are tested, whereas further breaks are searched from 1000 possibilities (10 candidates times 100 years).
Median
90%95%
Daily Stew Kickoff – 27. January 2011
Observations vs Random
After 4 breaks the gained variance
of the observations is comparable
to that found for random time series.
4 breaks are realistic for the
considered station.
95%Random 90% 50%
Observations
Daily Stew Kickoff – 27. January 2011
Leaving out one stationJa
nuar
y
Feb
ruar
y
Referencefromnearest 16 stations
ReferencewithoutBerlin-Dahlem