University of Amsterdam
Bsc Project
24/2/2020
Under supervision of John van Boxel
Modelling extreme
temperature events by
stochastic matrices
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Contents
Abstract ... 2 1. Introduction ... 3 1.1 ... 3 1.2 Research aim ... 5 2. Methods ... 6 2.1 Data ... 6 2.2 Anomaly analysis ... 6 2.3 Categories ... 8 2.4 Markov model ... 9 2.5 Trends ... 10 2.6 Kolmogorov-Smirnov Test ... 11 3. Results ... 133.1 Temperature and anomaly results ... 13
3.2 Kolmogorov-Smirnov test ... 14
3.3 Markov occurrence and probability values ... 16
3.4 Station results ... 17
3.5 Stations comparison ... 20
4. Discussion ... 21
4.1 On the model ... 21
4.2 Trend analysis and extrapolation ... 22
4.3 On the results ... 23
5. Conclusion ... 24
6. Literature list ... 25
Appendix 1: Occurrence and transition probability values for two stage transitions at 5 measuring stations in the Netherlands. ... 28
Appendix 2: Trend in transition probability data extrapolated from 1900 – 2100 at 5 measuring stations in The Netherlands. ... 33
Appendix 3: Outcome of Markov model. Predicted distribution of cold, normal and warm days in 1900, 2000 and 2100. ... 38
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Abstract
Increased occurrence of extreme weather events is one of the most threatening effects of climate change. Increasing variability of temperature anomalies could lead to more extreme events, but the climate change effects on this topic stay unclear. The aim of this research is to establish trends in the occurrence and persistence of extreme temperature anomalies in the Netherlands since 1906. Daily anomaly data of 5 measurement stations in the Netherlands was divided into three categories: cold, normal and warm. 2-stage transition matrices were examined for trends in yearly probability and occurrence. With use of a Markov-model, the effects of those trends were simulated. The analysis was also done on data detrended for the mean temperature increase, to establish changes in temperature variability.
The construction of transition matrices proved to be an insightful method of analysing daily anomaly data. The results showed a clear increase in daily mean temperature at all 5 weather stations. This will lead to more warm extreme events, longer streaks of warm days and a large decrease of cold days in the Netherlands. The detrended results show how the increased occurrence of warm events is not caused by an increasing temperature variability. Three of the five weather stations show a decrease in temperature variability, with an increasing amount of normal days, and a decreasing amount of cold and warm days. One weather stations shows no change and only one shows a small increase. Climate change leads to a higher risk of extreme summer heat events, but this study shows no signs of a higher temperature variability in het Netherlands.
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1. Introduction
1.1
The IPCC (2018) is warning policymakers about the effects of global warming. With high confidence they claim that anthropogenic emissions are causing a rise in surface temperature. At the same time changes in climatological patterns are predicted, such as the occurrence of heat waves, droughts or extreme precipitation (Klein Tank & Können, 2003). The amount of heat waves in Europe is expected to increase (Beniston et al., 2007). Periods of extreme cold are expected to occur less often on the other hand (Visser, 2005). With medium confidence is claimed that the amount and/or length of heatwaves have increased since the industrial revolution, and that it is very likely that this trend will
increase in the coming decades (Seneviratne et al., 2017). McGregor et al. (2005) claim that countries as France will experience the same level of heatwaves in 2100, as Sicily experiences right now. There is a lot of attention for such extreme weather events because of their impactful effects on nature and society. In the summer of 2003 Europe suffered from extreme heat and related events such as wildfires. The total economic losses of this heat were estimated at US$ 18,6 million (McGregor et al., 2005). Horton et al. (2016) emphasize that whereas an increase in mean temperature can lead to some positive effects, such as higher agriculture yields, research to the effects of extreme heat events did not lead to a single positive effect. Increasing amounts/lengths of heat waves ask for thorough measures by countries that are currently not properly prepared.
Already Katz and Brown (1992) asked the question whether the change in such patterns is caused by the average rise in temperature, or also by a change in temperature variability.
Whether the daily temperature anomaly variability is changing is still in wide debate. Hansen (2012) is the leading study to argue that the temperature variability is largely increasing. But also Schär et al. (2004) and Beniston et al. (2007) claim that this temperature variability is increasing.
Figure 1; An increased occurrence of extreme events can be caused by either an increase in mean or variance or a combination of both.
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But Perkins and Alexander (2013) argue that there is still no evidence that temperature variability has increased or will increase, when the methods are corrected for the way of the anomaly normalization and inhomogeneity in the data due to changes in observing stations. Multiple other studies such as Michaels et al. (1999), Rhines (2013) and Huntingford et al. (2013) agree with the statement that there is no proof for an increasing temperature variability. Different studies such as Brown, Ceasar & Ferro (2008) and Hoekstra et al. (1999) argue that there is an increase in warm extreme temperature events, and a decrease in cold extreme temperature events, without evidence for conclusions about changing temperature variability.
Then there is also the camp of studies that states that temperature variability is decreasing. Screen (2014) argues that over the mid- to high-latitude Northern Hemisphere the temperature variability has been decreasing over the past decades, of which Arctic amplification is one of the causes. Also Collow et al.(2019) and Lu et al. (2016) agree with the claim that daily temperature variability is decreasing. Researching temperature variability is very relevant, because it can lead to a better understanding of the character of climate change. Increasing amounts of extreme temperature events are not necessarily related to an increase of temperature variability. Figure 1 shows how extreme events can both be caused by an increasing variance as well as a shift in mean temperature. When predicting the future climate it is important to know the trends in temperature variability, apart from mean temperature trends.
When researching temperature variability the focus mostly lies on extreme temperatures; periods of extreme cold or heat for a certain region. But it is evident that such events mostly occur in respectively winter or summer. More information about temperature variability is available when looking at temperatures that are extreme for its moment in the year. The daily anomaly is suited for this topic, which is the difference between the daily temperature and the ‘normal’ temperature for that day. Periods of extreme anomalies happen throughout the whole year, both warm and cold. Trends in the occurrences of extreme anomalies can tell a lot about changes in temperature variability, but this is still a very poorly researched topic.
The chaotic behaviour of the climate is an aspect that troubles researchers in modelling robust predictions for the future (Oerlemans, 2008). A method to predict chaotic behaviour is by using Markov models. This stochastic model is commonly used in the economic department for different kinds of evaluations, predictions and decision making (Bowe et al., 1990). This varies from health care evaluation to stock market prediction, where so-called hidden Markov models are widely used (Bhar & Hamori, 2004). For weather prediction stochastic models are used as well, mostly to predict precipitation behaviour and crop
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yields (Wilks & Wilby, 1999). Whereas such models often generate day-to-day data, they must not be confused with weather forecasting models. The output data of Markov models are suited for statistical analysis of future climate behaviour, but the model is no algorithm that predicts the actual weather based on variables.
1.2 Research aim
The aim of this research is to establish trends in the occurrences and persistence of extreme temperature anomalies in the Netherlands since 1906 and to use stochastic transition matrices and a Markov model to predict anomaly trends and extreme temperatures in the future.
Research questions
This research will answer the following questions:
- Does global warming cause an increase in the amount and the length of clusters of unusually cold and warm days in the Netherlands?
- Is it possible to develop a Markov-like transition matrix based on the mean daily temperature data since 1951 in the Netherlands?
- What trends are there in the probability values of the stochastic transition matrix when basing it on daily temperature anomalies since 1951?
- What can be predicted about the clustering of extreme temperatures in the Netherlands by the use of this stochastic transition matrices?
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2. Methods
2.1 Data
This study will make use of stochastic transition matrices to research the trends in extreme temperatures. The stochastic transitions matrices tell about the chance that a transition to a certain temperature category will happen, when looking at the days before. They therefore provide information about the occurrence and persistence of those temperature categories. This model can be valuable for analysing different climate scenarios and their effects. In this section the different steps in this research will be explained, from collecting raw data to using the Markov model for analysis.
KNMI data
This research uses data delivered by the KNMI (2019). They provide open-source data of the daily mean temperature of their measuring stations, going back to 1901. Data of 5 different stations is used in this research: De Kooy, De Bilt, Eelde, Vlissingen and Maastricht. Those stations are equally spread over Netherlands. The measurements of these stations start between 1901 and 1908, going further back in time than most stations in the Netherlands. Because of the different start dates of those stations, only one data range is used in this research. This range started from 1 January 1907 till 31 December 2018, and only the daily mean temperature is used. All data analysis and storage was done in Microsoft Excel.
Homogeneity of the data
Changes in measuring techniques and relocations of the measuring stations can lead to inhomogeneities in the measurements (KNMI, 2019). Those inhomogeneities can be so large that they form an obstacle for comparing historical data to more recent data. For proper trend analysis it is important to have the data homogenized. The KNMI applied corrections to the data for the five stations used in this research, so the data were properly homogenized. The only missing values in the data were in the winter of 1944-1945. Because of the used research method, this didn’t lead to meaningful problems for the research.
2.2 Anomaly analysis
The analysis of this research is based on the anomaly of every day, which is the difference between the daily mean temperature and the ‘normal’ mean temperature for that day. For each individual day the anomaly will be calculated, before dividing them into different categories. To be able to calculate the exact values, the normal temperature is needed. Calculating the normal temperature is done in two different ways, each method getting its own results.
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Method I (normal)
The first method is done in a standard way. The normal temperature in this method is based on the average of the period 1960-1989, which is a conventional period to use for such practices (IPCC, 2013). For each month of the year the average temperature was calculated over the period 1960-1989. To transform these monthly averages into daily normal temperatures, these temperatures were used for the 15th day of each month, and then interpolated for the other days. So in this way the seasonal fluctuation gets cancelled, because each individual day has its own normal temperature. But there is no difference between years, the daily normal temperature is the same for the al the years in the research period.
Method II (detrended)
The second method adds a correction for the fluctuation between different years. The goal of this method is to take into account the trend of increasing mean temperature, of which the existence has been demonstrated multiple times (IPCC, 2018; Seneviratne et al., 2012). This trend would logically cause an increase in the occurrence of warm days, which is nothing new. The goal of this research is to find changes in the spread of the warm/normal/cold days, therefore a correction for the trend of global warming could give insightful results.
This could be done by subtracting the trend in a linear way, but figure 2 shows that the trend is not linear so this will distort the data. Therefore a 11 year running average is calculated, because it represents the actual trend better. This is the average temperature of a period of 11 years, with the concerning year in the middle of this 11-year period. Figure 2 shows the trend and variability of this running mean, which matches the shape of the actual trend better. Again this method is used to find the normal temperature for every month, which again was used for the 15th of each month and got interpolated for the days in between.
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Figure 2. The yearly average temperature of De Kooy and its 11-year running mean. The running mean shows to match the actual trend. The peak in 1945 is caused by missing data in the winter.
2.3 Categories
Using those two methods, the anomaly was calculated for every day in the period 1907-2018. The exact anomalies now had to be divided over three classes to perform stochastic analysis: cold, normal and warm. To determine the boundaries of these categories the 25th and 75th percentile of all anomaly data of De Bilt was determined. Those percentile borders are based on the detrended anomaly data, but also used for the normal analysis. It could have been possible to use values based on the normal analysis for the normal data, but the same values were used for both analyses, for a better comparison.
The same percentile borders were also used for the other stations. The distributions of the other stations were slightly different, but this is acceptable since for trend analysis the boundary can be set anywhere and the differences were small as well. Now for each daily mean temperature since 1907 a daily anomaly has been determined, using two different methods. This resulted in two different datasets of daily anomalies for the different stations. For those datasets the further analysis was the exact same.
Stochastic Markov matrices
To create stochastic transition matrices, not only the state of an individual day is taken into account but also the two days before. This way a so-called 2-stage transition matrix can be created, as shown in table 1. 7,00 8,00 9,00 10,00 11,00 12,00 13,00 1907 1911 1915 1919 1923 1927 1931 1935 1939 1943 1947 1951 1955 1959 1963 1967 1971 1975 1979 1983 1987 1991 1995 1999 2003 2007 2011 2015
Temperature (℃) De Kooy
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LAST 2 DAYS TODAY
Cold Normal Warm
CC CCC CCN CCW CN CNC CNN CNW CW CWC CWN CWW NC NCC NCN NCW NN NNC NNN NNW NW NWC NWN NWW WC WCC WNC WCW WN WNC WNN WNW WW WWC WWN WW
Table 1; The construction of a Markov Matrix. The fat marked letters are placed instead of values.
The values in this matrix represent the probability of a transition. For example, the value at NWC will give the probability of the transition to a cold day (C) after a normal and a warm day (NW). When using the matrix as a predictive model, the total sum of all values will add up to 900%, since the three values in every row always add up to 100%.
Those probability values were the subject of further analysis, and thus had to be constructed out of the anomaly data. For each year, both the occurrence of each transition (all values add up to 100%) and the probability of each transition (all values add up to 900%) were calculated. The second is not simply a scalar multiple of the first, since it predicts the probability of a certain transition but doesn’t take into account the occurrence of two days before. Thereby, to completely comprehend the changes in the data, both calculations are relevant.
2.4 Markov model
Now predictions can be made based on the probability values. A Markov-model was created, that can simulate a chosen number of years using the stochastic matrices. The only input in this model are the probability values of the stochastic matrices. Combining these values with a function that randomly generates a number between 0 and 1, this model simulates transitions and thereby the states of individual days. This simulation was run for a long period of 150 year to avoid the dominance of outliers.
This model also keeps track of streaks of states. By a streak is meant: an amount of days with the same state directly following each other. The length of every streak was measured and the amount of streaks with every length was counted. Note that when a day is not part of a longer streak, it was still measured
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as a streak with a length of 1 day. And days were never double counted, when a day is in a streak of 4 days, it was not counted as a combination of two streaks of 2 days or counted as part of a larger streak. This data was then summarized as the average amount of days per year in a streak with a certain length. See table 2. When transforming those distributions to a cumulative form, both the total days per year in each category, as the persistence of the category can be seen.
STREAK LENGTH IN DAYS
AVERAGE AMOUNT OF DAYS IN STREAK PER YEAR
COLD NORMAL WARM
1 7,30 12,77 8,01 2 7,50 17,71 8,68 3 8,54 17,50 9,15 4 7,44 16,86 7,83 5 7,33 16,04 7,85 6 6,25 14,04 6,58 7 5,98 10,89 4,91 8 3,50 9,33 4,94 9 4,06 6,25 2,13 10 2,15 5,21 2,57 (…)
Table 2; Example of a table in which the results of the Markov model are gathered. The average amount of days per year in a streak with a certain length is given.
2.5 Trends
Regression analysis
To search for trends in the values of the occurrence and probability matrices, an F-test with an significance of 5% (Burt et al., 2009) was performed on the occurrence and probability values of each transition, over the whole period of data.
Trend extrapolation
In order to answer the research questions, the trend in the transition matrices has been extrapolated to the years 1900, 2000 and 2100. This was done for both the normal and the detrended data. The probabilities of 1900, 2000 and 2100 were independently used as input for the Markov model, which simulated results as described above for each year. The results were summarized in a distribution as shown in table 2, and transformed to a cumulative distribution. For each category the results of 1900, 2000 and
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2100 were shown in one graph. Three categories in both normal and detrended data gives 6 different graphs in which all results are gathered.
2.6 Kolmogorov-Smirnov Test
The Kolmogorov-Smirnov test was used to validate the predictive value of the probability values and the Markov model. This non-parametric test can be used to compare a sample and a theoretical distribution, when both are cumulative frequency distributions (Burt et al., 2009). The test provides the probability that a deviation larger than the largest deviation can occur P(D>d), where d is the largest difference between both cumulative distributions.
The usual null-hypothesis using this test is that the sample (calibration data) has been drawn from the theoretical distribution (the validation data) and thus that the distributions are the same. In this case we want to prove that the two distributions are the same, so the null-hypothesis has to be that the two distributions are different, but the theory doesn’t provide the probability values, or ways to calculate them, for these interchanged hypotheses. So the p-values for the usual null-hypothesis were used, and the predicted distribution was accepted to be the same as the observed distribution when the probability value that both distributions are the same was 0.95 or larger.
In order to perform this test, the dataset was split up in even years and uneven years. The probability values of the stochastic matrices were calculated with only even years, we call this the calibration. The outcome of the Markov model using these calibrated matrices, was then compared to the actual data of the uneven years.
Therefore not only the states, but also the length of the streaks of states were determined. A streak is an amount of days with the same state directly following each other. The amount of days per year in a streak, was calculated for each different streak length. The cumulative distribution of days in all streaks was then compared between the calibration and validation data.
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Figure 3; The output of the Markov model based on the calibration data gets compared with the real-life validation data. Therefore the normalized cumulative distributions of the amount of warm days per year over the different streak lengths are plotted. The Kolmogorov-Smirnov test validates whether both distributions come from the same sample pool.
0 0,2 0,4 0,6 0,8 1 1 5 9 13 17 21 25 Po rtio n o f d ay s Length of streak
Cumulative amount of warm days in streak - normalized
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3. Results
3.1 Temperature and anomaly results
A quick analysis of the daily temperature data did confirm a trend of global warming. Based on the yearly mean annual temperature, all stations had a significant trend larger than 1,4 degrees Celsius per century with an average of 1,56 degrees Celsius, see table 3.
MEAN TEMPERATURE (℃) TREND PER CENTURY (℃) P(f) TREND De Kooy 9,50 1,57 1,1E-10 De Bilt 9,51 1,65 1,2E-16 Eelde 10,30 1,45 7,8E-11 Vlissingen 10,29 1,41 1,0E-10 Maastricht 9,66 1,72 2,9E-15
Table 3; Table of the annual mean temperature and its trend over the period 1907-2018 for the five measurement stations used in this research.
Based on the 25th and 75th percentile of the daily anomalies of De Bilt the categories of cold, warm and normal days were defined. Those limit values can be seen in table 5. So, days with an anomaly lower than -2,02℃ were defined as cold days. Days with anomalies between -2,02℃ and +2,14℃ were defined as normal days and days with anomalies larger than +2,14℃ as warm days. The different distributions can be seen in table 4.
COLD NORMAL WARM
DE BILT 25% 50% 25%
DE KOOY 22% 58% 20%
VLISSINGEN 22% 58% 20%
EELDE 22% 58% 20%
MAASTRICHT 29% 44% 27%
Table 4; The distribution of days in the states cold, normal and warm as defined by the borders of -2,02 and +2,14 degrees Celcius, based on the normal data.
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PERCENTILE ANOMALY NORMAL
25% -2,02
75% 2,14
Table 5; The anomaly values of the 25th and 75th percentile of all De Bilt daily anomalies.
3.2 Kolmogorov-Smirnov test
To validate the use of the Markov model, its predicting value was tested in a Kolmogorov-Smirnov test. This test compares the predicted distribution based on the even years with the distribution of the observed data of the uneven years. Based on the data of De Bilt, the amount of days in the three categories cold, normal and warm had to be compared. The distributions for the three states are given in figure 4.
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Figure 4; The normalized cumulative distributions of cold, normal and warm days of calibration and validation data in De Bilt. The Kolmogorov-Smirnov test validates whether both distributions can be drawn from the same sample.
0 0,2 0,4 0,6 0,8 1 1 5 9 13 17 21 25 29 33 37 41 45 49 53 Po rtio n o f t o ta l d ay s Length of streak
Amount of cold days in streak - normalized
Kalibration Validation 0 0,2 0,4 0,6 0,8 1 1 5 9 13 17 21 25 29 Po rtio n o f t o ta l d ay s Length of streak
Amount of normal days in streak - normalized
Kalibration Validation 0 0,2 0,4 0,6 0,8 1 1 5 9 13 17 21 25 Po rtio n o f t o ta l d ay s Length of streak
Amount of warm days in streak - normalized
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COLD
NORMAL
WARM
d 0,08339 0,03366 0,03439
n1 20 30 21
n2 54 31 25
P(D>d) 0,99996 1,00000 1,00000
Table 6; The Kolmogorov-Smirnov test confirms whether one sample distribution could be drawn from a theoretical distribution. The results validate that the Markov model predicts the same distribution as observed values.
The Kolmogorov-Smirnov test compared both distributions for each state. The results are given in table 6. The sample sizes differ from each other, whereas it stands for the longest streak of days that did occur. As discussed in the methods, a P(D>d) above 0,95 would mean that the model is validated. As can be seen, this is the case for each state and therefore the model was validated and used for further analysis.
3.3 Markov occurrence and probability values
For both normal and detrended data, the values of the occurrence and probability values are given in appendix 1. Appendix 2 gives the probability values of the extrapolated trend 1900, 2000 and 2100 probability values The output that the Markov given in appendix 3. In this appendix six graphics are given for each measuring station. The first row of three graphics is based on the raw data, the second is based on the detrended data. The three graphics in each row, are separate graphics for the three states cold, normal and warm. The graphic gives results for the output of the model in the years 1900, 2000 and 2100. This is based on the trend in the Markov matrices, as given in appendix 1. This trend is extrapolated to the three given years. The y-axis gives the cumulative amount of days per year, the x-axis the length of the streak of the given state. This way the graphic adds up to the total amount of days per year in the given state, and the shape of the line tells about the persistence of the state.
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Figure 5 shows the results of the normal state in the detrended data of station 235 (De Kooy). One could see how the amount of normal days increases slightly in 2100, with most days in a streak shorter than 15 days. The divergence between the three years mainly arises at streaks longer than 5 days.
3.4 Station results
In the next section, the results for each measuring stations will be discussed briefly.
De Kooy (235)
Normal
In the output of the cold data is an obvious shift in cold days to warm days. In an almost linear way, the cold days almost disappear between 1900 and 2100, and the warm days increase to 200 days per year in 2100. The amount of normal days stay around 200 days per year from 1900 to 2000, so the shift of normal days to warm days is about the same as the shift from cold to normal days. But the amount of normal days decreases to 140 days per year in 2100. The shape of the graph of warm days shows how the change from 1900 to 2000 mostly leads to more warm streaks of 5 to 10 days, and the change between 2000 and 2100 mostly leads to more streaks of 10+ days.
Detrended
In de results of the detrended data are less changes visible. The shapes of the distributions stay mainly the same. There is an small increase in normal days, that comes from a decrease in warm days. The amount of cold days mainly stays the same. In 2100 the amount of cold and warm days are equal at a 70 days per year, whereas the amount of normal days has increased to a 220 days per year. The small increase in
0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 30 Cumu lat iv e d ay s p er ye ar
Streak length (days)
NORMAL - 235 DETREND DATA
1900 2000 2100
Figure 5; Results of the Markov model are given in this format of the cumulative distribution of days per year over the different streak lengths.
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normal days indicates a decrease in temperature variability, but this increase normal days is only caused by a shift from warm days to normal days.
De Bilt (260)
Normal
When looking at the three graphs of the De Bilt normal data, immediately is visible how the amount of cold and normal days decrease largely, visually the same as in De Kooy. There is a large decline of cold and normal days, resulting in an incline of warm days. When looking at the graph of warm days, the different shape of the 2100 stands out. It rises slower than the 1900 and 2000 line in the 1-10 day streaks, and it looks as it is does not stabilise at the larger streaks. Starting with the latter, counting the amount of cold and normal days in 2100 (a total of 80 days) tells us that the line of warm days is only stabilised at 285 days. This confirms that this 2100 line isn’t yet fully stabilised and it means that streaks longer than 50 days are responsible for an additional 35 days per year. Looking at the small streaks it stands out how there are less streaks of <10 days than in 2000 and 1900, but the cumulative amount of warm days becomes larger. This is caused by a high persistence probability of warm days. The probability of a warm day following another warm day is so large, that short streaks of <10 occur less frequent.
Detrended
In contrast to the results of De Kooy, there is a small decline in normal days, whereas there is an increase in cold and warm days. Both cold and warm days increase from about 100 days per year in 1900 to an 80 days per year in 2100. The amount of normal days decreases from about 200 days per year in 1900 to about 160 days per year in 2100. The shift from normal to cold and warm days indicates an increasing temperature variability in De Bilt.
Eelde (280)
Normal
In Eelde a large shift from cold days to warm days can be seen. The amount of cold days almost disappears from about 100 days per year in 1900 to about 10 days per year in 2100. The amount of normal days stays about the same (200 days per year) from 1900 to 2000, but decreases with about 30 days per year between 2000 and 2100. The amount of warm days increases heavily from 55 days per year in 1900 to 110 days per year in 2000 and 185 days per year in 2100.
Detrended
In the detrended data a small shift from cold and warm days to normal days can be seen. The decline of warm days is larger than the decline of cold days, but the total amount of warm and cold days are around the same level of 60 days per year in 2100. Furthermore the shape of the normal graph stands out. It is
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remarkable more flat than the graphs of the cold and warm days, which indicates how normal days mainly occur in longer streaks. The shift to normal days indicates a trend of decreasing temperature variability in Eelde.
Vlissingen (310)
Normal
In the normal data of Vlissingen a large shift from cold days to warm days stands out as well. The amount of cold days almost disappears from 100 days per year in 1900 to 10 days per year in 2100. The amount of normal days stays the same just above 200 days per year between 1900 and 2000 but declines to 165 days per year in 2100. The amount of warm days increases linearly from about 55 days per year in 1900 to 185 days per year in 2100, only just stabilising at the streak length of 35 days.
Detrended
The shape of all three states stay the same. The amount of cold days decreases slightly from 70 days per year in 1900 to 65 days per year in 2000, but stays the same between 2000 and 2100. The amount of normal days increases slightly, from 210 days per year in 1900 to 235 days per year in 2100. The amount of warm days decreases from just above 80 days per year in 1900 to just above 60 days per year in 2100. The increase in normal days indicates a decreasing trend in temperature variability, although the decrease of warm days is four times larger than the decrease of cold days.
Maastricht (380)
Normal
In Maastricht a large shift from cold days to warm days occurs, but the change in normal days is smaller. The amount of cold days decreases from about 135 days per year in 1900 to just above 25 days per year in 2100. The amount of normal days decreases slightly from 160 days per year in 1900 to 140 days per year in 2100. The amount of warm days increases linearly between 1900 and 2100, from 75 days per year to just below 200 days per year. It is remarkable how the graph of the normal days is steeper than the graphs of the cold and warm days, which means that cold and warm days more often occur in longer streaks than normal days.
Detrended
Almost no changes are visible. One could say that cold days mainly occur in shorter streaks in 2100, but the change is small compared to other stations. Both cold and warm days stabilize just above 100 days per year, and normal days at 160 days per year. This indicates no change in temperature variability in Maastricht.
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3.5 Stations comparison
Table 7 summarizes the described Markov model output. It shows the average total amount of days in each state, based on the extrapolated transition probabilities for 1900, 2000 and 2100. The persistence of the temperature states is not visible in this table and therefore it does not show all results but it gives a good overview to compare the different stations.
It shows in the normal data how De Bilt with a big difference has the largest warming trend, followed by De Koo and Eelde. In the detrended data can be seen how Eelde and Vlissingen experience the least temperature variability, since their amount of normal days is the highest.
1900 2000 2100 STATION↓ STATE→ C N W C N W C N W De Kooy Normal 104 214 47 43 202 119 8 141 211 De Bilt Normal 129 188 48 57 173 136 11 70 284 Eelde Normal 102 209 55 42 213 110 10 169 186 Vlissingen Normal 101 210 53 42 210 113 11 167 186 Maastricht Normal 137 152 77 74 160 131 27 139 198 De Kooy Detrended 70 209 86 70 219 76 68 225 72 De Bilt Detrended 82 198 85 91 180 94 103 163 99 Eelde Detrended 72 207 85 66 223 76 64 241 60 Vlissingen Detrended 71 211 83 65 222 78 65 237 62 Maastricht Detrended 104 161 101 103 161 101 103 157 105
Table 7. The total average amount of days in each category as output of the Markov model. The trend in the probability matrices was extrapolated to create new probability values for 1900, 2000 and 2100. Those values were used as input for the Markov model to analyse the consequences of those trends.
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4. Discussion
4.1 On the model
The predicting value of models in climate change research is a complicated and debatable topic (Oerlemans, 2008; Perkins & Alexander, 2013). Climate change is not a linear process, it behaves quite chaotic, and the extrapolation of the trends is therefore complicated. The used Markov model creates insights into variability and the persistence of temperature anomalies, but this is all based on historical values and therefore not necessarily of predicting value. Furthermore, some choices made in the construction of the Markov model have impact on the results, and therefore in this section some of these choices will be discussed.
Detrending method
The method of detrending is the first subject to be discussed. The goal of detrending in this research was to correct for the increase in mean temperature due to climate change. Whereas this is possible with a linear or second/higher order correction, in this research is chosen to use an 11-year running mean. This could directly be combined with the correction for the seasonal cycle and was therefore a practical method as well. Furthermore the anomaly categorization also had a detrending effect, since the upper border for a cold day was defined as -2,04°C and the lower border for the warm category as +2.14°C. This is the case because those borders were defined on the 25th and 75th percentiles of normal daily data. In this research is chosen to hold on to the same category borders for both normal as detrended analysis, to be able to compare them fairly. In further research one should think about the desirability of such a double detrending measure, since the detrended results on itself could be more interesting than the comparison with the normal data.
Furthermore, the choice of category boundaries is somewhat arbitrary. Whereas Boersma (2019) chose to set the 33,33rd percentile and the 66,66th percentile as category boundaries, in this study is chosen for the 25th and the 75th percentile. The amount of categories could also be chosen differently. More research with different category boundaries could give different results and would therefore be insightful. To add two more categories (extremely warm and cold) could make the results more interesting.
Running length
Furthermore, conclusions about variability in the outcome of the model are very dependent on the running length of the model. A larger running length leads to less variability, since coincidental outliers will have less significance in the larger sample. But variability in the actual weather could be the result of coincidental outliers, the weather is not a perfect gaussian phenomenon. Further research could look into
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the optimal running length for such a model so that the results can be interpreted correctly. When the correct running length for the model is determined, more specific statements about changing temperature variability can be done.
4.2 Trend analysis and extrapolation
The major results of this study are the extrapolated temperature state results in the years 1900, 2000 and 2100. This extrapolation is based on the trends in the 27 transition matrix probability values. Three important remarks should be made about these trends, about the way of calculating these trend values and about the conclusions that can been drawn of the final results.
The first remark should be made about the results of the Markov model. The transition occurrence and probability values of the 27 transitions have individually been evaluated for trends and significance, but those transitions combined are used as input for the Markov model. This model provides distributions for the three different temperature categories as output. This research compared the 1900, 2000 and 2100 distributions but couldn’t test the statistical significance of the difference between those distributions. With the Kolmogorov-Smirnov test this research has verified how the created Markov model correctly predicted the historical values, but further verification on the variability results was not possible. Further research could improve predicting conclusions when coming up with a statistical test for this matter. The second remark is about the way of calculating the trends in the individual transition values. The trends in the transition probabilities are calculated in a linear way using a regression analysis on the yearly transition probabilities from 1907 till 2018. This results in a single trend value that has been extrapolated from 1900 to 2100. In appendix 2 can be seen how some of the probabilities turned into negative numbers in 1900 or 2100, which exemplifies how the extrapolation is too large. A higher order extrapolation could lead to more accurate results.
The third remark is about the statistical significance of the trends in the transition probabilities. For each of the 27 matrix values a trend value and a significance probability has been calculated, those are given in appendix 1.
In the normal data about 14 to 16 of the transition occurrence values and 10 to 16 of the transition probabilities have a significant trend at the different measuring stations. When looking at the occurrence percentages of the non-significant trends, it stands out how almost each of them has an occurrence rate of 0.5% or smaller. For those transitions there were not enough data points to calculate significant trends.
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But since their occurrence is so low, their relevance is small enough to accept the non-significance of the trends.
When looking at the detrended results of the different measuring stations, only 0 to 4 of the transition occurrence values and transition probabilities have a significant trend. For both De Kooy and Maastricht not one transition has a significant trend in their occurrence or transition probability. The results of the detrended data therefore will lead to less robust conclusions.
4.3 On the results
Normal data
All the measuring stations show a clear warming trend. The amount of cold and normal days decreases, and the amount of warm days increases heavily. The shapes of the graphs of the warm days show how more longer streaks of warm days occur. This indicates how the amount of extremely warm periods is increasing. But the increasing length of the warm periods cannot be interpreted without the large increase in the amount of warm days, and thereby does not necessarily show an increasing temperature variability.
Detrended data
When looking at the results of the detrended data, the results of this study agree with the view that there is no evidence for an increase in temperature variability. After detrending the data, and thereby correcting for the increase in mean temperature, the warming shift seen in the results of the normal data mainly disappeared. 3 of the 5 stations show an increase in normal days and a decrease in cold and warm days, which can be explained as a lower temperature variability. The detrended results of Maastricht show almost no change at all. Only the detrended results of De Bilt show a decrease in normal days, together with an evenly spread increase of cold and warm days, which indicates a small increase in temperature variability.
Study comparison
Whether the daily temperature anomaly variability is changing, is still in wide debate (Perkins & Alexander, 2013). This study disagrees with the claims that temperature variability in increasing (Hansen, 2012; Schär et al., 2004). Three of the stations even indicate a decrease in the temperature variability.
Differences in approaches make it hard to compare studies on the topic of extreme temperatures. Horton et al. (2016) argues in a review of research on extreme temperature events that there is a discrepancy between studies on increasing extreme temperature events and studies on detrended temperature variability.
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Regional differences make it hard to compare studies as well (Perkins & Alexander, 2013). Even within this research, the detrended results of De Bilt show a trend that is the opposite of the trend at the other stations. Such regional difference can also be the case on a larger scale, and thus it is hard to compare this local study to conclusions of global trend studies.
Within the Netherlands studies have focused only on mean temperature warming and increasing occurring extreme heat events (Visser, 2007; Planbureau voor de Leefomgeving, 2012), there are no known results on changing temperature variability to compare with.
Category distribution output
The method of calculating transition probability matrices to apply in a stochastic Markov model, is not a common method in the research of temperature extremes. A more technical and statistical approach is more common, leading to a quite technical debate as well. Because of the categorization, the output of the model is also given in those categories, which isn’t reversible to numeric temperature data. Therefore it is also hard to compare the findings in detail with other studies.
5. Conclusion
Based on the daily mean temperature data of 5 measuring stations in the Netherlands, this study successfully constructed Markov-like 2-stage transition matrices of both the transition occurrence and the transition probability of 3 anomaly categories. The output of the Markov model based on even years successfully simulated real life measurements of odd years. This has been verified by a Kolmogorov-Smirnov test.
There has been a clear increase in daily mean temperature at 5 weather stations in the Netherlands, this can be seen in the Markov model output based on the normal data. In this analysis the anomalies were defined relative to the averages of 1960 – 1989, and relative to this period, cold days will almost disappear whereas warm days will increase and will appear more and in longer streaks. This will lead to more and longer heat waves in the Dutch summers.
The analysis based on the data detrended for the annual mean temperature increase gave insights into the changing temperature variability. Both the transition matrix values and the model output were interesting to learn about the changing occurrence and persistence of the three states. The results show a clear increase of warm periods. This increase is mainly caused by an increase in mean annual temperature and not by an increasing temperature variability. Only one of the five stations (De Bilt) gave
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a result of a decreasing amount of normal days, and an increasing amount of cold and warm days. This could show larger temperature variability, but the increase was only small. The detrended results at station Maastricht showed almost no change at all in the distribution. The other three stations showed a decrease or in the amount of cold/warm days, with an increasing amount of normal days. Those changes differed in size between the 3 stations, but could be indicate a decreasing temperature variability at those stations. No major conclusions can be drawn about isolated change in persistence of extreme temperatures. Further research with a different analysis of the model output could give more results.
The model provides insight into the way more warm periods are occurring in the Netherlands. Contrary to the current belief temperature variability is not increasing. There are even indications that it is decreasing at 3 of the 5 stations.
6. Literature list
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Appendix 1: Occurrence and transition probability values
for two stage transitions at 5 measuring stations in the
Netherlands.
Station De Kooy (235)
Occurren ce tran sitio n Trend (p e r 1 00 y ear) P (f) trend P ro b ab ilit y tran sitio n Trend (p er 1 00 y ear) P (f) trend Occurren ce tran sitio n Trend (p er 1 00 y ear) P (f) trend P ro b ab ilit y tran sitio n Trend (p er 1 00 y ear) P (f) trend Normal Detrended CCC 9,5% -8,4% 0,0% 68,9% -9,5% 0,4% 10,1% -0,3% 86,9% 68,7% 0,0% 84,5% CCN 3,3% -2,4% 0,0% 29,4% 7,2% 1,9% 3,8% -0,2% 57,5% 30,1% 0,0% 95,6% CCW 0,1% 0,0% 86,6% 1,7% 2,3% 0,9% 0,1% 0,0% 57,5% 1,2% 0,0% 42,3% CNC 0,9% -0,9% 0,0% 16,1% -4,2% 12,4% 1,1% -0,4% 7,4% 17,3% -0,1% 6,3% CNN 3,9% -2,5% 0,0% 71,2% -7,4% 3,5% 4,3% 0,5% 16,1% 71,8% 0,1% 10,2% CNW 0,6% 0,0% 83,4% 12,7% 11,6% 0,0% 0,6% 0,0% 77,4% 10,8% 0,0% 92,7% CWC 0,0% 0,0% N/A 16,5% 2,3% 63,2% 0,0% 0,0% 34,0% 18,4% 0,0% 82,1% CWN 0,1% 0,0% 67,2% 32,3% -10,0% 24,4% 0,1% 0,0% 80,2% 32,9% 0,0% 94,9% CWW 0,1% 0,0% 77,3% 51,2% 7,7% 44,4% 0,1% 0,0% 77,5% 48,7% 0,0% 94,1% NCC 3,4% -2,4% 0,0% 60,1% -4,9% 19,2% 3,8% -0,1% 74,4% 62,0% 0,0% 20,7% NCN 2,2% -1,0% 0,0% 38,5% 4,1% 23,9% 2,3% 0,3% 28,8% 36,8% 0,0% 16,7% NCW 0,1% 0,0% 93,2% 1,4% 0,8% 38,1% 0,1% 0,0% 76,5% 1,2% 0,0% 75,8% NNC 4,2% -2,3% 0,0% 9,8% -4,9% 0,0% 4,6% 0,6% 11,5% 10,5% 0,0% 32,3% NNN 33,8% -4,9% 1,5% 77,4% -2,4% 5,3% 34,6% 1,0% 62,7% 77,6% 0,0% 57,4% NNW 5,4% 2,2% 0,0% 12,8% 7,3% 0,0% 5,1% 0,2% 64,0% 11,8% 0,0% 76,3% NWC 0,0% 0,0% 11,0% 0,2% 0,3% 23,4% 0,0% 0,0% 37,0% 0,3% 0,0% 42,7% NWN 2,8% 0,8% 0,5% 37,5% -4,2% 19,6% 2,7% 0,3% 30,4% 38,8% 0,0% 41,6% NWW 4,7% 2,4% 0,0% 62,3% 3,9% 22,6% 4,4% -0,3% 41,9% 60,9% 0,0% 37,4% WCC 0,0% 0,0% 69,0% 40,9% 7,9% 23,6% 0,0% 0,0% 19,5% 42,8% -0,1% 21,9% WCN 0,0% 0,0% 81,9% 30,2% -2,4% 57,6% 0,0% 0,0% 54,2% 30,4% 0,1% 16,5% WCW 0,0% 0,0% 34,4% 28,9% -5,5% 16,7% 0,0% 0,0% 34,4% 26,8% 0,0% 70,9% WNC 0,5% -0,2% 5,3% 7,0% -6,0% 0,0% 0,6% 0,0% 66,7% 8,1% 0,0% 62,2% WNN 5,6% 2,4% 0,0% 74,5% 1,2% 63,6% 5,3% 0,2% 51,7% 73,7% 0,0% 9,1% WNW 1,5% 1,0% 0,0% 18,5% 4,8% 5,2% 1,4% -0,2% 42,1% 18,2% 0,0% 15,2% WWC 0,0% 0,0% 40,3% 0,3% -0,5% 7,8% 0,0% 0,0% 9,0% 0,3% 0,0% 15,1% WWN 4,8% 2,4% 0,0% 31,4% -13,0% 0,0% 4,4% -0,3% 51,1% 32,3% 0,0% 53,6% WWW 12,4% 13,8% 0,0% 68,3% 13,5% 0,0% 10,3% -1,2% 40,6% 67,4% 0,0% 62,5%29
Station De Bilt (260)
Occurren ce tran sitio n Trend (p er 1 00 y ear) P (f) trend P ro b ab ilit y tran sitio n Trend (p er 1 00 y ear) P (f) trend Occurren ce tran sitio n Trend (p er 1 00 y ear) P (f) trend P ro b ab ilit y tran sitio n Trend (p er 1 00 y ear) P (f) trend Normal Detrended CCC 9,5% -9,3% 0,0% 69,0% -14,5% 0,0% 9,5% 1,5% 31,7% 70,2% -1,0% 63,9% CCN 3,3% -2,6% 0,0% 29,3% 11,4% 0,0% 3,3% 0,7% 1,5% 28,4% 0,6% 76,2% CCW 0,1% 0,0% 93,6% 1,7% 3,1% 0,8% 0,1% 0,1% 23,7% 1,4% 0,4% 35,9% CNC 0,9% -0,7% 0,0% 14,9% -4,0% 6,7% 0,9% 0,1% 77,0% 16,3% -0,3% 88,2% CNN 3,9% -2,8% 0,0% 68,9% -11,4% 0,1% 3,9% 0,4% 13,9% 68,9% 0,0% 99,0% CNW 0,6% 0,0% 84,4% 16,2% 15,4% 0,0% 0,6% 0,1% 50,7% 14,8% 0,4% 86,3% CWC 0,0% 0,0% 89,6% 15,0% 3,5% 43,4% 0,0% 0,0% 89,6% 10,7% -4,1% 33,0% CWN 0,1% 0,0% 90,1% 29,2% 2,6% 75,2% 0,1% 0,1% 2,2% 33,7% 18,2% 5,7% CWW 0,1% 0,0% 45,1% 55,9% -6,1% 53,7% 0,1% 0,0% 47,6% 55,6% -14,0% 18,2% NCC 3,4% -2,5% 0,0% 63,6% -9,2% 0,9% 3,4% 0,8% 0,6% 65,5% 4,8% 5,9% NCN 2,2% -1,0% 0,0% 35,7% 8,4% 1,5% 2,2% -0,1% 55,5% 33,8% -4,7% 6,2% NCW 0,1% 0,0% 8,9% 0,8% 0,9% 33,2% 0,1% 0,0% 62,9% 0,7% -0,1% 72,4% NNC 4,2% -2,7% 0,0% 14,3% -4,5% 0,0% 4,2% 0,4% 14,3% 14,4% 3,6% 0,0% NNN 33,8% -7,9% 0,0% 69,6% -4,2% 0,2% 33,8% -5,7% 0,0% 69,8% -5,6% 0,0% NNW 5,4% 0,2% 52,6% 16,1% 8,7% 0,0% 5,4% -0,1% 72,7% 15,7% 2,0% 1,6% NWC 0,0% 0,0% 99,4% 0,7% -0,2% 72,6% 0,0% 0,0% 83,5% 0,6% -0,1% 85,8% NWN 2,8% -0,5% 4,3% 34,3% -13,1% 0,0% 2,8% 0,0% 99,2% 34,7% -0,1% 98,3% NWW 4,7% 1,2% 0,1% 65,0% 13,3% 0,0% 4,7% 0,1% 77,5% 64,7% 0,1% 96,2% WCC 0,0% -0,1% 5,8% 52,3% -10,2% 26,1% 0,0% 0,0% 49,3% 55,1% -6,1% 53,6% WCN 0,0% 0,0% 27,1% 30,4% 1,5% 83,5% 0,0% 0,0% 41,0% 29,1% 0,8% 92,5% WCW 0,0% 0,0% N/A 17,2% 8,6% 5,6% 0,0% 0,0% 31,3% 15,8% 5,4% 28,2% WNC 0,5% -0,1% 39,2% 12,3% -3,5% 6,0% 0,5% 0,2% 16,9% 14,0% 2,4% 18,0% WNN 5,6% 0,4% 29,9% 71,2% -3,2% 25,8% 5,6% -0,1% 78,7% 69,5% -2,8% 25,8% WNW 1,5% 0,5% 1,1% 16,5% 6,7% 0,3% 1,5% 0,1% 68,2% 16,5% 0,4% 81,8% WWC 0,0% -0,1% 1,0% 1,0% -1,6% 0,0% 0,0% -0,1% 21,8% 1,0% -0,3% 35,8% WWN 4,8% 1,3% 0,1% 30,0% -16,0% 0,0% 4,8% 0,1% 75,3% 31,5% -3,2% 9,3% WWW 12,4% 26,8% 0,0% 69,0% 17,6% 0,0% 12,4% 1,7% 12,9% 67,5% 3,5% 7,8%30
Station Eelde (280)
Occurren ce tran si tio n Trend (p er 1 00 y ear) P (f) trend P ro b ab ilit y tran sitio n Trend (p er 1 00 y ear) P (f) trend Occurren ce tran sitio n Trend (p er 1 00 y ear) P (f) trend P ro b ab ilit y tran sitio n Trend (p er 1 00 y ear) P (f) trend Normal Detrended CCC 9,5% -8,1% 0,0% 67,7% -11,9% 0,1% 9,5% -0,9% 57,4% 68,0% -4,0% 21,3% CCN 3,3% -2,4% 0,0% 31,4% 11,1% 0,1% 3,3% 0,0% 97,4% 31,3% 4,3% 15,5% CCW 0,1% 0,0% 65,0% 0,9% 0,9% 46,4% 0,1% 0,0% 38,7% 0,7% -0,3% 71,9% CNC 0,9% -0,7% 0,0% 14,0% -5,8% 2,6% 0,9% 0,0% 83,8% 14,4% 0,4% 84,6% CNN 3,9% -2,1% 0,0% 75,6% 3,3% 29,1% 3,9% 0,2% 65,6% 75,6% 3,0% 31,6% CNW 0,6% -0,2% 6,4% 10,4% 2,5% 25,1% 0,6% -0,2% 16,7% 10,0% -3,4% 10,9% CWC 0,0% 0,0% 31,9% 25,7% -4,1% 42,4% 0,0% 0,0% 83,1% 25,7% -1,3% 78,0% CWN 0,1% 0,0% 65,8% 29,6% 2,6% 67,5% 0,1% 0,0% 11,5% 31,0% 9,1% 13,6% CWW 0,1% 0,0% 85,5% 44,7% 1,5% 86,2% 0,1% 0,0% 43,5% 43,4% -7,8% 33,1% NCC 3,4% -2,3% 0,0% 63,1% -7,1% 6,8% 3,4% 0,2% 64,6% 64,3% 1,4% 66,8% NCN 2,2% -0,6% 0,8% 35,8% 7,9% 3,4% 2,2% 0,1% 77,0% 34,9% -0,6% 86,1% NCW 0,1% 0,0% 29,3% 1,1% -0,8% 45,6% 0,1% 0,0% 38,0% 0,8% -0,9% 39,0% NNC 4,2% -1,8% 0,0% 9,3% -4,2% 0,0% 4,2% 0,5% 19,0% 10,0% 0,3% 73,0% NNN 33,8% -2,3% 29,1% 77,7% -0,4% 74,0% 33,8% 3,5% 6,7% 77,5% 1,6% 16,3% NNW 5,4% 1,8% 0,0% 12,9% 4,6% 0,0% 5,4% -0,1% 75,7% 12,5% -1,9% 6,8% NWC 0,0% 0,0% 18,0% 0,3% -0,5% 21,9% 0,0% 0,0% 23,9% 0,4% -0,5% 24,3% NWN 2,8% 0,2% 53,0% 35,2% -9,7% 0,3% 2,8% 0,1% 71,1% 37,5% 2,1% 53,7% NWW 4,7% 2,4% 0,0% 64,4% 10,2% 0,2% 4,7% -0,4% 32,9% 62,1% -1,6% 63,3% WCC 0,0% -0,1% 3,1% 43,4% -10,5% 16,0% 0,0% -0,1% 0,8% 43,8% -9,0% 23,8% WCN 0,0% 0,0% 40,5% 30,5% 1,8% 73,7% 0,0% 0,0% 29,3% 31,1% -0,1% 98,0% WCW 0,0% 0,0% 21,0% 26,1% 8,6% 3,3% 0,0% 0,0% N/A 25,1% 9,1% 2,8% WNC 0,5% -0,4% 0,0% 7,2% -8,5% 0,0% 0,5% -0,3% 0,3% 7,7% -5,1% 0,3% WNN 5,6% 2,1% 0,0% 75,3% 2,5% 35,3% 5,6% 0,2% 53,5% 76,0% 5,0% 6,2% WNW 1,5% 1,0% 0,0% 17,4% 6,0% 1,3% 1,5% 0,0% 82,1% 16,3% 0,1% 97,4% WWC 0,0% -0,1% 15,7% 0,5% -0,5% 12,0% 0,0% -0,1% 1,1% 0,6% -0,8% 7,9% WWN 4,8% 2,4% 0,0% 32,9% -10,7% 0,0% 4,8% -0,3% 47,2% 34,1% 3,7% 18,4% WWW 12,4% 11,3% 0,0% 66,6% 11,2% 0,0% 12,4% -2,3% 18,4% 65,3% -2,9% 29,5%31
Station Vlissingen (310)
Occurren ce tran sitio n Trend (p er 1 00 y ear) P (f) trend P ro b ab ilit y tran sitio n Trend (p er 1 00 y ear) P (f) trend Occurren ce tra n sitio n Trend (p er 1 00 y ear) P (f) trend P ro b ab ilit y tran sitio n Trend (p er 1 00 y ear) P (f) trend Normal Detrended CCC 9,2% -8,3% 0,0% 68,3% -9,6% 0,1% 9,9% -0,9% 58,8% 68,4% -2,4% 42,6% CCN 3,5% -2,4% 0,0% 31,3% 9,6% 0,2% 3,8% -0,1% 71,8% 31,3% 2,2% 46,7% CCW 0,0% 0,0% 95,8% 0,4% 0,1% 86,5% 0,0% 0,0% 36,1% 0,3% 0,2% 39,0% CNC 0,8% -0,7% 0,0% 14,0% -5,0% 3,7% 0,9% -0,1% 72,6% 14,4% -0,9% 70,9% CNN 4,1% -2,2% 0,0% 75,3% 0,6% 84,3% 4,5% 0,2% 57,9% 76,0% 3,4% 18,8% CNW 0,5% -0,2% 13,4% 10,7% 4,4% 6,6% 0,6% -0,1% 25,6% 9,6% -2,6% 22,6% CWC 0,0% 0,0% 31,9% 25,8% 1,2% 79,3% 0,0% 0,0% 69,7% 26,4% 1,0% 83,1% CWN 0,0% 0,0% 95,9% 29,8% 2,5% 67,1% 0,0% 0,0% 6,5% 29,1% 10,5% 5,7% CWW 0,1% 0,0% 38,6% 44,4% -3,8% 64,2% 0,1% 0,0% 30,0% 44,5% -11,5% 14,8% NCC 3,5% -2,4% 0,0% 63,7% -9,2% 2,1% 3,8% 0,0% 97,0% 64,3% -2,3% 49,8% NCN 1,9% -0,6% 2,2% 35,6% 9,8% 1,4% 2,1% 0,2% 50,3% 35,2% 2,8% 39,6% NCW 0,0% 0,0% 13,2% 0,7% -0,6% 27,1% 0,0% 0,0% 25,0% 0,5% -0,5% 33,6% NNC 4,1% -1,8% 0,0% 9,4% -4,3% 0,0% 4,5% 0,5% 12,8% 10,2% 0,4% 61,4% NNN 34,6% -1,9% 34,4% 77,5% -0,3% 82,0% 35,2% 3,2% 8,2% 77,6% 1,1% 32,3% NNW 5,6% 1,7% 0,0% 13,0% 4,6% 0,0% 5,4% 0,0% 99,3% 12,3% -1,6% 11,6% NWC 0,0% 0,0% 22,4% 0,3% -0,4% 20,8% 0,0% 0,0% 49,3% 0,4% -0,2% 68,5% NWN 2,6% 0,1% 74,9% 35,3% -10,3% 0,1% 2,7% 0,2% 51,6% 37,7% 2,5% 41,9% NWW 4,9% 2,4% 0,0% 64,5% 10,7% 0,1% 4,5% -0,4% 32,7% 62,0% -2,3% 44,6% WCC 0,1% -0,1% 3,5% 42,8% -7,0% 36,3% 0,1% -0,1% 5,9% 43,5% -3,8% 62,3% WCN 0,0% 0,0% 38,7% 31,9% 1,0% 86,1% 0,0% 0,0% 14,2% 32,0% -4,4% 46,8% WCW 0,0% 0,0% N/A 25,4% 5,9% 15,0% 0,0% 0,0% N/A 24,5% 8,3% 5,2% WNC 0,5% -0,4% 0,0% 7,0% -8,4% 0,0% 0,5% -0,4% 0,1% 7,7% -5,6% 0,1% WNN 5,6% 2,1% 0,0% 75,6% 3,5% 18,9% 5,4% 0,3% 34,5% 75,5% 6,1% 1,2% WNW 1,4% 0,9% 0,0% 17,4% 4,8% 5,1% 1,2% -0,1% 71,4% 16,8% -0,5% 82,4% WWC 0,1% -0,1% 8,2% 0,4% -0,6% 1,5% 0,1% -0,1% 1,8% 0,5% -0,6% 5,2% WWN 4,9% 2,4% 0,0% 32,7% -11,0% 0,0% 4,5% -0,3% 44,0% 33,8% 2,0% 47,6% WWW 12,0% 11,5% 0,0% 67,0% 11,6% 0,0% 10,0% -2,0% 19,5% 65,7% -1,4% 62,4%32
