12 PLATFORM VOLUME 5 NUMBER 4 2021 e-ISSN: 26369877
INTRODUCTION
Hydrological analysis and Water resources development largely depend on hydrological and meteorological data. To achieve reliable and quality results for practical applications, such data should be free from outliers, homogeneous, and consistent to carry out frequency analyses or simulate a hydrological system [1]-[2]. Long-term precipitation data may be suffered from non-climatic factors that result in inhomogeneity of historical records. The non-climatic factors that caused the variations of long-term time records are the location of the stations, instruments, formulae used to calculate the mean, observing practices, and the station environment [3].
There are two ways to ensure homogeneity of data records; direct and indirect methods [4]. The direct methods of homogenisation involve best
management practices in data recording and documentation. Indirect methods involve metadata analysis, creation of reference series, statistical tests for breakpoint detection, and adjustment of the data records [4]. To identify the effects of non−climatic factors in the observations over time, homogeneity tests are performed on time series data [5]. The homogeneity tests are of two types; absolute methods and comparative methods. In absolute homogeneity tests, each station is considered separately without considering the effect of neighbouring stations. In relative homogeneity tests, neighboring reference stations are considered. This research used absolute methods. Several researchers have either assumed the data to be homogeneous or used absolute homogeneity tests alone in their research [6].
Received: 7 March 2021, Accepted: 2 June 2021, Published: 31 December 2021, Publisher: UTP Press, Creative Commons: CC BY-NC-ND 4.0
OUTLIER AND HOMOGENEITY ANALYSIS OF EXTREME RAINFALL SERIES IN KANO, NIGERIA
Abdulrasheed Mohammed1*, Salisu Dan’Azumi1, Abubakar Ahmed Modibbo2
1Department of Civil Engineering, Bayero University, Kano, Nigeria
2Department of Civil Engineering, University of Maiduguri, Borno, Nigeria
*Email: [email protected]
ABSTRACT
Data preparation is one of the most important steps in the field of hydrology since the results of any research depend on whether the input data were reliable and rightly collected or not. Reliability and quality of long-term rainfall data are required before carrying out any study in hydrology. For this reason, outlier identification techniques and homogeneity tests have proved helpful. This study aimed at carrying out outliers and homogeneity analysis of extreme rainfall data series for the Kano rainfall gauging station. XLSTAT, 2019 was used to identify outliers through Grubb’s test for outliers/two-tailed test and Dixon test for outliers/two-tailed statistic test and Standard Normal Homogeneity Test (SNHT), Buishand Range (BR) Test, Pettitt Test, and Von Neumann Ratio (VNR) Test were used for homogeneity analysis. Four outliers were identified and treated accordingly. The results of homogeneity tests showed that the maximum daily rainfall data of the Kano gauging station is homogenous and classified as useful. The maximum daily rainfall data of Kano station can be used for any study in water resources and environmental engineerings, such as in Development of Intensity Duration Frequency (IDF) Curves and Rainfall- Runoff Simulations.
Keywords: Outliers test, homogeneity tests, rainfall data, XLSTAT 2019
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Outliers are data points that depart significantly from the trend of the remaining [7]. The retention or removal of these outliers can significantly affect the magnitude of statistical parameters computed from the data, especially for small samples. They are two types of outliers; high and low outliers [8]. Outlier values can arise from three different causes; an error in measurement or recording, an observation from a different population such as a flood triggered by a dam break rather than by rainfall, and a rare event from a single population that is somewhat skewed [7].
Homogeneity, on the other hand, is a critical issue to detect the variability of the data. In general, when the data is homogeneous, it means that the data measurements are taken simultaneously with the same instruments and environments.
However, it is hard to deal with rainfall data because it is always caused by changes in measurement techniques and observational procedures, environment characteristics and structures, and the location of stations [9].
There are many ways available in the literature to identify outliers and homogeneity tests on rainfall data series and other meteorological data. For example, Asikoglu [7] analyses outliers’ detection in extreme value series of 14 flow gauging stations in the River basin of Seyhan and Ceyhan in Turkey using five different methods: z-score method, Box- plot method, Stedinger test, Quality control test, and Grupps-Back test. Whitacre et al. [10] used statistical outlier detection methods by evolutionary algorithms.
Filzmoser and Hron [11] used robust methods to detect outliers for composite data. Wijngaard et al. [12] employed SNHT, BR, and VNR test to detect the reliability and homogeneity of daily precipitation and temperature data series and classified the data series into useful (class 1), doubtful (class 2) and suspect (class 3). Kang and Yusof [9] used SNHT, BR, Pettitt Test, and VNR tests to conduct Homogeneity Tests on Daily Rainfall Series in Peninsular, Malaysia.
Wijesekera and Perera [2]. Their studies highlighted key issues of Data and Data Checking for Hydrological Analyses - Case Study of Rainfall Data in the Attanagalu Oya Basin of Sri Lanka. Furthermore, Omar et al. [3] also used SNHT, BR, Pettitt Test, and VNR to carry out Homogeneity Analysis of Precipitation Series in North Iraq. Talaee et al. [13] tested the homogeneity of annual and monthly rainfall data in
Iran using the Bayesian, Cumulative Deviations, and VNR tests at the 5% level. The monthly precipitation data series for stations considered are homogeneous and classified as “useful”. In contrast, the monthly precipitation data series of 7 out of 41 stations considered are inhomogeneous according to the VNR test. Kahya et al. [14] checked the homogeneity of the monthly precipitation data sets at 160 meteorological stations in Turkey using the SNH, Pettitt, and BR tests.
For this study, XLSTAT, 2019 was used to identify outliers through Grubb’s test for outliers/two- tailed test and Dixon test for outliers/two-tailed statistic test. For homogeneity tests, methods employed by Wijesekera and Perera [2], Omar et al.
[3], Kang and Yusof [9], and Wijngaard et al. [12] were adopted though, through the application of XLSTAT, 2019 software.
MATERIALS AND METHODS The study area
Kano State is located in Northwestern Nigeria on latitude 12°N and longitude 8.30°E within the semi- arid Sudan savannah zone of West Africa. Kano has a mean height of about 481 m above sea level [19].
The temperature of Kano usually ranges between a maximum of 33°C and a minimum of 15.8°C, although sometimes, during the harmattan, it falls to as low as 10°C. Kano has two seasonal periods:
four to five months of the wet season and a long dry season lasting from October to April. The mean annual rainfall is about 800 mm around metropolitan Kano. Significant temporal variation occurs in rainfall received, and no two consecutive years recorded the same amount [19]. Kano gauging station is located at Mallam Aminu Kano Airport. Three large watersheds distinguish the hydrogeology of Kano; the outfall to River Challawa, the outfall to River Jakara, and the others are to River Wateri. These basins are considered as the primary receptacles of runoff from the city.
Kano city has a total area of 574 km2 where 43.41%
covered built-up area, 42.32% is bare land, 13.68%
is vegetation cover, and 0.59% is water body for the year 2018 Landuse/landcover (LULC) classification [20]. Figure 1 shows Kano metropolis LULC for the year 2018.
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Data
Daily maximum rainfall data for the 50 years used for this research was collected from Nigerian Meteorological Agency, Geography Department Bayero University Kano, and Center for Dryland Agriculture Bayero University, Kano.
Method
Data preparation is one of the most important steps since the results of the research depend on whether the input data were reliable and correctly collected or not. For this study, the outliers and homogeneity tests were carried out on the maximum daily rainfall series of Kano State.
Identifying outliers
Outliers are data points that depart significantly from the trend of the remaining data. The retention or removal of these outliers can significantly affect the magnitude of statistical parameters computed from the data, especially for small samples [8]. They are two outliers; high and low outliers. For this study, XLSTAT, 2019 was used to identify outliers through Grubb’s test for outliers/two-tailed test and Dixon test for outliers/
two-tailed statistic test.
Grubb’s test for outliers/two-tailed test
Grubb developed several tests to determine whether the sampled data have outliers or not. This test assumes that the data corresponds to a sample extracted from a population that follows a normal distribution [15].
Let x1, x2, x3,…xi,…xn be a sample extracted from a population that is assumed to be following a normal distribution N(μ, s2). Parameters μ and s2 were respectively estimated as given by:
μ,= 1
––m
Σ
mi=1xi (1)S2 = 1
m – 1–––
Σ
mi=1(xi – x–)2 (2)The Grubb’s test statistic is given as:
G = max(Gmin, Gmax) (3)
The hypothesis of Grubb’s test is stated as:
1. Ho: There is no outlier in the data
2. Ha: The maximum or minimum value is an outlier.
Figure 1 Kano metropolis land use/land cover classification map for the year 2018
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An approximation of the critical value Gcrit given the threshold above which, for a given 5% significance level, one must reject the null hypothesis is given by:
Gcrit(n, α) ≈ (n – 1)tn–2,1–α/k ––––––––––––––
√
n – 2 + t2n–2,1–∞/k(4)
where tn–2,1–α/k is the value of the inverse of the student cumulative distribution function at 1 – α/k with n-2 degrees of freedom, and where k equals 2n for the two-sided test. Gcrit was then compared with the value of G statistic computed for the sample and deduced if Gcrit is greater than G(Gmin or Gmax), Ho is accepted and rejected otherwise. From the Gcrit approximation, the p-value that corresponds to G was equally approximated. XLSTAT displayed all these results as well as the conclusion based on the 5% significance level given.
To help identify potential outliers, z-scores correspond to the standardised sample were computed using:
Zi = xi – x–
––––s , (i = 1, 2, ..., n) (5) The z-scores are set at the acceptable interval (typically –1.96 and 1.96 for a 95% interval), any value outside this range is considered suspicious. Furthermore, for a given n number of samples, the highest z-score is at most given by:
argmaxi=1,2 zi ≥ n – 1
––––√m (6)
Iglewicz and Hoaglin [21] recommended using a modified z-score to identify outliers as given in:
Zi = 0.6745 xi – x–
MAD––––, (i = 1, 2, ..., n) (7) where MAD is the Median Absolute Deviation. The acceptable interval is given by (–3.5;3.5), whatever n may be.
Dixon test for outliers/two-tailed statistic
The Dixon test was developed to help determine if the greatest value or lowest value of a sample, or the two largest values, or the two smallest ones can be considered outliers. This test assumes that the data correspond to a sample extracted from a population that follows a normal distribution [16].
The statistics used for the Dixon test and the corresponding ranges of several observations they should be used as given by:
–R21 = xn – xn–2 –––––––
xn – x2 , recommended for 4 ≤ n ≤ 100,
also named as N11 (8)
–R22 = xn – xn–2 –––––––
xn – x3 , recommended for 6 ≤ n ≤ 100,
also named as N13 (9)
The hypothesis of the Dixon test is stated as, 1. Ho: There is no outlier in the data
2. Ha: The maximum or minimum value is an outlier.
To compute a critical value and p-value for the Dixon test, literature provides more or less accurate approximations of the critical value beyond which the null hypothesis could be rejected for a given significance level. However, XLSTAT, 2019 provides an approximation of the critical values based on Monte Carlo simulations. The number of these approximations is by default set to 1000000, which is more reliable than those provided in the literature. XLSTAT also provides, based on these simulations, a p-value and the conclusion of the test based on the 5% significance level chosen. The z-scores for the Dixon test were also computed and interpreted the same as that of Grubb’s test.
Treating the outliers
The outliers were treated following a method proposed by Vukmirovic and Pavlovic as cited in [17],
1. The outlier is removed from the sample.
2. The outliers are replaced with the second largest value in the same year in which the outlier occurred.
3. The outlier remains in the sample.
Homogeneity test Using XLSTAT 2019
They are many homogeneity tests in existence.
However four are commonly used, e.g. [1],[9],[18]. The four homogeneity tests that were carried out for this study included:
1. Standard Normal Homogeneity Test (SNHT), 2. Buishand Range (BR) Test,
3. Pettitt Test, and
4. Von Neumann Ratio (VNR) Test.
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For the null hypothesis, the annual values Yi of the testing variables Y are independent and identically distributed. The series is considered homogeneous, while for the alternative hypothesis, SNHT, BR, and Pettitt tests assumed the series consisted of a break in the mean and considered inhomogeneous. These three tests can detect the year where the break occurs.
The VNR test cannot give information on the year break because the test assumes the series is not randomly distributed under the alternative hypothesis. The Standard Normal Homogeneity Test is more sensitive to detect inhomogeneity near the dataset’s beginning and end. BR test and Pettitt test are sensitive to identify the break in the middle of the series [9],[18].
For this study, the annual maximum values of the daily rainfall data of the Kano station were considered.
Given that Yi(i =1, 2,...,n) is the variable to be tested with Y, S, and k as the mean, standard deviation, and the last years of record with that of the last (n-1) years, respectively. The mathematical equations of the four tests are given below:
Standard normal homogeneity test (SNHT)
According to this test, a statistic TK is used to compare the mean of the first y years with the last (n-y) years and can be written as;
Tk = kz–1 + (n-k)z–2, k = 1, 2, 3, 4 (10)
z–1 = 1 –k
Σki=1(Yi – Y– ) –––––––––
S , (11)
z–2 = 1 ––––n – k
Σ
ki=1(
Yi – Y–)
––––––––––
S , (12)
The ith year consists of a break if the value of T is maximum. To reject the null hypothesis, the test statistic To = maxTi is greater than the critical value, depending on the sample size.
Buishand range test
According to this technique, the homogeneity test is based on the cumulative deviations from the mean or adjusted partial sums, which are given as:
S0 = 0 and Sk =
Σ
ki=1(
Yi – Y–)
, k = 1, 2, 3….n (13) When the series is homogeneous, the values of Sk will fluctuate around zero since there is no systematicdeviation of the Yi values concerning their mean.
The year Y breaks when Sk has reached a maximum (negative shift) or minimum (positive shift). Rescaled adjusted partial sums are obtained by dividing the values of Sk by the sample standard deviation, S.
Q = max0≤k≤n|Sk/S| (14)
Another test used is the range that computes the difference between the maximum and minimum value of the rescaled adjusted partial sums. The formula is given by:
R = Max(Sk/S) – Min(Sk/S) (15) ––Q
√
n and––R
√
n are then compared with the critical values given by Buishand.Pettitt test
The Pettitt test is used to detect a single breakpoint in a time series. The test is based on the rank of the Ri of the Yi and does not consider the normality of the series. The null hypothesis of this test is that data are independent and randomly distributed. This means that data follow the same distribution Omar et al. [3].
1. The observations Yi are ranked from 1 to N 2. The value of Vi,N is estimated from;
Vi = N + 1 – 2Ri i = 1, 2, 3, ...N (16) 3. The value of Ui is estimated from
Ui = Ui+1 + Vi, U1 = V1 4. The value of KN = max1≤i≤N|Ui|
5. The value of POA = 2e–
(
(N––––––6K3+N2N2))
(17)The null hypothesis is rejected when POA is less than ∝ where ∝ is the statistical significance level of the test.
Von Neumann ratio test
According to Kang and Yusof [9], the test used the ratio of mean square successive (year to year) difference to the variance. The test statistic is as follows:
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N =
Σ
ni –1=1(Yi – Yi+1)2 ––––––––––––
Σ
ni=1(
Yi – Y–)
2 , (18)When the sample or series is homogeneous, then the expected value E(N) = 2. When the sample has a break, then the value of N must be lower than 2. Otherwise, we can imply that the sample has rapid variation in the mean.
Evaluation of the Tests
Wijngaard et al. [12] classifi ed result of the tests as follows,
1. Class A: Useful
The series that rejects one or none null hypothesis under the four tests at a 5% significance level are considered. Under this class, the series is grouped as homogeneous and can be used for further analysis.
2. Class B: Doubtful
The series that reject two null hypotheses of the four tests at a 5% signifi cance level is placed in this class. In this class, the series has an inhomogeneous signal and should be critically inspected before further analysis.
3. Class C: Suspect
When three or all tests reject the null hypothesis at a 5% signifi cance level, the series is classifi ed into this category. In this category, the series can be deleted or ignored before further analysis.
RESULTS AND DISCUSSIONS
Grubb’s Test for Outliers/Two-tailed Test
The 50 years of maximum daily rainfall data of the Kano station was subjected to Grubb’s test for outlier identification using XLSTAT 2019 software. Table 1 shows the statistical result of this test.
Table 1 Grubb’s Test Result
G (Observed value) 3.445
G (Critical value) 3.128
p-value (Two-tailed) 0.012
Alpha (∝) 0.05
The p-value has been computed using 1000000 Monte Carlo simulations.
From Table 1, it is observed that the G (observed value) is greater than G (critical value), and the computed p-value is lower than the signifi cance level ∝. Therefore, the null hypothesis Ho is rejected, and the alternative hypothesis Ha is accepted. That is, the data has outliers, as indicated by the z-scores value shown in Figure 2.
From Figure 2, it can be seen that the data series has four (4) high outliers (bars in red color) and has no low outliers. The outliers identifi ed according to this test were 175.50, 163.80, 162.50, and 143.00 mm, which
Figure 2 z-score graph for Grubb’s outlier test
z-score
Observations
z-score Series2 Series3
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occurred in the years 2014, 2001, 1997, and 2012 respectively. It could be observed from Figure 2 that these values are greater than the acceptable limit of –1.96 and 1.96 for a 95% interval. Hence these values are considered suspicious (outliers).
Dixon test for outliers/two-tailed statistic The result of the Dixon test is shown in Table 2.
Table 2 Dixon test for outlier result
R22 (Observed value) 0.115
R22 (Critical value) 0.345
p-value (Two-tailed) 0.752
Alpha (∝) 0.05
The p-value was computed using 1000000 Monte Carlo simulations. From Table 2, the R22 (observed value) is less than R22 (critical value), and the computed
p-value is greater than the signifi cance level ∝, and as such, the null hypothesis Ho is accepted. That is, there are no outliers in the data. However, from Figure 3, it is observed that the data has four (4) high outliers, as equally indicated by Grubb’s test. The identifi ed high outliers by this test are the same as that of Grubb’s test.
Treating the outliers
For this paper, the outliers were replaced with the second largest value in the same year in which the outliers occurred. Then, the outliers tests were repeated with XLSTAT 2019, and the results are as presented in Figures 4 to 5.
From Figures 4 and 5, it is seen that the outliers have been successfully treated that is all the values lie within the acceptable limits. Hence, the Homogeneity test was further conducted on the annual maximum series of daily rainfall of Kano station.
Figure 3 z-score graph of Dixon outlier test
Figure 4 Grubb’s test z-score graph for repeated test
z-score
Observations
z-score Series2 Series3
Observations
4 3 2 1 0 -1 -2 -3 -4
Figure 4 Grubb’s test z-score graph for repeated test
Figure 5 Dixon test z-score graph for repeated test
-‐2.5 -‐2 -‐1.5 -‐1 -‐0.5 0 0.5 1 1.5 2 2.5
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
z-score
Observations
Z-‐score Series2 Series3
-‐2.5 -‐2 -‐1.5 -‐1 -‐0.5 0 0.5 1 1.5 2 2.5
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
z-score
Observations
Z-‐score Series2 Series3 z-score Series2 Series3
2.5 2.0 1.5 1.0 0.5 0 -0.5 -1.0 -1.5 -2.0 -2.5
z-score
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VOLUME 5 NUMBER 4 2021 e-ISSN: 26369877 PLATFORM
Homogeneity test results using XLSTAT 2019 The annual maximum series of daily rainfall data for Kano station was subjected to four homogeneity tests (Standard Normal homogeneity Test (SNHT), Buishand Range (BR) Test, Pettitt Test, and Von Neumann Ratio (VNR) Test) using XLSTAT 2019 software. Table 3 shows the results of the homogeneity tests carried out for the annual maximum series of daily rainfall data of Kano station.
From Table 3, it is seen that the p-values for the four homogeneity tests are greater than the significance level alpha = 0.05; hence, the null hypothesis is accepted, that is, the annual maximum daily rainfall data of Kano station are homogenous. Following the homogeneity test result by Wijngaard et al. [12], this data is classified as ‘useful’ because the series data do not reject any of the null hypotheses under the four tests at a 5% significance level. Hence, the Figure 5 Dixon test z-score graph for the repeated test
Table 3 Homogeneity tests result
SNHT Pettit’s BR VNR
To p-value Alpha k p-value Alpha Q p-value Alpha N p-value Alpha
3.601 0.499 0.050 178.00 0.679 0.050 6.755 0.247 0.05 1.869 0.322 0.050
Figure 6 Homogeneity test p-values chart for SNHT
z-score
Observations
z-score Series2 Series3
Figure 6 Homogeneity test p-values chart for SNHT
Figure 7 Homogeneity test p-values chart for Pettite test
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6
p-values
Rainfall data variable
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
p-‐value
Rainfall data variable
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.1 0.2 0.3
p-values
Figure 4Grubb’s test z-score graph for repeated test
Figure 5Dixon test z-score graph for repeated test
-‐2.5 -‐2 -‐1.5 -‐1 -‐0.5 0 0.5 1 1.5
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
z-score
Observations
Z-‐score Series2 Series3
-‐2.5 -‐2 -‐1.5 -‐1 -‐0.5 0 0.5 1 1.5 2 2.5
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
z-score
Observations
Z-‐score Series2 Series3
2.5 2.0 1.5 1.0 0.5 0 -0.5 -1.0 -1.5 -2.0 -2.5
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Figure 7 Homogeneity test p-values chart for Pettite test
Figure 8 Homogeneity test p-values chart for Buishand test
Figure 9 Homogeneity test p-values chart for Von Neumann
Figure 6 Homogeneity test p-values chart for SNHT
Figure 7 Homogeneity test p-values chart for Pettite test
Figure 8 Homogeneity test p-values chart for Buishand test
0 0.1 0.2 0.3 0.4 0.5
p-values
Rainfall data variable
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
p-‐value
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3
p-values
Rainfall data vaiable
Figure 6 Homogeneity test p-values chart for SNHT
Figure 7 Homogeneity test p-values chart for Pettite test
Figure 8 Homogeneity test p-values chart for Buishand test
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6
p-values
Rainfall data variable
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
p-‐value
Rainfall data variable
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3
p-values
Figure 9 Homogeneity test p-values chart for Von Neumann
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 0.1 0.2 0.3 0.4
p-values
Rainfall data variable
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Rainfall data variable Rainfall data variable
series data is homogeneous and suitable to be used for further analysis. Figures 6 to 9 show the graphical representation of four green colors, which means that the null hypothesis is not rejected. If the graphs are in red colours, it means the null hypothesis is rejected.
CONCLUSION
The outliers identified in the daily maximum rainfall data series of the Kano gauging station were treated accordingly, and the data was further subjected to homogeneity tests. According to the four tests carried out (SNHT, BRT, Pettitt Test, and VNRT), the daily maximum rainfall data series of Kano gauging station is homogenous and classified as useful and can therefore be used for hydrological studies and water resources management.
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