Forecasting solid waste generation in Juba Town, South Sudan using Artificial Neural Networks (ANNs) and Autoregressive Moving Averages (ARMA)

Forecasting weekly SW generation using ANNs and ARMA models in Juba Town, South Sudan
JEWM
Forecasting solid waste generation in Juba Town, South
Sudan using Artificial Neural Networks (ANNs) and
Autoregressive Moving Averages (ARMA)
1David Lomeling* and 2Santino Wani Kenyi
1,2Department of Agricultural Sciences, College of Natural Resources and Environmental Studies (CNRES), University of
Juba, P.O. Box 82, Juba, South Sudan
Prediction of solid waste generation is critical for any long term sustainable waste management,
especially of a fast-growing municipality. Lack of, or inaccurate solid waste generation records
poses unparalleled challenges in developing cohesive and workable waste management
strategies for any concerned authorities, as this is influenced by several interlinked demo-
graphic, economic, and socio-cultural factors. The objective of this study was to compare two
models in forecasting of MSW generation and how this would be built into an effective MSW
management program. Two models, the Autoregressive Moving Average (ARMA 1,1) and the
Artificial Neural Networks (ANNs) were tested for their ability to predict weekly waste generation
of 14 households in Juba Town, Central Equatoria State (CES), South Sudan. Results showed that
both the artificial intelligence model ANNs and the traditional ARMA model had good prediction
performances; where for ANNs the RMSE, MAPE and r² were 0.080, 10.64%, 0.238 respectively,
whereas for ARMA the RMSE, MAPE and r² were 0.102, 6.98% and 0.274 respectively. Both models
showed no significant differences and could be therefore be used for Solid Waste (SW)
forecasting. Based on the results, the weekly SW generated 52 weeks later (end of year) had
reached 0.365 kg/capita indicating a 18.2% rise from 0.3 kg/capita at the start of the study. Under
the current consumption rate, the weekly SW per capita in Juba Town is expected to reach 0.596
kg by 2020.
Keywords: Artificial Neural Networks, Autoregressive Moving Averages, Continuous Wavelet Transform, Waste
Generation Forecasting,
INTRODUCTION
South Sudan witnessed rapid economic growth rates with
Gross Domestic Product (GDP) estimates of over
$16billion (World Bank Report 2008) immediately after the
signing of the Comprehensive Peace Agreement, CPA in
2005. Huge oil revenues from crude oil sales attracted
investments and rapid population growth especially in
urban centers like Juba. Conversely, this economic and
population growth put enormous strain on the local
environment and on the availability of natural resources
(Lomeling et al., 2016) in terms of increased demand for
areas for settlement as well as depleting the forest
resources as cheap energy source. With the rapid
population increase in Juba, waste generation
inadvertently also increased. This meant that more land
and resources would have to be required for planned
waste disposal site in the short to long term, if serious
environmental pollution through indiscriminate waste
disposal is to be abated.
*Corresponding author: David Lomeling, Department of
Agricultural Sciences, College of Natural Resources and
Environmental Studies (CNRES), University of Juba, P.O.
Box 82, Juba, South Sudan. Email:
dr.david_lomeling@gmx.net
Journal of Environment and Waste Management
Vol. 4(2), pp. 211-223, August, 2017. © www.premierpublishers.org. ISSN: XXXX-XXXX
Research Article

X2
Xn
w1
w2
wn
y
b
Lomeling and Kenyi 212
Prediction or forecasting of solid waste generation in Juba
Town has become indispensable and crucial, if available
resources are to be effectively deployed in the sustainable
management of SW. Time series offers an important area
of stochastic forecasting in which past observations of a
specific variable are analyzed to develop a model that can
be used to make future projections. Over the past
decades, much effort has been undertaken in the
development and improvement of time series models that
can be applied for forecasting like the Artificial Neural
Networks (ANNs) and the Auto-Regressive Moving
Averages (ARMA) as compared to classical and traditional
methods like linear and multiple regressions or polynomial
functions. During the last two decades, several stochastic
models have been used to predict SW waste like the
support vector machine, (Abbasi et al., 2013); artificial
neural networks (Abdoli et al., 2011); (Antanasijevic et al.,
2013); hybrid procedure (Xu et al., 2013); time series
analysis, (Mwenda et al., 2014); multi-step chaotic model
(Song and He, 2014); principal component analysis and
gamma testing (Noori et al., 2010); grey fuzzy dynamic
modeling (Chen and Chang, 2010); Fourier series (Darko
et al., 2016); simulated annealing based hybrid forecast
(Song et al., 2014). In general, most ANN prediction
models have clearly outlined architecture with specific
number of input variables at the input layer and
corresponding number of expected outputs at the output
layer. Depending on the problem to be modeled, several
input variables may be chosen for a given number of
anticipated outputs. The choice of any one single, all-
purpose model under any prevailing conditions is therefore
unrealistic. Structurally, such a multi-purpose model would
require more complex algorithms capable of handling
several calculations simultaneously for some desired
number of input variables and then come up with optimal
predictions.
The objective of this study was to compare the
effectiveness of machine learning method (Artificial Neural
Networks ANNs) and the stochastic linear model
(Autoregressive Moving Average – ARMA) for medium to
long-term weekly forecasting of solid waste generation in
some households of Juba Town, South Sudan. The
integration of Continuous Wavelet Transform (CWT) with
both ANNs and ARMA models was primarily used for easy
visualization and interpretation of input signals as well as
frequencies in the time series. Using both models, a good
estimate of the weekly SW per capita was to be made
during the 2012-2020 forecasting period.
Forecasting Models
Artificial Neural Networks (ANN)
ANNs is basically a computational approach whose
architecture is mostly composed of three layers: an input,
hidden and output layers mimicking the way biological
neurons receive, transfer and output signals. Each neural
unit or perceptron is linked with many others and can either
be enforcing once the summation function has surpassed
some threshold value to be propagated or inhibitory in their
effect once below the summation function value. The
single neuron is illustrated by the McCulloch-Pitts Model
(1943).

(input e.g. SW (b, bias that increases (output or
data) or lowers the net input of forecasted
activation/sigmoid function) value)
Figure 1: A model of a three-layers perceptron.
Mathematically, the output variable, y is the sum of the
individual weighted variables and bias that influences the
activation function S:
(x1w1 + x2w2 + ⋯ xnwn)+b=y= 𝑺 ∑ (xj
n
j=1 wj + b)
Equation (1)
The underlying concept is to arrive at a function that
minimizes the error (E) between the input (actual) and
output (forecasted) variables thereby enhancing the
accuracy in the forecasting or prediction.
Emin(x,b)=𝑓[∑ x;w,b−yn
j=1 ]²
Equation (2)
Usually the sums of each input signal (x1, x2, …xn) and
intensity or weighted values (w1, w2,…wn) are passed on
through a non-linear function known as an activation or
transfer function that usually has a sigmoid shape, that is
bounded, and differentiable as:
𝑺(x) =
1
(1+e−x)
Equation (3)
Many theoretical and experimental works have shown that
a single hidden layer (with one or more several hidden
nodes) is sufficient for ANN to approximate any complex
nonlinear function (Dreiseitl and Ohno-Machado, 2002;
Chattopadhyay and Bandyopadhyay 2007; Matias et al.,
2013; Vishwakarma and Gupta, 2011; Aggarwal and
Kumar 2015). A more plausible argument is that, the low
number of nodes in the hidden layer directly linked to the
input neuron have a low bias (b) and would tend to
increase the input of the activation function. This in turn
would enhance large changes in their weights and learn
very quickly and so incur less errors as manifested by the
high correlation coefficients for both training and test sets.
In this study, a model based on a feedforward neural
network with a single hidden layer was used. Hereby, the
learning process in understanding hidden and strongly
non-linear dependencies in the time series of the observed
X1
 S

J. Environ. Waste Manag. 213
Figure 2a. Run plot sequence of weekly disposed solid waste showing seasonality in the time series
Figure 2b. Effects of alpha term (α) on exponential smoothing or the exponentially weighted moving average (EWMA)
and modeled data in the training and test were faster and
the forecasting made easier. However, such forecasts can
only be made for shorter prediction times and not for
extensively longer future times as the error would tend to
increase.
Auto-Regressive Moving Average, ARMA model
The first step in developing the ARMA model was
determining the stationarity of the time series in which case
the mean and variance are time invariable. The
autocorrelation function (ACF) may signify stationarity of
the time series, if it cuts off or decomposes quite rapidly
towards zero. Conversely, if the ACF decomposes very
slowly and gradually towards zero, this would indicate non-
stationarity and would need to be transformed or
differenced to obtain stationarity by stabilizing the variance
of a time series. Differencing helps stabilize the mean of a
time series by removing changes in the level of a time
series, and so eliminating trend and seasonality.
On the other hand, a time series that shows seasonality as
in Figure 2a can be exponentially smoothened by an
exponentially weighted non-parametric value (α) to
“smoothen” the value X_t to a new value X ̂ recursively
(Figure 2b):
X̂ = αXt + (1 − α)Xt−1 Equation (4)
where 0 ≤ α ≤ 1
Our data showed that the best seasonal exponential
smoothing was when α=0,2 with r²=0,26; α=0,5 with
r²=0,08 and α =0,8 with r²=0,04. The value α =0,2 best
approximated the mean value and regression constant of
the time series and therefore gave a better trend analysis.
Owing to its simplicity, the exponential smoothing only “de-
seasonalizes” a time series thereby assuming the nature
of a linear regression equation between two variables.
However, it´s predictive ability is inadequate in more
complex stochastic and non-linear processes with more
y = 0,0011x + 0,3077
r² = 0,04
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 8 15 22 29 36 43
KgofSWperhousehold
Time.: Weeks
Obs. a=0,2 a=0,5 a=0,8
y = 0,0011x + 0,3077
r² = 0,04
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 8 15 22 29 36 43 50
WeeklySW/household
Time. Weeks

than 3 or more input variables. Although some authors
(Mwenda et al., 2014; Petridis et al., 2016); Rimaityte et
al., 2011; Karpušenkaitė et al., 2016) have mentioned the
theory behind single exponential smoothing, however, not
much in terms of its practical applicability have yet been
reported.
The ARMA (p,q) model used herein is made up of an
Autoregressive AR(p) and Moving Average MA(q)
components. This means that, the forecast value of (X) at
time (t) in a time series is a function of both linear
combination of past X-innovations and a moving average
of series ( 𝜀t), known as white noise process characterized
by zero mean ( 𝜇 ) and variance (𝜎).
Xt = c + 𝜀t + ∑ αiXt−1 + ∑ βi
𝑞
i=1
𝑝
i=1 𝜀t−i
Equation (5)
With the values p and q identified from the ACF and PACF,
the model parameters (αi) and (βi) can then be estimated.
Once we had confirmed the stationarity of the time series,
the autocorrelation (ACF) and partial autocorrelation
functions (PACF) were used to determine the correlation
and model structure of the data.
MATERIALS AND METHOD
This study focused on solid waste generated by single
persons in 14 households in Kator residential area of Kator
Payam, Juba County of Central Equatoria State in South
Sudan. Collected waste was placed into a container whose
tare weight was initially determined using the hanging
scale. The net weight of the solid waste was then
determined. The data were collected weekly over a period
of 31 weeks as from June 2010 till January 2011. Each
household had on average 6 persons with monthly income
of about 650 SDG (Sudanese Guinee equivalent to
145$/month as of June 2010). About 40-60% of the
collected waste was made up of predominantly degradable
organic component consisting mainly of food residues and
partly cartons and newspapers. The rest was made up of
PET plastic water bottles. The waste was collected at the
end of each week, weighed and the daily amounts per
capita generated (kg/capita/week) was then calculated
This work attempted to predict the weekly waste generated
in the remaining 21 weeks till July 2011 based on the
previous data. For the training set, data from the first 20
weeks were used representing 90% of the actual data. The
rest 10 weeks representing 10% of the actual data were
then used as test set.
Data description and analysis
From the weekly solid waste data reports, the Excel-based
Alyuda Forecaster XL software was used to make future
projections in the time series. Its algorithm allows an easy
data preprocessing of the neural networks. Additionally,
the Continuous Wavelet Transfer (CWT) using the PAST3
software was used to illustrate through the spectral power
the peaks or spikes of weekly solid waste disposal in the
time series.
Model identification
The effect of model choice on both correlation functions is
shown in Figure 3 (a) and (b). The spikes presented in the
ACF and PACF showed a correlation in the data every 2
lag units. The model identification revealed that with the
cut-off at lag 1, the autoregressive of order p and the
moving average of order q was also 1 as the ACF was got
below zero after first lag. For illustrative purposes, the
ARMA (1,2) as opposed to ARMA (1,1) was also used to
compare the parameter values and how these influenced
the model choice. The ARMA (1,2) in Figure 4 (a) and (b)
as compared to ARMA (1,1) showed dissimilar AC and
PAC functions at lag 1 and was outside the 95%
confidence limits. The ARMA (1,1) was then chosen and
there was therefore no need for any differencing.
Training Set: From this, 32 data entries of the weekly solid
waste disposed per household were trained through a
process of finding values for the weights (w) and biases (b)
whereby the error between the measured and predicted
values was minimized. The back-propagation algorithm
was used here. The accuracy of the resulting training
process was then applied for making projections in neural
network model. Figure 5 shows the accuracy of the
training set between the observed and forecasted values
with a high r²=0.99.
Testing Set: This data set consisted of entries of the last
weekly solid waste disposed per household and was
used to test whether, or not the accuracy derived from
the training process would provide the most appropriate
solution and hence confirm the predictive power of the
network model. Similarly, the test set showed high
correlation coefficient of r²=0.99 as shown in the training
set (Table 2).
Table 2: Comparison of the learning ability (MSE) between the
training and test set of a ANN.
Training set Test set
Nr. of rows 32 12
Nr. of Good Forecast 30(94%) 12(100%)
Average MSE 8,91E-05 1,25E-05
r² 0,99 0,99
Validation Set: Generally, this data set is used to minimize
overfitting in the model. As in our case, given the high
correlation coefficients of both the training and test sets
and the high predictive power of the neural network model,
there was therefore no need for a validation set. Figure 6
shows simulation runs of both the training and test sets
and confirmed the accuracy of the ANN in predicting future
solid waste data.

Figure 3. Autocorrelation plot of weekly per capita solid waste disposed in Juba (red dotted lines are upper and lower 95% confidence
limits). ACF plot of residuals of ARMA (1,1) with α1 = -0,999 and β1 = -0,800 (a) and PACF plot of residuals of ARMA (1,1) with α1 = -
0,999 and β1 = -0,800 (b).
Figure 4: ACF and PACF plot of residuals of ARMA (1,2) with AR (1) at α1 = -0,999 and AM (2) at β1 = -0,526 and β2 = -0,800
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 1 2 3 4 5 6 7 8 9 10 11 12 13
AutocorrelationFunction,ACF
Lag (a)
ACF residuals ARMA (1,1)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13
PartialAutocorrelation
Function,PACF
Lag (b)
PACF residuals ARMA (1,1)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13
AutocorrelationFunction,
ACF
Lag (a)
acf residuals of ARMA (1,2)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10 11 12 13
AutocorrelationFunction,
ACF
Lag (b)
pacf residuals of ARMA (1,2)

Figure 5. Scatter plot of a ANN for both training and test set of a weekly per capita solid waste in Juba Town
Figure 6. Simulation of the training and test data sets
Figure 7. Error distribution of training and test sets
Figure 7 shows the error distribution of both the training
and test sets. Whereas the error margin by the training set
varied between -0,03 and 0,03, for the test set, this was
between -0,006 and 0,006. The error magnitude in the
training set was ten-fold more than that in the test set.
Presumably, the larger the data set, the larger the variance
between the observed and forecasted and hence the
larger is the error margin and vice versa.
RESULTS
ANN Model
Our simulation was based on a simple network
architecture that returned the smallest MSE and therefore
the best prediction accuracy. The MSE and (r²) recorded
for both the training and test sets have already been
y = 0,9798x + 0,0069
r² = 0,99
0.1
0.2
0.3
0.4
0.5
0.6
0.1 0.2 0.3 0.4 0.5 0.6
Forecasted
Actual
Training set
y = 0,9855x + 0,0069
r² = 0,99
0.43
0.45
0.47
0.49
0.51
0.43 0.48 0.53
Forecasted
Actual
Test set
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 4 7 10 13 16 19 22 25 28 31 34 37
Forescasted
Actual
Rows
Actual Forecasted
0.4
0.45
0.5
0.55
0.4
0.45
0.5
0.55
1 4 7 10 13
Forecasted
Actual
Rows
Actual Forecasted
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
1 7 13 19 25 31 37
Error
Rows
Error by training set n=40
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Error
Rows
Error by test set n=12

Table 3. Parameters of ARMA (1,1) and (1,2) models of weekly SW generated in Juba Town, South Sudan
Parameters Standard Error, SE t-statistics p-Value AIC LLF
ARMA
(1,1)
AR (1)
AM (1)
-0,999 (α1)
-0,888 (β1)
0,073 7,643 7,66E-10 -
96,56
50,28
ARMA
(1,2)
AR (1)
AM (1)
AM (2)
-0,999 (α1)
-0,526 (β1)
-0,278 (β2)
0,000 12,738 1,08E-16 -
96,34
51,17
ARMA
(1,3)
AR (1)
AM (1)
AM (2)
AM (3)
-0,999 (α 1)
-0,632 (β1)
-0,420 (β2)
0,289 (β3)
0,000 13,464 3,304E-17 -
99,05
53,53
presented in Table 2, from where we observed 1-1-1 (1
input layer, 1 hidden layer, and 1 output layer) gives an
accurate prediction of the weekly solid waste output.
Applying the rule-of-thumb method for estimating the
number of neurons in the hidden layer reported by
Karsoliya (2012), we can assume that the number of
neurons in the hidden layer is approximately 1. (or 70-
90%) of the input layer. Noticeably, the number of hidden
neurons is equal to the number of input nodes whereby the
larger number of neurons would ostensibly lead to
“overfitting” whereas with a relatively smaller number of
neurons would lead to “underfitting”. The good fit (r²) for
both training and test sets shows that the ANN modeled
the observed data quite accurately. There is generally no
“golden rule” for the number of hidden layers that is
applicable for all non-linear time series. Whereas during
the training process other problems are best predicted with
two hidden layers (Srinivasan et al., 1994; Zhang, 1994;
Baron, 1994). Other studies (Wanas, et al., 1998) showed
that the best performance of a neural network occurred
when the number of hidden neurons was equal to log (N),
where N is the number of training samples. Another study
conducted by Mishra and Desai (2006) showed that the
optimal number of hidden neurons is (2n+1), where n is the
number of input neurons.
ARMA Model
Although model identification as per Box-Jenkins
methodology clearly showed an ARMA (1,1)
autoregressive (p) and moving average (q), we
experimented with different q values to see to what extent
this influenced not only model estimation but also the
diagnostic checking and consequently the forecast. Three
different model MA values were varied while the AR was
kept constant. i.e. ARMA (1,1); (1,2) and (1,3) respectively.
The best model parameters were selected based on the
model that gave the least Akaike Information Criterion
(AIC) value and highest likelihood estimation here denoted
as Logarithmic Likelihood Function (LLF) Table 3.
Judging by the values of AIC and LLF, it is evident that
experimented values at ARMA (1,2) and (1,3) are close to
the actual ARMA (1,1) values and suggested the
adequacy of the ARMA (1,1) model.
The scatter plot in Figure 8 shows a positive trend with
several points around the trend line. The relatively low (r²)
values suggested that there was neither an under- or
overestimation for both ANN and ARMA models with most
points between the 0,3-0,4 kg/week for both the observed
and ANN-ARMA models. Whereas the ANN model had a
r²-value of 0,238 this was 0,274 for ARMA model, these r²-
values for both models were not significantly different from
each other. The comparatively lower r²-value of the ANN
would suggest the inability of a linear function in describing
an entirely non-linear time series data.
Performance evaluation
The performance of either of the models was determined
by measuring the difference between the observed and
predicted values in the time series. Best estimates were
those error values that were closer to zero, indicating less
differences between the measured and observed values.
Three accuracy measures were used as follows:
(1) Mean Absolute Percentage Error (MAPE):
MAPE
=
1
𝑛
∑ |
Pobs − Ppred
Pobs
| 100
𝑛
𝑡=1
Equation (6)
(2) Root Mean Square Error (RMSE):

Figure 8. Comparing the ARMA (p, q) and ANN for the observed and forecasted weekly SW per capita
RMSE = √
1
n
(Pobs − Ppred)² Equation (7)
Weekly SW generation for the first 32 weeks which
included both the training and test sets were later
combined with the lead time 20 weeks and the errors
between observed and predicted assessed as in Table 3.
Based on the MAPE, RMSE performance comparison
between both models, there was however no significant
difference between both models. The slightly higher MAPE
value for the ANN model could be due to inherent
drawback of the MAPE in overestimating error value
especially when the difference between observed and
predicted values is zero. (Pobs. = 0).
Table 4. Comparison of the model performances in terms of MAPE and
RMSE of both ANNS and ARMA models.
ANN ARMA
MAPE (%) 10,60 6,98
RMSE 0,080 0,102
Ideally, the ARMA model would have shown poor
performances in both MAPE and RMSE due to its inability
to model non-linear variables as the ANN model. In
complex and non-linear data in the time series, the ARMA
model would inevitably lose predictive accuracy as it is
unable to adequately capture the errors between the
observed and forecasted values in the time series and so
the prediction errors would increase (see Figure HH).
Conversely, the ANN model is capable of identifying
nonlinearity in the time series by having an additional
computation or hidden layer that allows for better curve-
fitting with minimal errors between the forecasted and
observed data.
Diagnostic checking
Based on the autocorrelation plot, the Ljung-Box (1978)
test attempts to establish the overall data randomness
within a time series at some chosen or predetermined lags.
Basically, the null hypothesis (H0) assumes that the data
are random or independently distributed whereas this is
the contrary for the alternative hypothesis (Ha) that
assumes the non-randomness nature of the data. We then
performed the Ljung-Box statistic test as:
Q (LB)= ∑
𝑝̂ 𝑘
2
𝑛−𝑘
𝑁 𝐾
𝑘=1
Equation (8)
Where n=sample size, 𝑝̂ 𝑘
2
=sample correlation at lag k,
𝑁 𝐾=number of lags being tested. Choosing the 𝑝̂ 𝑘
2
at lag 7
and testing at the p=0,05 confidence level, the Q(LB) value
at 0,645 was less than the (chi-square) at 2.013 thereby
reaffirming the H0 and adequacy of the ARMA (1,1) model.
As aforementioned the Q(LB) for randomness is reinforced
by the autocorrelation functions as in Figures 3 and 4
Hereby, all the autocorrelation at the subsequent lags fall
within the 95% confidence limits, other than that at lag 0.
Model verification
Model verification dealt with ascertaining whether the
residuals of the ARMA model as expressed by the ACF
and PACF had any discernible and systematic patterns
y = 0,3139x + 0,2314
r² = 0,274
y = 0,3355x + 0,2367
r² = 0,238
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
ARMA
ANN
Actual
ARMA Forecast ANN Forecast

Figure 9. Forecasting SW using the ANN and ARMA (1, 1) models
Figure 10. Weekly SW forecast of households in Juba Town using the ARMA (1,1) model
with respect to the lags. Our study showed that, none of
these correlations was significantly different from zero at
the 95% confidence limit indicating the goodness of fit and
appropriateness of the ARMA model.
Forecasting
The two forecasting models ARMA and ANNs presented
in this paper (Figures 9 and 10) allowed us to predict the
weekly generated SW with mean value of about 0.35
kg/household and upper and lower limits of about 0.63 and
0.16 kg/household respectively. Towards the 51st week,
the generated SW was well below the 0.4 kg level and
would under constant economic and political conditions
remain below the 0.5 kg level till 2020. This forecasting is
useful in mobilizing and optimizing available financial
resources and personnel needed for effective SW
management.
2.2 Continuous Wavelet Transform (CWT)
One of the main goals of a CWT is to enable the easy
visualization and interpretation of input signals and
frequencies as a function of time. The CWT decomposes
a continuous time function of a time series into the
components called wavelets each with a different localized
frequency. Although CWT are usually applied for non-
stationary signals, we tried to apply both the Morlet wavelet
transform and the Derivative of Gaussian (DOG) to
account for the high instantaneous amplitude or signal
outbursts in the time series as well as in the recognition of
inherent frequency patterns. The Morlet wavelet transform
(Goupillaud, et al., 1984) is given as:
y = 0,0011x + 0,3077
r² = 0,04
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 8 15 22 29 36 43 50
ANN
ARMA(1,1)
Time. Weeks
Obs. ARMA (1, 1) ANN
y = 0,0011x + 0,3033
r² = 0,06
-
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
1 6 11 16 21 26 31 36 41 46 51
Forecast
Actual
Time: Weeks
SW forecast using ARMA (1, 1) model
Actual Forecast

𝜗 𝑤0
(𝑡) = 𝐾𝑒 𝑖𝑤0 𝑡
𝑒
−𝑡2
2 Equation (9)
Where 𝑤0 is the non-dimensional frequency and the
vertical scale corresponds to the length of wavelet, i.e., the
number of time steps used for the CWT. The cone of
influence (COI) is the region where the wavelet power
spectra are limited due to the influence of the end points
of finite length signals also known as the e-folding time.
Here, the signal discontinuity drops by a factor of (e-2) and
ensures that edge effects are negligible beyond this point.
The DOG with m=derivative was set at 6 and expressed
as:
DOG =
(−1) 𝑚+1
√(𝑚+
1
2
)
𝑑 𝑚
𝑑 𝑚 (𝑒
−2
2 ) Equation (10)
Values of m=2 or 4 using DOG basis functions did not
describe the spectral decomposition in the time series
adequately. The choice of the wavelet used for time-
frequency decomposition in a time series is critical. The
use of the Morlet wavelet for example, showed that the
frequency resolution was “lumped” together and was
localized within a 95% confidence limit. On the contrary,
using Derivative of Gaussian (DOG) wavelet with m=6,
(Figure 11 a and b), the result was good time localization
with strong frequencies. The Morlet and Derivative of
Gaussian (DOG) wavelets are plotted below.
The choice for the type of wavelets in interpreting the
frequency and intensity of data entries (xn) in a time series
are shown in Figures 11 and 12. We used the forecasted
data from both models as input data to generate the
respective wavelets. For both models, (Figure 11a and b),
the dominant power spectrum was characterized by
smaller spikes between log scale 1.2 and 3.2 from week15
to 30. This time coincided with the highest weekly waste
generation around Christmas season where expectedly,
more households had much disposable incomes enabling
an increased consumption and so increased solid waste
generation. However, the ARMA as opposed to the ANNs
model showed certain areas outside the COI, clearly an
indication of model overestimation by the former. In both
cases, using the Morlet basis function (Figure 11c and d)
showed poor spectral decomposition as a function of time
than the DOG m=6.
As aforementioned with the Ljung-Box randomness test in
a time series with the ARMA model, the resulting power
spectrum using the CWT can as well exhibit coherent and
significant structures. A test of significance can be used to
distinguish between significant and random structures
(Mohr, 2003) in which case values in a power spectrum
may be considered as statistically significant at the 95%
level and therefore not random.
For the null hypothesis (H0), we assumed that the time
series had a mean power spectrum. Spikes or peaks in the
wavelet power spectrum above this background spectrum
were shown as black contoured spectrum “significant at
the 5% level” or equivalent to “the 95% confidence level”.
Therefore, if the peak in the power spectrum of the Morlet
and DOG wavelets are significantly larger than the
background spectrum, it is then assumed to be a true
feature. There was better interpretation of the spectral
decomposition for the observed data as well as for both
models when the DOG m=6 as opposed to Morlet.basis
function was applied.
SUMMARY AND CONCLUSION
In this study on time series models, we analyzed and
compared the Artificial Neural Networks (ANNs) and the
Autoregressive Moving Averages (ARMA) in forecasting
the weekly amounts of solid waste generated by single
persons in fourteen households of Kator residential area of
Juba town. For the ANNs model, the input training data
used were the average weekly amounts of solid waste
collected from during the month of June 2011. The test
data of July 2011 were then used to validate the ANNs
model. The result showed that ARMA (1,1) slightly
outperformed the ANNs model in terms of MAPE but not
in terms of the RMSE. However, considering both
performance indicators, there was no significant
differences between both models and so either model
could be used to forecast the amount of the solid waste
generation for the next weeks and years. Using both
models is, however, for comparative reasons imperative in
order qualify and quantify the extent of deviation of the
estimated values from the observed mean. The projected
values showed that by 2020, the weekly generation
according to both models will have reached about 0.596
kg/capita with a 95% confidence interval lying between
0.2-0.6 kg/capita. Such projections may be used by Juba
municipal or town council in the proper planning and
management of solid waste.
ACKNOWLEDGEMENTS
My special gratitude goes to Mr. Santino Wani Kenyi for
data collection as part of his BSc dissertation. We the
authors also wish to extend our thanks to the households
in Kator Payam for allowing the conduction of the
interviews.

(a) (b)
(c) (d)
Figure 11. (a) Weekly solid waste output signals for the ARMA model when using the DOG m=6 (b) for the ANNs models. Whereas when using the
Morlet basis function for ARMA (d) for ANNs model. Wavelet power spectrum showing cone of influence (COI) at the 95% confidence interval with areas
of intense peaks and signals in red contoured in (black) while those with poor signals in (blue).

(a) (b)
Figure 12. (a) Weekly solid waste output signals using the Morlet basis function for the observed data and (b) using the DOG m=6 basis function.
Wavelet power spectrum showing cone of influence (COI) at the 95% confidence interval with areas of intense peaks and signals in red contoured in
(black) while those with poor signals in (blue).
REFERENCES
Abbasi M, Abduli MA, Omidvar B, Baghvand A (2013).
Results uncertainty of support vector machine and
hybrid of wavelet transform-support vector machine
models for solid waste generation forecasting.
Environmental Progress and Sustainable Energy.
33:220.
Abdoli, MA, Nezhad MF, Sede RS, Sadegh B
(2011). Environmental Progress and Sustainable
Energy. 31 (4): 628–636.
Aggarwal R, Kumar K (2015). Effect of Training Functions
of Artificial Neural Networks (ANN) on Time Series
Forecasting. International Journal of Computer
Applications. (0975 – 8887), 109(3): 14-17.
Antanasijevic D, Pocajt, V, Popovic I, Redzic N, Ristic M
(2013). The forecasting of municipal waste generation
using neural networks and sustainability indicators.
Sustain Science. 8: 37-46.
Asante-Darko D, Adabor ES, Amponsah SK (2012). A
Fourier series model for forecasting solid waste
generation in the Kumasi metropolis of Ghana.
Transactions on Ecology and The Environment. 202:
173-185.
Barron, AR (1994). A comment on ‘‘Neural networks: A
review from a statistical perspective’’. Statistical
Science. 9 (1): 33–35.
Chattopadhyay, S and Bandyopadhyay, G (2007). Single
hidden layer artificial neural network models versus
multiple linear regression model in forecasting the time
series of total ozone. Int. J. Environ. Sci. Tech. 4 (1):
141-149.
Chen HW and Chang NB (2000). Prediction analysis of
solid waste generation based on grey fuzzy dynamic
modeling. Resour. Conserv. Recycling. 29, 1-18.

Dreiseitl S, Ohno-Machado L. (2002). Logistic regression
and artificial neural network classification models: a
methodology review. J Biomed Inform. 35: 352–359.
Foster G (1996). Wavelets for period analysis of unevenly
sampled time series. Astron J. 112: 1709-1729.
Frick P, Baliunas SL, Galyagin, D (1997). Wavelet analysis
of stellar chromospheric activity variations.
Astrophysics J. 483: 426-434.
Goupillaud, P., Grossman, A., Morlet, J (1984). Cycle-
octave and related transforms in seismic signal
analysis. Geo-exploration. 23: 85.102.
Karpušenkaitė A, Denafas G, Ruzgas T. (2016).
Forecasting Hazardous Waste Generation using Short
Data Sets: Case Study of Lithuania. Environmental
Protection Engineering. 8(4): 357–364.
Karsoliya S. (2012). Approximating Number of Hidden
layer neurons in Multiple Hidden Layer BPNN
Architecture. International Journal of Engineering
Trends and Technology. 3(6): 714-717.
Ljung, GM, Box, GEP (1978). On a Measure of a Lack of
Fit in Time Series Models”. Biometrika. 65(2), 297–
303.
Lomeling D, Modi, A. L., Kenyi, M. S., Kenyi, M. C.,
Silvestro, G. M., Yieb, J. L. L. (2016): Comparing the
Macroaggregate Stability of Two Tropical Soils: Clay
Soil (Eutric Vertisol) and Sandy Loam Soil (Eutric
Leptosol). International Journal of Agriculture and
Forestry. 6(4): 142-151.
Matias T, Souza F. Araújo R, Carlos AH (2013). Learning
of a single-hidden layer feedforward neural network
using an optimized extreme learning machine.
Neurocomputing. 129: 428–436.
McCulloch W and Pitts W (1943). A logical calculus of the
ideas immanent in nervous activity. Bulletin of
Mathematical Biophysics. 5:115–133.
Mishra AK, Desai VR. (2006). Drought forecasting using
feed-forward recursive neural network. Ecological
Modelling. 198 (1–2): 127–138.
Mohr, LB (2003). Understanding significance testing. Sage
Pub., Inc., 2003.
Mwenda A, Kuznetsov D, Mirau S (2014). Time series
forecasting of solid waste generation in Arusha city-
Tanzania. Mathematical Theory and Modelling. 4 (8):
29–39.
Noori R, Karbassi A, Salman SM. (2010). Evaluation of
PCA and Gamma test techniques on ANN operation
for weekly solid waste. J Environ Management.
91(3):767-71. doi: 10.1016/j.jenvman.2009.10.007.
Petridis NE, Stiakakis E, Petridis K, Dey P (2016).
Estimation of computer waste quantities using
forecasting techniques. Journal of Cleaner Production.
112: 3072-3085.
Rimaityte I, Ruzgas T, Denafas G, Racys V, Martuzevicius
D (2011). Application and evaluation of forecasting
methods for municipal solid waste generation in an
eastern-European city. Waste Management &
Research. 30(1) 89–98.
Song J, He J, Zhu M, Tan D, Zhang Y, Ye S, Shen D, Zou
P (2014). Simulated Annealing Based Hybrid Forecast
for Improving Daily Municipal Solid Waste Generation
Prediction. The Scientific World Journal. 1-7.
Song J and He J (2014). A Multistep Chaotic Model for
Municipal Solid Waste Generation Prediction.
Environmental Engineering Science. 31(8): 461-468.
Srinivasan D, Liew AC, Chang, CS. (1994). A neural
network short-term load forecaster. Electric Power
Systems Research. 28: 227–234.
Sweldens W. (1998). The lifting scheme, a construction of
second generation wavelets. SIAM J Math. Anal.
8(29): 511-546.
Vishwakarma VP and Gupta MN (2011). A New Learning
Algorithm for Single Hidden Layer Feedforward Neural
Networks. International Journal of Computer
Applications. (0975 – 8887), 28(6):26-33.
Wanas N, Auda G, Kamel MS, Karray F. (1998). On the
optimal number of hidden nodes in a neural network.
Proceedings of the IEEE Canadian Conference on
Electrical and Computer Engineering., 918–921.
Xu L, Gao P, Cui S, Liu C (2013). A hybrid procedure for
MSW generation forecasting at multiple time scales in
Xiamen City, China. Waste Management, 33(6):1324-
31.
Zhang X. (1994). Time series analysis and prediction by
networks. Optimization Methods and Software. 4:
151–170.
Accepted 23 August 2017
Citation: Lomeling D and Kenyi SW (2017) Forecasting
solid waste generation in Juba Town, South Sudan using
Artificial Neural Networks (ANNs) and Autoregressive
Moving Averages (ARMA). Journal of Environment and
Waste Management 4(2): 211-223.
Copyright: © 2017 Lomeling and Kenyi. This is an open-
access article distributed under the terms of the Creative
Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium,
provided the original author and source are cited.

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