Journal Search Engine
Search Advanced Search Adode Reader(link)
Download PDF Export Citaion korean bibliography PMC previewer
ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.18 No.3 pp.530-540
DOI : https://doi.org/10.7232/iems.2019.18.3.530

Forecasting Model of Gross Regional Domestic Product (GRDP) Using Backpropagation of Levenberg-Marguardt Method

Sukono, Betty Subartin, Ambarwati, Herlina Napitupulu, Jumadil Saputra*, Yuyun Hidayat
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Sumedang, Indonesia
School of Social and Economic Development, Universiti Malaysia Terengganu, Kuala Nerus, Terengganu, Malaysia
Department of Statistics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Indonesia
Corresponding Author, E-mail: jumadil.saputra@umt.edu.my
June 3, 2019 June 17, 2019 June 20, 2019

ABSTRACT


The GRDP is an important indicator to measure the economic growth of a region so that the GRDP forecast future needs. This paper intends to choose a better method for forecasting the GRDP of Bandung Regency. The method used in this study is the Autoregressive Integrated Moving Average (ARIMA) time series model and the Backpropagation of Levenberg-Marquardt or Feed Forward Neural Network (FFNN) method. To measure the accuracy of forecasting carried out using Mean Absolute Percentage Error (MAPE). The GRDP data obtained from the Bandung Regency Central Bureau of Statistics, in the years 2010-2016. The results of the analysis using the ARIMA model, obtained ARIMA (0, 1, 1) model, with MAPE of 3.90%. Meanwhile, analysis using Backpropagation of Levenberg-Marquardt obtained FFNN (1, 5, 1) model, with MAPE of 3.88%. Because the MAPE value in the FFNN (1, 5, 1) model is smaller than the MAPE value in the ARIMA (0, 1, 1) model, it can conclude that the Backpropagation of Levenberg-Marquardt method better used in forecasting the GRDP of Bandung Regency.



초록


    1. INTRODUCTION

    One of the most prominent issues in discussing the economy of a region is the issue of economic growth. Economic growth is the development of activities in the economy that cause goods and services produced in society to increase causing the welfare of society increases as well (Imansyah et al., 2017;Putratama et al., 2012). Economic growth is considered important in the context of the economy because it can be a measurement of economic achievement in the area. GRDP is an indicator that can be used to measure the economic growth of an area (Mbarek and Feki, 2013;Meltem et al., 2012). A year-to-year value of GRDP can give an idea of the achievement of regional economic performance in those years. Therefore, future GRDP values also need to be foreseen to know how regional economic developments are going forward (Saman, 2011;Vrbka, 2016;Singh et al., 2018, Jalali and Jaafar, 2019).

    The data of GRDP value movement follows a time series based on the order of data points within a certain time interval. Time series analysis can be applied to predict the probability of future circumstances. One time series model can be used in forecasting is ARIMA. The principle of the ARIMA model is to predict the value of observed data in the future based on a series of observations made in the past (Ili et al., 2016;Suwardo et al., 2010;Kodekova et al., 2018). In addition to the ARIMA model, Artificial Neural Networks (ANN) is also one of the methods that can use in forecasting. ANN is an artificial representation of the human brain that continuously learns to solve a problem. One of the learning methods in ANN is Backpropagation. In the learning process, ANN forms a model based on the pattern of training results of some data. Levenberg-Marquardt is one of the training algorithms that can accelerate learning from ANN (Claveria and Torra, 2013;Haider and Hanif, 2009;Jaramillo, 2018, Asad et al., 2018).

    Researches on forecasting using time series and ANN analysis have done (Ramli et al., 2018). Forecast the inflation in Ghana with the ANN method, using the data of monthly inflation from January 1991 to December 2010. The objective is to forecast and project inflation from January 2011 to December 2011. The results of ANN also compared with the time series model such as Autoregression AR (12) and Vector Autoregression VAR (14), which uses the same set of data variables. The basis of comparison is an out-of-sample estimated error (RMSFE). The results show that RMSFE of ANN is lower than that of the time series model. That is, with these prediction comparison predictions based on the ANN model are more accurate.

    Sukono et al. (2016), researched the modelling and forecasting of inflation data in Indonesia using time series models and feedforward neural networks. The purpose of the research is to develop ANN modelling in statistical structure and apply it to inflation data in Indonesia. The study also aims to optimise the development of the FFNN procedure model and its application. The FFNN model and time series model are evaluated based on Root Mean Square Error (RMSE), Mean Absolute Deviation (MAD) and MAPE. The best model criteria are those that have a minimum of RMSE, MAD, and MAPE. The result of comparative analysis between FFNN model and time series model shows that the FFNN model is better than the time series model.

    Chaudhuri and Ghosh (2016) were researching the application of ANN model and time series in the forecasting exchange rate of a currency. To forecast exchange rates, used two different classes of approaches, which based on ANN model and time series model. Multilayer Feed Forward Neural Network (MLFFNN) and Nonlinear Autoregressive model with Exogen Neural Network Input (NARX) is an approach used as an ANN model. Generalised Autoregressive Conditional Heteroscedastic (GARCH) and Exponential Generalized Autoregressive Conditional Heteroskedastic (EGARCH) used as time series models. The results show that although two different approaches are efficient in forecasting exchange rates, MLFNN and NARX are the most efficient.

    Based on these, this paper focuses on research on how to predict GRDP Bandung Regency Indonesia, using Artificial Neural Network Backpropagation Levenberg- Marquardt method. The goal is to form a model forecast of GRDP Bandung Regency Indonesia, using the ARIMA model, and the Artificial Neural Network Backpropagation Levenberg-Marquardt method, as well as to compare which of the approaches is better.

    2. MATERIALS AND METHODS

    This section discusses the object of research and the stages of research data analysis which includes: bootstrap resampling, ARIMA models, and models of Artificial Neural Networks (ANN) Backpropagation Levenberg- Marquardt method.

    2.1 Bootstrap Resampling

    According to Timmerman et al. (2009), in doing research, sometimes the sample data obtained is much less than the population. These causes the result of sample data processing to be less representative in giving a conclusion. The bootstrap method was introduced by Efron in 1993, with the aim of obtaining the best estimates based on minimal data with the help of computers. Bootstrap is a method of resampling the sample data with the condition of return on the data that can represent the actual population data. Bootstrap resampling can be done with the help of software R. However, raw data resampling using R software cannot be used directly in this research. The data used in this research is new data made based on the average value and standard deviation of the overall resampling result data.

    2.2 ARIMA Models

    There are two important representations in describing a time series data pattern, i.e. AR and Moving Average (MA) models. However, a new problem arises when one of the models used contains too many parameters (order is too large). The number of parameters that are too much can reduce the accuracy of the model in forecasting. To overcome these, there can be a combination of both AR and MA models into the ARMA model, which is a stationary time series model (Sukono et al., 2017). When p declares the order of the AR model, and q declares the order of the MA model, then the ARMA (p, q) model is expressed as follows:

    X t = c + φ 1 X t 1 + ... + φ p X t p + a t θ 1 a t 1 ... θ q a t q
    (1)

    with Xt random variable GRDP at time t ; φ 1 , ... , φ p parameters of the AR model; θ 1 , ... , θ q parameters of the MA model, and at residual value at time t.

    The ARIMA model is a non-stationary time series model denoted by ARIMA (p, d, q), where p declares the order of the AR model, q declares the order of the MA model, and d states the differencing order (Cadenas et al., 2016). The ARIMA model stated as follows:

    φ p ( B ) ( 1 B ) d X t = c + θ q ( B ) a t
    (2)

    where in AR operation φ p ( B ) = ( 1 φ 1 B φ 2 B ... φ p B p ) and MA operation θ q ( B ) = ( 1 θ 1 B θ 2 B ... θ q B q ) , with Xt random variable GRDP at the time t; B backshift operator; ( 1 B ) d X t time series is stationary on differencing to d; and at residual value at the time t.

    The steps of time series modelling are as follows: (i) Identification of the model, by determining the order value of p and q by autocorrelation function (ACF) and partial autocorrelation function (PACF) plot. (ii) Parameter Estimation by least squares method or maximum likelihood. (iii) Diagnostic test by white noise test and non-correlation serially against residual at. (iv) Forecasting, by deciding if the model is suitable so that it used for recursive prediction.

    2.3 Artificial Neural Network

    ANN is an artificial representation of the human brain that is continuously learning to solve a problem (Nakamura, 2005;Rădulescu and Banica, 2014). ANN mimics the workings of human biological neural net-works in processing information. The elements involved in this process are interconnected, and they operate in parallel. Similar to human neural networks, ANN is also composed of a set of neurons. In the process of sending information between neurons, these neurons connected by one or more networks that have weight values. The weight value obtained through the learning process, where one form of learning method ANN is Backpropagation. The simple form of neurons interpreted in Figure 1.

    In ANN, neurons are located and scattered in layers called neuron layers. There are three layers of the ANN constituent neurons: (i) Input Layer; the input neurons are in this layer. The input neuron receives input from outside. The input is a description of a problem. (ii) Hidden Layers; Hidden neurons are in this layer. The output of the hidden layer cannot observed directly. (iii) Output Layer; The output neurons are in this layer. The output of this layer is the output of artificial neural networks in the form of a solution to a problem.

    2.3.1 Activation Function

    The activation function is a function used in the processing of input and bias values to produce the output value of neurons (Mohammed et al., 2015;Ozbek et al., 2011). The following are two commonly used activation functions (equation 3). These Linear Functions; the linear function has an output value equal to its input value. The linear function formulated as follows:

    y = x
    (3)

    Binary sigmoid function; Binary sigmoid function has value in the range (0, 1). Therefore, this function used very commonly in ANN which usually produces output at intervals between 0 and 1. The binary sigmoid function formulated:

    y = f ( x ) = 1 1 + e x
    (4)

    This function has a derivative:

    f ' ( x ) = 1 ( 1 + e x ) 1 ( 1 + e x ) 2 alternatively , f ' ' ( x ) = f ( x ) [ 1 f ( x ) ]
    (5)

    2.3.2 Backpropagation Model

    Backpropagation is one of the learning methods of artificial neural networks. Backpropagation trains the network to strike a balance between the network's ability to recognise patterns during training with the network's ability to respond appropriately to the pat-terns used during the training (Naik and Pathan, 2012;Qi and Zhang, 2008). The activation function used in backpropagation is a binary sigmoid function. Back-propagation training divided into three phases: Phase I, Propagation Forward (feedforward); in this phase, the input signal xi is propagated (counted forward) by using the activation function to generate network output yk. These output results will be comparing with the expected target. The difference between the two is called an error. If the error is smaller than the specified tolerance, then iteration is stopped. However, if the opposite happens, then the weights on each line must be changed to minimise errors. Phase II, Backpropagation; in this phase, the error occurring is propagated backwards from the line directly related to the units in the output layer. Distribution error from output to the previous layer made by using delta function and Phase III, Change of Weight; in this phase, the weights of all layers are modified simultaneously. This change based on the delta neuron in its upper layer. The three phases of this training are done repeatedly until the conditions of termination of the iteration fulfilled.

    In the Backpropagation training phase, a condition is needed to stop the training process and as a benchmark for network accuracy in recognising the given pattern. One of the error calculations that used as a benchmark in the termination of training and the measurement of accuracy is the Mean Squared Error (MSE). MSE is the mean error of the square of the difference between network output and target output. The purpose of this training process is to get as minimal error value as possible by replacing the value of weights connected to all neurons on the network. The MSE formulated as follows:

    M S E = 1 2 k = 1 N ( t k y k ) 2
    (6)

    with k: input to k; tk: target output signal to k; and yk: output signal for network prediction to k.

    2.3.3 Levenberg-Marquardt Method

    The Levenberg-Marquardt algorithm is one type of ANN Backpropagation training algorithm. Levenberg-Marquardt is one solution to improve the performance of Backpropagation because the algorithm is designed to minimize the error sum of squares. The algorithm of Backpropagation Levenberg-Marquardt is described as follows:

    • Step 1: Initialize weights and biases with small ran-dom values and maximum epoch;

    • Step 2: Determining the required parameters, among others: (i) Initialization epoch = 0, (ii) Levenberg-Marquardt Parameters μ whose value must be greater than zero; and (iii) Factor parameters Beta β which is used as a parameter multiplied or divided by the Levenberg-Marquardt parameter;

    • Step 3: For each pair of training data, if epoch <epoch maximum or MSE> target error, do the following steps.

    Phase I: Propagation Forward (Feedforward)

    • Step 4: Each input unit xi, i = 1, 2, ..., n receives in-put signal xi and passes them to each unit in the hidden layer;

    • Step 5: Each hidden unit zi, j = 1, 2, ..., p sums the weighted input signal vij and bias voj, by formula:

      z _ i n j = v 0 j + i = 1 n x i v i j
      (7)

      Use the activation function to calculate the output signal:

      z j = f ( z _ i n j ) = 1 1 + e z _ i n j
      (8)

      Next, the signal zj will be sent to all units on the layer above it (output layer).

    • Step 6: Each unit of output yk, k = 1 , 2 , ... , m sums the weighted input signal wjk and bias w 0 k , with the formula

      y _ i n k = w 0 k + j = 1 p z j w j k
      (9)

      Use the activation function to calculate the output signal:

      y k = f ( y _ i n k ) = 1 1 + e y _ i n k
      (10)

      Next, the signal yk will be sending to all units on the layer above it.

    Phase II: Reverse Propagation (Backpropagation)

    • Step 7: Calculating error and MSE; the formula for calculating error is:

      e r r o r k = e k = t k y k
      (11)

      where k = output to k, so that MSE can be calculated using the formula:

      M S E = i = 1 n e r r o r k 2 n .
      (12)

      The formula to calculate the total error: e = ( e 1    e 2   ...   e n ) T

    • Step 8: Calculates an error on each unit of the output layer yk, k = 1 , 2 , ... , m with the formula:

      δ 2 k = e r r o r k × f ' ( y _ i n k ) ,
      (13)

      ϕ 2 j k = δ 2 k z j ; k = 1 , 2 , ... , m ; j = 0 , 1 , ... , p ,
      (14)

      β 2 k = δ 2 k ; k = 1 , 2 , ... , m .
      (15)

    • Step 9: Calculates errors on each hidden layer unit by z j , j = 1 , 2 , ... , p summing its input delta:

      δ _ i n j = k = 1 m δ 2 k w j k ,
      (16)

      So the error information:

      δ 1 j = δ _ i n j × f ' ( z _ i n j × [ z j ( 1 z j ) ] ,
      (17)

      ϕ 1 i j = δ 1 j x i ,
      (18)

      β 1 j = δ 1 j .
      (19)

    • Step 10: Fforming the matrix J

      J = [ ϕ 1 11 ... ϕ 1 n p   β 1 1 ... β 1 p   ϕ 2 11 ... ϕ 2 p m   β 2 1 ... β 2 m ] T .
      (20)

    Phase III: Change Weight

    • Step 11: Calculate new weights

      w n e w = w o l d [ J T J + μ I ] 1 J T e ,
      (21)

      with:

      w = [ w 1 1   ...   w p m   w 0 1   ...   w 0 m   v 11   ...    v n p   ...   v 0 1   ...   v 0 p ] T .

    • Step 12: Counting MSE

      If M S E n e w M S E o l d then,

      • μ ' = μ β with​ β = 0.7 p 1 n ,
        (22)

      • epoch = epoch + 1,

      • Go back to step 4,

        If M S E n e w > M S E o l d then:

      • μ ' = μ × β , with μ ' = μ × β .
        (23)

        Go back to step 11.

    • Step 13: Training stops when epoch = epoch is up.

    2.3.4 Normalisation and Denormalization of Data

    Normalisation was done with the aim to achieve a certain range using the input data and targets during the training phase and ANN testing. Normalisation min-max scales a given value into a new value between 0 and 1, where the scaling based on the maximum and minimum value of the data set. The formula in normalising the data is as follows:

    X n = 0.8 ( X X min ) X max X min + 0.1
    (24)

    with Xn normal data values; X actual data values; Xmin the minimum value of all actual data; and Xmax the maximum value of the actual data. Meanwhile, denormalisation of data is the process of returning data to the initial form before the normalisation. The following is the formula of denormalisation:

    X = ( X n 0.1 ) ( X max X min ) 0.8 + X min
    (25)

    with Xn normal data values; X actual data values; Xmin the minimum value of all actual data; and Xmin the maximum value of the actual data.

    2.3.5 Evaluate Error in Forecasting

    The difference between the forecast result and the actual data can be expressing in the Mean Absolute Percentage Error (MAPE), which formulated as follows:

    M A P E = ( 1 n t = 1 n | x t x ^ t x t | ) × 100 %
    (26)

    with n the amount of data; xt actual data on time t; and x ^ t output of forecast results at a time t. If the MAPE value is less than 10%, then the forecasting result is said to be very good. While if the value of MAPE is between 10-20%, then the forecasting results are said to be good.

    3. RESULT AND ANALYSIS

    This section discusses the result and analysis which includes: GRDP data of Bandung Regency Indonesia, data analysis using time series of ARIMA model, data analysis using ANN Backpropagation Levenberg-Marquardt method, and comparative analysis.

    3.1 GRDP Data of Bandung Regency, Indonesia

    GRDP data used in this study obtained from the official site of the Central Bureau of Statistics Bandung Regency Indonesia. The data taken is the GRDP data at constant prices based on the field of business from 2010 to 2016. The graph of the data series GRDP Bandung Regency Indonesia can be seen in Figure 2.

    Based on the plot in Figure 2, it can be seen that Bandung Regency GRDP has increased from year to year in 2010-2016. These means that the economy of Bandung Regency also progressed from year to year in 2010-2016. However, the available GRDP data is still too small to be processed because time series analysis requires much data in order to obtain a predictive model. Therefore, the data needs to be resampled using bootstrap. The data resampled 400 times, and the result of plot resampling of Bandung Regency GRDP data as shown in Figure 2.

    3.2 Data Analysis Using the ARIMA Time Series Model

    Before entering the analysis phase, 400 GRDP data of Bandung Regency resampling converted to GRDP data from September 1983 to December 2016.

    3.2.1 Stationary Test

    The stationary test needs to be done, to know sta-tionary GRDP data. The stationary test has done with Augmented Dickey-Fuller (ADF) test, with the help of Eviews 9.0 software. Hypothesis used is H0: data is not stationary, and H1: data is stationary. From the test results, a statistical value obtained is |tstatistic| with a GRDP data of tstatistic = -20.22669, while the critical value of statistics at a significance level of 5% is tcritical = -2.868565. Therefore, |tstatistic| > |tcritical| is applied, so the hypothesis of H0 the ADF test is rejected at 5% significance level, meaning that with 95% confidence the GRDP data has been stationary.

    3.2.2 Model Identification

    Identification of time series models can be made with Autocorrelation Function (ACF) and Partial Auto-correlation Function (PACF) plots, using Eviews 9.0 software. At level data, ACF and PACF plots do not indi-cate the order of a model, in the absence of a cut-off lag. Therefore, to overcome this problem, GRDP data is done in a different degree of one. Thus, the time series used is a model of ARIMA (p, 1, q). Furthermore, the plots of ACF and PACF of GRDP data of different degree of one can already indicate the order of a time series model. The ACF plot, cuts off significantly in lag 1, while the PACF plot decreases exponentially from lag 1. So there is an indication to model the GRDP data which can be done using ARIMA (0, 1, 1) model. However, time series modelling is also tried using ARIMA (1, 1, 1) and ARIMA (1, 1, 0) models.

    3.2.3 Selection of the Best Model

    In the previous model identification stage, it has said that three-time series models, i.e. ARIMA (0, 1, 1) models, ARIMA (1, 1, 1), and ARIMA (1, 1, 0) are attempted to considered. Choosing the best model can be done by using the Akaike Information Criterion (AIC), where the smallest AIC value shows the best model. Based on GRDP data processing with Eviews 9.0 software, obtained AIC value from each model is as follows: ARIMA(0, 1, 1) model with value of AIC = 32.59592; ARIMA (1, 1, 1) with value of AIC = 32.60034; and ARIMA(1, 1, 0) with value of AIC = 33.02344. The smallest AIC value is 32.59592 generated by the ARIMA (0, 1, 1) model, which makes the best model.

    3.2.4 Parameter Estimation

    Parameter estimation performed on ARIMA (0, 1, 1) tentative model by using Eviews 9.0 software. Estimation result of ARIMA (0, 1, 1) tentative parameter model first stage was obtained by constant estimator c of 1015,685, and parameter estimator for the coefficient of at-1 is -0.992332. However, the estimated value of the constant c of 1015,685 was not significant at a significance level of 5%. It is, therefore, necessary to estimate the second stage parameter, by not including the constants parameter c. Parameter estimators obtained the second stage estimation results for coefficients at-1 of -0.991942. Referring to equation (1), estimator model ARIMA (0, 1, 1) for Bandung Regency GRDP is

    X t = 1015.685 + 0.992332 a t 1 + a t
    (27)

    3.2.5 Parameter Significance Test (Test statistics t)

    Against the model estimator (27) it is necessary to test the significance of the parameters, using statistical tests t. In the significance test at equation (27) only the parameter estimator for the coefficients was performed at-1. The test hypothesis used is H0: θ1 = 0. Means parameter coefficient of at-1 had no significant effect on Xt, and Ht: θ1 ≠ 0, whereas means parameter coefficient at-1 had a significant effect on Xt. The results of tests conducted with the help of Eviews 9.0 software obtained statistical parameters for coefficients at-1 amount tstatistic = -267.9172. Meanwhile, at 5% significance level, the statistical critical value of t(0.025; 400-2) = -1.96, so |tstatistic| > |t(0.025; 400-2)| was applied. Hence, hypothesis H0 was rejected at a 5% significance level, meaning that with the confidence of 95%, parameter estimator for coefficients at-1 has a significant effect on Xt. Figure 3

    3.2.6 Test Validity (Test statistics F)

    The timing model estimate (27) also needs to be validated by statistical test F, with the aim of measuring the effect of parameters collectively in the model. The hypothesis used is H0: ϕ1 = θ1 = 0, meaning that all parameters in the model have significant influence together with Xt, and H 1 : φ 1 θ 1 0 meaning that there is at least one parameter that has a significant effect on Xt. Validation testing is done with the help of Eviews 9.0 software, with a generated value of F-statistic = 405.1586. While at a significance level of 5%, the critical value of statistic F(0.05; 2; 400-2-1) = 3, so Fstatistic > F(0.05; 2; 400-2-1) was applied. Hence, the hypothesis H0 is rejected at a 5% significance level. These means that with a confidence of 95% there is at least one parameter that has a significant effect on Xt.

    3.2.7 Diagnostic Test

    Once the model parameter estimator has been influenced significantly, further diagnostic testing of the residual model is necessary, in order to ensure that the model estimator is feasible for use in forecasting. Diag-nostic testing is performed with two treatments, i.e. as-sumption of residual normality, and white noise test. Residual Normality Test, conducted with the aim to determine the residual on the model of normal distribution. The hypothesis used is H0: at normally distributed, and Ht: at not normally distributed. The residual normality test on the model was performed using the Jarque-Bera (JB) statistical test, with the help of Eviews 9.0 software. Descriptive statistics of Bandung Regency GRDP, include 400 resampling data, with skewness of 0.202645 and kur-tosis of 3.411359, based on Jarque-Bera formula obtained value of JB = 5.55787. Jarque-Bera statistics are asymptotic to the Chi-Square distribution. Using the 5% significance level and the degrees of freedom 2, statistically critical values obtained were χ ( 0.05 ; 2 ) 2 = 5.99 , so J B < χ ( 0.05 ; 2 ) 2 , was applied. Thus hypothesis H0 was received at a 5% significance level, meaning that with the confidence of 95% residual at in ARIMA (0, 1, 1) the model estimator is a normal distribution.

    White Noise test is conducted to determine the serial correlation between lag on residual. The hypothesis used is H 0 : ρ 1 = ... = ρ m = 0 , meaning there is no residual correlation between lag, and H 1 : ρ k 0 for a k { 1 , ... , m } , meaning that there is a residual correlation between lag. White noise testing was performed using the Ljung-Box Q (m) test statistic where m is the degrees of freedom, through the residual correlogram plot of ARIMA (0, 1, 1) model, with the help of Eviews 9.0 software. Based on the results of the correlogram plot, it appears that all lag does not pass through the Bartlett line, and the value Pr [Q (m)] for every lag m with m = 1, 2,.., 20, has a greater value than the level of significance α = 0.05 . Therefore, it concluded that with 95% confidence, no residual corre-lation occurs between lag and residual in the model, which means that the ARIMA (0, 1, 1) model follows the white noise process.

    3.2.8 Forecasting

    After passing many tests of parameter significance, validation test, and diagnostic test, the best estimator of the ARIMA (0, 1, 1) model was obtained in the equation as follows:

    X t = 0.991942 a t 1
    (28)

    This model can used for forecasting Bandung Re-gency GRDP. The comparison between forecasting results data and actual data of Bandung Regency GRDP, during September 1983 - December 2016 can be seen in the graph in Figure 4

    Figure 4 above shows that the data forecast of Bandung Regency GRDP approached the actual data. The degree of difference or error occurring between these two data can be calculated using the Mean Absolute Percentage Error (MAPE) with equation (26), and a MAPE value of 3.90% obtained.

    3.2.9 Data Normalization

    Before performing data analysis stage using ANN Backpropagation Levenberg-Marquardt method, the data first of all needs to be normalised. These is because the activation function used is a binary sigmoid function, where it has a range value between 0 and 1. Normalisation of data done by applying equation (24). After data normalisation done, it was then used for the design of network structure, as follows.

    3.3 Data Analysis Using ANN Backpropagation Levenberg-Marquardt Method

    This section discusses data analysis using ANN Backpropagation Levenberg-Marquardt method, which includes: data normalisation, network structure design, training, and network testing, optimal network selection, data forecasting, and denomination.

    3.3.1 Data Normalization

    Before performing data analysis stage using ANN Backpropagation Levenberg-Marquardt method, the data first of all needs to be normalised. These is because the activation function used is a binary sigmoid function, where it has a range value between 0 and 1. Normalisation of data done by applying equation (24). After data normalisation done, it was then used for the design of network structure, as follows.

    3.3.2 Designing Network Structure

    After the data was normalised, the next step was to initialise the parameters used in ANN. These parameters can be shown in Table 1.

    Based on the parameter data in Table 1, there are five different combinations of inputs used in the Backpropagation process:

    1. Input on lag1 (Xt-1)

    2. Input on lag1 (Xt-1) and lag 2 (Xt-2)

    3. Input on lag1 (Xt-1), lag 2 (Xt-2), and lag 3 (Xt-3)

    4. Input on lag1 (Xt-1), lag 2 (Xt-1), lag 3 (Xt-3), and lag 4 (Xt-4)

    5. Input on lag1 (Xt-1), lag 2 (Xt-2), lag 3 (Xt-3), lag 4 (Xt-4), and lag 5 (Xt-5)

    A trial and error method conducted for each input, with some neurons between 1 to 5 to obtain optimal network in producing 1 GRDP data output. Networks were trained using Levenberg-Marquardt (trainlm) by applying the binary sigmoid activation function.

    3.3.3 Training and Testing Network

    Backpropagation attempts to strike a balance be-tween the recognition of proper training patterns, and a good response to other similar patterns. Therefore, the data are divided into two parts, the data pattern used as training, and the data used as the test. In this research, the training phase is done by using 280 GRDP data of Bandung Regency, whereas, in the testing phase, the data used were as many as 80 GRDP data. The remaining data used in the validation process. Afterwards, the data went through the training process, and network testing was done using the help toolbox in the MATLAB software. The results of the training process and backpropagation testing on the GRDP data using the Levenberg-Marquardt training method with one lag1 input (Xt-1), in generating one output with the number of neurons in the hidden layer between 1 and 5, is shown in Table 2.

    Table 2 shows the best selected FFNN (FFNN) Backpropagation or Feed Forward Neural Network (FFNN) model is the least likely to is rated MSE testing compared to other FFNN models. This FFNN model consists of 1 input, five neurons, and one output. The value of MSE FFNN (1, 5, 1) for the training stage is 0.0165924, and testing stages 0.0117982. Then, the results of the training process and Backpropagation testing on GRDP data, using the Levenberg-Marquardt training method with two inputs on lag1(Xt-1) and lag 2 (Xt-2) in producing one output with the number of neurons in the hidden layer between 1 and 5, is shown in Table 3.

    Table 3 depicts the best selected FFNN model is FFNN (2, 4, 1) because it has the smallest MSE testing value compared to other FFNN models. This FFNN model consists of 2 inputs, four neurons, and one output. The MSE value in the FFNN (2, 4, 1) model for the training stage is 0.0165924, whereas testing stage is 0.0157151. Furthermore, the results of the training process and Backpropagation testing on the PDRB data using the Levenberg-Marquardt training method with three inputs on lag 1(Xt-1), lag 2 (Xt-2), and lag 3 (Xt-3) in producing 1 output with the number of neurons in the hidden layer between 1 and 5, is shown in Table 4.

    Table 4 demonstrates that the best selected FFNN model is FFNN (3, 4, 1) because it has the smallest MSE testing value compared to other FFNN models. This FFNN model consists of 3 inputs, four neurons, and one output. The MSE value in the FFNN (3, 4, 1) model for the training stage is 0.0158606, whereas testing stage is 0.0135612. The results of the training process and Backpropagation testing on GRDP data using the Levenberg-Marquardt training method with four inputs on lag1 (Xt-1), lag 2 (Xt-2), lag 3 (Xt-3), and lag 4 (Xt-4) in generating 1 output with the number of neurons in the hidden layer between 1 and 5, is shown in Table 5.

    Table 5 indicates that the best selected FFNN model is FFNN (4.5.1) because it has the smallest MSE testing value compared to other FFNN models. This FFNN model consists of 4 inputs, five neurons, and one output. The MSE value in the FFNN (4.5.1) model for the training stage is 0.0165779, whereas testing stage is 0.0175038. Finally, the results of the training process and Backpropagation testing on GRDP data using the Levenberg-Marquardt training method with five inputs on lag 1 (Xt-1), lag 2 (Xt-2) , lag 3 (Xt-3), lag 4 (Xt-4), and lag 5 (Xt-5) in producing 1 output with the number of neurons in the hidden layer between 1 and 5, is shown in Table 6.

    Table 6 shows that the best selected FFNN model is FFNN (5, 2, 1) because it has the smallest MSE testing value compared to other FFNN models. This FFNN model consists of 5 inputs, two neurons, and one output. The MSE value in the FFNN (5, 2, 1) model for the training stage is 0.0166623, whereas testing stage is 0.015937.

    3.3.4 Selection of Optimal Network

    After training and testing the network using Back-propagation Levenberg-Marquardt is done, the best FFNN model from each combination of input and hidden neurons are obtained. The comparison of MSE values from these best FFNN models can be shown in Table 7.

    Table 7 shows the FFNN (1,5,1) model is the FFNN model which has the smallest MSE testing value com-pared to other FFNN model that is 0.0117982. Therefore, it can be concluding that FFNN (1.5.1) is the model with the most optimal network in predicting the GRDP data of Bandung Regency.

    3.3.5 Forecasting and Denormalization of Data

    The data of network output or forecast result using FFNN (1, 5, 1), then needed to go through denormalisation using equation (25). The objective is to return it to its original form before normalisation, in order to make a real difference to the actual data of GRDP. After the data is normalised, the comparison graph between the network output data and the actual GRDP data is formed as shown in Figure 5.

    Figure 5 displays the difference in network output data, and actual data of GRDP value is not much different. Data output network is the data whose value is relatively constant. The degree of difference or error between these two data can be calculated using MAPE with equation (26). Based on the calculation a MAPE value of 3.88% was obtained.

    3.4 Comparative Analysis of ARIMA and ANN Backpropagation Model Levenberg-Marquardt Method

    To find out which model is better to be used in pre-dicting the GRDP of Bandung Regency, both the ARIMA model and ANN Backpropagation Levenberg-Marquardt methods need to go through comparative analysis. Based on model estimation analysis that has been done, obtained MAPE for model ARIMA (0, 1, 1) is equal to 3.90%, while MAPE for FFNN (1, 5, 1) is equal to 3.88%. Because the MAPE value of FFNN (1, 5, 1) model is smaller, compared to MAPE ARIMA (0, 1, 1) model, it can be concluded that FFNN (1, 5, 1) model is a better model used in forecasting the GRDP of Bandung Regency. However, the MAPE values obtained from the ARIMA and FFNN models in this study are limited to the scope of 400 GRDP datasets of Bandung Regency resampling results using bootstrap. Therefore, it is possible to obtain different results if the ARIMA and FFNN models applied to other data.

    4 CONCLUSIONS

    In this paper, we have analysed the model of GRDP forecasting using ANN Backpropagation Levenberg-Marquardt method. Based on the results of the analysis, it can be concluding as follows: the best ARIMA time series model in forecasting of Bandung Regency GRDP is ARIMA (0, 1, 1). With MAPE error rate in of 3.90%, and the best model of ANN Backpropagation Levenberg-Marquardt method for forecasting of Bandung Regency GRDP is FFNN (1, 5, 1), having one input on lag-1, five hidden neurons, and one output, with MAPE error rate of 3.88%. Based on the MAPE error value of both models, a better model used in predicting of Bandung Regency GRDP is ANN Backpropagation Levenberg-Marquardt method, with the FFNN (1, 5, 1) model.

    ACKNOWLEDGEMENT

    Acknowledgements conveyed to the Rector, Director of Directorate of Research, Community Involvement and Innovation, and the Dean of Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, with whom the Internal Grant Program of Universitas Padjadjaran was made possible to fund this research. The grant is a means of enhancing research and publication activities for researchers at Universitas Padjadjaran.

    Figure

    IEMS-18-3-530_F1.gif

    Neurons in artificial neural networks.

    IEMS-18-3-530_F2.gif

    Bandung regency GRDP data.

    IEMS-18-3-530_F3.gif

    Plotting of resampling data.

    IEMS-18-3-530_F4.gif

    Graph of actual data and forecast results by ARIMA (0, 1, 1) model.

    IEMS-18-3-530_F5.gif

    Comparison of FFNN (1.5.1) network output data and GRDP actual data.

    Table

    Initialization of artificial neural networks

    Comparison of models with inputs on lag-1 ( Xt−1 )

    Comparison of models with inputs in lag 1 ( Xt−1 ) and lag 2 ( Xt−2 )

    Comparison of models with inputs in lag 1 ( Xt−1 ) Lag 2 ( Xt−2 ), and lag 3 ( Xt−3 )

    Comparison of Models with Inputs in Lag 1 ( Xt−1 ) Lag 2 ( Xt−2 ), lag 3 ( Xt−3 ), and lag 4 ( Xt−4 )

    Comparison of models with inputs on lag 1 ( Xt−1 ), lag 2 ( Xt−2 ), Lag 3 ( Xt−3 ), lag 4 ( Xt−4 ), and lag 5 ( Xt−5 )

    Comparison of MSE scores from the best FFNN models

    REFERENCES

    1. Asad, M. , Ahmad, I. , Haider, S. H. , and Salman, R. (2018), A crital review of islamic and conventional banking in digital era: A case of Pakistan, International Journal of Engineering & Technology, 7(4.7), 57-59.
    2. Cadenas, E. , Rivera, W. , Campos-Amezcua, R. , and Heard, C. (2016), Wind speed prediction using a univariate ARIMA model and a multivariate NARX model, Energies, 9(2), 1-15.
    3. Chaudhuri, T. D. and Ghosh, I. (2016), Artificial neural network and time series modeling based approach to forecasting the exchange rate in a multivariate framework, Journal of Insurance and Financial Management, 1(5), 92-123.
    4. Claveria, O. and Torra, S. (2013), Forecasting business surveys indicators: Neural networks vs. time series models, AQR Working Papers, 20, 1-28.
    5. Haider, A. and Hanif, M. N. (2009), Inflation forecasting in Pakistan using artificial neural networks, Pakistan Economic and Social Review, 47(1), 123-138.
    6. Ilić, I. , Jovanović, S. , and Janković-Milić, V. (2016), Forecasting corn production in Serbia using ARIMA model, Economics of Agriculture, 63(4), 1141-1156.
    7. Imansyah, M. H. , Suryani, S. , Nurhidayat, N. , and Muzdalifah, M. (2017), GDP Estimation and Slow Down Signal Model for Indonesia: An artificial neural network approach, Jurnal Keuangan dan Perbankan, 13(1), 77-93.
    8. Jalali, A. and Jaafar, M. (2019), The role of proactiveness as a mediatore between organizational-stakeholders relationship and SMEs performance, Journal of Southwest Jiaotong University, 54(3).
    9. Jaramillo, L. E. S. (2018), Malware detection and mitigation techniques: Lessons learned from Mirai DDOS attack, Journal of Information Systems Engineering & Management, 3(3), 19.
    10. Kodekova, G. , Mukatayeva, K. , Korvyakov, V. , and Auyezova, Z. (2018), Model of developing professional thinking in modern education conditions, Opción, 34(85-2), 458-478.
    11. Mbarek, M. B. and Feki, B. (2013), GDP and unemployment rate in Turkey: An empirical study using neural networks, International Journal of Economics, Finance and Management, 2(1), 154-159.
    12. Meltem, K. , Gamze, G. Y. , Hseyin, K. , and Suleyman, S. F. (2012), Using artificial neural networks to forecast GDP for Turkey, In 3rd International Symposium on Sustainable Development, 326-335.
    13. Mohammed, O. B. , Kolo, A. J. , and Solomon, A. A. (2015), Comparative study of inflation rates forecasting using feed-forward artificial neural networks and auto regressive (AR) models, International Journal of Computer Science Issue, 12(2), 260-266.
    14. Naik, A. R. and Pathan, S. K. (2012), Weather classification and forecasting using back propagation feed-forward neural network, International Journal of Scientific and Research Publications, 2(12), 1-3.
    15. Nakamura, E. (2005), Inflation forecasting using a neural network, Economics Letters, 86(3), 373-378.
    16. Ozbek, A. , Akalin, M. , Topuz, V. , and Sennaroglu, B. (2011), Prediction of Turkeys denim trousers export using artificial neural networks and the autoregressive integrated moving average model, Fibres and Textiles in Eastern Europe, 19(3), pp. 10-16.
    17. Putratama, L. A. , Muhayyatin, U. , and Kaloko, B. S. (2012), Forecasting of PDRB in Jember, east java using levenberg marquardt method, Academic Research International, 2(1), 30-34.
    18. Qi, M. and Zhang, G. P. (2008), Trend time-series modeling and forecasting with neural networks, IEEE Transactions on neural networks, 19(5), 808-816.
    19. Rădulescu, M. and Banica, L. (2014), Neural networks-based forecasting regarding the convergence process of CEE countries to the Eurozone, Transylvanian Review of Administrative Sciences, 42(E/2014), pp. 225-246.
    20. Ramli, A. , Shabbir, M. S. , Shukri, M. , Bakar, B. , Mohd, P. , Mohd, N. , and ahmad, I. (2018) . Mediating role of e- learning resources in developing entre¬pre¬neurial inclinations amongst undergraduate students at universiti utara Malaysia, International Journal of Engineering & Technology, 7, 51-56.
    21. Saman, C. (2011), Scenarios of the Romanian GDP evolution with neural models, Romanian Journal of Economic Forecasting, 14, 129-140.
    22. Singh, P. , Teoh, S. H. , Cheong, T. H. , Rasid, N. S. M. , Kor, L. K. , and Nasir, N. A. M. (2018), The use of problem-solving heuristics approach in enhancing STEM students development of mathematical thinking, International Electronic Journal of Mathematics Education, 13(3), 289-303.
    23. Sukono Hidayat, Y. , Suhartono Sutijo, B. , Bon, A. T. B. , and Supian, S. (2016), Indonesian financial data modeling and forecasting by using econometrics time series and neural network, Global Journal of Pure and Applied Mathematics, 12(4), 3745-3757.
    24. Sukono Lesmana, E. , Susanti, D. , Napitupulu, H. , and Hidayat, Y. (2017), Estimating the value-at-risk for some stocks at the capital market in Indonesia based on ARMA-FIGARCH models, Journal of Physics: Conference Series, 909(1), 012040.
    25. Suwardo, W. , Napiah, M. and Kamaruddin, I. (2010), ARIMA models for bus travel time prediction, Journal of the Institute of Engineers Malaysia, 2010, 71(2), 49-58.
    26. Timmerman, M. E. , Kiers, H. A. L. , Smilde, A. K. , Ceulemans, E. , and Stouten, J. (2009), Bootstrap confidence intervals in multi-level simultaneous component analysis, British Journal of Mathematical and Statistical Psychology, 62(2), 299-318.
    27. Vrbka, J. (2016), Predicting future GDP development by means of artificial intelligence, Littera Scripta, 9(3), 154-167.