Climate Risk Management 32 (2021) 100295
Contents lists available at ScienceDirect
Climate Risk Management
journal homepage: www.elsevier.com/locate/crm
Characterization of variability and trends in daily precipitation
and temperature extremes in the Horn of Africa
Emmanuel Afuecheta *, M. Hafidz Omar
Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia
A R T I C L E I N F O
A B S T R A C T
Keywords:
Extreme value theory
Return level
Extreme climatic factors
Food Security
Climate change
Modeling of extreme weather conditions is an important issue in environmental risk assessment,
management and protection. In this paper, Annual and monthly maxima of daily temperature and
rainfall data dating from 1901–2016 and 1950–2018 are characterized for nine countries within
the greater horn of Africa. The generalized extreme value (GEV) distribution is fitted to the data
sets from each of the nine countries using the method of maximum likelihood. Both the location
and scale parameters of the GEV is formulated as a function of time to account for variability and
trends in the extremes of temperature and rainfall so that their future behavior can be predicted.
Based on our results, We provide return levels for the years 2, 10, 50 100, and 200 which could be
used as measures of flood protection. To understand the spatial cross-correlation patterns on
different time scales in each country and how rainfall and temperature are related to agricultural
variables, we utilize the detrended cross-correlation coefficient(ρDCCA ) and its generalization,
(DMC2x ) to account for that, respectively. Given the results of the (DMC2x ) and the fact that climate
extremes pose a huge threat to agriculture and food security, we employed copula based models
to describe the structure of dependence between climatic variables and the crop related variables
such as yield and production quantity. The results from the copula analysis show that irrespective
of the country, climatic factors and the agricultural products(production/yield) the strongest
dependence is demonstrated by the pairs involving cereal crops, while the weakest dependence is
characterized by the pairs involving regular potato.
1. Introduction
Extreme weather conditions have significant environmental and socio-economic impacts. Understanding their patterns (trends and
variability) is of great importance since most of the global challenges such as food security is linked to extreme events induced by
changing climate (Xu et al., 2017). For example, it is estimated that one-ninth of the World’s population is suffering from chronic
hunger and malnutrition and the majority of the people in the world estimated to have little or no food as a result of climate change live
in developing countries (FAO IFAD UNICEF WFP WHO, 2017). Extreme weather-related disasters are increasing in intensity by the day
and have become a threat to household food security in different countries, especially in developing countries.
While extreme events(such as droughts and rainfall) are a natural part of our climate system, recent research indicates that their
destructive power, or intensity is growing, particularly in the Eastern part of Africa. Over the past few years, East Africa has been facing
frequent droughts and excessive rainfall events (Gebrechorkos et al., 2018; Hastenrath et al., 2010; Omondi et al., 2013). More than 10
* Corresponding author.
E-mail address: emmanuel.afuecheta@kfupm.edu.sa (E. Afuecheta).
https://doi.org/10.1016/j.crm.2021.100295
Received 19 January 2021; Received in revised form 25 February 2021; Accepted 3 March 2021
Available online 11 March 2021
2212-0963/© 2021 The Author(s).
Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Climate Risk Management 32 (2021) 100295
E. Afuecheta and M.H. Omar
million people in this subregion are dangerously hungry and are desperately in need of humanitarian assistance due to devastating
drought induced by El-Nĩ
no, which hit them in early 2017 (Oxfam Media Briefing, 2017). Consequently, East Africa is arguably the
most vulnerable region of Africa to extreme weather and climate events (Gebrechorkos et al., 2018). These estimates have shown the
magnitude of the challenge facing the subregion and Africa at large in meeting the 2030 agenda for Sustainable Development Goals
(SDGs), which was adopted by the United Nations General Assembly in 2015, targeted towards achieving a world without hunger (SDG
Target 2.1) and malnutrition (SDG Target 2.2).
Among the key factors driving the recent increase in food insecurity in the East Africa are conflicts, overexploitation of natural
resources and adverse climatic conditions such as extreme rainfall, heat-waves, and drought which are motivated by climate change
(Climate Change 2014: Synthesis Report,). As the magnitude and impact of climate change increase, the more the communities and
governments in this region are less able to absorb and adapt, making them increasingly vulnerable to future shocks. Climate anomalies
pose risks to human existence. For example, climate variability may affect or trigger outbreaks of infectious diseases like Ross River
virus disease, visceral leishmaniasis, etc. Recently, several studies have provided evidence of a link between vector-borne disease
outbreaks and El Niño driven climate anomalies. It is noteworthy to mention that a disease (infectious disease) crisis in one country can
spread economic pain to other countries, for example, Ebola, MERS-CoV, COVID-19, etc (see Shocket et al. (2018), Morand et al.
(2013) and Tong et al. (2002, 2008)). Hence, it is pertinent to study these variations in frequency or intensity of extreme weather
conditions (Climate variability and trends), having observed the increasing need to achieve sustainable development goals. This will
help to gain an understanding of their patterns and their possible occurrence and dimensions in the present and in a future climate. It
will in turn address the climate risks to food security, dynamics and resilience against shocks as well as mitigation strategies to avoid
both human and financial losses. To this end, this work is geared towards characterizing the annual maximum/minimum daily var­
iabilities and trends of two climatic factors, namely: extreme precipitation and extreme warm temperature in some of the East African
countries using extreme value theory. We shall focus more on the maximums since the potential consequences of frequent to extreme
rainfall and warm temperature are flooding and drought, respectively. This does not necessarily mean that rainfall and warm tem­
perature will automatically lead to flooding and drought. But the most immediate impact of frequent to extreme rainfall and warm
temperature is the likelihood of flooding and drought.
Extreme value theory or extreme value analysis (EVA) deals with the stochastic behaviour of the extreme values in a process
(Nadarajah and Choi, 2003). EVA and its techniques have a long history with a wide range of applications, particularly in general risk
management, hydrology and related areas; for example, see Gettinby et al. (2006), Karmakar and Shukla (2014), Embrechts et al.
(1997), Reiss and Thomas (1999), McNeil (1998), Beirlant et al. (1996), Embrechts et al. (1998), Lux (2000), Karimi and Voia (2015).
Basically, this technique is used to characterize rare events by estimating beyond the original observation range, so as to give prob­
ability estimates of events occurring at unobserved levels (Sharkey and Winter, 2019). Along these lines, forecasts of future extremes
can be obtained by extrapolating the behavior of the process using asymptotically justified limit model. In practice, under the
Fig. 1. Map of East Africa showing the nine countries under study. Source: Google map.
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E. Afuecheta and M.H. Omar
conventional type of statistics, several methods could be employed within the parlance of extreme value theory to statistically analyze
extreme values. These methods are generally grouped into two fundamental approaches, namely: the block maxima and the peaksover-threshold (POT) methods. The block maxima entails partitioning the observation of interest into non-overlapping time in­
tervals of equal length and then carrying out inference under the assumption that the maximum and or the minimum in each interval
follows the generalised Extreme Value (GEV) distribution. The form of the GEV distribution is presented later in the Methods section of
this paper. While the peaks-over-threshold (POT) method entails using a threshold to seclude observations considered extreme with
respect to the rest of the data and then perform inference under the assumption that those selected observations that exceed a certain
high threshold is approximately a generalized Pareto distribution-GPD (Jacob et al., 2020) and Pickands (1975). The form of the GPD
will also be presented later in the Methods section.
The GEV distribution and GPD are natural class of models for describing extreme events in both univariate and multivariate
extreme value theory settings and have become the most popular models for climatic data. Some recent applications include: Mak­
konen and Tikanmäki (2019), Dadi et al. (2017), Jonathan and Ewans (2013), Chikobvu and Sigauke (2013) and Eastoe and Tawn
(2012). See also Leadbetter et al. (1983), Koutsoyiannis and Baloutsos (2000), Resnick (2007), Dixon (1950), Pickands (1975);
Hosking and Wallis (1987), Coles (2001), Beirlant et al. (2004), Castillo et al., 2005, McNeil et al. (2005), Reiss and Thomas (2007). In
our study, we shall employ both methods to study the variability and trends in the annual maximum/minimum daily precipitation and
temperature of some countries in the horn of Africa. Data sparsity is one of the major challenges often encountered when one is trying
to analyze extremes (Casson and Coles, 2000) and this hampers statistical inference. However, since our datasets are substantial; more
that 100 years for each country based on annual maximum/minimum, we are not discouraged. Ideally, if there is no sufficient data, one
could employ other sources of information via prior distribution in Bayesian context which could improve the statistical inference on
the extremes. Accordingly, we shall exploit GEV and GPD methods(compare them) to characterize the annual maximum/minimum
daily rainfall and temperature for the years from 1901 to 2016 for the nine East African Countries. These countries include: Burundi
(BDI), Djibouti(DJI), Eritrea(ERI), Ethiopia(ETH), Kenya(KEN), Rwanda(RWA), Somalia(SOM), Tanzania(TZA) and Uganda(UGA).
The choice of these nine countries is that they provide a good geographical representation of the subregion with complete data sets. In
addition to that, they represent the top nine wealthiest East African countries according to the gross domestic product (GDP) with the
agricultural sector among the main driver of their respective growth (East Africa Economic Outlook, 2019), see Fig. 1.
The aim of this paper is threefold. Firstly, to model the annual maximum daily precipitations and extreme warm temperature for the
nine East African countries using both the GEV and GPD. Then to provide return levels for various years and to derive estimates of these
return levels outside the scope of our data using the better of the GEV and GPD models. Secondly, to estimate the cross-correlation
levels between temperature in each capital of the countries and those from other administrative regions or provinces and how they
vary based on different time scales. Thirdly, to quantify the dependencies between extreme values of rainfall/temperature and crop
related variables such as yield and production quantity within the study region. This is necessary since large variations of climatic
factors have severe impact on crop related variables. We believe that understanding the cross-correlation patterns as well as quan­
tifying how they relate with agricultural variables within the study region will provide economic changes and reduce risks to food
security and biodiversity loss. It will also help the policy makers to make informed evidence-based decisions at the most appropriate
times as climate changes.
The rest of this paper is organized as follows. The nine countries and the corresponding datasets are described in Section 2. The data
used are the annual maximum/minimum daily rainfall as well as maximum/minimum daily warm temperature from 1901 to 2016.
The models and the fitting procedures are described in Section 3. We use the method of maximum likelihood for estimation of the
models. We determine the better of the two distributions(GEV and GPD) by information criteria such as the AIC (Akaike information
criterion), BIC (Bayesian information criterion), AICC (Akaike information criterion correction) and HQIC (Hannan-Quinn information
criterion). The distribution with the smaller values of these criteria is deemed the better model. The detrended cross-correlation
technique and its generalization as well as the copula models used are also described in Section 3. The results of the fitted models
and their implications are discussed in Section 4 and estimates are given for the 2, 10, 50, 100, and 200 year return levels. Finally, some
conclusions and recommendations for future studies are given in Section 5.
2. Data
Three different data sets collected from multiple sources are used for this study. Our first data consist of annual maximum(min­
imum) daily rainfall and temperature for the years from 1901 to 2016 for the nine countries denoted in Fig. 1. The data were obtained
from the websitehttps://climateknowledgeportal.worldbank.org/ of climate change knowledge portal by World Bank Group for
development of practitioners and policy makers. Our data and analyses are limited to nine of the East African countries. The countries
were chosen carefully to give a good geographical representation of the subregion. Historically, according to the United Nations
scheme of geographic regions(M49), there are 20 nations that make up East African countries (United Nations Standard Country Code,
1998). But there are only six countries (Tanzania, Kenya, Uganda, Rwanda, Burundi and South Sudan) that are members of the East
African Community (EAC). Out of these six countries, there are only five that have data going back to 1901, the earliest year for which
data are available. We have also included Djibouti, Eritrea, Ethiopia and Somalia which are collectively known as the Horn of Africa.
They are sometimes considered a separate region from the East Africa as they are the easternmost extension of African land. For some
reasons, the rainfall data for Djibouti was not available in the database( https://climateknowledgeportal.worldbank.org/) at the time
of data collection.
The second data sets are high-resolution gridded datasets from 1950 to 2018 for the nine countries denoted in Fig. 1. We obtained
the data from CRU TS(Climate Research Unit gridded Time Series) available onhttps://crudata.uea.ac.uk/cru/data/hrg/. CRU TS is a
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E. Afuecheta and M.H. Omar
climate dataset on a 0.5◦ by 0.5◦ degrees grid. It is derived by the interpolation of monthly climate anomalies from a network of
weather station observations taken from across the world. The data was downloaded and pre-processed for the extractions. Our ex­
tractions were conducted for each variable(monthly/annual maximum temperature, monthly/annual minimum temperature,
monthly/annual rainfall) at different administrative boundaries(including: country (administrative level 0), state (administrative level
1) and district (administrative level 2) levels).
The third data sets are crop-related variables, including crops yield and production quantity associated with Burundi, Eritrea,
Ethiopia, Kenya, Rwanda, Somalia, Tanzania and Uganda over different periods. The data were obtained from Food and Agriculture
Organization of United Nations-FAO available on ( http://www.fao.org/faostat/en/#home). The data sets comprise of yields and
production quantities of principal food and cash crops from each of the countries. Details of the selected crops with respect to each
country are tabulated in Table 1. These data are used to assess the impact of climate extremes on crops with the assumption that a low
harvest(both in yields and production quantities) could be due to excessive rainfall and or very high temperature.
In this section, we employ extremes based on our first and second data sets to study the underlying pattern in the data. Boxplots of
the monthly maximums/minimums(temperature) versus years at administrative level 1 for each of the capital of the nine countries are
shown in Figs. 2,3.
We can observe an increasing trend in the location of the maximums/minimums(temperature) for each capital except possibly for
Mogadishu-Somalia. Also, an increasing trend in the variability of the maximums/minimums for each of the capital of the nine
countries is observed except possibly for Mogadishu-Somalia. Similar results were obtained for the monthly minima/maxima(tem­
perature) versus years at administrative level 1 for other cities/regions/provinces/counties in each country. But the figures are not
provided due to space concerns and they can be made available to any reader directly upon reasonable request.
The scatter plots of the annual maximum/minimum daily temperature and rainfall for the countries under study are given in Fig. 4.
Fig. (4) shows how the annual maxima(minima) daily temperature and rainfall varied from 1901 to 2016 for the nine countries.
Generally, the variation for all the countries with the few exceptions under rainfall appears to exhibit some kind of non-stationarity.
This is not surprising as it is common with environmental variables and processes. This could be attributed to the fact that climate
patterns of different season varies, or maybe due to more long term trends as a result of climate change (Cheng et al., 2014). To confirm
this, we tested for the presence of trends and non-stationarity in extremes (maxima/minima) using Mann–Kendall trend test (Kendall,
1976; Mann, 1945) at 5% level of significance. Mann–Kendall test (Kendall, 1976; Mann, 1945) for trend gave the p values of 2.22 ×
10− 16 for daily annual maximum temperature from Burundi, 7.74 × 10− 2 for daily annual maximum temperature from Djibouti,
6.86 × 10− 3 for daily annual maximum temperature from Eritrea, 1.19 × 10− 7 for daily annual maximum temperature from Ethiopia,
2.22 × 10− 16 for daily annual maximum temperature from Kenya, 2.22 × 10− 16 for daily annual maximum temperature from Rwanda,
2.34 × 10− 4 for daily annual maximum temperature from Somalia, 2.22 × 10− 16 for daily annual maximum temperature from
Tanzania, 2.22 × 10− 16 for daily annual maximum temperature from Uganda, 2.22 × 10− 16 for daily annual minimum temperature
from Burundi, 3.28 × 10− 3 for daily annual minimum temperature from Djibouti, 1.19 × 10− 3 for daily annual minimum temperature
from Eritrea, 2.86 × 10− 6 for daily annual minimum temperature from Ethiopia, 2.22 × 10− 16 for daily annual minimum temperature
from Kenya, 2.22 × 10− 16 for daily annual minimum temperature from Rwanda, 1.19 × 10− 7 for daily annual minimum temperature
from Somalia, 2.22 × 10− 16 for daily annual minimum temperature from Tanzania, 2.22 × 10− 16 for daily annual minimum tem­
perature from Uganda, 0.03638 for daily annual maximum rainfall from Burundi, 0.35331 for daily annual maximum rainfall from
Eritrea, 0.05910 for daily annual maximum rainfall from Ethiopia, 0.19346 for daily annual maximum rainfall from Kenya, 0.16133
for daily annual maximum rainfall from Rwanda, 0.53973 for daily annual maximum rainfall from Somalia, 0.79112 for daily annual
maximum rainfall from Tanzania, 0.00346 for daily annual maximum rainfall from Uganda, 0.51018 for daily annual minimum
rainfall from Burundi, 0.00033 for daily annual minimum rainfall from Eritrea, 0.67278 for daily annual minimum rainfall from
Ethiopia, 0.04900 for daily annual minimum rainfall from Kenya, 0.64858 for daily annual minimum rainfall from Rwanda, 0.41584
for daily annual minimum rainfall from Somalia, 0.01606 for daily annual minimum rainfall from Tanzania, 0.27343 for daily annual
minimum rainfall from Uganda. Similar results were also obtained for the monthly minimums/minimums(temperature) at adminis­
trative level 1 for each of the cities/regions/provinces/counties in every country. These results confirm our earlier observations based
on Fig. 2–4. This implies that the null hypothesis of no trend is rejected at 5% level of significance for all the countries with the
exception of few countries under rainfall. Given this evidence of monotonic (increasing or decreasing) trend in the location and
variability, the parameters of our models will be estimated under the non-stationary assumption. This is necessary so as to obtain more
realistic estimates of return values that are consistent with the behaviour of climatic extremes (Cheng et al., 2014).
Table 1
Description of food and cash crops variables from different countries used in this Study.
Country
Crops-[Yield and Production quantity]
Burundi
Eritrea
Ethiopia
Kenya
Rwanda
Somalia
Tanzania
Uganda
Bananas, Beans, Cassava, Sorghum, Sweet potatoes, Tea, Coffee.
Barley, Millet, Sorghum, Wheat.
Barley, Beans, Maize, Millet, Sorghum, Coffee, Tea.
Beans, Maize, Rice, Sugarcane, Coffee, Tea.
Beans, Cassava, Potatoes, Sorghum, Sweet potatoes, Coffee, Tea.
Bananas, Maize, Sorghum, Sugarcane.
Maize, Millet, Rice, Seed cotton, Sisal, Sorghum, Sweet potatoes.
Bananas, Cassava, Coffee, Millet, Plantains and others, Sweet potatoes.
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E. Afuecheta and M.H. Omar
Fig. 2. Boxplots of monthly maximum temperature for each of the capital of the 9 countries based on administrative level 1 from 1950 to 2018.
The following summary statistics based on our first data sets and their corresponding latitude and longitude are noted in Tables 2–5,
respectively, for temperature(maxima and minima) and rainfall(maxima and minima): minimum, maximum, median, mean, inter
quartile range (IQR), standard deviation.
From Tables 2–5, we observe that Djibouti has the highest annual maximum temperature followed by Eritrea and Somalia. Rwanda
has the least annual maximum temperature. Then for rainfall, Rwanda has the highest annual maximum rainfall followed by Ethiopia
and Uganda. Also, we can observe heterogeneity in precipitation patterns from the coefficient of variation(CV) given in Tables 4 and 5.
For the maxima, the coefficient of variation is smallest for Tanzania and largest for the Eritrea. This shows that Eritrea has the most
variable annual maximum rainfall followed by somalia. For the minima, the coefficient of variation of rainfall is relatively higher for all
the countries, showing that most countries in the Greater Horn of Africa are (GHA) drought-prone due to high variability of seasonal
and annual rainfall across time and space. This is in consistence with other studies, for example, see Ghebrezgabher et al. (2016),
Seleshi and Demarée (1995), Mpelasoka et al. (2018), Nicholson et al. (2018), Muthoni et al. (2019)) and references therein. This
variability is partly associated with the interaction of climate variability drivers such as the El Niño-Southern Oscillation(ENSO), the
Indian Ocean Dipole (IOD), Inter-decadal Pacific Oscillation (IPO), the North Atlantic Oscillation (NAO) and other phenomenons alike.
For instance, the El Niño-Southern Oscillation(ENSO) which is a cyclical variation in the surface temperature of the tropical eastern
Pacific Ocean occurs when the ocean surface in this region is warmer than the average. Whereas its counterpart La Niña occurs when
the ocean surface is cooler than the average (Daron, 2014).
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E. Afuecheta and M.H. Omar
Fig. 3. Boxplots of monthly minimum temperature for each of the capital of the 9 countries based on administrative level 1 from 1950 to 2018.
3. Methodology
Here we give a brief description of the models used. We begin with the GEV and GPD approaches. Suppose X1 ,X2 ,⋯are independent
and identically distributed (iid) random variables with common cumulative distribution function (CDF) F. If we let Mn =
max{X1 , ⋯, Xn } define the maximum of the first n random variables with w(F) = sup{x : F(x) < 1} denoting the upper end point of F.
Now if there exists sequences of constants an > 0 and bn and a nondegenerate G such that
)
)
(
Pr Mna−n bn ⩽x = Fn (an x + bn ) → G(x , as n → ∞, then G is a generalised extreme value (GEV) distribution with distribution
function
( )
{ [
(x − μ) ]−
G x = exp − 1 + ξ
σ
1/ξ
}
(1)
+
and F is in the domain of attraction of G, written as F ∈ D(G), where x+ = max(x, 0), σ > 0 denote scale parameter, μ is the location
parameter and ξis the shape parameter which governs the tail behaviour of G(x). This implies that if ξ > 0,ξ < 0and ξ = 0, G(x) reduces
to Fréchet, negative-Weibull and Gumbel distributions, respectively.
As noted before, instead of using the generalized extreme value distribution, one can model the values that exceed a sufficiently
high threshold (μ) using the generalized Pareto distribution (GPD) due to Pickands (1975). According to Pickand’s asymptotic theory
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Fig. 4. Annual maxima/minima daily temperature/rainfall for the countries under study from 1901 to 2016.
Table 2
Descriptive statistics of the annual daily maximum temperature for Burundi, Djibouti, Eritrea, Ethiopia, Kenya, Rwanda, Somali, Tanzania and
Uganda.
Country
Longitude
Latitude
N
Min
Max
Median
IQR
Mean
SD
CV
BDI
29◦ 20′
3◦ 21′
116
18.7249
23.1367
21.0959
0.7904
21.1584
0.7279
3.4404 × 10−
2
DJI
43◦ 8′
11◦ 35′
116
30.6974
34.2116
32.2428
0.9265
32.1904
0.7287
2.2638 × 10−
2
ERI
38◦ 35′
15◦ 40′
116
27.4914
31.2356
29.5157
0.8473
29.5685
0.6770
2.2898 × 10−
2
ETH
40◦ 29′
9◦ 8′
116
22.2775
26.4227
24.3311
4.1452
24.3038
0.7274
2.9932 × 10−
2
KEN
37 54
◦ ′
0 1
116
23.2372
27.7507
26.0107
1.0462
25.9723
0.8482
3.2658 × 10
RWA
30◦ 4′
1◦ 56′
116
17.7900
22.5970
19.8809
0.9491
19.9550
0.8026
4.0222 × 10−
2
SOM
45◦ 19′
2◦ 2′
116
26.8586
29.8682
28.6588
0.9301
28.6071
0.6477
2.2642 × 10−
2
TZA
39◦ 16′
6◦ 48′
116
21.3141
25.2944
23.7886
0.6792
23.7709
0.6234
2.6227 × 10−
2
UGA
32 34
0 20
116
20.5982
26.3056
23.7939
1.1442
23.8162
1.0157
◦
◦
′
′
◦
′
4.2650 × 10
− 2
− 2
for the extremes X, if certain regularity conditions are satisfied and μ is sufficiently large then
[
x ]−
Pr(X > x + μ|X > μ) ≈ 1 + ξ
1/ξ
σ
where σ > 0 is the scale parameter, ξ is the shape parameter similar to that of the GEV distribution, 1 + ξx/σ > 0. If we rearrange (1),
then we can write the CDF as
( )
[
x − μ]− 1/ξ
F x = Pr(X < x) ≈ 1 − p 1 + ξ
(2)
σ
where μ⩽x < ∞ if ξ⩾0 and μ⩽x⩽μ +σ/ξ if ξ < 0, p = Pr(X > μ).Given its advantages(e.g. more extreme data points are used) over the
block maxima method, the POT approach is often preferred in practical situations. However, our results show that the GEV distribution
gives the best fits. In our study, each of these distributions was fitted by the method of maximum likelihood. Discrimination among
them was performed using some information criteria, including the AIC (Akaike information criterion), BIC (Bayesian information
criterion), AICC (Akaike information criterion correction) and HQIC (Hannan-Quinn information criterion). The goodness of fit of
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E. Afuecheta and M.H. Omar
Table 3
Descriptive statistics of the annual daily minimum temperature for Burundi, Djibouti, Eritrea, Ethiopia, Kenya, Rwanda, Somali, Tanzania and
Uganda.
Country
Longitude
Latitude
N
Min
Max
Median
IQR
Mean
SD
CV
BDI
29 20
3 21
116
16.3209
20.5470
19.0025
0.8374
18.8584
0.8792
4.6625 × 10−
DJI
43◦ 8′
11◦ 35′
116
20.9952
24.6512
23.3921
0.8325
23.3855
0.654783
2.7999 × 10
ERI
38◦ 35′
15◦ 40′
116
19.3320
24.7275
22.1884
1.3405
22.1552
0.9516
4.2954 × 10−
2
ETH
40◦ 29′
9◦ 8′
116
18.4097
22.1883
20.8637
0.883125
20.7811
0.7209
3.4693 × 10−
2
KEN
37◦ 54′
0◦ 1′
116
20.3745
24.3910
22.6506
1.1273
22.5335
0.89366
3.9659 × 10−
2
RWA
30 4
1 56
116
15.5510
19.9799
18.3897
0.8389
18.1791
0.9006
4.9544 × 10
SOM
45◦ 19′
2◦ 2′
116
23.0591
26.4897
24.8479
0.62545
24.8032
0.5664
2.2837 × 10−
2
TZA
39◦ 16′
6◦ 48′
116
17.7403
21.2949
19.9074
0.8611
19.8273
0.7786
3.9273 × 10−
2
UGA
32◦ 34′
0◦ 20′
116
18.9005
23.3885
21.2874
1.1715
21.1840
0.9526
4.4970 × 10−
2
◦
′
◦ ′
◦
◦
′
′
2
− 2
− 2
Table 4
Descriptive statistics of the annual daily maximum rainfall for Burundi, Eritrea, Ethiopia, Kenya, Rwanda, Somali, Tanzania and Uganda.
Country
Longitude
Latitude
N
Min
Max
Median
IQR
Mean
SD
CV
BDI
29◦ 20′
3◦ 21′
116
99.5632
344.8180
212.2755
60.591
215.2596
42.9922
0.1997
ERI
38◦ 35′
15◦ 40′
116
44.8327
230.0850
92.9472
45.5360
102.5567
40.9593
0.3993
ETH
40◦ 29′
9◦ 8′
116
102.8570
361.6590
151.9515
24.7077
156.2100
30.4945
0.1952
KEN
37◦ 54′
0◦ 1′
116
76.8925
311.4080
134.6315
50.7480
143.8168
37.6292
0.2616
RWA
30◦ 4′
1◦ 56′
116
129.8950
507.0000
183.5100
38.092
188.1206
43.2743
0.2300
SOM
45◦ 19′
2◦ 2′
116
35.0054
146.5910
67.2985
33.9607
72.8562
25.4523
0.3493
TZA
39◦ 16′
6◦ 48′
116
137.8810
299.8660
204.2725
45.586
206.5834
33.6694
0.1629
UGA
32◦ 34′
0◦ 20′
116
120.7750
350.7410
174.2305
29.4235
177.4839
30.1333
0.1697
Table 5
Descriptive statistics of the annual daily minimum rainfall for Burundi, Eritrea, Ethiopia, Kenya, Rwanda, Somali, Tanzania and Uganda.
Country
Longitude
Latitude
N
Min
Max
Median
IQR
Mean
SD
CV
BDI
29◦ 20′
3◦ 21′
116
0.0000
18.1027
2.2104
3.229435
3.1587
3.3660
1.0656
ERI
38◦ 35′
15◦ 40′
116
0.0003
5.1347
1.0103
1.4374
1.2909
1.0279
0.7962
ETH
40◦ 29′
9◦ 8′
116
0.0590
21.6296
4.2027
4.9776
5.3266
4.0868
0.7672
KEN
37◦ 54′
0◦ 1′
116
1.1576
32.6564
12.7887
8.86159
13.1061
6.4548
0.4925
RWA
30◦ 4′
1◦ 56′
116
0.1027
53.6346
8.9600
10.6408
11.2947
8.8315
0.7819
SOM
45◦ 19′
2◦ 2′
116
0.3095
4.0084
1.5009
1.6811
1.6672
1.0214
0.6126
TZA
39◦ 16′
6◦ 48′
116
0.9602
13.8259
5.5327
3.45097
5.5566
2.3021
0.4142
UGA
32◦ 34′
0◦ 20′
116
7.3637
64.0128
26.5870
56.64902
28.5337
12.1613
0.4262
these two distributions was examined by P-P plots. Because of space concerns, details of the GPD fitting are not given but can be
obtained from the corresponding author upon request. Obviously, Figs. 2–4 suggest that extreme temperature and rainfall could
possibly exhibit some trends with respect to time. We explore this in the subsection that follows.
3.1. Time varying Models based on the GEV
In view of the fact that GEV (1) fits our data better than GPD (2) and also that Figs. 2–4 suggest that extreme temperature and
rainfall could possibly exhibit some trends with respect to time. We employ the following variations of (1) to study these trends and
variabilities at annual level.
1.
2.
3.
4.
5.
6.
7.
TA = GEV1,1,1 (μ, σ, ξ) : μ, σ, ξ
TB = GEV2,1,1 (μt , σ, ξ) : μt = a + b × (TY − t0 + 1), σ , ξ
TC = GEV3,1,1 (μt , σ, ξ) : μt = a + b × (TY − t0 + 1) + c × (TY − t0 + 1)2 , σ, ξ
TD = GEV1,2,1 (μ, σt , ξ) : μ, logσ t = a + b × (TY − t0 + 1), ξ
TE = GEV1,3,1 (μ, σt , ξ) : μ, logσ t = a + b × (TM/Y ) + c × (TY − t0 + 1)2 , ξ
TF = GEV2,2,1 (μt , σt , ξ) : μt = a + b × (TY − t0 + 1), logσt = c + d × (TY − t0 + 1), ξ
TG = GEV3,2,1 (μt , σt , ξ) : μt = a + b × (TY − t0 + 1) + e × (TY − t0 + 1)2 , logσ t = c + d × (TY − t0 + 1), ξ
8
Climate Risk Management 32 (2021) 100295
E. Afuecheta and M.H. Omar
8. TH = GEV2,3,1 (μt , σ t , ξ) : μt = a + b × (TY − t0 + 1), logσ t = c + d × (TY − t0 + 1) + e × (TY − t0 + 1)2 , ξ
9. TI = GEV3,3,1 (μt , σ t , ξ) : μt = a + b × (TY − t0 + 1) + e × (TY − t0 + 1)2 , logσt = c + d × (TY − t0 + 1) + f × (TY − t0 + 1)2 , ξ,
where TY is time in year and t0 symbolizes the year the record started. In each of these GEV based models; except for TA = GEV1,1,1 (μ,
σ, ξ) which is the same as (1), we assume that the location parameter and or the log of the scale parameter can be allowed to vary
polynomially(linearly and or quadratically) with respect to time. We note that one could possibly explore other higher structural
polynomial models but one of the remarkable disadvantages of these higher order polynomial models is that they do not give accurate
forcast as they fluctuate excessively, Nadarajah(2005).
Then, to see if seasonality is significant as depicted by the boxplots given in Fig. 2, we also fitted the following GEV based models to
monthly maximum/minimum temperature from 18 administrative provinces in Burundi; 5 administrative regions in Djibouti; 6
administrative divisions in Eritrea; 11 administrative regional states and chartered cities in Ethiopia; 47 administrative Counties in
Kenya; 5 administrative counties in Rwanda; 18 administrative regions in Somalia; 30 administrative regions in Tanzania and 58
administrative districts in Uganda.:
1.
2.
3.
4.
5.
6.
SA = GEV1,1,1 (μ, σ, ξ) : μ, σ , ξ
SB = GEV3,1,1 (μ, σ , ξ) : μ = a + bsin(πM/12) + ccos(πM/12), σ , ξ;
SC = GEVS1,3,1 (μ, σ, ξ) : μ, σ = exp[a + bsin(πM/12) + ccos(πM/12)], ξ;
SD = GEVS1,1,3 (μ, σ, ξ) : μ, σ, ξ = a + bsin(πM/12) + ccos(π M/12);
SE = GEVS3,3,1 (μ, σ, ξ) : μ = a + bsin(πM/12) + ccos(πM/12), σ = exp[d + esin(πM/12) + fcos(π M/12)], ξ;
SF = GEVS3,3,3 (μ, σ , ξ) : μ = a + bsin(πM/12) + ccos(πM/12), σ = exp[d + esin(πM/12) + fcos(πM/12)], ξ = g + hsin(πM/12) +
icos(πM/12),
where M represents the month number(Jan= 1 and Dec = 12). Again each of these models was fitted by the method of maximum
likelihood. Here, apart from SA = GEV1,1,1 (μ, σ , ξ) which is the same as (1), the location parameter and or the logarithm of the scale
parameter of other models are allowed to vary.
3.2. Spatial cross-correlation patterns and dependence
Given the complexity and highly non-stationary nature of climatic variables, we also introduce the following tools to analyze the
data described in Section 2: (i) detrended cross-correlation coefficient(ρDCCA ) due to Zebende (2011), (ii) the generalization of
detrended cross-correlation coefficient (DMC2x ) due to Zebende and da Silva Filhoa (2018) and (iii) Copula based models. Detrended
cross-correlation coefficient is basically employed to study the spatial cross-correlation patterns of temperature on different time scales
in every country. We achieved this by independently taking the capital of each country as the center and then calculate the ρDCCA
patterns between the monthly extreme-temperature in the capitals and those in other state administrative levels. The DMC2x and copula
based models are used to describe total association and complex dependences between extreme climatic variables and the crop-related
variables, such as production quantity and yield from these countries, respectively. This is vital as extreme climatic conditions have the
potential to affect and deteriorate the production quantity and or quality as well as crop yield of notable food crops. These three
techniques have been previously used in other papers too, see, for example, Yuan and Fu (2014), Alidoost et al. (2019), Brito et al.
(2019)).
3.2.1. DCCA and DMC cross-correlation coefficients
Suppose X1 , X2 , ⋯, Xn and Y1 , Y2 , ⋯, Yn are two nonstationary time series data on X and Y, respectively, then the DCCA crosscorrelation coefficient is defined by
( )
( )
F 2DCCA n
ρDCCA = ρxi ,yj n =
,
(3)
FDFA (n) FDFA (n)
{xi }
{yi }
for − 1⩽ρxi ,yj ⩽1, where n represents the time scale, F2DCCA (n) and DFA{} (n) denote detrended covariance function and detrended
variance function, respectively (Zebende and da Silva Filhoa, 2018). One notable advantage of DCCA is that it is insensitive to trend(i.e
not affected by trend) and also has the potential to produce results at varying time scalesn. A cross-correlation coefficient value of
0 implies no cross-correlation; 1 implies perfect cross-correlation and − 1 is perfect anti-cross-correlation. The generalization of (3),
also known as DCCA multiple cross-correlation due to Zebende and da Silva Filhoa (2018) is defined by
( )
( )
( )
DMC2x n = ϒy,x (n)T ρ− 1 n ϒy,x n
(4)
where ϒy,x (n)T ≡ {ρy,x1 (n), ρy,x2 (n), ⋯, ρy,xp (n)}gives the vector of the detrended cross-correlation involving the independent variables
(xp ) and the dependent variable (y); ρ(n) denotes the detrended cross-correlation matrix of the independent variables whose inverse is
given by ρ− 1 (n) and is defined as:
9
Climate Risk Management 32 (2021) 100295
E. Afuecheta and M.H. Omar
⎛
ρ(x1 , x2 )(n) ρ(x1 , x3 )(n)
⎜ ρ(x1 , x2 )(n) 1
ρ(x2 , x3 )(n)
⎜
ρ− 1 (n) ≡ ⎝
1
⋮(
ρ x1 , xp
)
⋮(
ρ x2 , xp
)
⋮(
)( )
ρ x3 , xp n
⋯
⋯
⋯
⋯
(
)( ) ⎞−
⋮
1
⎠
ρ( x1 , xp )( n)
ρ x1 , xp n ⎟
⎟
1
Notice that transpose of ρxi ,xj (n) is ρxj ,xi (n) implying that the detrended cross-correlation matrix of the independent variables is
symmetric and ρxj ,xj (n) = 1, hence DMC2x (n) ∈ [0, 1]. The strength of the multiple cross correlation increases as DMC2x (n) moves away
from 0 toward 1. Following Almeida (Brito et al., 2019), we classified the levels as: DMC2x (n) ∈ [0, 0.2] means very weak multiple cross
correlation; DMC2x (n) ∈ [0.2, 0.4] means weak multiple cross correlation; DMC2x (n) ∈ [0.4, 0.6] means medium multiple cross correla­
tion; DMC2x (n) ∈ [0.6, 0.8] means strong multiple cross correlation and DMC2x (n) ∈ [0.8, 1] implies very strong multiple cross correla­
tion. In our own case, maximum crop yield or production quantity from each of the nine countries will be regarded as y while annual
maximum/minimum temperature and rainfall will be regarded as, x1 and x2 , subsequently DMC2x (n) is given by
( )
( )
( )
( )
( )
( )
ρ2y,x1 n + ρ2y,x2 n − 2ρ2y,x1 n ρ2y,x2 n ρ2x1 ,x2 n
2
( )
DMCx n =
(5)
1 − ρ2x1 ,x2 n
3.2.2. Copula models
Given the results of the DMC2x , which suggest possible existence of relationship/dependence between climate extremes and agri­
cultural variables such as crop yield and production, we used copula based models to describe the joint behavior(relationship) of
annual maximum temperature/rainfall with maximum crop yield and production quantity in each country. We selected crop yield and
production based on notable food and cash crops that are peculiar to every country. Copula models have been previously used in other
climate change and variation work, see, for example, Alidoost et al. (2019) and Miao et al. (2016) and references therein. However,
none of these studies have investigated the effect of climate extremes on crop yields and productions in the horn of Africa. The focus on
the joint behavior of extreme climatic indices on crops is appropriate as this could potentially play a vital role supporting information
advisory services such as irrigation service management (Alidoost et al., 2019). Copulas are multivariate distributions whose margins
are uniform and they are particularly used to specify dependence between multiple random variables. For example, a two dimensional
copula is a function C : [0, 1]2 → [0, 1] that satisfies C(u,0) = C(0,v) = 0, C(u, 1) = u, C(1, v) = v, ∂C(u, v)/∂u⩾0 for all u and v, and ∂C(u,
v)/∂v⩾0 for all u and v. The following eleven copula models are considered: Gumbel, Galambos, Gaussian, Hüsler-Reiss, Tawn,
t-Student extreme(t-EV), Clayton, Frank, Joe, Plackett, Student’s t. Their distributions (where θ signifies the level of dependence
between random variates U and V) are specified as follows:
{
}
• Gumbel: C(u, v) = exp − [( − lnu)θ + ( − lnv)θ ]1/θ , θ⩾1, where θ = 1 (independence) and θ → ∞ (complete dependence).
}
{
• Galambos: C(u, v) = uvexp [( − lnu)− θ + ( − lnv)− θ ]− 1/θ , 0⩽θ < ∞, where θ = 0 (independence).
(
)
• Gaussian: C(u, v) = Φθ Φ− 1 (u), Φ− 1 (v) , − 1⩽θ ≤ 1, where Φ is the CDF of a standard normal random variable, Φθ is the joint CDF
of a bivariate normal random vector with zero means, unit variances and correlation θ. θ = 1 (Complete dependence) and θ = 0
(independence)
{
[
[
( )]
( )]}
• Hüsler-Reiss: C(u, v) = exp − ̃
uΦ 1θ + 12 θln ̃u
− ̃
vΦ 1θ + 12 θln ̃v
, θ⩾0, where ̃
u = − lnu, ̃
ν = − lnv and Φ(.) represents the
̃v
̃u
standard normal distribution function. θ = 0 (independence) and θ → ∞ (Complete dependence).
{
}
• Tawn: C(u, v) = uvexp − θ ln(u)ln(v)
, 0⩽θ⩽1, where θ = 0 (independence).
ln(uv)
[
[
{
( ) ]
( ) ] }
• t-EV: C(u, v) = exp Tν+1 −
ρ
θ
+ 1θ
lnu
lnv
1/ν
lnu + Tν+1 −
ρ
θ
+ 1θ
lnu
lnv
1/ν
lnv , where ν and ρ are the degrees of freedom and cor­
2
ρ
relation coefficient ρ ∈ ( − 1, 1), respectively. The parameter θ depends on the value of ρ and ν such that θ2 = 1−ν+1
.
• Clayton: C(u, v) = [u− θ + v− θ − 1]− 1/θ , − ∞ < θ < ∞, where θ = 0(independence) and θ → ∞ (Complete dependence)
[
]
− θu
− θv
− 1)
, − ∞⩽θ⩽∞, where θ = 0 (independence)
• Frank: C(u, v) = − 1θ ln 1 + (e − e−1)(e
θ− 1
• Joe: C(u, v) = 1 − [(1 − u)θ + (1 − v)θ − (1 − u)θ (1 − v)θ ]1/θ , 1⩽θ < ∞, where θ = 1 (independence)
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
1+(θ− 1)(u+v)−
[1+(θ− 1)(u+v) ]2 − 4θ(θ− 1)uv
• Plackett: C(u, v) =
, θ > 0, where θ = 1 (independence) and if θ → ∞ the Plackett copula be­
2(θ− 1)
comes the Fréchet-Hoeffding upper bound.
)
(
• Student’s t : C(u, v) = Tθ,ν T −ν 1 (u), T −ν 1 (v) , − 1⩽θ ≤ 1, ν > 2, Tν is the cumulative distribution function of a Student’s t random
variable with degrees of freedom ν and Tθ,ν represents the joint CDF of a bivariate t random vector with zero means, correlation θ
and degrees of freedom ν.
10
Climate Risk Management 32 (2021) 100295
E. Afuecheta and M.H. Omar
In fitting the copula models, we apply the following procedure: suppose x1 , x2 , ⋯, xn denote the extreme indices(annual maximum
temperature/rainfall) for one of the countries and let y1 , y2 , ⋯, yn denote the corresponding crop yield and production quantity for the
same country, where n denotes the sample size of each variable for the country under study. We transformed each of the data sets as
( )
ui = rank(xi )/(n +1) and vi = rank yi /(n +1) and then employed the method of maximum likelihood to fit the copula models. Then the
parameters of each copula model are the values maximizing the likelihood
n
∏
L(Θ) =
c(ui , vi ; Θ)
i=1
or the log-likelihood
n
∑
lnL(Θ) =
lnc(ui , vi ; Θ),
i=1
′
′
̂ = (̂
where Θ = (θ1 , θ2 , …, θk ) is a vector of parameters specifying c(⋅,⋅). Here, Θ
θ1 , ̂
θ2 , …, ̂
θk ) is the maximum likelihood estimate of
Θ. The maximization was done using the routine optim in the R software package (R Development Core Team, 2020).
4. Results and discussion
The GEV based models described in Section 3.1 were fitted to the datasets described in Section 2. Every one of these models was
fitted by the method of maximum likelihood. Being that these models are nested, we employed the standard likelihood ratio test due to
Cox and Hinkley (1974) to discriminate between them and to establish the best model. That is, given two models say model A and B,
where B is a sub-model of A with na and nb number of parameters, respectively. If LA and LB are the respective maximum likelihoods for
the two competing models. Then under the null hypothesis that the sub-model is preferred, the test statistic K = − 2log(LB /LA ) is
distributed according to a chi-square distribution with degrees of freedom equal to the difference in dimensionality (na − nb ) of the two
competing models, such that K = − 2log(LB /LA ) < χ 2na − nb ,0.95 . The parameter estimates along with the log likelihood values associated
with the best performing models are given in Table 7 for the annual maximum(minimum) daily temperature/rainfall from 1901 to
2016 for the nine countries. For annual maximum(minimum) daily temperature, the best model among TA to TI is the TB with varying
location parameters. This shows that there is a significant upward trend in the mean of maximum temperature in the countries under
study. This is also evidence that climate is changing. For the annual maximum(minimum) daily rainfall, the best performing model
Table 7
Parameters of the fitted models and their associated standard errors in parenthesis based on the best fitting distribution for the monthly maximum
temperature in each of the capitals of the nine countries.
Country
Parameter estimates
Likelihood
Model
Burundi-Gitega
̂
μ = 25.3527(0.0359), ̂a = − 0.0356(0.0028),
1204.696
SC
2190.159
SA
1449.567
SB
1616.158
SA
1627.151
SC
1010.471
SB
1055.038
SA
1495.103
SA
̂
a = 29.4676(0.0676), ̂
b = − 2.1174(0.0908),
1002.459
SB
̂c = 0.7437(0.0422), ̂
σ = − 0.23110(0.0269),
̂
ξ = − 0.2388(0.0223)
̂
b = 0.2861(0.0385) ,̂c = 0.0995(0.0337),
̂
ξ = − 0.1613(0.0277)
Djibouti-Djibouti
̂
μ = 32.2209(0.1480), ̂σ = 1.2383(0.0353),
̂
ξ = − 0.3299(0.0450) ,
Eritrea-Maekel
̂
a = 24.738(0.1152), ̂
b = 2.8589(0.1573),
̂c = 0.7829(0.0756), ̂
σ = 0.40487(0.02606),
̂
ξ = − 0.3788(0.0143),
Ethiopia-Addis Ababa
̂
μ = 22.8010(0.0636), ̂σ = 0.51598(0.0092),
̂
ξ = − 0.2462(0.0195),
Kenya-Nairobi
̂
μ = 23.8487(0.0679), ̂a = 0.6159(0.0296),
̂
b = − 0.1477(0.0269), ̂c = − 0.2542(.0313),
̂
ξ = − 0.4017(0.0251)
sRwanda-Kigali
̂
a = 24.6713(0.0664), ̂
b = 0.4736(0.0934),
̂c = − 0.1453(0.0432), ̂
σ = − 0.2332(0.0265),
̂
ξ = − 0.2127(0.0204),
Somalia-Mogadisho
̂
μ = 26.3908(0.0317), ̂σ = − 0.1818(0.0264),
̂
ξ = − 0.2092(0.0203),
Tanzania-Dodoma
̂
μ = 27.3675(0.0594), ̂σ = 0.4395(0.0288),
̂
ξ = − 0.3830(0.0244),
Uganda-Kampala
11
Climate Risk Management 32 (2021) 100295
E. Afuecheta and M.H. Omar
among TA to TI is the basic model-TA ,GEV 1,1,1 . This shows that there is no significant trend in the location or the scale of the maximum
rainfall within the countries under study. The probability plots serving as the goodness of fit of the best fitting models for the annual
maximum temperature and rainfall for these countries are shown in Fig. 5. These plots represent the visualization of the observed
probabilities against probabilities predicted by the best performing models. For instance, under GEV 1,1,1 which is the best performing
{ [
]− 1/̂ξ }
model for annual maximum(minimum) rainfall, we plotted exp − 1 + ̂
ξ(x(j) − ̂
μ )/̂
σ
against (j − 0.375)/(n + 0.25), where x(j)
represent the sorted values of the observed extremes, for j = 1⋯n and n is the sample size (Table 6).
We can observe from these plots that each of the best performing models describe our data well as all the points are reasonably close
to the diagonal line. Also, the return level plots for the annual maximum temperature and rainfall for 2, 10, 50, 100 and 200 year
period are provided in Figs. 6 and 7, respectively. The corresponding estimate of these return levels for rainfall are also given with 95%
confidence intervals in Table 8. From these estimates, we can observe that the associated confidence intervals are not very wide,
suggesting that reasonable sample was used for the analysis. This also guarantees the certainty surrounding the shape parameters
obtained for all the countries, which in turn indicate that our return level results are physically realistic and believable.
For the monthly maximums, the parameter estimates along with the log likelihood values and associated standard errors obtained
Fig. 5. Probability plots of the best fitting model:Annual maxima/minima temperature/rainfall.
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Climate Risk Management 32 (2021) 100295
E. Afuecheta and M.H. Omar
Table 6
Parameters of the fitted models based on the best fitting distribution for annual maximum temperature and rainfall.
Model
Country
Parameters
Loglikelihood
TB -Annual max temperature
Burundi
̂
a = 20.9090, ̂
b = 1.6025, ̂
σ = exp( − 0.2735), ̂ξ = − 0.3066
128.5223
Djibouti
̂
a = 31.9110, ̂
b = 0.47708, ̂
σ = exp( − 0.3631), ̂ξ = − 0.2112
126.7488
Eritrea
̂
a = 29.3345, ̂
b = 0.6042295, ̂
σ = exp( − 0.3752), ̂ξ = − 0.2941
119.1918
TA -Annual max rainfall
Ethiopia
̂
a = 24.0386, ̂
b = 24.0386, ̂
σ = exp( − 0.3125), ̂ξ = − 0.2576
128.142
Kenya
̂
a = 25.7278, ̂
a = 1.9749, ̂
σ = exp( − 0.1065), ̂ξ = − 0.4151
142.2617
139.7722
Rwanda
̂
a = 19.6487, ̂
b = 2.2285, ̂
σ = exp( − 0.2318), ̂ξ = − 0.2173
Somalia
̂
a = 28.4149, ̂
b = 0.77588, ̂
σ = exp( − 0.3949), ̂ξ = − 0.3955
111.5868
Tanzania
̂
a = 23.5747, ̂
b = 1.1773, ̂
σ = exp( − 0.3976), ̂ξ = − 0.3678
108.9583
Uganda
̂
a = 23.4778, ̂
b = 2.488414, ̂
σ = exp(0.0482), ̂ξ = − 0.3223
166.1377
Burundi
̂
μ = 198.0672, ̂σ = exp(3.6769), ̂ξ = − 0.1618
598.5048
Eritrea
̂
μ = 82.6302, ̂
σ = exp(3.3507), ̂ξ = 0.1134
579.618
Ethiopia
̂
μ = 143.5768, ̂σ = exp(3.0260), ̂ξ = 0.0429
535.3501
Kenya
̂
μ = 127.0894, ̂σ = exp(3.4020), ̂ξ = − 0.0202
576.1678,
Rwanda
̂
μ = 170.4430, ̂σ = exp(3.2716), ̂ξ = 0.0869
567.1471,
Somalia
̂
μ = 60.3538, ̂
σ = exp(2.8912), ̂ξ = 0.1095
526.1622,
Tanzania
̂
μ = 193.2224, ̂σ = exp(3.4362), ̂ξ = − 0.1753
570.2447,
Uganda
̂
μ = 164.7626, ̂σ = exp(3.0891), ̂ξ = 0.0137
540.7103,
by inverting the observed information matrices are given in Table 7 for the capitals of each country only.
Results for other regions/counties in each country are grouped, and summarized here according to the best model among SA to SF
without tabulating their parameter estimates due to space concern. These parameters are available from the corresponding author
upon request. From these results, we can observe that there is no one best fitting model jointly for all the regions/counties or provinces
in each of the nine countries. However, for the majority of the regions or provinces within the nine countries, the best model among SA
to SF is the model SB . For example, Model SB gives the best fit for 3 out the 6 administrative divisions in Eritrea; 31 out of the 47
administrative Counties in Kenya; 5 out of the 5 administrative provinces in Rwanda including Kigali; 13 out of the 30 administrative
regions in Tanzania; 21 out of the 58 districts considered in Uganda. This implies that there is a significant upward trend in the location
of the maximum temperature values from these administrative regions/provinces. Then model SC gives the best fit for 10 out of the 18
provinces in Burundi. That is, a significant upward trend can be seen in the scale of the maximum temperature values from these ten
provinces in Burundi. Model SA gives the best fit for the 18 administrative regions in Somalia and the 5 administrative regions in
Djibouti. This shows that there is neither a significant upward trend in the scale nor in the location(mean) of the maximum temperature
values in those 18 and 5 regions in Somalia and Djibouti, respectively. The probability plots associated with the best fitting models for
the capitals of each country are shown in Fig. 8. Again, we can observe from these plots that each of the best performing models with
respect to the capital of each country describe the data well as all the points are reasonably close to the diagonal line.
Then, from the analysis of the detrended cross-correlation coefficients (ρDCCA ) between the monthly maximum temperatures in each
capital of the nine countries and that in all other administrative regions/provinces/states, we observe varying spatial patterns with
respect to each country. Depending on the capital of the country’s specificity and the location of the corresponding regions and or
provinces, the (ρDCCA ) may show positive(perfect) cross correlations, anti-cross correlations(negative) and zero cross correlations.
From Fig. 9, a notable observation is that the cross-correlation coefficients (ρDCCA ) in each capital of the nine countries and that in most
other administrative regions/provinces/states are generally high(> 0.5), except possibly for few regions in Somalia, Ethiopia, Eritrea
and Tanzania. For eaxmple, in Ethiopia the (ρDCCA ) between Addis Ababa and Afar showed a significant downward trend(from 0.3 to
− 0.20) at small time scale before stabilizing and fluctuating around zero. Similar patterns were observe for Dire Dawa but the
downward trend observe for this case is not as steeped as Afar region. Somalia is the only country among these countries where
negative cross-correlation is observed. The regions with anti-cross correlations are mostly in the North-western or Central part of the
country. For example Awdal(north west), Togdheer(north west), Woqooyi Galbeed(north west) and Shabeellha Dhexe(central) are
geographically positioned in the North western or Central part of Somalia. These parts of the country are perhaps the rainiest regions
which are situated in the extreme offshoot of the Ethiopian highlands, and precipitation gets close to 400–500 mm per year as a result
of the altitude. The weather in the north-western part of Somalia is a sharp contrast of what is obtainable in the Mogadishu which is the
capital situated in the southern part of the east coast, just north of the Equator. Hence the possible reason for the anti-cross correlations.
In general, for Somalia, the maximum average of DCCA cross-correlations is for Shabeellaha Hoose region, the minimum average of
DCCA cross-correlations is for Awdal region, the maximum and minimum standard deviations of DCCA cross-correlations are for Sool
and Shabeellaha Hoose regions, respectively, the maximum and minimum coefficient of variations of DCCA cross-correlations are for
Nugaal and Sool regions, respectively. For Burundi, the maximum temperature in all the administrative provinces exhibit high strong
positive cross-correlation level with that in Gitega. The cross-correlations values are all greater than 9 at every time scale. This suggest
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Fig. 6. The return level plots for maximum temperature when model TB is assumed for the nine countries.
that the climate of the provinces vary considerably from mild to warm depending on the altitude since the country is located just south
of the Equator. Notably, the maximum average of DCCA cross-correlation is jointly for Bujumbur, Ruyigi and Bururi provinces, the
minimum average of DCCA cross-correlations is for Kirundo, the maximum standard deviation of DCCA cross-correlation is for Kirudo
which gives also the maximum coefficient of variations of DCCA cross-correlations. For Djibouti, the DCCA cross-correlation coeffi­
cient between the maximum temperature in Djibouti city-the capital and that in other administrative divisions are all >9 over different
time scale. This suggest evidence of high temperature across the entire country. The maximum average of DCCA cross-correlations is
for Ali Sabieh region and the minimum is for Dikhil. Obok region has the maximum range of cross-correlation. Every region appears to
have similar coefficient of variations and standard deviation of cross-correlations, except for Ali Sabieh. Summarizing the plots of the
(ρDCCA ) given in Fig. 9, we can say that there are mostly positive cross-correlation pattern between the monthly maximum temperatures
in each capital of the nine countries and that in their respective administrative regions/provinces/state. For virtually all the countries,
these cross-correlations exhibit pronounced seasonal patterns on a small to medium time scale. However, these seasonality patterns
level off on a large time scale for some countries. This could be due to varying synoptic scale disturbances (Yuan and Fu, 2014).
Tables 9 and 10 present the mean and median values of detrended multiple cross-correlation coefficient, DMC2x (n) for annual
maximum temperature, annual maximum rainfall and largest recorded crops production quantity and crop yields, respectively. From
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Fig. 7. The return level plots for maximum rainfall when model TA is assumed for the nine countries with 95% pointwise normal approximation
confidence intervals.
these results, we can observe that crop yields and production quantities are locally and respectively related with annual maximum
temperature and rainfall from the countries under study. The computed multiple cross-correlations range from very weak to weak
depending on the country, time scale(n) and variables. For instance, under production quantity; we can observe the following: for
Burundi, among the seven crops considered the largest mean value of DMC2x (n) is for Coffee green and smallest is for dry beans; for
Eritrea the largest mean value of DMC2x (n) is for Millet and the smallest is for Wheat; for Ethiopia, the largest mean of DMC2x (n) is for
Sorghum while the smallest is for millet; for Kenya, the largest mean value of DMC2x (n) is for Rice and the smallest is for Sugarcane; for
Rwanda, the largest mean value of DMC2x (n) is for regular potatoes and the smallest is for tea; for Somalia, sorghum gives the largest
mean value of DMC2x (n) while the smallest is for sugarcane; for Tanzania, the largest mean value of DMC2x (n) is for seed cotton and the
smallest is for sorghum; for Uganda, the largest mean value of DMC2x (n) is for Cassava and the smallest is for Banana. However, under
crops yield; the results are different for different crops. For example, under crop yield: for Burundi, the largest mean value of DMC2x (n)
is for dry beans and the smallest is for sorghum. Similar result is observed for other countries. Generally, under the crop production
quantity and yield, the cross-correlation coefficients DMC2x (n) are all less than 0.5. The results are similar for different time scale on
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Table 8
Estimates of the 2-year, 10-year, 50-year, 100-year and 200-year return levels with 95% pointwise normal approximation confidence intervals for
annual maximum rainfall.
Country
Return level
Lower CI
Estimate
Upper CI
Burundi
2-year
10-year
50-year
100-year
200-year
203.8126
260.5380
292.2287
301.3397
308.4495
212.1101
272.5974
312.4199
326.3112
338.6783
220.4076
284.6568
332.6110
351.2828
368.9072
Eritrea
2-year
10-year
50-year
100-year
200-year
86.3448
138.4346
175.1977
186.3546
194.2442
93.29069
155.75013
222.65715
254.94242
289.76024
100.2366
173.0656
270.1166
323.5302
385.2762
Ethiopia
2-year
10-year
50-year
100-year
200-year
146.6107
182.3828
210.6278
221.2676
231.0043
151.1961
192.2954
231.1628
248.4406
266.1776
155.7814
202.2080
251.6979
275.6135
301.3509
Kenya
2-year
10-year
50-year
100-year
200-year
131.3395
180.2898
212.9111
223.3894
231.9809
138.0754
193.1640
239.7517
258.9805
277.8692
144.8112
206.0381
266.5923
294.5716
323.7574
Rwanda
2-year
10-year
50-year
100-year
200-year
174.2149
221.7593
260.4569
275.0695
288.3233
180.2545
235.9485
292.8763
319.5136
347.7124
186.2941
250.1378
325.2958
363.9578
407.1016
Somalia
2-year
10-year
50-year
100-year
200-year
62.67268
95.42373
117.81037
124.28933
128.58518
67.09106
106.33018
148.06561
168.11092
189.66786
71.50944
117.23663
178.32084
211.93251
250.75053
Tanzania
2-year
10-year
50-year
100-year
200-year
197.5657
241.6616
264.6905
270.7896
275.2821
204.2174
250.9706
281.0254
291.3412
300.4398
210.8691
260.2796
297.3602
311.8928
325.5974
Uganda
2-year
10-year
50-year
100-year
200-year
168.0181
205.1125
233.1812
243.4603
252.7232
172.8455
214.9771
252.8138
269.0673
285.4160
177.6730
224.8416
272.4464
294.6743
318.1088
annual basis but could differ on inter-seasonal scale. Very weak to weak DCCA multiple cross-correlations could be due to the fact that
there may be other possible factors/variables in the form of cofounders, which is not considered and that may attenuate the estimated
relationship. For example, Soil fertility and disease or pest could also negatively affect crop production quantity as well as yield. A good
number of researchers have documented this, see for example, Feng et al. (2018), Horn et al. (2015), Horn and Shimelis (2020), Tadele
(2017) and references therein. East African countries have been severally plagued by swarms of various locust species and these have
resulted to serious socio-economic problems as a consequence of their widespread destruction (Alomenu, 1985). Remarkably, the most
recent locust outbreak in 2019 which mainly affected Kenya, Somalia and Ethiopia is regarded as the worst in the last 2 decades and
have resulted in a decline in crop yield and production. For example, the aggregated cereal production in 2019 declined by roughly 8%
in the region because of reduction in harvests in countries like Tanzania, Uganda and Kenya due to severe early season dryness, erratic
weather and pest attacks (FAO, 2020). In addition, most countries within the horn of Africa (perhaps Africa at large) are still widely
characterized by weak data collection, harmonization and monitoring of agricultural sectors trends (Warinda et al., 2020). This has
resulted to disconnection between realities, such as actual, estimated and forecasting of growth rate within the agricultural sectors.
Hence, this may also be a factor behind very weak to weak DCCA multiple cross-correlations.
In addition to the results of the DCCA multiple cross-correlations, we explored copula based models to understand the respective
impact of annual maximum temperature and rainfall on the crop yield and production quantity. The eleven copula models in Section 3
were sequentially fitted to the annual maximum temperature and crop yield/production quantity as well as to annual maximum
rainfall and crop yield/production quantity from one of the countries at a time. There are nine countries in all, however only eight of
the countries have data on crop yield and production quantity. The selected crop is different for every country. So, the eleven models
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Fig. 8. Probability plots for the monthly maximums temperature associated with the capitals of each of the nine country.
were fitted to a total of
• 28 pairs from Burundi (including, yield/temperature(Tmax): Tmax vs banana, Tmax vs dry beans, Tmax vs cassava, Tmax vs
sorghum, Tmax vs sweat potatoes, Tmax vs tea, Tmax vs coffee; production/temperature(Tmax): Tmax vs banana, Tmax vs dry
beans, Tmax vs cassava, Tmax vs sorghum, Tmax vs sweat potatoes, Tmax vs tea, Tmax vs coffee; yield/rainfall(Rmax): Tmax vs
banana, Tmax vs dry beans, Tmax vs cassava, Tmax vs sorghum, Tmax vs sweat potatoes, Tmax vs tea, Tmax vs coffee; production/
rainfall(Rmax): Tmax vs banana, Tmax vs dry beans, Tmax vs cassava, Tmax vs sorghum, Tmax vs sweat potatoes, Tmax vs tea,
Tmax vs coffee);
• 16 pairs from Eritrea (including, yield/temperature(Tmax): Tmax vs barley, Tmax vs millet, Tmax vs sorghum, Tmax vs wheat,
production/temperature(Tmax): Tmax vs barley, Tmax vs millet, Tmax vs sorghum, Tmax vs wheat; yield/rainfall(Rmax): Rmax vs
barley, Rmax vs millet, Rmax vs sorghum, Rmax vs wheat; production/rainfall(Rmax): Rmax vs barley, Rmax vs millet, Rmax vs
sorghum, Rmax vs wheat);
• 28 pairs from Ethiopia (including, yield/temperature(Tmax): Tmax vs barley, Tmax vs dry beans, Tmax vs maize, Tmax vs millet,
Tmax vs sorghum, Tmax vs coffee green, Tmax vs tea; production/temperature(Tmax): Tmax vs barley, Tmax vs dry beans, Tmax vs
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Fig. 9. DCCA cross-correlation coefficient ρDCCA between the monthly maximum temperature in each of the capitals of the nine countries and that in
other administrative regions,respectively.
maize, Tmax vs millet, Tmax vs sorghum, Tmax vs coffee green, Tmax vs tea; yield/rainfall(Rmax): Rmax vs barley, Rmax vs dry
beans, Rmax vs maize, Rmax vs millet, Rmax vs sorghum, Rmax vs coffee green, Rmax vs tea; production/rainfall(Rmax): Rmax vs
barley, Rmax vs dry beans, Rmax vs maize, Rmax vs millet, Rmax vs sorghum, Rmax vs coffee green, Rmax vs tea);
• 24 pairs from Kenya (including, yield/temperature(Tmax): Tmax vs dry beans, Tmax vs maize, Tmax vs rice, Tmax vs sugarcane,
Tmax vs coffee green, Tmax vs tea; production/temperature(Tmax): Tmax vs dry beans, Tmax vs maize, Tmax vs rice, Tmax vs
sugarcane, Tmax vs coffee green, Tmax vs tea; yield/rainfall(Rmax): Rmax vs dry beans, Rmax vs maize, Rmax vs rice, Rmax vs
sugarcane, Rmax vs coffee green, Rmax vs tea; production/rainfall(Rmax): Rmax vs dry beans, Rmax vs maize, Rmax vs rice, Rmax
vs sugarcane, Rmax vs coffee green, Rmax vs tea);
• 28 pairs from Rwanda (including, yield/temperature(Tmax): Tmax vs dry beans, Tmax vs cassava, Tmax vs potatoes, Tmax vs
sorghum, Tmax vs sweet potatoes, Tmax vs coffee green, Tmax vs tea; production/temperature(Tmax): Tmax vs dry beans, Tmax vs
cassava, Tmax vs potatoes, Tmax vs sorghum, Tmax vs sweet potatoes, Tmax vs coffee green, Tmax vs tea; yield/rainfall(Rmax):
Rmax vs dry beans, Rmax vs cassava, Rmax vs potatoes, Rmax vs sorghum, Rmax vs sweet potatoes, Rmax vs coffee green, Rmax vs
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Table 9
Mean and median values of DMC2x (n) between annual maximum temperature, rainfall and largest recorded crops production quantity in each country.
Country
Crops(Production Quantity)
Burundi
Banana
Dry beans
Cassava
Coffee green
Sorghum
Sweet potatoes
Tea
0.0267311
0.01586473
0.06542337
0.08879556
0.03984648
0.06217045
0.03162235
0.02850016
0.0122015
0.04285919
0.07306103
0.02496417
0.06263571
0.01136495
Eritrea
Barley
Millet
Sorghum
Wheat
0.1008377
0.3566572
0.3144881
0.08315373
0.1008377
0.3566572
0.3144881
0.08315373
Ethiopia
Barley
Dry beans
Coffee green
Maize
Millet
Sorghum
Tea
Wheat
0.2404406
0.1082153
0.08899601
0.2015268
0.09437968
0.3667605
0.222472
0.1484381
0.2404406
0.1082153
0.08899601
0.2015268
0.09437968
0.3667605
0.222472
0.1484381
Kenya
Dry beans
Coffee green
Maize
Rice
Sugarcane
Tea
Wheat
0.03984774
0.04218437
0.04424917
0.07738944
0.01363724
0.05024468
0.0295696
0.03822387
0.03518542
0.05260844
0.07751266
0.009289968
0.04617821
0.02296226
Rwanda
Dry beans
Cassava
Coffee green
Potatoes(regular)
Sorghum
Sweet potatoes
Tea
0.1021505
0.07485318
0.0459572
0.1034671
0.05170472
0.03678845
0.03580734
0.08291592
0.04231879
0.0471506
0.0912652
0.03191529
0.01643704
0.03133111
Somalia
Banana
Maize
Sorghum
Sugarcane
0.06582099
0.07249685
0.1678232
0.0373044
0.05991377
0.07079406
0.1865772
0.03066796
Tanzania
Maize
Millet
Rice
Seed cotton
Sisal
Sorghum
Sweet potatoes
0.05457564
0.08895918
0.08453628
0.09812685
0.04328987
0.04032166
0.09730964
0.01483128
0.1047692
0.03971663
0.07807331
0.02956949
0.0331527
0.08191026
Uganda
Banana
Cassava
Coffee green
Millet
Plantains
Sweet potatoes
0.02968093
0.1686789
0.04845207
0.06089387
0.06767799
0.1012706
0.01646794
0.1603874
0.03241015
0.07818331
0.07937095
0.0789819
Mean values of DMC2x (n)
Median values of DMC2x (n)
tea; production/rainfall(Rmax): Rmax vs dry beans, Rmax vs cassava, Rmax vs potatoes, Rmax vs sorghum, Rmax vs sweet po­
tatoes, Rmax vs coffee green, Rmax vs tea);
• 16 pairs from Somalia (including, yield/temperature(Tmax): Tmax vs banana, Tmax vs maize, Tmax vs sorghum, Tmax vs sug­
arcane; production/temperature(Tmax): Tmax vs banana, Tmax vs maize, Tmax vs sorghum, Tmax vs sugarcane; yield/rainfall
(Rmax): Rmax vs banana, Rmax vs maize, Rmax vs sorghum, Rmax vs sugarcane; production/rainfall(Rmax): Rmax vs banana,
Rmax vs maize, Rmax vs sorghum, Rmax vs sugarcane);
• 28 pairs from Tanzania (including, yield/temperature(Tmax): Tmax vs maize, Tmax vs millet, Tmax vs rice, Tmax vs seed cotton,
Tmax vs sisal, Tmax vs sorghum, Tmax vs sweet potatoes; production/temperature(Tmax): Tmax vs maize, Tmax vs millet, Tmax vs
rice, Tmax vs seed cotton, Tmax vs sisal, Tmax vs sorghum, Tmax vs sweet potatoes; yield/rainfall(Rmax): Rmax vs maize, Rmax vs
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Table 10
Mean and median values of DMC2x (n) between annual maximum temperature, rainfall and largest recorded crops yield in each country.
Country
Crops(Yield)
Mean values of DMC2x (n)
Median values of DMC2x (n)
Burundi
Banana
Dry beans
Cassava
Coffee green
Sorghum
Sweet potato
Tea
0.04724334
0.08254537
0.04492253
0.05460166
0.02503005
0.0457423
0.02710484
0.04290667
0.05181167
0.02455955
0.03376634
0.02080968
0.03616615
0.02886359
Eritrea
Barley
Millet
Sorghum
Wheat
0.08997237
0.2375969
0.3162197
0.03049508
0.08997237
0.2375969
0.3162197
0.03049508
Ethiopia
Barley
Dry beans
Coffee green
Maize
Millet
Sorghum
Tea
Wheat
0.2496309
0.1302107
0.06154357
0.1416701
0.2001502
0.3931963
0.2604967
0.2381488
0.2496309
0.1302107
0.06154357
0.1416701
0.2001502
0.3931963
0.2604967
0.2381488
Kenya
Dry beans
Coffee green
Maize
Rice
Sugarcane
Tea
Wheat
0.01240797
0.03721756
0.05551024
0.00938076
0.01210361
0.0422591
0.01303708
0.008578291
0.02234529
0.05523383
0.00776347
0.007225772
0.01068856
0.005704828
Rwanda
Dry beans
Cassava
Coffee green
Potatoes(regular)
Sorghum
Sweet potatoes
Tea
0.07086354
0.03892904
0.06557167
0.09397602
0.01804413
0.06189837
0.04659045
0.06098257
0.02660257
0.05956364
0.04511187
0.01963569
0.05649181
0.03239431
Somalia
Banana
Maize
Sorghum
Sugarcane
0.008109014
0.130706
0.03703666
0.1088082
0.006477799
0.1424412
0.03610817
0.1081269
Tanzania
Maize
Millet
Rice
Seed cotton
Sisal
Sorghum
Sweet potatoes
0.04213929
0.0168461
0.01897522
0.07392475
0.02767052
0.04101571
0.03874897
0.03633243
0.007698343
0.01611845
0.07443533
0.006006027
0.03087147
0.03871002
Uganda
Banana
Cassava
Coffee green
Millet
Plantains
Sorghum
0.02026953
0.1460173
0.09187126
0.0441338
0.05814964
0.02331391
0.01996173
0.1521111
0.07926838
0.03330717
0.04563948
0.01960571
millet, Rmax vs rice, Rmax vs seed cotton, Rmax vs sisal, Rmax vs sorghum, Rmax vs sweet potatoes; production/rainfall(Rmax):
Rmax vs maize, Rmax vs millet, Rmax vs rice, Rmax vs seed cotton, Rmax vs sisal, Rmax vs sorghum, Rmax vs sweet potatoes);
• 24 pairs from Uganda (including, yield/temperature(Tmax): Tmax vs banana, Tmax vs cassava, Tmax vs coffee green, Tmax vs
millet, Tmax vs plantain, Tmax vs sweet potatoes; production/temperature(Tmax):Tmax vs banana, Tmax vs cassava, Tmax vs
coffee green, Tmax vs millet, Tmax vs plantain, Tmax vs sweet potatoes; yield/rainfall(Rmax): Rmax vs banana, Rmax vs cassava,
Rmax vs coffee green, Rmax vs millet, Rmax vs plantain, Rmax vs sweet potatoes; production/rainfall(Rmax):Rmax vs banana,
Rmax vs cassava, Rmax vs coffee green, Rmax vs millet, Rmax vs plantain, Rmax vs sweet potatoes).
The method of maximum likelihood was used to estimate the parameters of the copula models. We used the log-likelihood values,
the AIC values, the BIC values, the CAIC values, the AICc values and the HQC values obtained from the fitted copula models to
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discriminate between them. However, owing to space concern, the values of these criteria are not provided. From the results, we
observe that the eleven models are viable for different contexts in each country as there is no one best fitting copula model jointly for all
the eight countries and yield/production considered. For instance, with respect to Burundi, the clayton and Frank copula give the
smallest values for the AIC, the BIC, the CAIC, the AICc and the HQC for most of the pairs, except for Rmax vs banana pair(normal
copula) under production quantity. With respect to Eritrea, the Clayton copula gives the smallest values for the AIC, the BIC, the CAIC,
the AICc and the HQC for most of the pairs, except for the Rmax vs barley under production. With respect to Ethiopia, the Joe copula
gives the smallest values for the AIC, the BIC, the CAIC, the AICc and the HQC for most of the pairs, except for Tmax vs millet(Frank
Fig. 10. Contour plots of some randomly selected pairs, from Left to right: top panel(Clayton and Frank), middle panel(Clayton) and bottom
panel(Joe).
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copula) under production quantity. With respect to Kenya, the Plackett copula gives the smallest values for the AIC, the BIC, the CAIC,
the AICc and the HQC for most of the pairs, except for Tmax vs dry beans(Clayton copula) and Tmax vs dry beans(Student’s t copula)
under production and yield, respectively. With respect to Rwanda, the Joe copula gives the smallest values for the AIC, the BIC, the
CAIC, the AICc and the HQC for most of the pairs, except for Rmax vs sweat potatoes(Tawn copula) and Tmax vs sweat potatoes
(Clayton copula) under production and yield, respectively. With respect to Somalia, the Plackett copula gives the smallest values for
the AIC, the BIC, the CAIC, the AICc and the HQC for most of the pairs, except for Tmax vs maize(Frank copula) under yield. With
respect to Tanzania, the Joe copula gives the smallest values for the AIC, the BIC, the CAIC, the AICc and the HQC for all the tem­
perature(Tmax) pairs under production and yield, while the Plackett copula gives the smallest values for the AIC, the BIC, the CAIC, the
AICc and the HQC for all the rainfall(Rmax) pairs under production and yield. With respect to Uganda, the Plackett copula gives the
smallest values for the AIC, the BIC, the CAIC, the AICc and the HQC for most of the pairs, except for Tmax vs coffee green(Joe copula)
and Rmax vs coffee green(Hüsler-Reiss copula) under production. We performed the goodness of fit of the best fitting copula models
using the Cramer-von Mises and Kolmogorov–Smirnov statistics. The contour plots of randomly selected pairs of some of the best
performing copula density in each country are shown in Fig. 10.
A close examination of the plots show that the Clayton and Joe copulas have more probabilities concentrated in the left and right
tails, respectively and they appear to be asymmetric. This is not surprising as these two copulas are respectively well known for
capturing left and right tail dependencies. These results are general for all the pairs where the Clayton and Joe give the best fits. In
contrast, the Frank copula cannot capture either left or right tail dependence. It is also radially symmetric. Using the estimated pa­
rameters, the associated Kendall’s tau and tail dependence coefficients(where possible) for the best copula models from the pairs of the
variables are obtained. It is interesting to observe that majority of the pairs involving food crops, especially cereals give high
dependence parameters. Taking the results of Burundi and Eritrea for illustration. Out of the 28 pairs under Burundi, where the Clayton
copula gives the best fit with the smallest values for the selection in 24 pairs. The strongest dependence in both extremes with crop
production and yield is found for sorghum: Tmax vs sorghum(production) − θ = 0.945, Tmax vs sorghum(yield) − θ = 0.932, Rmax vs
sorghum(production) − θ = 0.815 and Rmax vs sorghum(production) − θ = 0.798; with their corresponding Kendall’s τ, respectively
given as: 0.3208829, 0.3178718, 0.2895204 and 0.285203. Similarly, Out of the 16 pairs under Eritrea, though they are all cereal; it is
again interesting to observe that none of the pairs have Kendall’s τ < 3. The Clayton copula exhibits greater dependence in the left tail
with the associated lower tail given by λL = 2− 1/θ . For Ethiopia, Rwanda and Tanzania where the Joe copula gives the best, the lower
and upper tail dependence coefficients are obtained using λL = 0 and λU = 2 − 21/θ . In general, regardless of the country, climatic
factors and the agricultural products(production/yield), similar results were obtained for other pairs. That is, the strongest dependence
is demonstrated by the pairs involving cereal crops, while the weakest dependence is characterized by the pairs involving regular
potato. This result could be attributed to the fact that cereals are the main staples within the region and Africa at large. For example, as
noted by FAO (2020) and Biswas et al. (1987), cereal crops account for more than 50% of cultivation land in Ethiopia. In addition,
cereal crops like millet and sorghum are mostly grown in areas that can withstand drought as well as heat. And they are among the
notable staple foods for millions of people within the horn of Africa.
5. Conclusion
In this paper, we have conducted an extreme value analysis of both monthly and annual extremes of rainfall and temperature for
nine countries within the greater horn of Africa. Selected countries were carefully chosen to give a good geographical representation of
the region. For each of these countries, we have identified a model which best describe the behavior of these two climatic factors. Based
on the best performing model for each country, estimates of the return levels of the maximum rainfall and temperature are noted with
their corresponding 95% confidence intervals. The estimated return levels could be used as a measure for flood assessment, man­
agement and protection within the region.
Given the complexity and highly non-stationary nature of climatic variables, particularly temperature; we employed detrended
cross-correlation coefficient (ρDCCA ) and its generalization (DMC2x ) to study the spatial cross-correlation patterns of temperature on
different time scales in every country. Taking the capital of each country as a center, the cross-correlation coefficients (ρDCCA ) between
the temperatures in other regions and those in each capital were computed. Positive cross-correlations with pronounced seasonal
patterns are observed for small to medium time scale between the monthly maximum temperatures in each capital of the nine countries
and that in their respective administrative regions/provinces/state, except for Somalia. However, these seasonality patterns appear to
level off on a large time scale for some countries. This finding has a number of implications for risk assessment and management. For
example, based on this quantification of cross-correlation pattern between region’s temperature and their capitals, climate network
could be developed which may be useful for climate diagnosis and prediction in every location within the horn of Africa.
Furthermore, eleven copula models (including Gumbel, Galambos, Gaussian, Hüsler-Reiss, Tawn, t-Student extreme (t-EV),
Clayton, Frank, Joe, Plackett, Student’s t.) were used to characterized the dependence and effects of climate extremes on crop yield and
production. Selected crops are principal food and cash crops peculiar to each country. Based on the selection criteria, which include the
log-likelihood values, the AIC values, the BIC values, the CAIC values, the AICc values and the HQC values we observe that the eleven
copula models are viable for different contexts in each country as there is no one best fitting copula model jointly for all the eight
countries considered. However, regardless of the country, climatic factors and the agricultural products(production/yield) the
strongest dependence is demonstrated by the pairs involving cereal crops, while the weakest dependence is characterized by the pairs
involving regular potato.
Some future work are to: (1) employ DCCA in a more elegant fashion such that we can model the cross-correlations of two tem
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Climate Risk Management 32 (2021) 100295
E. Afuecheta and M.H. Omar
perature series and at same time find the factors that regulate the pattern; (2) fit copulas with d > 2 to describe the dependence among
climatic factors and the agricultural products(production/yield) of these countries; (3) account for other factors such as socialeconomic conditions and technologies on the yield as well as production which were neglected due non-availability of data; (4)
divide the data into different seasons such that the impact of period like winter, summer, etc will be properly accounted for.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgments
The authors would like to acknowledge the support provided by Deanship of Scientific Research (DSR) at King Fahd University of
Petroleum and Minerals for funding this work through project No. SR191030. The authors would also like to thank the Editor and the
two referees for careful reading and comments which greatly improved the paper.
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