Khalil Ahmad

1 Time Series Seasonal Forecasting Using Box-Jenkins and Other Recent Statistical Techniques By Khalil Ahmad Senior Statistician and Data Analyst 3. Research Methodology Let Y1 , Y2 , Y3 ,..., Yt ,... be the elements of time series and the mean and variance at time t are given by t = E Yt  2  t2 = E (Yt − t )    The covariance of Yt , Ys by Cov(Yt , Ys ) = E (Yt − t )(Ys − s ) Definition 3.1 If the mean, variance and covariances are independent of time the time series is called stationary, mathematically, t =  ,  t = , t ,s = t −s , t = 1, 2, 3,... t = 1, 2, 3,... ts WhatsApp - - 2 Definition 3.2 In time series autocorrelation is used instead of covariances and it may be defined as  = t ,t +  E (Yt −  )(Yt + −  )  = = 2 0 0 E (Yt −  )    Autoregressive Process AR(1) 3.3 Let  t be white noise and Yt =Yt −1 +  t Yt where (  is an AR(1) process if 1) Yt =  t +  (Yt − 2 +  t −1 ) Yt =  t +  t −1 +  2Yt − 2 Yt =  t +  t −1 +  2 (Yt −3 +  t − 2 ) Yt =  t +  t −1 +  2 t − 2 +  3Yt −3 . . . . . . . . . . . . . . . Yt =  t +  t −1 +  2 t −2 +  3 t −3 + ... E Yt  = 0 ; It is independent of time. Also, the autocorrelation is k = E Yt ,Yt + k    k = E   i = 0  i t − i   i = 0  i t −i     k =  i = 0  i k + i 2  WhatsApp - - 3 k =  k + i 2  i = 0  2i Inf k =  2 k = k k 1− 2 where k 0 k = k k = 0,1,2,3,... k = k k = 0, 1, 2, 3,... which is independent of time. Thus AR(1) is stationary process. Notation of Lag Operators 3.4 Consider the time series Y1 , Y2 , Y3 ,..., Yt then we define the lag operator represented by L as given below: if  ( L ) =1 − 1L −  2 L2 − 3 L3 − ... −  p Lp LYt =Yt −1 Then we may define AR(1) process as follow: Yt =1Yt −1 + 2Yt −2 + 3Yt −3 + ... +  pYt − p +  t where  t is white noise. So, using lag operator, it may be written as Yt =1LYt +  2 L2Yt + 3 L3Yt + ... +  p LpYt +  t (1 −  L −  L −  L − ... −  L )Y =  2 1  ( L ) Yt =  t 3 2 p 3 where  p t t 1 Autoregressive Process AR(2) 3.5 WhatsApp - - 4 AR(2) process may be written as Yt = 1Yt −1 +  2Yt − 2 +  t Using lag operators, we have (1 −  L −  L )Y =  2 1 2 t t Also the process may be write as Yt =  ( L )  t Yt = (1 +  1L +  2 L2 +  3 L3 + ...)  t Where (1 −  L −  L ) = (1 +  L +  2 −1 1 2 1 2 L2 +  3 L3 + ...) (1 −  L −  L )(1 +  L +  L +  L + ...) 1 2 1 2 2 1 3 2 3 Equating coefficients, we have L1 : −1 +  1 = 0   1 = 1 L2 : −2 + 1 1 +  2 = 0   1 = 12 + 1 L3 : −2 + 12 1 +  2 = 0   1 = 13 + 212 . . . . . . . . . . . . . . . . WhatsApp - - 5 . . Lj :  j = 1 j −1 + 2 j − 2 . . . . . . All the weights can be determined recursively. Autoregressive Process AR(p) 3.6 AR(p) is defined as Yt − 1Yt −1 − 2Yt − 2 − ... −  pYt − p =  t (1 −  L −  L −  L − ... −  L )Y =  2 1 3 2  ( L ) Yt =  t p 3 p t t where  ( L ) = (1 − 1L − 2 L2 − 3 L3 − ... −  p Lp ) With the following condition of stationarity that can be set out as follows by writing  ( L ) = (1 − h1L )(1 − h2 L )(1 − h3 L ) ...(1 − hp L ) hi 1 for i =1,2,3,..., p Alternatively, we may write hi−1 all lie outside the unit circle. The autocorrelation will follow a difference equation of the form  ( L ) k = 0 for k =1,2,3,... Which has the solution in the form k = A1h1k + A2h2k + A3h3k + ... + Ap hpk Moving Average Process MA(1) 3.7 Here an MA(1) process can be defined as WhatsApp - - 6 Yt =  t +  t −1 Where  t is the white noise E Yt  = 0 2 Var Yt  = E (  t +  t −1 )    Var Yt  = E  t2 +  2 E ( t2 ) Var Yt  = (1 +  2 ) 2 1 = E YY t t −1  1 = E ( t +  t −1 )( t −1 +  t − 2 ) 1 = E  t2−1  1 = 2 So, we have 1=  1+ 2 2 = E ( t +  t −1 )( t − 2 +  t −3 ) 2 = 0 Generally,  j  0 for j 2 . So Moving Average process MA(1) is stationary irrespective of the value of  . Moving Average MA(q) process can be defined in the following fashion. Yt =  t + 1 t −1 + 2 t −2 + 3 t −3 + ... + q t −q E Yt  = 0 WhatsApp - - 7 Var Yt  = (1 + 12 + 22 + 32 + ... + q2 ) 2 k = Cov YY t t −k    t + 1 t −1 +  2 t − 2 + 3 t −3 + ... +  k  t − k      k = E  + k +1 t − k −1 +  k + 2 t − k − 2 +  k + 3 t − k − 3 + ... +  q t − q      (  t − k + 1 t − k −1 +  2 t − k − 2 + 3 t − k − 3 + ... +  q − k  t − q + ...)  k = (k + k +11 + k + 22 + ... + qq − k ) 2 and k = k Var Yt  Also, for moving average process to be stationary regardless of the values of the  ,s . n−k  (i ,i + k )  i =0  k =  (1 + 12 +  22 + 32 + ... +  q2 )  0 , kq , k q Autoregressive Moving Average ARMA(p,q) Process 3.8 It is mixture of AR(p) and MA(q) processes of order p,q and it is represented as ARMA(p,q) if Yt =1Yt −1 + 2Yt −2 + 3Yt −3 + ... +  t + 1 t −1 + 2 t −2 + 3 t −3 + ... + q t −q (1 −  L −  L −  L − ... −  L )Y = (1 +  L +  L +  L + ... +  L ) 2 1 Or 3 2 3 p p 2 t 1 2 3 3 q q t  ( L ) Yt =  ( L )  t Where the polynomials  and  of degree p and q respectively in L. Box-Jenkins Methodology 3.9 WhatsApp - - 8 Box-Jenkins methodology is very famous and widely used method of univariate time series analysis, which is a hybrid of the AR and MA models Yt =1Yt −1 + 2Yt −2 + 3Yt −3 + ... +  t + 1 t −1 + 2 t −2 + 3 t −3 + ... + q t −q where the terms in the equation have the same meaning as given for the AR and MA model. To apply the Box-Jenkins method, it is assumed that the time series is stationary. If the time series data is non-stationary then Box and Jenkins recommend differencing one or more times to achieve stationarity which produces an ARIMA model, where the "I" stands for "Integrated". To include the seasonal autoregressive and seasonal moving average terms, we extends the Box-Jenkins models which may complicate the notations and mathematics of the model. Box-Jenkins model commonly includes difference operators, autoregressive terms, moving average terms, seasonal autoregressive terms, seasonal difference operators, , and seasonal moving average terms. The four stages of methodology are: ➢ Identification ➢ Estimation ➢ Diagnostic check on Model adequacy ➢ Forecasting These steps may be shown by a logic flow diagram as below: WhatsApp - - 9 Box-Jenkins Methodology stage of Identification 3.9.1 For the implementation of the Box-Jenkins model, the initial first step is to determine if the series is stationary. Dickey-Fuller Test In statistics and econometrics , an augmented Dickey–Fuller test ( ADF ) tests the null hypothesis that a unit root is present in a time series sample . The alternative hypothesis is different depending on which version of the test is used, but is usually stationarity or trend- stationarity . It is an augmented version of the Dickey–Fuller test for a larger and more complicated set of time series models. The augmented Dickey–Fuller (ADF) statistic, used in the test, is a negative number. The more negative it is, the stronger the rejection of the hypothesis that there is a unit root at some level of confidence. Making the ADF test for the four cases: a) Levels WhatsApp - - 10 b) First difference c) Logs d) Difference of log Only the partial autocorrelation function computed from sample is generally not helpful for identifying the order of the moving average process. Here are the basic guideline are listed below for the identification of AR(p) and MA(q) processes. Box-Jenkins Methodology stage of Estimation 3.9.2 The step for the implementation of Box-Jenkins methodology is the estimating the parameters. Therefore, the parameter estimation is accomplished using sophisticated and highquality software program that successfully implement Box-Jenkins models. Box-Jenkins Methodology stage of Diagnostics 3.9.3 When some candidate models are listed then next step is to check diagnostics for BoxJenkins models. It means that the error term as assumed to follow the assumptions for a stationary univariate process, That is residuals should be white noise, that is having a constant mean and WhatsApp - - 11 variance. If the residuals satisfy these assumptions, then it is expected that Box-Jenkins model is a good model for the data and will forecast the time series better. Box-Jenkins Methodology stage of Forecasting 3.9.4 When best is selected then it will be used for forecasting. Seasonal ARIMA Model 3.9.5 If there exists, the seasonal trend after k period of then we try to remove the seasonality from the series and produce a modified time series which may not be seasonal. After that we apply the Box-Jenkins methodology of ARIMA model. Mathematically, let vt be the no seasonal time series, then the proposed the seasonal ARIMA (SARIMA) model filter as k ( Lk )(1 − Lk ) Yt = k ( Lk ) vt D Where k ( Lk ) = 1 − 1k Lk − 2k L2k − 3k L3k − ... − Pk LPk k ( Lk ) = 1 − 1k Lk −  2k L2k −  3k L3k − ... −  Qk LQk Also vt is the ARIMA(p,d,q) which is the estimated using the ARIMA model as follow  ( L )(1 − L ) vt =  ( L )  t d Putting for vt , we have  ( Lk )(1 − Lk ) k ( Lk )(1 − Lk ) =  ( L ) k ( Lk )  t D D Which is SARIMA. Use of Artificial Neural Networks in Time series Analysis 3.10 The alternative approach for the forecasting a time series is artificial neural networks (ANNs) which is efficiently used to meet this objective effectively. It became very popular in last WhatsApp - - 12 few years. The basic objective of ANNs was to construct a model for mirroring the intelligence of human brain into machine. ANNs, are quite favorite for time series analysis and forecasting. Architecture of Artificial Neural Network 3.10.1 Multi-layer perceptrons is the most widely used ANNs for forecasting problems of time series. Multi-layer perceptrons used a single hidden layer feed forward network (FNN) to forecast the future values based on training data and the multi-layer feed forward architecture of ANN models can be diagrammatically depicted as below: WhatsApp - - 13 Time Lagged ANN 3.10.2 In the Feedforward Neural Network formulation, mentioned above, the nodes for input are the data values of the time series, recorded successively i.e. the target Y is a function of the values Yt −i , ( i = 1,2.3,..., p ) where p is the number of input nodes. In Time Lagged Neural Network, the input nodes are the time series values at some lags. WhatsApp - - 14 Seasonal ANN 3.10.3 The Seasonal Artificial Neural Network (SANN) model was first proposed by C. Hamzacebi to enhance the performance of performance using ANNs in case of seasonal time series data. As there does not need of preprocessing of the raw data for the application of the proposed Seasonal Artificial Neural Network (SANN) model. Graphically the structure of an SANN can be shown as below: WhatsApp - - 15 3.11 Bayesian Forecasting Bayesian Statistics is not just another inference technique. It is a statistical theory with its own methods and techniques derived from a unique strategy for the solution of any inference problem. In fact, this strategy (maximizing expected utility) arises as a consequence of adopting a set of axioms. A model is most often recognized as Bayesian when a probability distribution is used to describe uncertainty regarding the unknown parameters and when Bayes Theorem is applied. Bayes theorem is used to update a prior distribution (probabilities specified prior to data analysis) into a posterior distribution (the probabilities following data analysis) by incorporating the information, called likelihoods, provided by the observed data. A full Bayesian analysis can lead to the optimal choice among a set of alternative inferences, taking into account all sources of uncertainty in the problem and the consequences of every possible selection. For forecasting problems, Bayesian analysis generates point and interval forecasts by combining all the information and sources of uncertainty into a predictive distribution for the WhatsApp - - 16 future values. It does so with a function that measures the loss to the forecaster that will result from a particular choice of forecasts. 3.11.1 Bayesian Analysis for A Seasonal Series When forecasts are required for the year-end total of a short time series, the use of the yearly ratios of part-year totals to whole-year totals for previous years can play an important role. A series can be said to have a stable seasonal pattern when the expected proportion of events that occur in a given fraction of the year is constant over time. Under this assumption, the whole-year total for the variable in question can be forecast, given a few (say two) years of data and the corresponding part-year figures (up to a given month) of the current year. To determine whether seasonality is stable, you can calculate for each year of history the cumulative proportion of the part-year total occurring through the current month. If seasonality is stable, then these yearly-to-date proportions should all be close to each other. When a short series is being analyzed, it is important to make use of the simplest possible models. Specifically, the number of unknown parameters must be kept at a minimum. Binomial, normal and log-normal distributions have been considered in the literature for this purpose. When a Bayesian analysis is conducted, inferences about the unknown parameters are derived from the posterior distribution. This is a probability model which describes the knowledge gained after observing a set of data. The procedure to obtain the posterior distribution is known as Bayes Theorem. Bayes Theorem calculates the posterior distribution as proportional to the product of a prior distribution and the likelihood function. The prior distribution is a probability model describing the knowledge about the parameters before observing the currently available data. It can be elicited from past information or expert judgment. Alternatively, it can be chosen to WhatsApp - - 17 represent a state of relative ignorance; in which case, the prior distribution is said to be neutral or no informative and the resulting posterior distribution is mostly dependent on the observed data 3.12 Time series forecasting with KNN regression Time series forecasting has been performed traditionally using statistical methods such as ARIMA models or exponential smoothing. However, the last decades have witnessed the use of computational intelligence techniques to forecast time series. KNN regression simply holds a collection of training instances. The ith training instance consists of a vector of n features: ( g1i , g 2i , g3i ,..., g ni ) , describing the instance and an associated target vector of m attributes: ( t1i , t2i , t3i ..., tni ) . Given a new instance, whose features are known ( q ,q ,q ...,q ) but whose target is unknown, the features of the new instance are used to find its i 1 i 2 i 3 i n k most similar training instances according to the vectors of features and a similarity or distance metric. For example, assuming that the similarity metric is the Euclidean distance, the distance between the new instance and the ith training instance is computed as follows: ( g n x =1 i x − qx ) 2 The k training instances that are closest to the new instance are considered their k most similar instances or k nearest neighbors. KNN is based on learning by analogy. Given a new instance, we think that the targets of its nearest neighbors are probably similar to its unknown target. This way, the targets of the nearest neighbors are aggregated to predict the target of the new instance. For example, assuming that the targets or the k nearest neighbors are the vectors: t1 , t 2 , t 3 ,..., t k , they can be averaged to predict the target of the new instance as: WhatsApp - - 18 k ti k i =1 In short, KNN stores a collection of training instances described by n features. Each training instance represents a point in an n-dimensional space. Given a new instance, KNN finds its k closest instances in the n-dimensional space in the hope that their targets are similar to its unknown target. 3.13 Research Design: Appropriate quantitative techniques will be applied for the finding of this study. The method will be compare to find the adequate and appropriate technique to forecast the values of the water outflow data. 3.14 Data Collection and analysis Secondary time series data of water outflow is used from 2015 to 2019 recorded on daily basis. Appropriate and adequate statistical method have been applied using R, a statistical software. WhatsApp - - 19 Chapter No. 4 Results This chapter contains the analysis of the data and presents the results. The chapter, further; contains a detailed discussion of the findings drawn from data analysis. Seasonal Autoregressive Integrated Moving Average (SARIMA) Model 4.1 Figure 4.1:- Time series plot of Water Outflow data From figure 4.1, it is shown that there is seasonality in the data as within a year it may be repeated in a similar way. It looks like a non-stationary time series as the mean and variance are not same with respect to time. We may note that there is gradual decrease in water outflow. For more elaboration we have other plots as given below. Stationarity WhatsApp - - 20 The above ACF and PACF have showed that given time series is not stationary. For further evidence, ADF test for unit is applied. WhatsApp - - 21 Augmented Dickey-Fuller Test Dickey-Fuller Lag Order P-value -2.6024 0 0.3233 Alternative hypothesis: Stationary From the result of unit root test at level, it is showed that the time series is non stationary. Now, after taking first difference the again the ADF test is applied. Augmented Dickey-Fuller Test Dickey-Fuller Lag Order P-value -32.786 1 0.01 Alternative hypothesis: Stationary From the result of unit root test at first difference, it is showed that the time series became stationary after taking the first difference. Now, SARIMA can be applied. Figure 4.2:- The First Difference Plot of Water Outflow From figure 4.2, it is shown the mean of the given series becomes the constant with respect to time after taking the first difference of the series. It can also be seen that there is a seasonal trend the series and it can be further exploring as below: WhatsApp - - 22 Figure 4.3:- Seasonal Plot of the Water outflow From figure 4.3, The seasonal plot of the time series represents that there is almost same pattern during all the year which emphasis that there is seasonal trend the time series of the plot of the water outflow. It can be further explore by displaying the season subplot of Water Outflow data. Figure 4.4:- Season Subplot of the Water Outflow data WhatsApp - - 23 From figure 4.4, The season subplot of the time series signifies that there is roughly same pattern during all the year which focus on seasonal trend the time series of the plot of the water outflow. This suggest that the given time series should be decompose. Figure 4.5:- Decomposition of the Water Outflow time series From figure 4.5, Obviously, there is seasonal trend in the given time series of Water Outflow, it also shows that there is a decreasing trend. WhatsApp - - 24 Figure 4.6:- Correlogram of the Water Outflow series with ACF From figure 4.6, it is shown that the process is memory driven process. WhatsApp - - 25 Figure 4.7:- Correlogram of the Water Outflow series with PACF From figure 4.7, it is shown that the process is also moving average WhatsApp - - 26 Figure 4.8:- Correlogram of the difference of Water Outflow series with ACF WhatsApp - - 27 Figure 4.9:- Correlogram of the difference of Water Outflow series with PACF WhatsApp - - 28 Table 4.1 Candidate SARIMA Models Model AIC Model AIC ARIMA(0,1,0)(0,1,0) - ARIMA(1,1,4)(0,1,0) - ARIMA(0,1,1)(0,1,0) - ARIMA(2,1,0)(0,1,0) - ARIMA(0,1,2)(0,1,0) - ARIMA(2,1,1)(0,1,0) - ARIMA(0,1,3)(0,1,0) - ARIMA(3,1,0)(0,1,0) - ARIMA(0,1,4)(0,1,0) - ARIMA(3,1,1)(0,1,0) - ARIMA(0,1,5)(0,1,0) - ARIMA(4,1,0)(0,1,0) - ARIMA(1,1,0)(0,1,0) - ARIMA(4,1,1)(0,1,0) - ARIMA(1,1,1)(0,1,0) - ARIMA(5,1,0)(0,1,0) - In the above table the candidate models for forecasting the Water Outflow data are tabulated, by studying the correlogram with respect to ACF and PACF. Table 4.2 Summary of Seasonal ARIMA Model ARIMA(1,1,4)(0,1,0) AR(1) MA(1) MA(2) MA(3) MA(4) Estimate -0.2741 0.4444 -0.0502 -0.1378 -0.2082 S.E 0.1608 0.1588 0.0428 0.0370 0.0308 Table 4.3 Diagnostics of SARIMA AIC AICc BIC RMSE 8756.43 8756.5 8786.48 10.8925 WhatsApp - - 29 Table 4.4 Ljung-Box test Q-statistic df p-value 353.11 289 0.0059 SARIMA Figure 4.10:- Graphical Presentation of the Actual and Forecast WhatsApp - - 30 Table 4.5 Forecast Values Estimated through SARIMA Model Date Point Date Forecast Point Forecast Date Point Forecast 1/1/2019 2.87 1/24/2019 23.02 2/16/2019 33.02 1/2/2019 3.01 1/25/2019 30.02 2/17/2019 32.02 1/3/2019 2.98 1/26/2019 30.02 2/18/2019 29.32 1/4/2019 3.03 1/27/2019 33.02 2/19/2019 18.72 1/5/2019 15.92 1/28/2019 33.02 2/20/2019 18.72 1/6/2019 16.02 1/29/2019 40.02 2/21/2019 22.12 1/7/2019 16.02 1/30/2019 43.02 2/22/2019 17.22 1/8/2019 18.02 1/31/2019 43.02 2/23/2019 15.42 1/9/2019 20.02 2/1/2019 42.12 2/24/2019 14.22 1/10/2019 20.02 2/2/2019 38.02 2/25/2019 12.72 1/11/2019 20.02 2/3/2019 38.02 2/26/2019 12.72 1/12/2019 20.02 2/4/2019 38.02 2/27/2019 12.92 1/13/2019 20.02 2/5/2019 38.02 2/28/2019 12.32 1/14/2019 20.02 2/6/2019 38.02 3/1/2019 12.22 1/15/2019 20.02 2/7/2019 38.02 3/2/2019 11.82 1/16/2019 20.22 2/8/2019 38.02 3/3/2019 13.22 1/17/2019 19.82 2/9/2019 38.02 3/4/2019 24.42 1/18/2019 20.02 2/10/2019 38.02 3/5/2019 23.52 1/19/2019 20.02 2/11/2019 38.02 3/6/2019 18.82 1/20/2019 20.02 2/12/2019 38.02 3/7/2019 17.02 1/21/2019 23.02 2/13/2019 38.02 3/8/2019 15.72 1/22/2019 23.02 2/14/2019 33.02 3/9/2019 14.62 1/23/2019 23.02 2/15/2019 33.02 3/10/2019 13.72 Continued WhatsApp - - 31 Date Point Date Forecast Point Forecast Date Point Forecast 3/11/2019 15.02 4/3/2019 25.52 4/26/2019 35.72 3/12/2019 14.52 4/4/2019 30.52 4/27/2019 35.42 3/13/2019 14.22 4/5/2019 28.02 4/28/2019 34.42 3/14/2019 14.22 4/6/2019 28.02 4/29/2019 33.42 3/15/2019 13.82 4/7/2019 28.02 4/30/2019 31.52 3/16/2019 14.32 4/8/2019 28.02 5/1/2019 32.52 3/17/2019 14.12 4/9/2019 28.02 5/2/2019 38.02 3/18/2019 14.42 4/10/2019 20.02 5/3/2019 38.02 3/19/2019 14.42 4/11/2019 23.02 5/4/2019 33.02 3/20/2019 14.92 4/12/2019 23.02 5/5/2019 33.02 3/21/2019 14.82 4/13/2019 23.02 5/6/2019 32.22 3/22/2019 15.52 4/14/2019 28.02 5/7/2019 38.02 3/23/2019 18.32 4/15/2019 38.02 5/8/2019 38.02 3/24/2019 20.42 4/16/2019 43.02 5/9/2019 38.02 3/25/2019 19.02 4/17/2019 48.02 5/10/2019 43.02 3/26/2019 16.02 4/18/2019 48.02 5/11/2019 48.02 3/27/2019 16.02 4/19/2019 43.02 5/12/2019 48.02 3/28/2019 16.02 4/20/2019 38.02 5/13/2019 48.02 3/29/2019 16.02 4/21/2019 38.02 5/14/2019 43.02 3/30/2019 16.02 4/22/2019 38.02 5/15/2019 43.02 3/31/2019 16.02 4/23/2019 38.02 5/16/2019 43.02 4/1/2019 16.02 4/24/2019 38.02 5/17/2019 43.02 4/2/2019 21.02 4/25/2019 38.02 5/18/2019 53.02 Continued WhatsApp - - 32 Date Point Forecast Date Point Forecast Date Point Forecast 5/19/2019 58.02 6/11/2019 171.42 7/4/2019 138.02 5/20/2019 58.02 6/12/2019 170.92 7/5/2019 138.02 5/21/2019 58.02 6/13/2019 165.62 7/6/2019 138.02 5/22/2019 58.02 6/14/2019 157.12 7/7/2019 138.02 5/23/2019 58.02 6/15/2019 153.02 7/8/2019 138.02 5/24/2019 68.02 6/16/2019 151.82 7/9/2019 128.02 5/25/2019 68.02 6/17/2019 136.22 7/10/2019 106.62 5/26/2019 68.02 6/18/2019 140.52 7/11/2019 108.22 5/27/2019 78.02 6/19/2019 136.52 7/12/2019 98.02 5/28/2019 92.62 6/20/2019 135.22 7/13/2019 98.02 5/29/2019 98.02 6/21/2019 139.22 7/14/2019 78.02 5/30/2019 97.52 6/22/2019 144.62 7/15/2019 78.02 5/31/2019 98.52 6/23/2019 161.72 7/16/2019 78.02 6/1/2019 110.12 6/24/2019 160.92 7/17/2019 153.32 6/2/2019 140.92 6/25/2019 148.02 7/18/2019 180.02 6/3/2019 138.02 6/26/2019 143.22 7/19/2019 111.72 6/4/2019 138.02 6/27/2019 121.72 7/20/2019 133.02 6/5/2019 138.02 6/28/2019 118.62 7/21/2019 133.02 6/6/2019 136.12 6/29/2019 120.82 7/22/2019 148.02 6/7/2019 139.92 6/30/2019 136.52 7/23/2019 153.02 6/8/2019 167.12 7/1/2019 148.02 7/24/2019 163.02 6/9/2019 178.02 7/2/2019 148.02 7/25/2019 163.02 6/10/2019 178.02 7/3/2019 138.02 7/26/2019 168.02 Continued WhatsApp - - 33 Date Point Date Forecast Point Forecast Date Point Forecast 7/27/2019 168.02 8/19/2019 143.22 9/11/2019 128.02 7/28/2019 148.02 8/20/2019 152.32 9/12/2019 128.02 7/29/2019 138.02 8/21/2019 160.82 9/13/2019 128.02 7/30/2019 200.42 8/22/2019 159.82 9/14/2019 118.02 7/31/2019 202.32 8/23/2019 163.62 9/15/2019 113.02 8/1/2019 195.42 8/24/2019 161.82 9/16/2019 113.02 8/2/2019 204.32 8/25/2019 151.92 9/17/2019 103.02 8/3/2019 223.32 8/26/2019 141.82 9/18/2019 98.02 8/4/2019 212.02 8/27/2019 130.92 9/19/2019 98.02 8/5/2019 180.62 8/28/2019 158.02 9/20/2019 98.02 8/6/2019 155.12 8/29/2019 158.02 9/21/2019 98.02 8/7/2019 157.02 8/30/2019 168.02 9/22/2019 98.02 8/8/2019 176.02 8/31/2019 168.02 9/23/2019 83.02 8/9/2019 179.12 9/1/2019 178.02 9/24/2019 83.02 8/10/2019 207.92 9/2/2019 178.02 9/25/2019 78.02 8/11/2019 204.12 9/3/2019 178.02 9/26/2019 78.02 8/12/2019 222.82 9/4/2019 148.02 9/27/2019 78.02 8/13/2019 228.22 9/5/2019 148.02 9/28/2019 68.02 8/14/2019 232.22 9/6/2019 148.02 9/29/2019 58.02 8/15/2019 230.72 9/7/2019 148.02 9/30/2019 48.02 8/16/2019 208.02 9/8/2019 135.62 10/1/2019 48.02 8/17/2019 187.52 9/9/2019 128.02 10/2/2019 38.02 8/18/2019 159.52 9/10/2019 128.02 10/3/2019 33.02 Continued WhatsApp - - 34 Date Point Forecast Date Point Forecast Date Point Forecast 10/4/2019 33.02 10/27/2019 39.12 11/19/2019 48.02 10/5/2019 33.02 10/28/2019 43.02 11/20/2019 48.02 10/6/2019 33.02 10/29/2019 53.02 11/21/2019 43.02 10/7/2019 33.02 10/30/2019 57.02 11/22/2019 43.02 10/8/2019 33.02 10/31/2019 57.02 11/23/2019 43.02 10/9/2019 28.02 11/1/2019 57.02 11/24/2019 38.02 10/10/2019 28.02 11/2/2019 57.02 11/25/2019 33.02 10/11/2019 28.02 11/3/2019 57.02 11/26/2019 26.02 10/12/2019 23.02 11/4/2019 57.02 11/27/2019 26.02 10/13/2019 23.02 11/5/2019 54.02 11/28/2019 26.02 10/14/2019 23.02 11/6/2019 54.02 11/29/2019 24.92 10/15/2019 23.02 11/7/2019 53.02 11/30/2019 23.02 10/16/2019 23.02 11/8/2019 53.02 12/1/2019 23.02 10/17/2019 23.02 11/9/2019 53.02 12/2/2019 21.02 10/18/2019 33.02 11/10/2019 53.02 12/3/2019 26.02 10/19/2019 43.02 11/11/2019 53.02 12/4/2019 26.02 10/20/2019 43.02 11/12/2019 53.02 12/5/2019 26.02 10/21/2019 46.02 11/13/2019 48.02 12/6/2019 26.02 10/22/2019 46.02 11/14/2019 48.02 12/7/2019 21.02 10/23/2019 43.02 11/15/2019 48.02 12/8/2019 21.02 10/24/2019 38.02 11/16/2019 48.02 12/9/2019 18.02 10/25/2019 38.02 11/17/2019 48.02 12/10/2019 18.02 10/26/2019 38.02 11/18/2019 48.02 12/11/2019 13.02 Continued WhatsApp - - 35 Date Point Forecast 12/12/2019 13.02 12/13/2019 8.02 12/14/2019 8.02 12/15/2019 8.02 12/16/2019 8.02 12/17/2019 8.02 12/18/2019 8.02 12/19/2019 8.02 12/20/2019 8.02 12/21/2019 3.12 12/22/2019 3.02 12/23/2019 3.02 12/24/2019 3.02 12/25/2019 1.72 12/26/2019 1.02 12/27/2019 0.52 12/28/2019 1.02 12/29/2019 1.02 12/30/2019 1.02 12/31/2019 1.02 Forecasting using Non-parametric Technique 4.2 WhatsApp - - 36 Figure 4.11:- T Figure 4.5:- Decomposition of the Water Outflow time series From figure 4.5, Obviously, there is seasonal trend in the given time series of Water Outflow, it also shows that there is a decreasing trend. Table 4.6 Mean Square Error of Non-parametric Method MSE Non-parametric Method- Table 4.7 Forecast Values Estimated through Non-parametric Method WhatsApp - - 37 Date Point Forecast Date Point Forecast Date Point Forecast 1/1/2019 3.00 1/24/2019 22.00 2/16/2019 40.00 1/2/2019 3.00 1/25/2019 22.00 2/17/2019 40.00 1/3/2019 3.00 1/26/2019 22.00 2/18/2019 40.00 1/4/2019 3.00 1/27/2019 22.10 2/19/2019 40.00 1/5/2019 3.00 1/28/2019 22.00 2/20/2019 40.00 1/6/2019 3.00 1/29/2019 21.90 2/21/2019 40.00 1/7/2019 3.00 1/30/2019 22.00 2/22/2019 40.00 1/8/2019 3.00 1/31/2019 22.00 2/23/2019 40.00 1/9/2019 3.00 2/1/2019 23.50 2/24/2019 40.00 1/10/2019 3.00 2/2/2019 25.00 2/25/2019 37.50 1/11/2019 3.00 2/3/2019 25.00 2/26/2019 35.00 1/12/2019 4.00 2/4/2019 25.00 2/27/2019 35.00 1/13/2019 5.00 2/5/2019 28.50 2/28/2019 34.50 1/14/2019 5.00 2/6/2019 31.00 3/1/2019 32.65 1/15/2019 5.00 2/7/2019 33.50 3/2/2019 26.00 1/16/2019 11.45 2/8/2019 35.00 3/3/2019 20.70 1/17/2019 17.95 2/9/2019 38.50 3/4/2019 22.40 1/18/2019 18.00 2/10/2019 43.50 3/5/2019 21.65 1/19/2019 19.00 2/11/2019 45.00 3/6/2019 18.30 1/20/2019 21.00 2/12/2019 44.55 3/7/2019 16.80 1/21/2019 22.00 2/13/2019 42.05 3/8/2019 15.45 1/22/2019 22.00 2/14/2019 40.00 3/9/2019 14.70 1/23/2019 22.00 2/15/2019 40.00 3/10/2019 14.80 Date Point Date Point Continued Date Point WhatsApp - - 38 Forecast Forecast Forecast 3/11/2019 14.60 4/3/2019 18.90 4/26/2019 35.00 3/12/2019 14.25 4/4/2019 21.35 4/27/2019 42.00 3/13/2019 14.00 4/5/2019 21.70 4/28/2019 47.50 3/14/2019 14.50 4/6/2019 19.50 4/29/2019 50.00 3/15/2019 20.80 4/7/2019 18.00 4/30/2019 47.50 3/16/2019 25.95 4/8/2019 18.00 5/1/2019 42.50 3/17/2019 23.15 4/9/2019 18.00 5/2/2019 40.00 3/18/2019 19.90 4/10/2019 18.00 5/3/2019 40.00 3/19/2019 18.35 4/11/2019 18.00 5/4/2019 40.00 3/20/2019 17.15 4/12/2019 18.00 5/5/2019 40.00 3/21/2019 16.15 4/13/2019 20.50 5/6/2019 40.00 3/22/2019 16.35 4/14/2019 25.25 5/7/2019 38.85 3/23/2019 16.75 4/15/2019 30.00 5/8/2019 37.55 3/24/2019 16.35 4/16/2019 31.25 5/9/2019 36.90 3/25/2019 16.20 4/17/2019 30.00 5/10/2019 35.90 3/26/2019 16.00 4/18/2019 30.00 5/11/2019 34.45 3/27/2019 16.05 4/19/2019 30.00 5/12/2019 34.00 3/28/2019 16.20 4/20/2019 30.00 5/13/2019 37.25 3/29/2019 16.25 4/21/2019 26.00 5/14/2019 40.00 3/30/2019 16.40 4/22/2019 23.50 5/15/2019 37.50 3/31/2019 16.65 4/23/2019 25.00 5/16/2019 35.00 4/1/2019 16.85 4/24/2019 25.00 5/17/2019 34.60 4/2/2019 17.15 4/25/2019 27.50 5/18/2019 37.10 Continued Date Point Forecast WhatsApp - Date Point Forecast Date Point Forecast - 39 5/19/2019 40.00 6/11/2019 100.00 7/4/2019 143.90 5/20/2019 42.50 6/12/2019 106.30 7/5/2019 155.15 5/21/2019 42.50 6/13/2019 127.50 7/6/2019 163.30 5/22/2019 47.50 6/14/2019 141.45 7/7/2019 156.45 5/23/2019 50.00 6/15/2019 140.00 7/8/2019 147.60 5/24/2019 50.00 6/16/2019 140.00 7/9/2019 134.45 5/25/2019 47.50 6/17/2019 139.00 7/10/2019 122.15 5/26/2019 45.00 6/18/2019 139.05 7/11/2019 121.70 5/27/2019 45.00 6/19/2019 140.00 7/12/2019 130.65 5/28/2019 45.00 6/20/2019 155.50 7/13/2019 144.25 5/29/2019 50.00 6/21/2019 174.55 7/14/2019 150.00 5/30/2019 57.50 6/22/2019 180.00 7/15/2019 15.00 5/31/2019 60.00 6/23/2019 176.70 7/16/2019 145.00 6/1/2019 60.00 6/24/2019 173.15 7/17/2019 140.00 6/2/2019 60.00 6/25/2019 170.25 7/18/2019 140.00 6/3/2019 65.00 6/26/2019 163.35 7/19/2019 140.00 6/4/2019 70.00 6/27/2019 157.05 7/20/2019 140.00 6/5/2019 70.00 6/28/2019 154.40 7/21/2019 135.00 6/6/2019 75.00 6/29/2019 146.00 7/22/2019 119.30 6/7/2019 87.60 6/30/2019 140.35 7/23/2019 109.40 6/8/2019 97.60 7/1/2019 140.50 7/24/2019 105.10 6/9/2019 97.30 7/2/2019 137.85 7/25/2019 100.00 6/10/2019 99.75 7/3/2019 139.20 7/26/2019 90.00 Continued Date 7/27/2019 Point Date Forecast 80.00 8/19/2019 WhatsApp - Point Forecast 158.05 Date 9/11/2019 Point Forecast 146.45 - 40 7/28/2019 80.00 8/20/2019 168.50 9/12/2019 160.00 7/29/2019 177.65 8/21/2019 176.55 9/13/2019 165.00 7/30/2019 168.65 8/22/2019 179.55 9/14/2019 170.00 7/31/2019 147.85 8/23/2019 195.50 9/15/2019 175.00 8/1/2019 165.00 8/24/2019 208.00 9/16/2019 180.00 8/2/2019 167.50 8/25/2019 215.45 9/17/2019 180.00 8/3/2019 167.50 8/26/2019 227.50 9/18/2019 165.00 8/4/2019 160.00 8/27/2019 232.20 9/19/2019 150.00 8/5/2019 165.00 8/28/2019 233.45 9/20/2019 150.00 8/6/2019 167.50 8/29/2019 221.35 9/21/2019 150.00 8/7/2019 167.50 8/30/2019 199.75 9/22/2019 133.80 8/8/2019 170.00 8/31/2019 175.00 9/23/2019 130.00 8/9/2019 160.00 9/1/2019 175.50 9/24/2019 130.00 8/10/2019 145.00 9/2/2019 153.35 9/25/2019 130.00 8/11/2019 171.20 9/3/2019 149.75 9/26/2019 125.00 8/12/2019 203.35 9/4/2019 158.55 9/27/2019 117.50 8/13/2019 200.85 9/5/2019 162.30 9/28/2019 115.00 8/14/2019 201.85 9/6/2019 163.70 9/29/2019 110.00 8/15/2019 215.80 9/7/2019 164.70 9/30/2019 102.50 8/16/2019 219.65 9/8/2019 158.85 10/1/2019 100.00 8/17/2019 198.30 9/9/2019 148.85 10/2/2019 100.00 8/18/2019 169.85 9/10/2019 138.35 10/3/2019 100.00 Continued Date 10/4/2019 Point Date Forecast 100.00 10/27/2019 WhatsApp - Point Forecast 25.00 Point Date Forecast 11/19/2019 55.00 - 41 10/5/2019 92.50 10/28/2019 25.00 11/20/2019 55.00 10/6/2019 85.00 10/29/2019 25.00 11/21/2019 55.00 10/7/2019 82.00 10/30/2019 30.00 11/22/2019 55.00 10/8/2019 80.00 10/31/2019 40.00 11/23/2019 55.00 10/9/2019 80.00 11/1/2019 45.00 11/24/2019 55.00 10/10/2019 75.00 11/2/2019 46.50 11/25/2019 52.50 10/11/2019 65.00 11/3/2019 48.00 11/26/2019 50.00 10/12/2019 55.00 11/4/2019 46.50 11/27/2019 50.00 10/13/2019 50.00 11/5/2019 42.50 11/28/2019 50.00 10/14/2019 45.00 11/6/2019 40.00 11/29/2019 50.00 10/15/2019 37.50 11/7/2019 40.00 11/30/2019 50.00 10/16/2019 35.00 11/8/2019 40.55 12/1/2019 50.00 10/17/2019 35.00 11/9/2019 43.05 12/2/2019 50.00 10/18/2019 35.00 11/10/2019 50.00 12/3/2019 47.50 10/19/2019 35.00 11/11/2019 57.00 12/4/2019 45.00 10/20/2019 35.00 11/12/2019 59.00 12/5/2019 45.00 10/21/2019 32.50 11/13/2019 59.00 12/6/2019 42.50 10/22/2019 30.00 11/14/2019 59.00 12/7/2019 37.50 10/23/2019 30.00 11/15/2019 59.00 12/8/2019 31.50 10/24/2019 27.50 11/16/2019 59.00 12/9/2019 28.00 10/25/2019 25.00 11/17/2019 57.50 12/10/2019 28.00 10/26/2019 25.00 11/18/2019 56.00 12/11/2019 27.45 Continued Date Point Forecast 12/12/2019 25.95 12/13/2019 25.00 WhatsApp - - 42 12/14/2019 24.00 12/15/2019 25.50 12/16/2019 28.00 12/17/2019 28.00 12/18/2019 28.00 12/19/2019 25.50 12/20/2019 23.00 12/21/2019 21.50 12/22/2019 20.00 12/23/2019 17.50 12/24/2019 15.00 12/25/2019 12.50 12/26/2019 10.00 12/27/2019 10.00 12/28/2019 10.00 12/29/2019 10.00 12/30/2019 10.00 12/31/2019 10.00 WhatsApp - - 43 Figure 4.12:- Graph of Forecast Table 4.8 MSE of Artificial Neural Network WhatsApp - - 44 Method MSE MLP with 5 Hidden nodes 11.0876 Figure 4.13:- Graphical presentation of Multilayer Perceptron Table 4.9 Forecast Values Estimated through ANN with 5 Hidden nodes Date Point Forecast Date2 WhatsApp - Point Forecast3 Date4 Point Forecast5- 45 1/1/2019 1/2/2019 1/3/2019 1/4/2019 1/5/2019 1/6/2019 1/7/2019 1/8/2019 1/9/2019 1/10/2019 1/11/2019 1/12/2019 1/13/2019 1/14/2019 1/15/2019 1/16/2019 1/17/2019 1/18/2019 1/19/2019 1/20/2019 1/21/2019 1/22/2019 1/23/2019 Continued Date 3/11/2019 - Point Forecast 7.52 1/24/2019 1/25/2019 1/26/2019 1/27/2019 1/28/2019 1/29/2019 1/30/2019 1/31/2019 2/1/2019 2/2/2019 2/3/2019 2/4/2019 2/5/2019 2/6/2019 2/7/2019 2/8/2019 2/9/2019 2/10/2019 2/11/2019 2/12/2019 2/13/2019 2/14/2019 2/15/2019 Date2 4/3/2019 WhatsApp - - 2/16/2019 2/17/2019 2/18/2019 2/19/2019 2/20/2019 2/21/2019 2/22/2019 2/23/2019 2/24/2019 2/25/2019 2/26/2019 2/27/2019 2/28/2019 3/1/2019 3/2/2019 3/3/2019 3/4/2019 3/5/2019 3/6/2019 3/7/2019 3/8/2019 3/9/2019 3/10/2019 Point Date4 Forecast3 17.94 4/26/2019 - Point Forecast5 58.35- 46 3/12/2019 3/13/2019 3/14/2019 3/15/2019 3/16/2019 3/17/2019 3/18/2019 3/19/2019 3/20/2019 3/21/2019 3/22/2019 3/23/2019 3/24/2019 3/25/2019 3/26/2019 3/27/2019 3/28/2019 3/29/2019 3/30/2019 3/31/2019 4/1/2019 4/2/2019 Continued Date 5/19/2019 - Point Forecast 86.52 4/4/2019 4/5/2019 4/6/2019 4/7/2019 4/8/2019 4/9/2019 4/10/2019 4/11/2019 4/12/2019 4/13/2019 4/14/2019 4/15/2019 4/16/2019 4/17/2019 4/18/2019 4/19/2019 4/20/2019 4/21/2019 4/22/2019 4/23/2019 4/24/2019 4/25/2019 - Point Forecast 6/11/- Date WhatsApp - 4/27/2019 4/28/2019 4/29/2019 4/30/2019 5/1/2019 5/2/2019 5/3/2019 5/4/2019 5/5/2019 5/6/2019 5/7/2019 5/8/2019 5/9/2019 5/10/2019 5/11/2019 5/12/2019 5/13/2019 5/14/2019 5/15/2019 5/16/2019 5/17/2019 5/18/2019 Date 7/4/2019 - Point Forecast 182.00- 47 5/20/2019 5/21/2019 5/22/2019 5/23/2019 5/24/2019 5/25/2019 5/26/2019 5/27/2019 5/28/2019 5/29/2019 5/30/2019 5/31/2019 6/1/2019 6/2/2019 6/3/2019 6/4/2019 6/5/2019 6/6/2019 6/7/2019 6/8/2019 6/9/2019 6/10/2019 Continued Date 7/27/2019 - Point Forecast 243.49 6/12/2019 6/13/2019 6/14/2019 6/15/2019 6/16/2019 6/17/2019 6/18/2019 6/19/2019 6/20/2019 6/21/2019 6/22/2019 6/23/2019 6/24/2019 6/25/2019 6/26/2019 6/27/2019 6/28/2019 6/29/2019 6/30/2019 7/1/2019 7/2/2019 7/3/2019 - Date 8/19/2019 WhatsApp - 7/5/2019 7/6/2019 7/7/2019 7/8/2019 7/9/2019 7/10/2019 7/11/2019 7/12/2019 7/13/2019 7/14/2019 7/15/2019 7/16/2019 7/17/2019 7/18/2019 7/19/2019 7/20/2019 7/21/2019 7/22/2019 7/23/2019 7/24/2019 7/25/2019 7/26/2019 Point Forecast 155.49 - Date 9/11/2019 Point Forecast 123.78 - 48 7/28/2019 7/29/2019 7/30/2019 7/31/2019 8/1/2019 8/2/2019 8/3/2019 8/4/2019 8/5/2019 8/6/2019 8/7/2019 8/8/2019 8/9/2019 8/10/2019 8/11/2019 8/12/2019 8/13/2019 8/14/2019 8/15/2019 8/16/2019 8/17/2019 8/18/2019 Continued Date 10/4/2019 - 8/20/2019 8/21/2019 8/22/2019 8/23/2019 8/24/2019 8/25/2019 8/26/2019 8/27/2019 8/28/2019 8/29/2019 8/30/2019 8/31/2019 9/1/2019 9/2/2019 9/3/2019 9/4/2019 9/5/2019 9/6/2019 9/7/2019 9/8/2019 9/9/2019 9/10/2019 Point Forecast 49.12 WhatsApp - Date 10/27/2019 - Point Forecast 30.40 9/12/2019 9/13/2019 9/14/2019 9/15/2019 9/16/2019 9/17/2019 9/18/2019 9/19/2019 9/20/2019 9/21/2019 9/22/2019 9/23/2019 9/24/2019 9/25/2019 9/26/2019 9/27/2019 9/28/2019 9/29/2019 9/30/2019 10/1/2019 10/2/2019 10/3/2019 Date 11/19/2019 - Point Forecast 39.51 - 49 10/5/2019 10/6/2019 10/7/2019 10/8/2019 10/9/2019 10/10/2019 10/11/2019 10/12/2019 10/13/2019 10/14/2019 10/15/2019 10/16/2019 10/17/2019 10/18/2019 10/19/2019 10/20/2019 10/21/2019 10/22/2019 10/23/2019 10/24/2019 10/25/2019 10/26/2019 Continued Date 12/12/2019 - 10/28/2019 10/29/2019 10/30/2019 10/31/2019 11/1/2019 11/2/2019 11/3/2019 11/4/2019 11/5/2019 11/6/2019 11/7/2019 11/8/2019 11/9/2019 11/10/2019 11/11/2019 11/12/2019 11/13/2019 11/14/2019 11/15/2019 11/16/2019 11/17/2019 11/18/2019 - 11/20/2019 11/21/2019 11/22/2019 11/23/2019 11/24/2019 11/25/2019 11/26/2019 11/27/2019 11/28/2019 11/29/2019 11/30/2019 12/1/2019 12/2/2019 12/3/2019 12/4/2019 12/5/2019 12/6/2019 12/7/2019 12/8/2019 12/9/2019 12/10/2019 12/11/2019 - Point Forecast 20.99 WhatsApp - - 50 12/13/2019 12/14/2019 12/15/2019 12/16/2019 12/17/2019 12/18/2019 12/19/2019 12/20/2019 12/21/2019 12/22/2019 12/23/2019 12/24/2019 12/25/2019 12/26/2019 12/27/2019 12/28/2019 12/29/2019 12/30/2019 12/31/2019 - WhatsApp - - 51 Figure 4.14:- Graph of Forecast using Artificial Neural Network WhatsApp - - 52 Table 4.10 Mean Square Error of Artificial Neural Network Method MSE ANN fit with (10,5) hidden nodes 3.4394 Figure 4.10:- Graphical presentation of Artificial Neural Network WhatsApp - - 53 Table 4.11 Forecast Values Estimated through ANN with (10,5) hidden nodes Date 1/1/2019 1/2/2019 1/3/2019 1/4/2019 1/5/2019 1/6/2019 1/7/2019 1/8/2019 1/9/2019 1/10/2019 1/11/2019 1/12/2019 1/13/2019 1/14/2019 1/15/2019 1/16/2019 1/17/2019 1/18/2019 1/19/2019 1/20/2019 1/21/2019 1/22/2019 1/23/2019 Continued Point Forecast- Date2 1/24/2019 1/25/2019 1/26/2019 1/27/2019 1/28/2019 1/29/2019 1/30/2019 1/31/2019 2/1/2019 2/2/2019 2/3/2019 2/4/2019 2/5/2019 2/6/2019 2/7/2019 2/8/2019 2/9/2019 2/10/2019 2/11/2019 2/12/2019 2/13/2019 2/14/2019 2/15/2019 WhatsApp - Point Forecast- Date4 2/16/2019 2/17/2019 2/18/2019 2/19/2019 2/20/2019 2/21/2019 2/22/2019 2/23/2019 2/24/2019 2/25/2019 2/26/2019 2/27/2019 2/28/2019 3/1/2019 3/2/2019 3/3/2019 3/4/2019 3/5/2019 3/6/2019 3/7/2019 3/8/2019 3/9/2019 3/10/2019 Point Forecast- - 54 Date 3/11/2019 3/12/2019 3/13/2019 3/14/2019 3/15/2019 3/16/2019 3/17/2019 3/18/2019 3/19/2019 3/20/2019 3/21/2019 3/22/2019 3/23/2019 3/24/2019 3/25/2019 3/26/2019 3/27/2019 3/28/2019 3/29/2019 3/30/2019 3/31/2019 4/1/2019 4/2/2019 Continued Point Forecast- Date2 4/3/2019 4/4/2019 4/5/2019 4/6/2019 4/7/2019 4/8/2019 4/9/2019 4/10/2019 4/11/2019 4/12/2019 4/13/2019 4/14/2019 4/15/2019 4/16/2019 4/17/2019 4/18/2019 4/19/2019 4/20/2019 4/21/2019 4/22/2019 4/23/2019 4/24/2019 4/25/2019 WhatsApp - Point Forecast- Date4 4/26/2019 4/27/2019 4/28/2019 4/29/2019 4/30/2019 5/1/2019 5/2/2019 5/3/2019 5/4/2019 5/5/2019 5/6/2019 5/7/2019 5/8/2019 5/9/2019 5/10/2019 5/11/2019 5/12/2019 5/13/2019 5/14/2019 5/15/2019 5/16/2019 5/17/2019 5/18/2019 Point Forecast- - 55 Date 5/19/2019 5/20/2019 5/21/2019 5/22/2019 5/23/2019 5/24/2019 5/25/2019 5/26/2019 5/27/2019 5/28/2019 5/29/2019 5/30/2019 5/31/2019 6/1/2019 6/2/2019 6/3/2019 6/4/2019 6/5/2019 6/6/2019 6/7/2019 6/8/2019 6/9/2019 6/10/2019 Continued Point Forecast- Date 6/11/2019 6/12/2019 6/13/2019 6/14/2019 6/15/2019 6/16/2019 6/17/2019 6/18/2019 6/19/2019 6/20/2019 6/21/2019 6/22/2019 6/23/2019 6/24/2019 6/25/2019 6/26/2019 6/27/2019 6/28/2019 6/29/2019 6/30/2019 7/1/2019 7/2/2019 7/3/2019 WhatsApp - Point Forecast- Date 7/4/2019 7/5/2019 7/6/2019 7/7/2019 7/8/2019 7/9/2019 7/10/2019 7/11/2019 7/12/2019 7/13/2019 7/14/2019 7/15/2019 7/16/2019 7/17/2019 7/18/2019 7/19/2019 7/20/2019 7/21/2019 7/22/2019 7/23/2019 7/24/2019 7/25/2019 7/26/2019 Point Forecast- - 56 Date 7/27/2019 7/28/2019 7/29/2019 7/30/2019 7/31/2019 8/1/2019 8/2/2019 8/3/2019 8/4/2019 8/5/2019 8/6/2019 8/7/2019 8/8/2019 8/9/2019 8/10/2019 8/11/2019 8/12/2019 8/13/2019 8/14/2019 8/15/2019 8/16/2019 8/17/2019 8/18/2019 Continued Point Forecast- Date 8/19/2019 8/20/2019 8/21/2019 8/22/2019 8/23/2019 8/24/2019 8/25/2019 8/26/2019 8/27/2019 8/28/2019 8/29/2019 8/30/2019 8/31/2019 9/1/2019 9/2/2019 9/3/2019 9/4/2019 9/5/2019 9/6/2019 9/7/2019 9/8/2019 9/9/2019 9/10/2019 WhatsApp - Point Forecast- Date 9/11/2019 9/12/2019 9/13/2019 9/14/2019 9/15/2019 9/16/2019 9/17/2019 9/18/2019 9/19/2019 9/20/2019 9/21/2019 9/22/2019 9/23/2019 9/24/2019 9/25/2019 9/26/2019 9/27/2019 9/28/2019 9/29/2019 9/30/2019 10/1/2019 10/2/2019 10/3/2019 Point Forecast- - 57 Date 10/4/2019 10/5/2019 10/6/2019 10/7/2019 10/8/2019 10/9/2019 10/10/2019 10/11/2019 10/12/2019 10/13/2019 10/14/2019 10/15/2019 10/16/2019 10/17/2019 10/18/2019 10/19/2019 10/20/2019 10/21/2019 10/22/2019 10/23/2019 10/24/2019 10/25/2019 10/26/2019 Continued Point Forecast- WhatsApp - Date 10/27/2019 10/28/2019 10/29/2019 10/30/2019 10/31/2019 11/1/2019 11/2/2019 11/3/2019 11/4/2019 11/5/2019 11/6/2019 11/7/2019 11/8/2019 11/9/2019 11/10/2019 11/11/2019 11/12/2019 11/13/2019 11/14/2019 11/15/2019 11/16/2019 11/17/2019 11/18/2019 Point Forecast- Date 11/19/2019 11/20/2019 11/21/2019 11/22/2019 11/23/2019 11/24/2019 11/25/2019 11/26/2019 11/27/2019 11/28/2019 11/29/2019 11/30/2019 12/1/2019 12/2/2019 12/3/2019 12/4/2019 12/5/2019 12/6/2019 12/7/2019 12/8/2019 12/9/2019 12/10/2019 12/11/2019 Point Forecast- - 58 Date 12/12/2019 12/13/2019 12/14/2019 12/15/2019 12/16/2019 12/17/2019 12/18/2019 12/19/2019 12/20/2019 12/21/2019 12/22/2019 12/23/2019 12/24/2019 12/25/2019 12/26/2019 12/27/2019 12/28/2019 12/29/2019 12/30/2019 12/31/2019 Point Forecast- WhatsApp - - 59 Figure 4.16:- Graph of Forecast using Artificial Neural Network WhatsApp - - 60 Time Series Analysis and Forecasting using Bayesian Approach 4.3 Figure 4.15:- T WhatsApp - - 61 Figure 4.16:- T WhatsApp - - 62 Local Trend with Seasonality through MSE Bayesian Approach - Figure 4.17:- T WhatsApp - - 63 Table 4.5 Forecast Values Estimated through Bayesian Approach Date 1/1/2019 1/2/2019 1/3/2019 1/4/2019 1/5/2019 1/6/2019 1/7/2019 1/8/2019 1/9/2019 1/10/2019 1/11/2019 1/12/2019 1/13/2019 1/14/2019 1/15/2019 1/16/2019 1/17/2019 1/18/2019 1/19/2019 1/20/2019 1/21/2019 1/22/2019 1/23/2019 Continued Point Forecast- Date 1/24/2019 1/25/2019 1/26/2019 1/27/2019 1/28/2019 1/29/2019 1/30/2019 1/31/2019 2/1/2019 2/2/2019 2/3/2019 2/4/2019 2/5/2019 2/6/2019 2/7/2019 2/8/2019 2/9/2019 2/10/2019 2/11/2019 2/12/2019 2/13/2019 2/14/2019 2/15/2019 WhatsApp - Point Forecast- Date 2/16/2019 2/17/2019 2/18/2019 2/19/2019 2/20/2019 2/21/2019 2/22/2019 2/23/2019 2/24/2019 2/25/2019 2/26/2019 2/27/2019 2/28/2019 3/1/2019 3/2/2019 3/3/2019 3/4/2019 3/5/2019 3/6/2019 3/7/2019 3/8/2019 3/9/2019 3/10/2019 Point Forecast- - 64 Date 3/11/2019 3/12/2019 3/13/2019 3/14/2019 3/15/2019 3/16/2019 3/17/2019 3/18/2019 3/19/2019 3/20/2019 3/21/2019 3/22/2019 3/23/2019 3/24/2019 3/25/2019 3/26/2019 3/27/2019 3/28/2019 3/29/2019 3/30/2019 3/31/2019 4/1/2019 4/2/2019 Continued Point Forecast- Date 4/3/2019 4/4/2019 4/5/2019 4/6/2019 4/7/2019 4/8/2019 4/9/2019 4/10/2019 4/11/2019 4/12/2019 4/13/2019 4/14/2019 4/15/2019 4/16/2019 4/17/2019 4/18/2019 4/19/2019 4/20/2019 4/21/2019 4/22/2019 4/23/2019 4/24/2019 4/25/2019 WhatsApp - Point Forecast- Date 4/26/2019 4/27/2019 4/28/2019 4/29/2019 4/30/2019 5/1/2019 5/2/2019 5/3/2019 5/4/2019 5/5/2019 5/6/2019 5/7/2019 5/8/2019 5/9/2019 5/10/2019 5/11/2019 5/12/2019 5/13/2019 5/14/2019 5/15/2019 5/16/2019 5/17/2019 5/18/2019 Point Forecast- - 65 Date 5/19/2019 5/20/2019 5/21/2019 5/22/2019 5/23/2019 5/24/2019 5/25/2019 5/26/2019 5/27/2019 5/28/2019 5/29/2019 5/30/2019 5/31/2019 6/1/2019 6/2/2019 6/3/2019 6/4/2019 6/5/2019 6/6/2019 6/7/2019 6/8/2019 6/9/2019 6/10/2019 Continued Point Forecast- Date 6/11/2019 6/12/2019 6/13/2019 6/14/2019 6/15/2019 6/16/2019 6/17/2019 6/18/2019 6/19/2019 6/20/2019 6/21/2019 6/22/2019 6/23/2019 6/24/2019 6/25/2019 6/26/2019 6/27/2019 6/28/2019 6/29/2019 6/30/2019 7/1/2019 7/2/2019 7/3/2019 WhatsApp - Point Forecast- Date 7/4/2019 7/5/2019 7/6/2019 7/7/2019 7/8/2019 7/9/2019 7/10/2019 7/11/2019 7/12/2019 7/13/2019 7/14/2019 7/15/2019 7/16/2019 7/17/2019 7/18/2019 7/19/2019 7/20/2019 7/21/2019 7/22/2019 7/23/2019 7/24/2019 7/25/2019 7/26/2019 Point Forecast- - 66 Date 7/27/2019 7/28/2019 7/29/2019 7/30/2019 7/31/2019 8/1/2019 8/2/2019 8/3/2019 8/4/2019 8/5/2019 8/6/2019 8/7/2019 8/8/2019 8/9/2019 8/10/2019 8/11/2019 8/12/2019 8/13/2019 8/14/2019 8/15/2019 8/16/2019 8/17/2019 8/18/2019 Continued Point Forecast- Date 8/19/2019 8/20/2019 8/21/2019 8/22/2019 8/23/2019 8/24/2019 8/25/2019 8/26/2019 8/27/2019 8/28/2019 8/29/2019 8/30/2019 8/31/2019 9/1/2019 9/2/2019 9/3/2019 9/4/2019 9/5/2019 9/6/2019 9/7/2019 9/8/2019 9/9/2019 9/10/2019 WhatsApp - Point Forecast- Date 9/11/2019 9/12/2019 9/13/2019 9/14/2019 9/15/2019 9/16/2019 9/17/2019 9/18/2019 9/19/2019 9/20/2019 9/21/2019 9/22/2019 9/23/2019 9/24/2019 9/25/2019 9/26/2019 9/27/2019 9/28/2019 9/29/2019 9/30/2019 10/1/2019 10/2/2019 10/3/2019 Point Forecast- - 67 Date 10/4/2019 10/5/2019 10/6/2019 10/7/2019 10/8/2019 10/9/2019 10/10/2019 10/11/2019 10/12/2019 10/13/2019 10/14/2019 10/15/2019 10/16/2019 10/17/2019 10/18/2019 10/19/2019 10/20/2019 10/21/2019 10/22/2019 10/23/2019 10/24/2019 10/25/2019 10/26/2019 Continued Date 12/12/2019 12/13/2019 12/14/2019 12/15/2019 12/16/2019 12/17/2019 12/18/2019 12/19/2019 12/20/2019 12/21/2019 Point Forecast- Date 10/27/2019 10/28/2019 10/29/2019 10/30/2019 10/31/2019 11/1/2019 11/2/2019 11/3/2019 11/4/2019 11/5/2019 11/6/2019 11/7/2019 11/8/2019 11/9/2019 11/10/2019 11/11/2019 11/12/2019 11/13/2019 11/14/2019 11/15/2019 11/16/2019 11/17/2019 11/18/2019 Point Forecast- Date 11/19/2019 11/20/2019 11/21/2019 11/22/2019 11/23/2019 11/24/2019 11/25/2019 11/26/2019 11/27/2019 11/28/2019 11/29/2019 11/30/2019 12/1/2019 12/2/2019 12/3/2019 12/4/2019 12/5/2019 12/6/2019 12/7/2019 12/8/2019 12/9/2019 12/10/2019 12/11/2019 Point Forecast- Point Forecast- WhatsApp - - 68 12/22/2019 12/23/2019 12/24/2019 12/25/2019 12/26/2019 12/27/2019 12/28/2019 12/29/2019 12/30/2019 12/31/2019 - WhatsApp - - 69 Chapter No. 5 Table 5.1 Conclusions and Recommendations Forecasting Methods RMSE SARIMA Model 10.8925 Bayesian Approach - Non-parametric Method KNN - ANN with 5 Hidden nodes 11.0876 ANN fit with (10,5) hidden nodes 3.4394 Generally, it has often been seen that the adequate selection of the order of the SARIMA model and the number of input, hidden and output neurons is very much crucial for the effective and successful prediction of the values. We have listed the Mean Square Error (MSE) for the comparison of the models. From the Table 5.1 the minimum MSE is attained by Artificial Neural Network ANN with (10,5) hidden nodes and declared as the best one while on the change of the number of hidden nodes and layers it becomes the 11.0876 which is not the least. As Seasonal Autoregressive Integrated Moving Average (SARIMA) model has the MSE 10.8925 which the which is the 2nd least value. As SARIMA model is a parametric technique and has reasonable low MSE, therefore, we conclude it the best one model for the forecasting of the water outflow data. Also, if someone has excellent art of selection of the number of nodes and hidden layers, the ANN is also the adequate technique in this respect. References WhatsApp - - 70 Adhikari K., R., & R.K., A. (2013). An Introductory Study on Time Series Modeling and Forecasting Ratnadip Adhikari R. K. Agrawal. ArXiv Preprint ArXiv:-. https://doi.org/10.1210/jc- Archer, D. R., & Fowler, H. J. (2008). Using meteorological data to forecast seasonal runoff on the River Jhelum, Pakistan. Journal of Hydrology, 361(1–2), 10–23. https://doi.org/10.1016/j.jhydrol- Aussem, A., Murtagh, F., Sarazin, M., Aussem, A., Murtagh, F., & Sarazin, M. (n.d.). Dynamical Recurrent Neural Networks and Pattern Recognition Methods for Time Series Prediction : Application to Seeing and Temperature Forecasting in the Context of ESO ’ s VLT Astronomical Weather Station Dynamical Recurrent Neural Networks and Pattern R. 1–39. Cai, H., Lye, L. M., & Khan, A. (2009). Flood forecasting on the Humber river using an artificial neural network approach (Vol. 2). Chatfield, C. (1996). Model Uncertainty and Forecast Accuracy. Journal of Forecasting, 15(July), 495–508. Collantes-duarte, J., & Rivas-echeverría, F. (n.d.). Time Series Forecasting using ARIMA , Neural Networks and Neo Fuzzy Neurons 2 Time Series : ARIMA Model 2 Artificial Neural Networks. Elganainy, M. A., & Eldwer, A. E. (2018). Stochastic Forecasting Models of the Monthly Streamflow for the Blue Nile at Eldiem Station 1. 45(3), 326–327. https://doi.org/10.1134/S- Hipel, K. W. (1994). TIME SERIES MODELLING OF WATER RESOURCES AND ENVIRONMENTAL SYSTEMS. 1994. Jam, F. A. (2013). Time Series Model to Forecast Area of Mangoes from Pakistan : An Application of Univariate Arima Model. (December 2012), 10–15. WhatsApp - - 71 Jones, A. L., & Smart, P. L. (2005). Spatial and temporal changes in the structure of groundwater nitrate concentration time series -) as demonstrated by autoregressive modelling. Journal of Hydrology, 310(1–4), 201–215. https://doi.org/10.1016/j.jhydrol- Klose, C., Pircher, M., & Sharma, S. (2004). Univariate time series Forecasting. -). Merkuryeva, G. V., & Kornevs, M. (2014). Water Flow Forecasting and River Simulation for Flood Risk Analysis. Information Technology and Management Science, 16(1). https://doi.org/10.2478/itms- Mills, T. C. (2015). Time Series Econometrics: A Concise Introduction. https://doi.org/10.1016/S- Naveena, K., Singh, S., Rathod, S., & Singh, A. (2017a). Hybrid ARIMA-ANN Modelling for Forecasting the Price of Robusta Coffee in Hybrid ARIMA-ANN Modelling for Forecasting the Price of Robusta Coffee in India. International Journal of Current Microbiology and Applied Sciences, 6(7),-. https://doi.org/-/ijcmas- Naveena, K., Singh, S., Rathod, S., & Singh, A. (2017b). Hybrid ARIMA-ANN Modelling for Forecasting the Price of Robusta Coffee in India. 6(7),-. Pal, A., & Prakash, P. (2017). Practical Time Series Analysis. In Packt Publishing. Retrieved from https://www.packtpub.com/big-data-and-business-intelligence/practical-time-series-analysis Rbunaru, A. B. Ă. C. Ă., & Cescu, L. M. C. Ă. (2013). Methods Used in the Seasonal Variations Analysis Of Time Series. Revista Română de Statistică, 61(3), 12–18. Sarjinder Singh. (2003). Advanced Sampling Theory with Applications. https://doi.org/10.1017/CBO- Seymour, L., Brockwell, P. J., & Davis, R. A. (1997). Introduction to Time Series and Forecasting. In Journal of the American Statistical Association (Vol. 92). https://doi.org/10.2307/- WhatsApp - - 72 Shmueli, G. (2011). Practical Time Series Forecasting: A Hands-On Guide. 202. Retrieved from http://galitshmueli.com/practical-time-series-forecasting-book Since, I., & Model, J. (2008). Improving artificial neural networks ’ performance in seasonal time series forecasting. Information Sciences, 178, -. https://doi.org/10.1016/j.ins- Tealab, A. (2018). Time series forecasting using artificial neural networks methodologies : A systematic review. Future Computing and Informatics Journal, 3(2), 334–340. https://doi.org/10.1016/j.fcij- Valipour, M., Banihabib, M. E., Mahmood, S., & Behbahani, R. (2013). Comparison of the ARMA , ARIMA , and the autoregressive artificial neural network models in forecasting the monthly inflow of Dez dam reservoir. Journal of Hydrology, 476, 433–441. https://doi.org/10.1016/j.jhydrol- Wong, F. S. (1990). Time series forecasting using backpropagation neural networks. Neurocomputing, 2, 147–159. Zhang, G. P. (2003). Time series forecasting using a hybrid ARIMA and neural network model. Neurocomputing, 50, 159–175. Zhang, G., Patuwo, B. E., & Hu, M. Y. (1998). Forecasting with artificial neural networks : The state of the art. International Journal of Forecasting, 14, 35–62. Aussem, A., Murtagh, F., & Sarazin, M. (1994). Dynamical recurrent neural networks and pattern recognition methods for time series prediction: application to seeing and temperature forecasting in the context of ESO's VLT Astronomical Weather Station. Vistas in Astronomy, 38, 357-374. WhatsApp - - 73 Aussem, A., Murtagh, F., & Sarazin, M. (1995). Dynamical recurrent neural networks—towards environmental time series prediction. International Journal of Neural Systems, 6(02), 145-170. F. E S. Wong, ―Time Series Forecasting Using Backpropagation Neural Networks,‖ Nuralcomputing, Vol. 2, pp. 147-159, 1990. Fan, J., & Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods Springer-Verlag. New York. G. E. P. Box, G. M. Jenkins and G. C. Reinsel, Time series Analysis: Forecasting and Control (third edition), Englewood Cliffs, NJ: Prentice Hall, 1994. G. Tsibouris, and M. Zeidenberg, ―Testing the Efficient Markets Hypotheses with Gradient Descent Algorithms,‖ A. P. Refenes (ed.): Neural Networks in the Capital Markets, Wiley, pp. 127-136, 1995. H. White, ―Economic Prediction Using Neural Networks: The Case of IBM Daily Stock Returns,‖ Proceedings of the IEEE International Conference on Neural Networks II, pp. 451-458, 1988. Herbrich, R., Keilbach, M., Graepel, T., Bollmann-Sdorra, P., & Obermayer, K. (1999). Neural networks in economics. In Computational techniques for modelling learning in economics (pp. 169-196). Springer US. Hipel, K. W., & McLeod, A. I. (1994). Time series modelling of water resources and environmental systems (Vol. 45). Elsevier. Jones, A. L., & Smart, P. L. (2005). Spatial and temporal changes in the structure of groundwater nitrate concentration time series -) as demonstrated by autoregressive modelling. Journal of Hydrology, 310(1), 201-215. WhatsApp - - 74 Maguire, L. P., & Campbell, J. G. (1995). Fuzzy reasoning using a three layer neural network. Proc. 6th IFSA, Sao Paulo, Brazil, 2, 627-630. Maguire, L. P., Roche, B., McGinnity, T. M., & McDaid, L. J. (1998). Predicting Mozer, M. C. (1993, February). Neural net architectures for temporal sequence processing. In Santa Fe Institute Studies In The Sciences Of Complexity-Proceedings Volume- (Vol. 15, Pp. 243243). Addison-Wesley Publishing Co. Sibanda, W., & Pretorius, P. (2012). Artificial neural networks-a review of applications of neural networks in the modeling of hiv epidemic. International Journal of Computer Applications, 44(16), 1-9. W. E. Bosarge, ―Adaptive Processes to Exploit the Nonlinear Structure of Financial WhatsApp - - 75 Market,‖ R. R. Trippi and E. Turban (eds.): Neural Networks in Finance and Investing, Probus Publishing, pp. 371-402, 1993. Zhang, G. P. (2003). Time series forecasting using a hybrid ARIMA and neural network model. Neurocomputing, 50, 159-175. Zhang, T., & Fukushige, A. (2002). Forecasting time series by Bayesian neural networks. In Neural Networks, 2002. IJCNN'02. Proceedings of the 2002 International Joint Conference on (Vol. 1, pp. 382-387). IEEE. WhatsApp - -