Table 2. Adjusted period and Pearson correlation coefficients (P.C.C.) for observed mean annual temperatures at the five observatories for the reference period 1971-1980 and for the whole period of study 1958-2010 after adjustments.

Figure 3. Regressions of adjusted long-term time series of mean annual temperature data (1958-2010) for the five observatories.

detection of abrupt changes. Regression analysis was performed for the trends and the Mann-Kendall Rank Statistic Test was used for the examination of their significance.

3.2.1. Cumulative Sum Charts (CUSUM) and Bootstrapping

The cumulative sum charts (CUSUM) and bootstrapping procedure was performed as suggested by Taylor [54]. Let represent N data points of a time series, are iteratively computed as follows:

1) The average of is given by

(1)

2) Let be equal to zero 3) Compute recursively as follows:

(2)

A section of the CUSUM chart with an increasing slope will indicate a period of time where the values tend to be above the overall average. Likewise, a section with a decreasing slope will indicate a period of time where the values tend to be below the overall average. The confidence level can be determined by performing bootstrap analysis [55-57].

3.2.2. Mann Kendal Rank Statistic

There are a number of statistical tests that are used for the analysis of trend such as Spearman rank statistic test, Cramer’s test, Pearson’s test, Pettit’s test, Buishand’s test, Von Neumann’s test, Standard normal homogeneity test (SNHT), Mann Kendall’s Rank Statistic test. The latest is considered to be the most appropriate for the analysis of climatic changes in climatological time series or for the detection of climatic abrupt changes [58].

The data were analyzed in order to identify significant long-term trends using sequential version of the MannKendall rank statistics, the effective application of which includes the following steps in sequence:

i) The values of the original series are replaced by their ranks arranged in ascending order

ii) The magnitudes of, () are compared with, (). At each comparison number of cases is counted and denoted by.

iii) A statistic is, therefore, defined as follows:

(3)

iv) The distribution of the test statistic has a mean and a variance as

Figure 4. Time series of adjusted mean annual temperature (˚C) for the period 1958-2010.

and

(5)

v) The sequential values of the statistic are then computed as

(6)

Herein, is a standardized variable that has zero mean and unit standard deviation, and there-fore, its sequential behavior fluctuates around zero level. Furthermore, follows a Gaussian normal distribution.

vi) Similarly, the values of are computed backward starting from the end of the series.

The intersection of the curves and localizes the change (in fact this change represents an abrupt climatic change) and allows the identification of the year when a trend or change initiates. When the values of are higher—in absolute value—than 1.96, it implies a trend or a change in the time series. In the absence of any trend in the series, the graphical representation of and gives curves which overlap several times.

3.2.3. Kendall Rank Correlation

Let be a set of joint observations from two random variables X and Y respectively, such that all the values of the couple (x_{i}, y_{i}) are unique. Any pair of observations (x_{i}, y_{i}) and (x_{j}, y_{j}) are said to be concordant if the ranks for both elements agree: that is, if both x_{i} > x_{j} and y_{i} > y_{j} or if both x_{i} < x_{j} and y_{i} < y_{j}. They are said to be discordant, if x_{i} > x_{j} and y_{i} < y_{j} or if x_{i} < x_{j} and y_{i} > y_{j}. If x_{i} = x_{j} or y_{i} = y_{j}, the pair is neither concordant, nor discordant. Kendall’s rank correlation measures the strength of monotonic association between the vectors X and Y.

In the case of no ties in the X and Y variables, if N represents the number of concordant pairs and M represents the number of discordant pairs, Kendall’s rank correlation coefficient τ is defined as:

(7)

or

(8)

where S, is the Kendall score given by:

(9)

where denotes the sign function and D is the maximum possible value of S. In the case where there are no ties among either the (x_{i}) or the (y_{i}),

(10)

D is called the denominator and corresponds to the total number of pairs, so the coefficient must be in the range [–1, +1]. If the agreement between the two rankings is perfect (i.e., the two rankings are the same) the coefficient has value 1. If the disagreement between the two rankings is perfect (i.e., one ranking is the reverse of the other) the coefficient has value –1. If X and Y are independent, then we would expect the coefficient to be approximately zero.

More general, if there are distinct ties of extent, among the (x_{i}) and distinct ties of extent, among the (y_{i}) then:

(11)

where

(12)

and

(13)

In the case where there are no ties in either ranking, it is well known that under the null hypothesis that there is no trend in the data i.e. no correlation between considered variable X and Y, each ordering of the dataset is equally likely, the statistic S may be well approximated by a normal distribution with the mean and the variance as follows:

(14)

(15)

provided that.

For situation where ties occur, then is extended to a more complicated including form adjusting terms for tied or censored data:

(16)

For trend test, the variable Y can be time. The presence of statistically significant trend is evaluated using the Z value. This statistic is used to test the null hypothesis such that no trend exists. The standardized test statistic Z is given by:

(17)

A positive Z indicates an increasing trend in the timeseries, while a negative Z indicates a decreasing trend. To test for either increasing or decreasing monotonic trend at P significant level, the null hypothesis is rejected if the absolute value of Z is greater than, where is obtained from the standard normal cumulative distribution tables and represents the standard normal deviates and p is the significant level for the test. The test of the null hypothesis H_{0}:τ = 0 is equivalent to testing H_{0}:Z = 0.

3.2.4. Regression Analysis

A simple linear regression model was applied to show the long-term annual trends in temperature for the periods of study. Change in temperature over the study period given by the estimated slope of the regression line and the coefficient of determination () were computed from the regression model [59]. The Mann-Kendall Rank Statistic Test was used for the examination of their significance [60].

4. Results and Discussion

The results of a change-point analysis for the annual mean temperature for the five observatories are presented in Table 3. Each table gives a level associated with each change; the level is an indication of the importance of the change. Using the Change Point Analyzer no departure from the independent error structure and no outlier’s assumptions were found. The CUSUM charts are presented in Figure 5. This analysis shows on one side that the central plateau observatories of KIGALI, RUBONA and GITARAMA exhibit an abrupt change in the year 1977 both at a confidence level of 100%, respectively at level 2, 1 and 2. Prior to the change in 1977, annual mean temperatures for the three stations were respectively 20.32˚C, 19.05˚C and 18.15˚C, while after the change the temperatures became 20.90˚C, 19.76˚C and 18.77˚C, respectively. On the other side, the western side observatories of GISENYI and KIBUYE exhibit an abrupt change in the year 1979 both at a confidence level of

(a)(b)(c)(d)(e)

Table 3. Change-point analysis of mean annual temperature for Kigali (a), RUBONA (b), GITARAMA (c), GISENYI (d), KIBUYE (e).

100%, respectively at level 1 and 2. Prior to the change in 1979, annual mean temperatures for the two stations were respectively 19.66˚C and 21.27˚C, while after the change the temperatures became 20.44˚C and 21.64˚C, respectively. The Mann Kendall’s Rank Statistic was furthermore used for the detection of abrupt change and trends. Results presented in Figure 6, were similar to those obtained using the cumulative sum charts (CUSUM) and bootstrapping method.

It was important to analyze the two sub-periods separately based on the most important change point detected in 1977-1979. Results obtained from the Mann-Kendall test revealed a not very significant cooling trend during the period ranging from 1958 to 1977-1979 for all observatories. A statistically significant positive trend at p < 10^{–4} level was observed after 1977-1979. Mann-Kendall statistics for the five observatories are presented in Table 4. Trends of annual mean temperatures for the five observatories are presented in Figure 7. Slope and regression coefficient R^{2} for the trends of mean annual temperature for the five observatories corresponding to the periods 1958-1977, 1978-2010 and 1958-2010 are also presented in Table 5. Kigali, the Capital City of Rwanda, presented for the period 1978-2010 the highest values of the slope (0.0455˚C/year) with high value of coefficient of determination (R^{2} = 0.6798), the Kendall’s tau statistic (M-K = 0.62), the Kendall Score (S = 328) with a twosided p-value far less than the confidence level α of 5%).

Trends of Rwanda air temperature anomalies for the

Figure 5. CUSUM chart for the mean annual maximum temperature for the five observatories.

Figure 6. Trends of mean annual temperature (1958-2010) at the five observatories.

period 1958-2010 have been compared to trends of Global air temperature anomalies for the period 1958-2010 as it is shown in Figure 8. Results indicate that the trends observed in the increasing of temperature are roughly similar. The observed warming is most likely explained by the growing population accompanied by the increasing emission of green house gases, and the escalating urbanization and industrialization the country has experienced during the last decades, especially the capital City Kigali, during the last decades.

5. Conclusion

Long term time series of annual mean temperature has been compiled and analyzed. The data set has been carefully quality controlled and passed an intensive homogeneity assessment. Long term time series have been adjusted for further statistical analysis. There is a clear indication that climate change has occurred in Rwanda. Statistically significant abrupt changes and trends have been detected. The major change point in the annual mean temperature occurred around 1977 for all observa-

Figure 7. Annual trends of mean annual temperature (1958-2010) at the five observatories.

Table 4. Mann-Kendall statistics for the five observatories. M-K, S and p are respectively the Kendall’s tau statistic, the Kendall score and the two-sided p-value. (+ and – indicate respectively positive trend and negative trend, ^{+}, ^{*} indicate respectively significant and not significant at confidence level α = 0.05).

Table 5. Slope (˚C/year) and regression coefficient R^{2} for the trends of mean annual temperature for the five observatories corresponding to the periods 1958-1977, 1978-2010 and 1958-2010.

Figure 8. Global air temperature 2010 anomalies + 0.47˚C (left) compared to Rwanda air temperature 2010 anomalies + 0.97˚C (right) for the period 1958-2010. Global data are extracted from Phil Jones http://www.cru.uea.ac.uk/cru/info/warming/.

tories. The analysis of the annual mean temperature showed a not very significant cooling trend during the period ranging from 1958 to 1977 for all observatories. A significant warming trend was furthermore observed for the period after the year 1977 for all observatories where Kigali, the Capital of Rwanda, presented the highest values of the slope (0.0455/year) with high value of coefficient of determination (R^{2} = 0.6798), the Kendall’s tau statistic (M-K = 0.62), the Kendall Score (S = 328) with a two-sided p-value far less than the confidence level α of 5%). The observed warming is most likely explained by the growing population accompanied by the increasing emission of green house gases, and the escalating urbanization and industrialization the country has experienced, especially the capital City Kigali, during the last decades.

6. Acknowledgements

The author is grateful to the National Meteorological Service of the Ministry of Infrastructure (Rwanda), for providing relevant information for this article. The present study has been supported by the Research Commission of the National University of Rwanda (NUR) through a partnership with the Swedish International Agency SIDA/SAREC.

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