FORECASTING OF LAKE MALAWI WATER LEVEL FLUCTUATIONS USING STOCHASTIC MODELS

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1 FORECASTING OF LAKE MALAWI WATER LEVEL FLUCTUATIONS USING STOCHASTIC MODELS Mulumpwa, M. ab, Jere, W.W.L. b and Lazaro, M. b. a Senga Bay Fisheries Research Unit, P.O. Box 316, Salima, Malawi. b Aquaculture and Fisheries Science Department, Lilongwe University of Agriculture and Natural Resources, Bunda Campus, P. O. Box 219, Lilongwe, Malawi. Corresponding author: mulumpwa.mexford@gmail.com ABSTRACT The study considered Seasonal Autoregressive Integrated Moving Average (SARIMA) processes to select an appropriate stochastic model for forecasting the monthly data from the Lake Malawi water levels for the period 1986 through The appropriate model was chosen based on SARIMA (p, d, q)(p, D, Q)S. The Autocorrelation function (ACF), Partial autocorrelation (PACF), Akaike Information Criteria (AIC), Bayesian Information Criterion (BIC), Box Ljung statistics, correlogram and distribution of residual errors were estimated. The selected model was SARIMA (1, 1, 0) (1, 1, 1)12 for forecasting the monthly data of the Lake Malawi water levels from August, 2015 to December, The plotted time series showed that the Lake Malawi water levels are decreasing since 2010 to date but not as much as was the case in 1995 through The future forecast of the Lake Malawi water levels until 2021 showed a mean of m ranging from to meters with a confidence interval of 80% and 90% against registered mean of m in 1997 and m in 1989 which was the lowest and highest water levels in the lake respectively since These results suggest that the Lake Malawi water level may not likely going to be lower than that recorded in Therefore, utilisation and management of water related activities and programs among others on the lake should provide room for such scenarios. The findings suggest a need to manage the Lake Malawi jointly and prudently with other

2 stakeholders starting from the catchment area to influence its resilience to climate change impacts and reduce impacts of anthropogenic activities on the lake s water quality, levels and both aquatic and adjacent terrestrial ecosystems. Key words: forecasting, SARIMA, Lake Malawi, water level fluctuation, climate change, anthropogenic activities 1. Introduction Malawi has 118,484 km 2 covered by surface water representing 20% of the total surface area (Department of Fisheries, 2012) of which Lake Malawi has a surface area of 29,000 km 2. The lake has a drainage system made up of a number of rivers such as Shire, Lithipe, Bua, Dwangwa, Songwe, North Rukuru and South Rukuru among others. The Lake Malawi is third largest lake in Africa with an average depth of 292 m, bordered by three countries namely Malawi, Mozambique and Tanzania and is situated in the Great African Rift Valley between 9 30'S and 14 30'S (Patterson & Kachinjika, 1995). The most productive areas on the lake are the shallow areas found in in the southeast and southwest arms of the lake (Kanyerere, 2001). The depth of Lake Malawi is influenced by the activities of its basement tectonics and climatic. The climate influence is due to the long dry seasons caused by subtropical climate and the small dimensions of the hydrological catchment area. The lake dried out almost completely at the beginning of the Pleistocene, as a consequence of stable tectonic conditions and dry climate. It is reported that the tectonic lowering of the overflow sill, through subsidence of the rift floor, combined with erosional incision currently being accelerated by anthropogenic activities lowered the water level by 40m since the late Pleistocene. Of late climate change and anthropogenic activities in the catchment areas have resulted into fluctuation of the water levels of Lake Malawi as the case with other water bodies in Africa. However, there is a level

3 at which these two factors can be controlled unlike the stability of the tectonic conditions. This challenge underpins the importance of modelling and forecasting Lake Malawi water levels to appreciate how they will behave in the future using available data. This is very crucial to policy makers responsible of different user-groups of the Lake Malawi whether directly or indirectly to develop strategies that counteract impacts of climate change and anthropogenic activities to Lake Malawi water levels. Forecasting plays a central role in management as it precedes planning which, in turn, precedes decision making (Makridakis et al., 1983). Forecasting is used by policy makers to select an appropriate policy option to meet anticipated goals and objectives (Stergiou and Christou, 1996). Forecasting has been used successfully in metrological services to forecast weather patterns hence advise farmers accordingly, advice on impending natural disasters such as earthquake to save lives. It is importance that the trend of the Lake Malawi water levels be modelled and forecasted in the face of climate change to provide a possible picture of how water levels will behave in the lake in the years to come. Prediction models are necessary for water resource managers in planning for the future. This is very important to policy makers to device tools to sustainably manage the water resources for the benefit of the nation. The study employed stochastic models to model the Lake Malawi water levels using available times series of the same covering a number of previous days and months from 1986 to Since the data available is of time series nature, the models which have been used extensively in modelling such data are autoregressive (AR), moving average (MA), autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models. The models are applicable to

4 stationary data. Differencing among other methods is used to transform the data series that is not stationary. This study has employed SARIMA models to forecast the Lake Malawi water levels as the data was seasonal. Description of the time series used analysed The data that have been used in this study are univariate time series of Lake Malawi water levels from 1986 to 2015 obtained from the Department of Water Resources in Malawi. The unit of measurement used in this study was in meters and refers to the water level at the water level collection point. The study was aimed at modelling and forecasting patterns of Lake Malawi water levels in Malawi. The time series of Lake Malawi water levels short term forecasts were made by employing SARIMA model as the series showed seasonality and correlation with the water levels from the previous months as proved by the plotted autocorrelogram and/or partial autocorrelogram. Forecasting using SARIMA model The SARIMA has been widely applied to forecast time series data with seasonality. The following procedure was used in SARIMA model application in this study; a. Data was plotted to check if it required differencing. The Dickey-Fuller test was also used to test if the series was stationary or not. b. Differenced to remove trend, to find d and later differenced to remove seasonality, D. c. Examination of ACF and PACF of differenced series to find P and Q first, by examining just at lags s, 2s, 3s, etc. and to find p and q by examining between seasonal lags.

5 d. Fit SARIMA (p, d, q) x (P, D, Q)s model to original data. e. Check model diagnostics. f. Forecast for short term was made. The SARIMA model works where the data is or made stationary and deseasonalised. Therefore, in this study, Lake Malawi water levels were tested for stationary using two methods namely graphical analysis method and the Dickey Fuller test. The data was found significant to non stationarity hence they were differenced to make them stationary. The differencing of the series was achieved by a model: Where: Zt = Yt Yt-1 (1) Zt Yt is differenced value of the new differenced series at time t is the value of the original series at time t Yt-1 is the value of the original series at time t-1 The differenced data was later plotted and subjected to stationarity test using the same graphical analysis method and the Dickey Fuller test and this time they were found stationary meaning the data was ready for modelling. Selecting a candidate SARIMA model The stationary differenced data for the Lake Malawi water levels four species was used to come up with correlogram and partial correlogram in order to identify an appropriate model for the lake water levels. This process is called model identification. It simply involved finding the most appropriate values of p and q for an ARIMA (p, d, q) model by examining the correlogram and partial correlogram of the stationary time series.

6 The autocorrelation function ρ(k) at lag k was denoted by: (2) Where γ(k) is the autocovariance function at lag k of a stationary random function {Y (t)} given by: (3) When the PACF had a cut-off at p while the ACF tails off, it gave an autoregressive (AR) of order p. When also the ACF had a non-zero lag at q it gives a moving-average (MA) of order q. However, when there was non-zero lag(s) on both ACF and PACF, it implied that the application of the autoregressive moving-average of order p and q was possible. It also meant that since the data was differenced once, then even the autoregressive integrated movingaverage (ARIMA) p, d, q was possible. Where d is number of times the data has been differenced to remove the noise within it. An autoregressive model (AR) of a time series {Xt} is a regression model of that time series on its previous history (Craine, 2005). Autoregressive process of order (p) was found by using the following model; (4)

7 A moving average (MA) model of a time series was aimed at averaging out previous error steps of a time series {Xt} to attempt to smooth the process or make the time series stationary. Moving Average process of order (q) was found by using the following model; (5) The combination of linear autoregressive and moving average properties results into the autoregressive moving average (ARMA): ARMA of order (p, q) is, = (6) The general form of ARIMA model of order (p, d, q) is (7) Where: is the original data series or differenced of degree d of the original data at time t; is the white noise at time t.,,, are the autoregressive parameters. p is the autoregressive order.,,., are the moving average parameters. q is the moving average order.

8 Then, the general form of SARIMA model (p, d, q)(p,d,q)s is donated as: Model parameter estimation When the models had been identified to be AR, MA, ARMA, ARIMA or SARIMA the next step was to estimate the best possible parameters of the identified models. This is called model fitting. These best possible parameters were found using Akaike Information Criteria (Akaike 1973). The best model is obtained on the basis of minimum value of Akaike Information Criteria (AIC) (Satya et al. 2007). The AIC was found by AIC = 2 log L + 2m (9) Where: m L is p + q is the likelihood function

9 The AIC was used to obtain a model that well represent the data on the basis of minimum value of AIC. Forecasting Once the appropriate best candidate SARIMA (p, d, q)(p, D, Q)s model was selected for Lake Malawi water levels time series data for the species then the parameters of the selected SARIMA model was estimated. The fitted SARIMA model was then used as a predictive model for making forecasts for the future (next seven (7) years) of Lake Malawi water levels fluctuations. Diagnostic checks After the forecast were made, diagnostic tests were carried out to check to what extent the forecast could be trusted. The diagnostic tests were done by using method of autocorrelation of the residuals and the Ljung-Box test. A good forecast should come from an SARIMA model with forecast errors that have a mean of zero, with no significant correlations between successive forecast errors and have constant variance. Once the model was found to be inappropriate, the process was restarted through the four steps in the SARIMA modelling until the diagnostic checks validates the model as fitting. 2. Results and discussion Identifications of models The plotted original series was visibly not stationary as shown in Figure 1. The original series was fluctuating so much over the years hence did not have a constant variance.

10 Figure 1: Lake Malawi water level fluctuation from 1985 to Autocorrelation function (ACF) plot in Figure 2 showed several non-zero lags that tailed off at lag 4 hence proving that the original series was indeed not stationary.

11 Figure 2: Autocorrelation function generated from Lake Malawi Water Levels from 1986 to Dickey-Fuller test on the original series of the Lake Malawi water levels in Table 1 showed that the original data was not stationary and required some form of transformation. The Dickey-Fuller test proved this by giving a p-value of which demanded the rejection of the stationarity as an alternative hypothesis. The data had also seasonality trend as shown by the decomposing analysis in Figure 3.

12 Figure 3: Decomposed Lake Malawi water levels time series from 1985 to The removal of the non-stationarity and seasonality was done through differencing. The data was differenced once for trend and resultant time series proved to be stationary and ready for SARIMA modelling. The seasonality shown in Figure 3 was removed by employing a seasonal differencing at every 12 months. The graphical analysis method on the plotted differenced time series data showed stationarity as shown in Figure 3, as it showed that it had a constant variance and a mean of zero.

13 Figure 4: Stationary differenced Lake Malawi water levels from 1985 to The Dickey-Fuller test proved the stationarity in the differenced time series data by the smaller p-value of 0.00 in Table 1. The Dickey-Fuller test results in Table 1 also showed that the difference time series data was no longer explosive as indicated by the higher p-value of 0.99 hence the differenced time series was indeed stationary. The differencing made the new time series values to vary over time with a constant mean and constant variance hence stationary as shown in Figure 4. This implied that the differenced data was ready for modelling and forecasting.

14 Table 1: Table showing Dickey-Fuller and Augmented Dickey-Fuller Test results on Lake Malawi water levels time series generated from monthly Lake Malawi water levels in meters from 1985 to Type of data Alternative Dickey-Fuller Lag order p-value Hypothesis statistic Non-differenced data Stationarity Difference data Stationarity Difference data Explosiveness p 0.05 data is not stationary p 0.05 data is not explosive hence stationary Since the new series was successfully proved to be stationary by the Dickey-Fuller test and the graphical analysis method, then autocorrelogram and partial autocorrelogram were plotted to determine the values of p and q in the ARIMA models. The plotted partial autocorrelation function showed second-order autoregressive (AR) model as shown in Figure 6 while the plotted autocorrelation function showed second-order moving average (MA) model as shown in Figure 5. The autocorrelogram and partial autocorrelogram, in Figures 5 and 6 were used to identify various competing model.

15 Figure 5: Autocorrelation function of differenced Lake Malawi water levels showing secondorder moving average (MA) generated from monthly Lake Malawi water levels in meters from 1985 to Figure 6: Partial autocorrelation function of differenced Lake Malawi water levels showing second-order autoregressive (AR) model generated from monthly Lake Malawi water levels in from 1986 to 2015.

16 The most competing models identified together with their corresponding fit statistics are shown in Table 2. The model in the SARIMA family with the lowest AIC values was selected. The value of the AIC of the selected SARIMA model was as also shown in the Table 2. Owing to that, the most suitable model for forecasting Lake Malawi water levels is SARIMA (1, 1, 0) (1, 1, 1)12, as this model had the lowest AIC values. Table 2: Fit statistics for various competing SARIMA models generated from monthly water bodies in meters from 1986 to 2015 from artisanal fishers from Lake Malawi in Mangochi District. SARIMA (p, d, q) (P, D, Q)12 AIC SARIMA (0, 1, 1)(1, 1, 1) SARIMA (0, 1, 2)(1, 1, 1) SARIMA (1, 1, 0)(1, 1, 1) SARIMA (1, 1, 1)(1, 1, 1) SARIMA (1, 1, 0)(2, 1, 1) Model with the lowest AIC and BIC is the best fit The Box Pierce (and Ljung Box) test also proved that model (1, 1, 0) (1, 1, 1)12 was found to be among the best fitting models as shown in Figures 7, 8, 9, 10 and 11.

17 Figure 7: The Box Pierce (and Ljung Box) test out-put for SARIMA (0, 1, 1)(1, 1, 1)12 generated from monthly Lake Malawi water levels in meters from 1986 to 2015.

18 Figure 8: The Box Pierce (and Ljung Box) test out-put for SARIMA (0, 1, 2)(1, 1, 1)12 generated from monthly Lake Malawi water levels in meters from 1986 to Figure 9: The Box Pierce (and Ljung Box) test out-put for SARIMA (1, 1, 0)(1, 1, 1)12 generated from monthly Lake Malawi water levels in meters from 1986 to 2015.

19 Figure 10: The Box Pierce (and Ljung Box) test out-put for SARIMA (1, 1, 1)(1, 1, 1)12 generated from monthly Lake Malawi water levels in meters from 1986 to 2015.

20 Figure 11: The Box Pierce (and Ljung Box) test out-put for SARIMA (1, 1, 0)(2, 1, 1)12 generated from monthly Lake Malawi water levels in meters from 1986 to This implied SARIMA (1, 1, 0)(1, 1, 1)12 model was still outstanding among the five (5). The Box Pierce test basically examines the Null of independently distributed residual errors, derived from the idea that the residual errors of a correctly specified model are independently distributed. In a case where the residual errors are not independently distributed, then it indicates that they come from a miss-specified model. Model Estimation All the analyses of the time series in this study were done by R software version ( ). The software was used to estimate the parameters of the selected models as shown in Table 3.

21 Table 3: Selected competing models parameters with their AIC generated from monthly Lake Malawi water levels in meters from 1986 to SARIMA SARIMA SARIMA SARIMA SARIMA (0, 1, 1) (0, 1, 2) (1, 1, 0) (1, 1, 1) (1, 1, 0) (1, 1, 1)12 (1, 1, 1)12 (1, 1, 1)12 (1, 1, 1)12 (2, 1, 1)12 Constant L1. AR L1. MA L2. MA L1. SAR L2. SAR L SMA AIC Model with the lowest AIC is the best fit The model with significant coefficients parameters with least AIC is better in terms of forecasting performance than the one with insignificant coefficients parameters with large AIC (Guti errez-estrada et al. 2004, Czerwinski et al. 2007). All these tests and examinations proved that the SARIMA (1, 1, 0)(1, 1, 1)12 model is the best model to forecasting of the future of Lake Malawi water levels. Diagnostic Checks After identifying SARIMA (1, 1, 0)(1, 1, 1)12 as the best fitting model, the next step was to forecast the future Lake Malawi water levels. However, several diagnostic checks were made

22 on the identified model before the actual forecasting such as examination of the residuals of the model to identify any systematic structure still in it requiring improvement of the selected SARIMA (1, 1, 0)(1, 1, 1)12 model ( Singini et al. 2012, Lazaro and Jere, 2013). The diagnostic checks were made by examining the autocorrelations of the residual errors of various orders. In this regard, the Box Pierce (and Ljung Box) test and residual errors plots were made to see if the residual errors had a mean of zero. ACF for residual errors was plotted as shown in Figures 9, and showed that there was no non-zero lags. This indicated that there were no significant autocorrelations among the residual errors to exceed the 95% significance bounds. The Box Pierce (and Ljung Box) test also showed that the model fitted the series very well as the p-value was close to one (1) as shown in the Ljung Box statistic in Figure 9. The time plot of the forecast errors shown in Figures 9 proves that the forecast errors has a constant variance. These diagnostic tests proved that the selected SARIMA (1, 1, 0)(1, 1, 1)12 model was indeed an appropriate model for forecasting Lake Malawi water levels. Forecasting The fitted model was used to forecasts for Lake Malawi water levels from September, 2016 to 2022 at a confidence interval of 80% and 95% and they included a zero (0) as shown in Table 4. The ability of the model to forecast was tested to check the level of accuracy on the post sample forecasting. The graph in Figure 14 shows actual catches and the forecasted trend with their confidence interval of 80% and 95%. A good model should have a low forecasting error as the case with SARIMA (1, 1, 0)(1, 1, 1)12, therefore when the distance between the forecasted and actual values are low then the model has a good forecasting power (Czerwinski et al. 2007, Singini 2012, Lazaro and Jere, 2013).

23 Table 4: Forecast catches of Lake Malawi water levels with 80% and 95% confidence intervals. Year Forecasted Lake Malawi water level January February March April May June July August September October November December Where the confidence interval includes a zero (0); extinction of the species cannot be ruled out

24 Figure 12: Forecasted Lake Malawi Water levels using SARIMA (1, 1, 0)(1, 1, 1)12 generated from monthly Lake Malawi water levels in meters from 1986 to The forecast for Lake Malawi water levels have a mean of meters by the year 2021 and the mean of the actual recorded depth was meters which is below the actual recorded mean by 0.57 meter and 0.69 meter from the maximum ( meters) ever recorded Lake Malawi water level. These results are a clear demonstration that the Lake Malawi water levels will drop as compared to the mean of recorded water levels of the lake. This has not come as a surprise as Lake Malawi water mark on the shore has dropped with 35 meters into the lake since 2010 at Senga Bay Fisheries Research Unit in Salima District (Personal observation). Other lakes in the tropical region are reportedly experiencing water level fluctuation due to climate change. The fluctuation and drop of water level of Lake Malawi in the time series and the forecast could be as a result of the climate change and tectonics of the lake bed. It is therefore crucial that all direct or indirect water users should take into consideration the result of this study for continued and sustainable use of the water resource. 3. Conclusion The SARIMA (1, 1, 0)(1, 1, 1)12 was generated successfully to forecast the Lake Malawi water levels from the September, 2015 to December, The forecast for Lake Malawi water levels showed that the water levels will relatively drop by 0.57 meters as compared to the mean water levels of the record in the previous years. This will have negative implications over use of Lake Malawi and rivers that flow out of it for irrigation, pumping of water for domestic use and hydroelectric power generation among others.

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