Shuffled complex evolution and multi-linear approaches to flow prediction in the equatorial Nile
basin.
Jean-Marie Kileshye Onema a, b, Zacharia Katambara a, Akpofure Taigbenu a
(a)Water Research Group, School of Civil and Environmental Engineering, University of
Witwatersrand, Private Bag 3, WITS 2050, South Africa
(b)
Ecole Supérieure des Ingénieurs, Université de Lubumbashi, PO Box 1825, République
Démocratique du Congo
Corresponding author email : jmkilo3@yahoo.co.uk
Abstract
Continuous development and investigation in flow prediction is of interest in watershed hydrology
especially where watercourses are poorly gauged and data are scarce like in most parts of Africa.
Thus, this paper reports on the potential and limitations of two approaches in forecasting discharges
for the Semliki River. The Semliki River is part of the upper drainage of the Albert Nile. With an
average annual local runoff of 4.622km3, the Semliki watershed contributes up to 20% of the flows of
the White Nile. The watershed was sub-divided in 21 subcatchments (S3 to S23); eight physiographic
attributes for each subcatchment were generated and used to forecast flows. Multi linear and shuffled
complex evolution approaches were used to predict the discharges. The predictions were validated and
calibrated using the limited historical flows records on the river.
The statistics of prediction
performance, namely the Nash-Sutcliffe efficiency (NSE), the percent bias (PBIAS) and the RMSEobservations standard deviation ratio (RSR) were performed. The linearity assumption proved to be
adequate in capturing the interactions between catchments descriptors and the discharges.
Subsequently the flows predicted were more accurate. The shuffled complex evolution embedded in a
Delphi programme provided a less precise combination of parameters for flow prediction.
Additionally, no physical meaning could be linked to those parameters due to the black box approach
associated with the shuffled complex evolution.
Key words: flow prediction, Nile, Semliki, shuffled complex evolution
1. Introduction
Numerous approaches exist for flow prediction in natural river reaches. Flow forecasting has
significant interest both from research as well as from an operational point of view. The choice of
methods depends on data availability and the type of application. While continuous developments
strive at enhancing our predictive capability for streamflow, we are often facing the problem of
predictions in ungauged basin (Sivapalan et al., 2003). Reliable and accurate estimates of hydrologic
components are not only important for water resources planning and management but are also
increasingly relevant to environmental studies (Schröder, 2006). Several studies have reported on the
use of catchment descriptors and regionalization of parameters for flow prediction in ungauged basins.
Among the most recent studies Sefton and Howard (1998), Mwakalila (2003), Xu (2003), Merz and
Blöschl (2004), McIntyre et al., (2005), Sanborn and Bledsoe (2006), Yadav et al.,(2007), Sharda et
al., (2008) Kwon et al., (2009) and Shao et al., (2009) have dealt with the subject. In their comparison
of linear regression with artificial neural network, Heuvelmans et al., (2006) indicated the need of
well-informed choice of physical catchment descriptors as a first condition for a successful parameter
regionalization. Cheng et al., (2006) reported on the importance and usefulness of parsimonious
models for runoff prediction in data-poor environment as these models are characterized by small
number of parameters. Reducing uncertainty associated with predictions in ungauged basin is critical
as reported by Uhlenbrook and Siebert (2003) Koutsoyiannis, (2005a, 2005b) as well as Zhang et al.,
(2008).
Lately, Koutsoyiannis et al., (2008) indicated that analogue modeling techniques for simulation are
also used for prediction with impressive performance due to the advances achieved on non linear
dynamical systems (chaotic systems). The major drawback is the fact that these approaches are data
intensive and work as black box, thus no process insight is provided.
Relevant spatial and temporal scale to flow prediction continues to be a subject of discussions and
investigations in watershed hydrology (Kundzewicz, 2007). Thus, this paper reports on the potential
and limitations of multi-regression and shuffled complex optimizations when it comes to flow
prediction in a medium size and data-scarce watershed of the equatorial Nile region.
2
2. Study area
These analyses are conducted within the Semliki watershed of the equatorial Nile region (Fig1.). The
catchment covers an estimated area of 23,621.0 km2.
Figure 1: Semliki watershed
The Semliki drains the basins of lakes Edwards and George, and a contributing area downstream that
includes the western slopes of the Ruwenzori range. The watershed receives an average rainfall of
1245mm per annum, with peaks occurring in May (95mm) and October (205mm). An average annual
local runoff of 4.622km3 has been estimated from records at Bweramule (Sutcliffe and Parks, 1999).
The elevations comprise flat areas and ice-caped mounts climbing up to 4862m above the sea level.
The flora and the fauna of the watershed constitute one of the unique and distinct ecosystem of the
Albertine Rift ecosystem. The vegetation predominantly comprises medium altitude moist evergreen
to semi deciduous forest. Five distinct vegetation zones have been documented under the mount
Ruwenzori and they occur with changes in altitude. Detailed information on landscape physiographic
attributes is reported in Kileshye Onema and Taigbenu (2009).
3.
Methods and materials
The landscape on any catchment is made up of several combinations of physiographic attributes.
These combinations are usually variable among catchment, giving rise to different hydrological
responses. Table 1 presents the eight physiographic attributes extracted from the 21 subcatchments
that form the Semliki watershed (S3 - S23) (Fig 2).
Table 1: Physiographic attributes generated for subcatchments (S3-S23)
3
Physiographic attribute
Stream Length
Drainage density
unit
m
km.km-2
Abbreviation
Strm_len
Drainage
Mean Stream slope
%
avg_slope
Max elevation of the subcatchment
m
Max_elev
Min elevation of the subcatchment
Weighted average elevation of the
area
Mean monthly precipitation
m
Min_elev
m
avg_elev
mm/month
monthly__prec
Mean monthly NDVI
-
monthly_NDVI
Figure 2: Semliki Subcatchments (S3 to S23).
The Principal component analysis (PCA) as Indirect Gradient Analysis was used as the exploratory
technique to study the structure of the data. Multi-regression was performed on the eight
physiographic selected to derive the model as illustrated in figure 3.
4
Descriptive Statistics
Section
Predicted Values with
Confidence Limits of
Means
Normality Tests
Section
Residual Report
Analysis of Variance
Regression Equation
Section
Estimated Model
Figure 3: Multiple regression analysis
The shuffled complex evolution embedded in a Delphi programme was used on the other hand for the
purpose of determining the optimum parameters of the predictive equation. The method combines the
strength of the downhill simplex procedure with the concepts of controlled random search competitive
evolution and complex shuffling. The optimization approach used is further documented in Katambara
and Ndiritu (2009).
The performance rating for prediction accuracy used was the following dimensionless evaluation
statistics: The Nash-Sutcliffe efficiency (NSE), the Percent bias (PBIAS) and the RMSE-observations
standard deviation ratio (RSR). They were computed respectively as shown in equations (1) (2) and
(3) (Moriasi et al., 2007).
 n
obs
sim 2 
  (Yi  Yi ) 

NSE  1   in1
obs
mean 2 

(Yi  Y
)
 

i 1
Equation 1
 n

obs
sim
  (Yi  Yi ) *100 

PBIAS   i 1
n
obs


(Yi )



i 1
Equation 2
5
RSR 
RMSE
STDEVobs
 n

obs
sim 2
  (Yi  Yi ) 
 i 1

 
 n

obs
mean 2
) 
  (Yi  Y
 i 1

Equation 3
4. Results and discussions
4.1. Principal Components Analysis (PCA)
The descriptive statistics below (Table 2) show that the variables included in the PCA are measured at
significantly varying scales, however this does not affect the results of the analysis as the matrix
analyzed is the scale invariant correlation matrix of the variables (as opposed to the covariance
matrix).
Table 2: Descriptive statistics of variables
Variables
Stream Length
Drainage
Average Slope
Maximum elevation
Minimum elevation
Average elevation
Monthly
precipitation
Monthly NDVI
Count Mean
Standard
Deviation
21
21
21
21
21
21
29.46
7.85E-02
0.2
2577.9
733.18
1164.82
20.66
3.52E-02
0.23
1273.84
99.78
392.54
21
21
101.72
0.6
6.97
7.52E-02
The correlations between the variables are summarized in table 3. There are some high correlations
(greater than 0.5), implying that there is a correlation structure that can potentially be modeled or
further explored using PCA. If all the correlations were low there would be no need to try to model
the correlation structure using principal components analysis.
The value of phi for this data (0.4) (Table 4), suggests that there is considerable redundancy or
complexity in the group of variables which warrants further examination using PCA. Bartlett’s
sphericity test is used to test the null hypothesis that the correlation matrix of the group of variables is
a zero identity matrix i.e. none of the variables are correlated. If we obtain a p-value for the Bartlett’s
test which is greater than 0.05 we should not carry out PCA. The p-value obtained is very low
indicating that we can carry out the PCA (Table 4).
Table 3: Coefficients of correlations between variables
Stream Drainage Average Maximum Minimum Average
Monthly
Monthly
6
Length
Stream
Length
Drainage
Average
Slope
Maximum
elevation
Minimum
elevation
Average
elevation
Monthly
precipitation
Monthly
NDVI
Slope
elevation
elevation
elevation precipitation NDVI
1
0.344
-0.2
-0.05
-0.29
-0.29
-0.08
0.07
0.34
1
-0.19
-0.46
-0.24
-0.45
0.07
0.16
-0.2
-0.19
1
0.37
0.13
0.54
0.08
-0.15
-0.05
0.47
0.36
1
0.23
0.87
-0.5
-0.63
-0.28
-0.24
0.13
0.226
1
0.46
-0.25
-0.23
-0.28
-0.45
0.54
0.87
0.46
1
-0.45
-0.64
-0.09
0.07
0.08
-0.5
-0.25
-0.45
1
0.78
-0.07
0.16
-0.15
-0.63
-0.23
-0.64
0.78
1
Table 4: Bartlett test and Glaeson – Staelin (Phi)
Bartlett
DF P-Value Glaeson –Staelin
Test
(Phi)
81.98 28 0.00000
0.395415
According to the Kaiser criterion when the principal components have been calculated using
correlation coefficients is to retain the principal components with an eigenvalue greater than 1.
Therefore in this case we would retain the first 3 principal components. These 3 principal components
account for 76% of the variation in the data (table 5).
Table 5: Eigenvalues of components
No.
Eigenvalue
1
2
3
4
5
6
7
8
3.49
1.56
1.02
0.78
0.64
0.30
0.17
0.05
Individual
Percent
43.59
19.46
12.71
9.80
7.97
3.74
2.13
0.62
Cumulative
Percent
43.59
63.04
75.75
85.55
93.51
97.25
99.38
100.00
Scree Plot
|||||||||
||||
|||
||
||
|
|
|
The Eigenvectors are the coefficients that relate the scaled original variables to the derived factors.
The scaled original variables are defined as follows:
xi 
X i  i
i
Equation 4
where; xi = the scaled variable; X i = the original variable;  i = the mean of the original variable and
7
 i = the standard deviation of the original variable
For instance, the first principal component is:
Factor1 = -0.157886(Strm_len) -0.283358(Drainage) + 0.241135(avg_slope) +
0.468021(Max_elev)
+
0.268251(Min_elev)+
0.505941(avg_elev)
0.339627(monthly__prec) - 0.417503(monthly_NDVI)
-
Equation 5
The italics are there to emphasise that we are referring to the scaled original variables and not their
original values.
Inspection of the eigenvectors shows that the first factor is a contrast of avg_slope, max_elev,
min_elev and avg_elev to strm_len, drainage, monthly__prec and monthly_NDVI. This factor
explains 44% of the variation in the data. The eigenvectors of the three factors that have been retained
are shown in table 6 below:
Table 6: Eigenvectors of principal components
Variables
Factor1 Factor2 Factor3
Strm_len
-0.16
-0.52
-0.42
Drainage
-0.28
-0.36
-0.10
avg_slope
0.24
0.37
-0.60
Max_elev
0.47
-0.08
-0.27
Min_elev
0.27
0.17
0.56
avg_elev
0.51
0.10
-0.14
monthly__prec
-0.34
0.52
-0.22
monthly_NDVI
-0.42
0.37
-0.05
The factor loadings are the correlations between the variables and the factors. Factor 1 is most highly
correlated to the maximum elevation and the average elevation; whereas factor 3 is most highly
correlated to the average slope and the minimum elevation (Table 7).
Table 7: Factor loadings of principal components
Variables
Factor1 Factor2 Factor3
Strm_len
-0.29
-0.65
-0.42
Drainage
-0.53
-0.45
-0.10
avg_slope
0.45
0.46
-0.60
Max_elev
0.87
-0.10
-0.27
Min_elev
0.50
0.21
0.56
avg_elev
0.94
0.12
-0.14
monthly__prec
-0.63
0.65
-0.22
monthly_NDVI
-0.78
0.46
-0.05
4.2. Multi-regression and shuffled complex evolution
Multi-regression and shuffled complex evolution were the optimization approaches used for the
determination of runoff predicting equations.
Several normality tests were performed, results are reported in table 8, the Anderson Darling test was
the only one that rejected the null hypothesis at 20%.
8
Table 8: Normality test
Reject H0
Test name
Test value
Prob level
At Alpha = 20%
Shapiro Wilk
0.96
0.44
No
Anderson Darling
0.53
0.18
Yes
D'Agostino Skewness
0.64
0.52
No
D'Agostino Kurtosis
0.35
0.73
No
D'Agostino Omnibus
0.54
0.76
No
The estimated model generated from the multiple regression is represented in equations (4). The
optimum parameters established with the shuffled complex evolution are illustrated in equation (5).
Table 9 and figure 4 a-b indicate that the multi-regression outperformed the shuffled complex
evolution in the optimization of parameters. While shuffled complex evolution has been documented
to provide optimum parameters this study illustrates the fact the approach is data-driven and limited
performance can be achieved from it in data-poor environment.
Table 9: Performances statistics
Method
Multi-regression
Shuffled complex
45
Flow (Mm3 month-1)
Flow (Mm3 month-1)
evolution
Multi Regression
40
Observed
Simulated
35
30
NSE
0.90
PBIAS
-2.8E-15
RSR
0.31
-0.50
42.43
1.22
45
Shuffled Complex
40
30
25
25
20
20
15
15
10
10
5
5
0
Observed
Simulated
35
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21
Subbasin
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21
(a)
Subbasin
(b)
Figure 4. Predicted vs observed: (a) multi-regression, (b) shuffled complex evolution
Q=10349036.065+ 1663.615*avg_elev-276576.251*avg_slope42571909.54*Drainage-556.309*Max_elev+ 293.425*Min_elev+
7178929.965*monthly_NDVI-100711.884*monthly__prec+ 168414.558*Strm_len
Equation 4
Q=0.2*Strm_len1.12Drainage0.3Max_el0.6Min_el0.1avg_el0.4monthly_Prec0.2NDVI0.1
Equation 5
9
5. Conclusions
This study undertaken in the data-scarce Semliki watershed of the equatorial Nile reported on the use
of two optimisation approaches for the prediction of flows. The principal Component analysis
performed identified variables that explained most of the variability in the dataset investigated. The
dimensionless statistics on the predictions indicated that the multi-regression approach outperformed
the shuffled complex evolution approach. While the later approach has been documented as one of the
germane approach in the determination of optimum parameters, this study illustrated the fact the
shuffled complex approach is data-driven and limited performance can be achieved from it in datapoor environments.
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