Owusu-Sekyere, E., Harris, E. and Bonyah, E. (2013), âForecasting and ... Kwame Nkrumah University of Science and. Technology (KNUST) and PhD in.
Municipal Solid Waste Estimation and Landfill Lifespan Prediction- A Case Study T. Akyen†, C. B. Boye† and Y. Y. Ziggah † Akyen†, T., Boye, C. B., and Ziggah, Y. Y. (2016), “Municipal Solid Waste Estimation and Landfill Lifespan Prediction”, 4th UMaT Biennial International Mining and Mineral Conference, pp. GG127-133.
Abstract Municipal Solid Waste (MSW) management has been a global challenge facing most developing countries. This is because of the multi-faceted factors/considerations required for effective management; failure to which may pose health implications on humans, flora and fauna as a result of possible pollution of land, water and air. Though standards exist for regulating solid waste management, the inability of some municipalities to adhere to these strict social, safety and environmental regulations coupled with the scarcity of land due to growing population has made it difficult to acquire land for landfills. Most researchers have therefore resorted to the use of forecasting models such as time series in the management of MSW. It is also acknowledged that accuracy of time series forecasting plays a crucial role in MSW management. Therefore, the selection of a particular model for forecasting is an important factor that will influence the performance of the forecasting accuracy. In this study, the volume of solid waste generated in the Tarkwa Municipality was estimated from secondary data source over seven-year quarterly intervals. The data was analysed using ARIMA time series modelling technique to generate a suitable model for estimation and prediction of MSW for the municipality. The results revealed that ARIMA (2, 1, 2) had the least Akiake Information Criterion value and thus was selected as the best model suitable for predicting waste generation in the Tarkwa Municipality. The life span of the existing Tarkwa Nsuaem Municipal Assembly landfill located at Abosso was also determined giving the MSW management of the municipality ample time to plan ahead for the future.
Keywords: Municipal Solid Waste, ARIMA Time Series, Prediction and Forecasting.
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time now. This is as a result of increase in human population which continues to impact on waste generation and management. Also, absence of engineered landfills, operated below recommended standards of regulations or sanitation practices, is a challenge in the management of waste. For this reason, the issue of MSW generation in the Municipality has attracted the attention of opinion leaders including members in academia. It is therefore important to know the amount of MSW generated in the Municipality in order to do an accurate prediction of solid wastes thus, facilitating effective solid waste management.
Introduction
Municipal Solid Waste (MSW) is commonly referred to as waste that are generated daily at households resulting from everyday activities regarded as useless and are disposed of at landfills (Zade and Noari, 2007). MSW poses a challenge which leads to land and water pollution, this is particularly true where the waste is dumped openly in an uncontrolled and unhygienic manner. This type of dumping system allows biodegradable materials to decompose and produce foul smell and sometimes served as breeding grounds for insects and infectious organisms. MSW is a global issue which continues to pose health risks to many cities of the world including Africa (Foray, 2012; Xiao et al., 2006; Roy et al., 2013).
However, forecasting of MSW at landfills is a challenging task due to: insufficient data, inconsistencies in the data collected and the selection of suitable forecasting methods (Intharathirat et al., 2015). It is worth mentioning that the selection of a technique to forecast future MSW depends on the context of the forecast, the relevance and availability of statistical data, the cost/benefit of the forecast, easiness of interpretation among others (Mwenda et al., 2014).
In spite of the innovation in science and technologies, accurate estimation of MSW is still a fundamental problem facing most urban cities of developing countries (Ansah, 2014). Ghana is one of such developing countries where acquisition of land for landfills, the cost of managing waste and collection of the waste is a major concern to stakeholders. In Tarkwa and its environs the aforementioned challenges still prevails and continues to be the prime objective of the Municipal Assembly. The quantity of MSW generated in the Tarkwa Municipality has been rising consistently for some
For these reasons, several methods such as artificial neural network, time series analysis and parametric techniques have been employed by researchers in estimating the amount of MSW generated. It has been established by researchers (Owusu-Sekyere et al., 2013; Chung, 2010; Momani, 2009, Fishman et
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al., 2016) that Auto Regressive Integrated Moving Average (ARIMA) time series model is among the viable techniques that could yield accurate predictions of MSW generated. The ARIMA has the advantages of predicting both long and short term based on the nature of the historic data obtained (Mondal et al., 2014)).
2.2.2 Model Identification At the model identification stage, the auto correlation function (ACF) and partial auto correlation function (PACF) of the differenced data were determined. This was necessary to estimate the values of AR (p) and MA (q) respectively. Both ACF and PACF were used to check whether the model selected was appropriate.
This study therefore, applied the ARIMA time series model to forecast quarterly MSW generated in Tarkwa and its environs.
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A model with one order of differencing assumes the original ARIMA time series has a constant average trend.
Resources and Methods Used
2.1 Materials
2.2.3 Parameter Estimation
This study applied secondary data of MSW generated from 2007 to 2014 in Tarkwa and its environs. The data was obtained from the Environmental Health and Sanitation Unit of the Tarkwa Nsuaem Municipal Assembly. The R software was adopted for the ARIMA modelling.
The essence of the parameter estimation was to test the normality assumptions of the residuals. To accomplish such a task, the residuals were standardized and the ACF plot of the residuals was also done. Likewise, the Ljung-Box statistic was set at p-value of 0.05. This implied that p-values below 0.05 signified that the model was not accurate.
2.2 ARIMA Modelling Technique
2.3.4 Diagnostic Check
The Box-Jenkins (ARIMA) approach which is one of the most widely used techniques was adopted for the modelling of MSW time series data. The basic steps in the Box-Jenkins methodology are: stationarity check, model identification, parameter estimation and diagnostic check.
The diagnostic checking was done to assess the accuracy of the ARIMA model. Thus, to verify whether the estimated parameters selected adequately fitted the data. This was done by checking the residual autocorrelation in the ACF and PACF plots for white noise properties. In the case of evidence of autocorrelation, it becomes necessary to go back to the identification stage and reassigned the model by adding more lags.
The general mathematical expression for the ARIMA model is given by Eq. (1) as: X t X t 1 1(X t 1 X t 2 ) 2 (X t 2 X t 3 ) ε t θ1ε t 1 θ 2ε t 2
(1)
2.2
A description of the approach is given in the following subsections.
Model Accuracy Assessment
In order to assess the adequacy of the ARIMA model developed, the residuals estimated between the desired outputs and the outputs produced by the ARIMA model were utilised. Hence, to make an objective assessment, Performance Criteria Indices (PCI) of Akaike Information Criterion (AIC), Mean Error (ME) and Mean Percentage Error (MPE) were applied. Their mathematical expressions are given by Eqs. (2) to (4) respectively.
2.2.1 Stationarity Check Stationarity check was rigorously done to check whether the ARIMA time series is stationary or not. The following are the procedure followed in differencing to achieve stationarity: If the series has positive autocorrelation out of a high order of lags, the ARIMA needs a higher order of differencing. If the lag-1 autocorrelation is zero or negative or the autocorrelations are small, the series does not need a higher order of differencing. If the autocorrelation is -0.5 or more negative, the ARIMA series may be overdifferenced. The optimal number of differencing is the order of differencing at which the standard deviation is lowest.
n AIC n.log(SSR) 2k (2) n k 1 Where, SSR is the sum of squared residuals n is the number of observations k is the number of parameters ME
2
1 n (O Pi ) n i1 i
(3)
3 Results and Discussion 3.1 Stationarity Check
26000 20000 25000 18000 -1000 0 0
(4)
-1500
random seasonal trend observed
Decomposition of additive time series
100 n O i Pi MPE ) ( n i1 O i Where, Oi is the observed value Pi is the predicted value
2008
2010
2012
2014
Time
A total of 32 observations were made of which the mean, standard deviation, variance, minimum and maximum values were estimated (Table 3.1).
Fig. 3.2 Decomposition Waste Generated Series After decomposition, it was observed clearly that the data exhibits a non-systematic linear trend but the existence of seasonality showed that the MSW data was non-stationary. This was also confirmed by the shape of the ACF and PACF as shown in Figs. 3.3 and 3.4 respectively.
Fig. 3.1 shows the pattern of quarterly waste generated (in tonnage) with zero order differencing. That is the original time series obtained between 2007 and 2014. It was observed from Fig. 3.1 that the series displayed considerable variations that seem to have a repeated cycle, which is an indication that quarterly waste generation data shows a nonstationary trend, with evidence of seasonality.
0
2
4
6
8
Lag
22000
Fig. 3.3 ACF Plot of Series The autocorrelation function (ACF) plot (Fig. 3.3) shows a very slow linear decay which is typical of a nonstationary time series.
18000
PACF Plot
2008
2010
2012
2014
Partial ACF
Year
-0.4
Fig. 3.1 Waste Generation in Tarkwa Municipality
0.0 0.2 0.4 0.6 0.8
Tonnage
26000
-0.4 0.0
Waste Collection In Tarkwa Municipal
0.4
ACF
0.8
ACF Plot
0
Moreover, the presence of trend was observed in the data as shown in Fig. 3.1. Thus, the left hand side of the plot is lower than the right hand side. This is because there is no evidence of seasonal components since there was no regular peaks and troughs displayed. From Fig. 3.2 the irregular pattern exhibit fluctuations.
2
4
6
8
Lag
Fig. 3.4 PACF Plot of Series The PACF (Fig. 3.4) of the series indicated that without differencing of the data, an Auto Regressive (AR) (1) model should fit, which will turn out to be equivalent to taking a first difference. Alternatively, the series needed an order of differencing to be stationarised.
This further support the evidence of trend in its characteristic nature based on which it is assumed that the data is non-seasonal.
Table 3.1 Descriptive Statistics of Waste Generation Data (Tonnage) N Mean Std Deviation Variance Minimum 32 22780.75 2788.686 7776772 18120
3
Maximum 27735
With the MSW data being non-stationary, a stationary series was attained after taking one nonseasonal difference. That is, fitting an ARIMA (0, 1, 0) model with constant AR and moving average (MA) as shown in Fig 3.5.
From the ACF plot (Fig. 3.6), it was noticed that the series displays a cutoff at four (4) successive significant lag, that is, at lag one (1), and there after cuts inconsistently at lag five (5), nine (9) and lag thirteen (13).
data
-0.4 -0.2 0.0 0.2
2000
Partial ACF
4000
PACF Plot
0
2
4
6
0
Lag
-2000
Fig. 3.7 PACF Plot of First Differencing of Series 2008
2010
2012
Also the PACF plot (Fig. 3.7), displays a cutoff at two (2) successive significant lag, that is, at lag two (2), and lag three (3). Lag two (2) is however, considered not significant since this may be due to chance, considering the fact that lag one (1), which comes right before it is significantly not far from zero.
2014
Time
Fig. 3.5 First Difference of Waste Collection Data From Fig. 3.5, the series appears approximately stationary with no long term trend. It exhibits a definite tendency to converge to its mean. The ACF (Fig 3.6) and PACF (Fig. 3.7) confirms a slight amount of positive autocorrelation.
Table 3.2 presents the various ARIMA models that were considered during the model building. Table 3.2 Identification of the Best ARIMA Model Model (p,d=1,q)
σ2
Log likelihood
AIC
(0,1,0)
2517694
-272.44
546.88
(0,1,2)
2105704
-269.76
545.58
(0,1,3)
2081198
-269.65
547.29
(0,1,4)
1662926
-266.63
543.26
(1,1,0)
2487739
-272.26
548.52
(0,1,1)
2313569
-271.24
546.49
(1,1,1)
2235352
-270.70
547.39
(1,1,2)
2103898
-269.78
547.56
1.0
(1,1,3)
1880294
-268.39
546.77
0.5
(1,1,4)
1646251
-266.52
545.05
0.0
The series at this stage is stationary since the trend (definite tendency to converge to its mean) has been completely eliminated and the amount of autocorrelation remaining is small. Since the series was only differenced once to attain stationarity, we can therefore conclude that our data is nonseasonal. This is because for non-seasonal data, at most a first order differencing is usually sufficient to attain apparent stationarity.
(2,1,0)
1890478
-268.29
542.57
(2,1,1)
1842158
-267.91
543.82
(2,1,2)
1034960
-261.77
533.55
3.3 Model Identification In order to select the appropriate model and also make more accurate forecasts, several feasible ARIMA models were observed by making reference to the ACF (Fig. 3.6) and PACF (Fig. 3.7) of the first difference data series. Since the data was differenced ones, the fitted ARIMA models would be of order (p, d=1, q).
-0.5
ACF
ACF Plot
0
2
4
6
Lag
Fig. 3.6 ACF Plot of First Differenced Series
4
(2,1,3)
1015883
-261.48
534.96
(2,1,4)
918569
-260.86
535.72
(3,1,0)
1757837
-267.26
542.53
accumulated residual autocorrelation from lag 1 up to and including the lag on the horizontal axis. The dashed blue line is at 0.05. All p-values above the dashed blue line indicate a good and adequate model.
(3,1,1)
1542252
-265.55
541.11
3.4.1 ARIMA (2, 1, 2) Model
(3,1,2)
1004639
-261.33
543.66
(3,1,3)
928264
-260.47
534.93
Table 3.4 shows the estimated parameters of the ARIMA (2, 1, 2) model. The coefficients show the AR and MA terms of the ARIMA model whilst S.E denotes standard error determined.
(3,1,4)
926875
-260.44
536.89
(4,1,4)
877380
-260.03
538.06
(1,1,4)
1646251
-266.52
545.05
Table 3.4 Estimated Parameters for ARIMA (2,1,2) Model AR1 AR2 MA1 MA2 Coefficients 0.0103 1 0.9966 0.1077 S.E 0.0361 0.0096 0.1319 0.1337
From Table 3.2, preliminary analysis on the feasible models was computed and in order to select the best model, the model with the least Akaike Information Criterion (AIC) was taken into consideration. Table 3.3 shows the best three models considered to have the least AIC.
From Fig. 3.8, the standardized residual plot displayed normally distributed residuals. The points on the plot indicate zero trace of trend, no outliers, and in general, no changing variance across time.
Table 3.3 Best Three Models under Consideration
σ2
Log likelihood
AIC
(2,1,2)
1034960
-261.77
533.55
(2,1,3)
1015883
-261.48
534.96
(3,1,3)
928264
-260.47
534.93
-1
Model (p,d=1,q)
1 2
3
Standardized Residuals
2008
2010
2012
2014
Time
ACF
-0.4
On the basis of the results in Table 3.3, ARIMA (2 1,2) had the least the AIC value. Hence, was adjudged the optimal ARIMA model for MSW prediction within Tarkwa and its environs.
0.2
0.8
ACF of Residuals
0
1
2
3
Lag
3.4 Estimation of Parameters and Diagnostic Checking 0.8 0.0
The parameters were estimated and investigated to ascertain whether the residuals of the selected ARIMA models were normally distributed and also whether there were no correlations between successive residuals (randomness of residuals). In order to check for correlations between successive residuals, a standardized residual plot and ACF plot of residuals were carried out. If the residuals were normally distributed, the points on the plot would indicate zero trace on trend, no outliers, and in general, no changing variance across time.
0.4
p value
p values for Ljung-Box statistic
2
4
6
8
10
lag
Fig. 3.8 Diagnostic Plot of ARIMA (2, 1, 2) Model Also, the ACF of residuals shown in Fig. 3.8, shows that only one out of the sixteen lags of the series residuals exceeded the significant bounds, that is, the single spike at lag one. These lags can be ignored, since the probability of a spike being significant by chance is about one in sixteen. The ACF dies down after lag two with most lags getting
Also, the p-value for Ljung-Box statistic was plotted to further ascertain the adequacy of the model’s residual. This statistic considered the
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close enough to zero. This simply gives an indication of non-significant autocorrelation, since it would be expected at most one out of sixteen sample autocorrelations to exceed the 95% significance bounds.
2022 2023 2024
28245.10 28245.12 28244.64
26832.53 26856.55 26880.40
26137.65 26137.88 26138.60
27538.26 27514.33 27490.56
Forecasts from ARIMA(2,1,2)
30000
35000
40000
The p-value for Ljung-box statistic plot in Fig. 3.8 show that all p-values (the accumulated residual autocorrelation from lag 1 up to the 10th lag) are positioned above the dashed blue line (0.05) which is an indication of an adequate model.
25000
3.5 Model Selection
Table 3.5 Performance Criteria Model ME (2,1,2) 252.9934 (2,1,3) 282.8924 (3,1,3) 256.5223
15000
20000
Although the AIC was used in the model selection as previously discussed, statistic indicators (Table 3.5) like Mean Error (ME) and Mean Percentage Error (MPE) were also used in assess the adequacy for the ARIMA (2, 1, 2) model.
2010
2020
2025
Fig. 3.9 Forecast for Waste Generation Plot for the next 10 years
MPE 0.9689 1.0936 0.9953
The graph of projected waste using ARIMA Time Series as used to forecast the amount of MSW generated in Tarkwa and its Environs. With reference to Fig. 3.9, the quarterly waste generated showed some evidence of non-seasonality. Again trend was observed on the left hand side of the plot which is much lower than the right hand side. However there were no evidence of non-seasonal components observed at the right side of the MSW graph since the peaks and troughs that show up were regular in nature. Based on this, it was concluded that the MSW data plotted showed some seasonality.
From Table 3.5, it is clear that ARIMA (2, 1, 2) model is the optimum model for forecasting since its values were better than that of the other competing models. Therefore, the ARIMA chosen model for estimating the MSW is of the form:
X t X t 1(1 1) X t 2 (1 2 ) 2X t 3 ε t θ1ε t 1 θ2ε t 2 X t X t 1 (1 0.0103) X t 2 (0.0103 0.9966)
4
0.9966(X t 3 ) ε t 0.1077ε t 1 (1)ε t 2
3.6
2015
Conclusions and Recommendations
The following are the conclusions drawn at the end of the study: Time series model formulation and forecasting of MSW generation in Tarkwa and its environs has been produced. The use of ARIMA time series model to predict MSW in the Municipality was found to be an alternative as it gives satisfactory results in terms of its prediction and forecasting of MSW data recorded. The importance of time series analysis and forecasting has been evaluated and discussed in this study. It is recommended that the ARIMA time series model be used by the Assembly to estimate MSW in the Municipality, since the selection of an appropriate model affects the decision making process. It was observed from extensive review of literature that the systematic recording of MSW data yields accurate results and reliable
Forecasting
The ARIMA (2, 1, 2) model was then applied to forecasting MSW that could be generated for next ten (10) years as shown in Table 3.6. Fig. 5.8 show forecast for waste generation plot for the next ten years in Tarkwa and its environs. Table 3.6 Forecast Values for the Next Ten (10) Years Year Qtr1 Qtr2 Qtr3 Qtr4 2015 28230.70 26660.34 26150.24 27709.99 2016 28234.33 26685.30 26146.87 27685.08 2017 28237.43 26710.16 26144.04 27660.28 2018 28240.00 26734.90 26141.74 27635.60 2019 28242.05 26759.52 26139.95 27611.05 2020 28243.57 26784.00 26138.68 27586.63 2021 28244.59 26808.34 26137.92 27562.37
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prediction. Therefore, frantic efforts must be made by the Municipality to collect MSW data regularly to support such studies. The estimation of waste generation rates could also be used by engineers and planners to determine the type, size, design and location of facilities, routes from the waste sources to the landfill site. Personnel needs and equipment requirement are essential to effectively carried out proper management system of the landfill site.
Mondal, P., Labani, S. and Gaswani, S. (2014), Study of Effective of Time Series Modeling (ARIMA) in forecasting Stock Prices, International Journal of Computer Science, Engineering and Application, Vol. 4, No. 2, 27 p. Owusu-Sekyere, E., Harris, E. and Bonyah, E. (2013), “Forecasting and Planning of Solid Waste Generation in the Kumasi Metropolitan Area of Ghana”, An ARIMA Time Series Approach”, International Journal of Science, Vol. 2, Issue April 2013. Xiao, Y., Xuemei, B., Zhiyun, O., Hua, Z. and Fangfang, X. (2006), “The Composition, Trend and Impact of Urban Solid Waste in Beijing”, Zade, J., G. and Noori, R. (2007), “Prediction of Municipal Solid Waste Generation by Use of Artificial Neural Network”: A Case Study of Mashhad: International Journal of Environmental Research; 2008, Vol. 4 Issue. Zaini, S. and Gerrard, S. (2013), “The Development of Predictive Model for Waste Generation Rates in Malaysia” Research Journal of Applied Sciences, Engineering and Technology 5(5): pp.1774-1780.
Acknowledgements The authors are grateful to the Environmental Health and Sanitation Unit of the Tarkwa Nsuaem Municipal Assembly for providing us with the necessary data. We are also thankful to staff and Postgraduate students of UMaT for their support.
References Ansah, B. (2014), “Characterization of Municipal Solid Waste in Three Communities in the Tarkwa Nsuaem Municipalities in Ghana”, Publish Chung, S. S. (2010), “Projecting Municipal Solid Waste: The Case of Hong Kong SAR”. Resources, Conservation and Recycling 54.11 (2010): pp. 759-768. Foray, J. (2012), Solid Waste Management in Ghana: A Comprehensive case for West Africa, African Reality. Fisherman, T., Schandl, H. and Tanikawa, H. (2016), “Stochastic Analysis and Forecasting of the Patterns of Speed, Acceleration and Levels of Materials Stock Accumulation in Society”, Journal of Environmental Science and Technology, 50 (7), pp. 3727-3737. Intharathirat, R., Salam, P. A. and Kumar, A. U. (20I5), “Forecasting of Municipal Solid Waste Quantity in Developing Country Using Mult-Varate Grey Models”, Journal of Waste Management, Vol. 39, May 2015, pp. 3-14. Mwenda, A. (2014),“Time Series Forecasting of Solid Waste Generation in Arusha CityTanzania”, Journal of Mathematical Theory and Modeling, Vol. 4, No. 8, 2014. Roy, S., Rafizul, I. M., Didarulm, Asma, U. H., Shohel, M. R. and Hasibul, M. H. (2013), “Prediction of Municipal Solid Waste Generation of Khulna City Using Artificial Neural Network: A Case Study. Momani, P. E. N. (2009), “Time series Analysis for Rainfall Data in Jordan: Case study for Using Time series”, American Journal of Environmental Sciences 5(5): pp. 599-604.
Authors THOMAS AKYEN is an MSc Geomatic Engineering student at the university of Mines and Technology, (UMaT), Tarkwa. He holds Bsc in Geomatic Engineering from UmaT and HND in Civil Engineering from Accra Polytechnic. His research interest include time series forecasting and analysis and civil engineering surveying. Dr. B. Cynthia is a senior lecturer of the Geomatic Engineering at the University of Mines and Technology, (UMaT). She holds Bsc in Geodetic Engineeing from the Kwame Nkrumah University of Science and Technology (KNUST) and PhD in Oceanography from the University of Ghana, Legon. Her research interest include Engineering surveying, Geographic Information System (GIS) and Remote Sensing. Yao Yevenyo Ziggah is an Assistant Lecturer at the Geomatic Engineering Department of the University of Mines and Technology (UMaT). He holds a BSc in Geomatic Engineering and a MEng in Geodesy and Survey Engineering from the China University of Geosciences (Wuhan). His research interests include geodetic deformation modelling, geoid modelling, geodetic coordinate transformation and application of Artificial Intelligence Techniques in Geodesy.
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