Accurate prediction of the short time series with highly irregular behavior is a challenging task found in many areas of modern science. Such data fluctuations are not systematic and hardly predictable. In recent years, artificial neural networks have widely been exploited for those purposes. Although it is possible to model nonlinear behavior of short time series by using ANNs, very often they are not able to handle all events equally well. Therefore, alternative approaches have to be applied. In this study, a new, concurrent, performancebased methodology that combines best ANN topologies in order to decrease the forecasting errors and increase the forecasting certainty is proposed. The proposed approach is verified on three different data sets: the Serbian Gross National Income time series, the municipal traffic flow for a particular observation point, and the daily electric load consumption time series. It is shown that the method can significantly increase the forecasting accuracy of the individual networks, regardless of their topologies, which makes the methodology more applicable. For quantitative comparison of the accuracy of the proposed methodology with that of similar methodologies, a series of additional forecasting experiments that include a stateoftheart ARIMA modelling and a combination of ANN and linear regression forecasting have been conducted.
Prediction is a process that uses data from the present and the past in order to estimate future. The result of this process is the information about probable events in the future and their effects and outcomes. Making good forecasts is essential for making good decisions and planning in all areas of life. Although it does not have to reduce uncertainties and difficulties of the future, it can increase the certainty and the level of the preparedness for challenges and environmental changes that future events bring.
The need for development of prediction methods occurs in almost every area of life—technology, engineering, industry, science, politics, economy, business, sport, medicine, etc. Good forecasts can ensure lower cost of the services and products, increased customer/client satisfaction, and significant competitive advantage [
Every daily activity begins with planning. The planning begins with a prediction [
Prediction methods described in the literature can be roughly categorized into two large groups: traditional and modern. Each of them has advantages and disadvantages. None of them is superior to all others if we consider all possible criteria of evaluation [
Despite numerous ways to predict the future mathematically, there are many cases of variables that could not reliably be predicted. Causes for this limitation could be found in the randomness of the events and the lack of significant relationship in data. When factors considered during forecasting of a certain variable are not well known or understood, prediction becomes imprecise or mistaken. Sometimes, there is simply not enough data about everything that affects the forecasted variable. The prediction process relies on some specific hypothesis. If they are set wrong, due to bad judgment, i.e., human error, the prediction will be mistaken. Although the forecasting is based on past events, no one can guarantee that the history will repeat every time in the same way. Therefore, forecasts are subject to human error.
A time series can be defined as a sequence of numerical data occurring in regular intervals over a period of time, collected in a successive order. Short time series are characterized by a lack of trend information, randomness and periodicity, and demands for such forecasting represent a challenging problem [
The usual step in development of forecasting ANN is to train many networks, while changing the number of neurons in some particular layer. The ANN with the most accurate forecasting wins. Nevertheless, if we observe the forecasts of all obtained networks, we can conclude that sometimes different networks predict different directions of the trend change in the next forecasting step. In this point, one cannot determine which one is the correct. This is particularly noticeable when dealing with volatile data series. Therefore, incorporating more than one network in the forecasting decision could make better predictions of the future events. The methodology that is suggested in this paper improves the forecasting accuracy of the ANN in the sense that it concurrently exploits several most accurate networks instead of the winner one. In this way, the forecasting accuracy can be significantly improved, as well as the confidence of the prediction. The performance of the proposed method is verified on an example of Serbian Gross National Income (GNI) data series, using Feed Forward Accommodated for Prediction (FFAP) neural networks’ topology. The results demonstrate higher forecasting accuracy compared to individual FFAP networks.
The rest of the paper is organized in the following manner. In Section
In general, neural networkbased computational and forecasting methods developed from the desire to reveal, realize, and emulate the capability of the brain to process information [
Consider a time series denoted by
In the past decades, ANNs have been developed as a tool that has great capabilities for recognizing and modelling data patterns that are not easily identifiable by traditional methods. However, one may notice a common feature in all existing ANN applications in forecasting. It is the necessity for a relatively long time series in order to achieve high accuracy. Usually, there should be at least 50 data points to consider [
General structure of a feed forward neural network is illustrated in Figure
Basic fully connected feed forward neural network (one hidden layer and multiple outputs in the output layer).
In the case of short time series prediction problem, a set of observables (samples) is given (per time instant) meaning that only one input signal is available, the discretized time [
Having in mind that
Exploiting the basic topology shown in Figure
FFAP—feed forward accommodated for prediction ANN structure.
The FFAP architecture is depicted in Figure
We can express the functionality of the network as
Our task here was to do onestepahead prediction. Using the already predicted value as input data for multistepahead prediction leads to accumulation of the prediction error what we demonstrated in [
In this way, an efficient network topology is created, that uses in the test phase, only matrix
The methodology for increasing the accuracy and certainty of short time series ANN forecasting that we are suggesting is depicted in Figure
Algorithm for obtaining highaccuracy predictions.
We start with a turbulent short time series, expressing a certain variable for a given period of time. In order to facilitate their learning, these data sets have to be properly adjusted (shifted and/or normalized). After that, we evaluate the effect of changing the number of neurons in the hidden layer of the FFAP on the forecast accuracy. We begin with generating and training the FFAP network that has 3 neurons in the hidden layer. The experiment repeats with increasing number of hidden neurons. The construction of networks ends when FFAP with 10 neurons in the hidden layer is obtained and trained. This gives 8 different neural networks.
When a single network is trained, it requires a minimal learning set of first 13 samples (in our case, this is shown to be enough) in order to create the first forecast at the output, that is the extrapolated value of the trend curve. Since this is still training, one can also calculate the first forecasting error for the particular network. By entering every further sample, the network can better learn and predict the trend and its change for the input variable. As already mentioned, this process repeats for the entire group of networks. At the end of this process, all networks are trained with the entire time series, representing the past and the present. The result of this process can be systematized in a form of a forecasting matrix, with columns representing the number of neurons in a particular network, while rows represent time instances for which the forecast were made. The number of columns goes from three to ten, while number of rows is equal to the number of input samples.
In the next step, we analyze the forecasting matrix and search for three topologies that have the best performance measures. The reason for choosing three topologies lies in the fact that the time series can be irregular. Two best networks can predict totally opposite trend change for the next predicting step.
Because no one can tell which one is more correct, we introduce the forecast of the third best network to perform a kind of arbitration. In this way, the confidence of the prediction as well as its accuracy can be improved. In this analysis, we calculate mean average error (MAE), mean square error (MSE), rootmeansquared error (RMSE), and mean absolute percentage error (MAPE), for each column, using equations (
The process ends by adopting the final prediction on a test set data calculated using three best topologies simultaneously. Four cases of concurrent forecasting were investigated. First, we simply calculate an average of the forecasts for three most accurate networks. Namely, if the two of three predictions are supporting each other in forecasting the future trend, they make the largest impact to the final prediction, decreasing the importance of the third that predicts the opposite change of the trend. In that way, none may be qualified as the better one. So, the average is the best representative.
Nevertheless, from our experience, creating a linear combination of three best forecasts, where weights or a multiplying factor for each network, corresponds to a reciprocal of its forecasting error, can further improve the forecasting accuracy. The one that was the closest to the correct value gets the largest weight in the equation for the final forecast, as shown in the following equations:
Gross National Income (GNI) is defined as an estimate of the income from goods and services produced by an economy and received by a country both domestically and from abroad. This measure of the size of an economy is highly important and has large economic, political, and social implications. For politically and economically turbulent developing countries, it is very difficult to predict it due to mutual influence of many known or unknown factors. The methodology that can be considered for such predictions exploits artificial neural networks. This kind of time series appears superb for verifying the methodology that we propose. In this study, we will try enhancing the neural network approaches, described in Section
For the analyzed series, time is discretized at year long periods and reduced by 1989, as given by (
Although this time series covers a long period of time (1990–2017), the GNI is the economic variable that is obtained annually. It was first introduced by economic scientists in 1990s. Because of that, this time series can be considered as a very short. The time series data are obtained from the World Bank National Accounts data and OECD National Accounts data files.
After the initial training of 8 networks using the test set that contains GNI data for a period 1990–2012, we have calculated their performance measures, i.e., MAE, RMSE, and MSE. Three most accurate FFAP networks have 7, 9, and 10 neurons in the hidden layer. Corresponding weights were calculated based on the values of the networks’ errors in order to be used in combined predictions of the test set (2013–2017). Values are listed in Table
Forecasting weight coefficients based on the networks’ training set accuracy in GNI forecasting.
Weight coefficients  7 neurons, 1^{st} most accurate  9 neurons, 2^{nd} most accurate  10 neurons, 3^{rd} most accurate 

Averaged  33.33  33.33  33.33 
MAEbased  40.63  31.03  28.34 
RMSEbased  51.26  28.83  19.91 
MSEbased  42.14  31.60  26.26 
Combined weights of three best networks predicting GNI, corresponding to particular error types.
Table
GNI ANN forecasting for three most accurate FFAPs.
Year  True value  7 neurons  9 neurons  10 neurons 

2013  43.64  56.45  30.17  15.42 
2014  42.42  49.20  50.12  28.56 
2015  35.33  38.33  42.30  34.45 
2016  36.06  36.97  37.33  30.28 
2017  38.55  34.73  55.25  35.57 
GNI forecasting of three most accurate FFAP ANNs.
Combined predictions of the GNI: true, averaged, MAEweighted, RMSEweighted, and MSEweighted values.
As a final measure of performances, for the proposed methodology, we have calculated MAE, MSE, RMSE, and MAPE for all GNI predictions of the test set for the period 2013–2017, and these are shown in Table
GNI forecasting errors of different ANN approaches.
Error type  7 neurons FFAP  9 neurons FFAP  10 neurons FFAP  Aver. FFAP  MAE weighted  RMSE weighted  MSE weighted 

MAE  5.47  9.22  10.34  3.47  3.00  2.95  2.36 
RMSE  6.85  10.68  14.36  4.78  3.70  3.20  2.60 
MSE  46.92  114.02  206.23  22.84  13.68  12.27  6.77 
MAPE  13.26  23.12  24.71  8.60  7.47  7.39  5.99 
An economic variable that is very similar to the GNI is the GNI per capita. It is an economy measure that is obtained when the value of the GNI is divided by the number of residents in a country. It should be emphasized that values for the GNI and the GNI per capita show very similar trends for particular time interval in the case of the Republic of Serbia. This could be explained by the fact that the natality for the Republic of Serbia is varying very slow over time. The authors have performed the similar forecasting procedure on a GNI per capita time series for the same time period, and it has shown very similar forecasting performance measures.
To the best of our knowledge, analyzing GNI time series in the case of Serbian economy is the task that has not been studied or published yet, and no comparative analysis of the forecasting accuracies with some alternative forecasting methods can be conducted. To overcome this problem alternative stateoftheart, i.e., autoregressive integrated moving average forecasting methodology (ARIMA) was performed on the same data set. The theoretical background on this topic as well as its implementation strategies can be found in the literature [
The model fitting process for GNI training set will be very briefly explained here, since this task is out of the scope of this study. To achieve the adequate ARIMA(
In forecasting the Serbian GNI over the period of five years (2013–2017), using the obtained ARIMA(0, 1, 1) model, values for MAE, RMSE, MSE, and MAPE were 5.98, 6.52, 42.51, and 14.92, respectively.
We believed that it would be interesting to extend existing experiments with additional predictions that combine ANN weights with those obtained if an appropriate weight for linear regression of the observed time series is added into a final linear combination. We have constructed a new forecasting system that exploits the performance measures of both most accurate individual ANN topologies with most accurate linear regression model. According to their performances, new weights have been calculated in order to obtain modified forecasting linear combinations of four terms. The newly obtained forecasting results are shown in Figure
Combined ANN and linear regression models for predictions of the GNI: true, MAEweighted, RMSEweighted, and MSEweighted values.
Corresponding performance measures in GNI forecasting in the case of linearly combined best ANN topologies and their extensions with linear regression models, outperformed the traditional ARIMA modelling. This is systematized in Table
GNI forecasting performances of different approaches.
Method  Error type  

MAE  RMSE  MSE  MAPE  
Single ANN  5.47  6.85  46.92  13.26 


Combined ANN  
Averaged  3.47  4.78  22.84  8.60 
MAE weighted  3.00  3.70  13.68  7.47 
RMSE weighted  2.95  3.20  12.27  7.39 
MSE weighted  2.36  2.60  6.77  5.99 


Combined ANNLIN.REG.  
MAE weighted  2.71  3.64  13.30  6.78 
RMSE weighted  2.62  3.40  11.54  6.60 
MSE weighted  2.24  2.51  6.29  5.73 


ARIMA  13.26  23.12  24.71  8.60 
We define the traffic flow as the number of vehicles that pass a particular observation point per unit of time (usually 15 minutes). There can be various ways to “measure” the volume of the traffic and various sources of data such as simulations, sensors, taxi GPS, floating cars, and similar. In shortterm prediction, which is our goal, the traffic is predicted in the next moments (usually 15 minutes) on the basis of realtime historical data.
The traffic flow time series consists of about 70 measurements, collected in collaboration with the Faculty of Transport and Traffic Engineering of the University of Belgrade, for one of the Belgrade’s busiest roundabouts. This time series can also be considered as short. Here, we assume that the traffic from the immediate past has the greatest impact on the future value which, in turn, is produced for a moment in the near future. In that sense, a large series of consecutive values, we believe, can obscure the information needed for prediction. That stands especially for specific days such as state holidays (very low urban traffic) or football matches (very high urban traffic) for which the information older than several hours (at most 24 or so) has no significance. This is the reason for shortening the initial time series to 70 samples of interest.
Traffic data are accommodated and constant M (here, 140) selected so as to reduce the relative difference between output values. Although this transformation creates negative values in the training set, it is also the way to make the training process numerically better conditioned.
The initial training of 8 networks using the test set was performed. We have calculated their performance measures, i.e., MAE, RMSE, and MSE. Three most accurate FFAP networks have 5, 3, and 4 neurons in the hidden layer. Based on the values of the networks’ errors, corresponding weights were calculated for the combined predictions of the test set. Combined weights of three best networks, corresponding to particular error types, are shown in Table
Forecasting traffic flow weight coefficients based on the networks’ training set accuracy.
Weights coefficients  5 neurons, 1^{st} most accurate  3 neurons, 2^{nd} most accurate  4 neurons, 3^{rd} most accurate 

Averaged  33.33  33.33  33.33 
MAE based  35.34  33.73  30.93 
RMSE based  39.07  34.33  26.60 
MSE based  44.73  34.54  20.73 
Combined weights of three best networks predicting traffic flow, corresponding to particular error types.
Table
Traffic ANN forecasting for three most accurate FFAPs.
Sample number  True value  3 neurons  4 neurons  5 neurons 

1  12.00  13.20  16.27  −23.84 
2  3.67  –26.15  9.22  1.16 
3  2.67  26.61  –34.38  −25.58 
4  –5.33  –92.84  –13.90  −34.12 
5  –7.00  –2.21  26.67  −40.55 
6  –12.67  –14.42  –7.51  −37.49 
7  –11.67  –18.03  –25.44  –17.15 
8  –19.67  –17.49  –22.77  3.46 
9  –30.00  –59.27  –72.45  –55.44 
10  –40.33  –94.37  –94.53  –49.01 
11  –42.33  –37.72  –91.69  –33.72 
Traffic flow forecasting of three most accurate ANNs.
Predictions of the traffic density: true, averaged, MAEweighted, RMSEweighted, and MSEweighted values.
We have again calculated errors (MAE, MSE, RMSE, and MAPE), i.e., performance measures of most accurate networks and of the concurrent, performancebased predictions for the next 11 time instances, and these are shown in Table
Traffic flow forecasting errors of different ANN approaches.
Error type  5 neurons FFAP  4 neurons FFAP  3 neurons FFAP  Aver. FFAP  MAE weighted  RMSE weighted  MSE weighted 

MAE  20.46  23.38  22.32  16.60  16.53  16.44  18.17 
RMSE  23.39  30.20  34.37  21.32  21.02  20.54  22.31 
MSE  547.34  911.99  1181.01  454.45  441.86  421.84  497.88 
MAPE  266.59  253.19  341.13  186.23  186.41  204.37  187.39 
We have again performed an ARIMA fitting process in order to get the best forecasting model. The ARIMA(8,1,2) model appeared to have minimal values of AIC and BIC.
In forecasting the municipal traffic flow for 11 samples from the test set with 95% confidence limits, using ARIMA(8,1,2) model, values obtained for MAE, RMSE, MSE, and MAPE were 47.20, 24.00, 575.83, and 389.60, respectively. Corresponding performance measures in the case of linearly combined best ANN topologies and their extensions with linear regression model performed better than the traditional ARIMA modelling, considering different types of forecasting errors. This is systematized in Table
Traffic flow forecasting performances of different approaches.
Method  Error type  

MAE  RMSE  MSE  MAPE  
Single ANN  20.46  23.39  547.34  266.59 


Combined ANN  
Averaged  16.60  21.32  454.45  186.23 
MAE weighted  16.53  21.02  441.86  186.41 
RMSE weighted  16.44  20.54  421.84  204.37 
MSE weighted  18.17  22.31  497.88  187.39 


Combined ANNLIN.REG.  
MAE weighted  10.21  13.00  168.67  102.84 
RMSE weighted  10.22  12.39  153.41  98.40 
MSE weighted  9.67  11.35  128.84  105.95 
ARIMA  47.20  24.00  575.83  389.60 
Combined ANN and linear regression models for predictions of traffic flow: true, MAEweighted, RMSEweighted, and MSEweighted values.
We define values of the electric load consumption as an average power consumed (in kWh) for a period of one day, at a particular suburban measuring point. Data for the implementation of the method are acquired from the EUNITE 2001 competition file [
Using the training set data, 8 FFAP ANNs were trained with rising number of hidden neurons. The best performance measures are obtained for ANNs with 3, 5, and 7 neurons in the hidden layer. Combined weights of three best networks, corresponding to particular error types, are listed in Table
Electric load consumption forecasting weight coefficients based on the networks’ training set accuracy.
Weights coefficients  3 neurons, 1^{st} most accurate  5 neurons, 2^{nd} most accurate  7 neurons, 3^{rd} most accurate 

Averaged  33.33  33.33  33.33 
MAE based  34.12  33.99  31.89 
RMSE based  35.13  33.91  30.95 
MSE based  36.93  34.41  28.66 
Combined weights of three best networks predicting electric load consumption, corresponding to particular error types.
Table
Electric load consumption ANN forecasting for three most accurate FFAPs.
Sample number  True value  3 neurons  5 neurons  7 neurons 

1  131.54  149.29  138.41  146.25 
2  107.41  136.26  145.27  143.92 
3  65.44  100.88  97.84  104.24 
4  115.08  63.43  78.41  80.12 
5  118.29  24.84  29.32  82.51 
6  122.52  69.39  54.85  206.74 
7  92.98  109.1  92.36  91.51 
8  51.89  88.85  141.45  145.32 
9  50.12  51.37  40.14  9.38 
10  54.35  91.64  57.9  78.68 
11  88.39  81.68  59.03  88.35 
Electric load consumption forecasting of three most accurate ANNs.
Predictions of the electric load consumption: true, averaged, MAEweighted, RMSEweighted, and MSEweighted values.
For this forecasting process, we have introduced additional accuracy metric, i.e., maximal error of estimate
Calculated errors (MAE, MSE, RMSE, MAPE, and maximal error of estimate) for prediction of the test set with 11 time instances are shown in Table
Electric load consumption forecasting errors of different ANN approaches.
Error type  3 neurons FFAP  7 neurons FFAP  5 neurons FFAP  Aver. FFAP  MAE weighted  RMSE weighted  MSE weighted 

MAE  34.42  36.82  36.68  30.67  30.55  30.53  29.72 
RMSE  42.31  46.35  48.02  38.18  38.05  38.00  37.28 
MSE  1789.95  2148.44  2305.90  1457.48  1448.13  1443.75  1389.59 
MAPE  39.00  49.23  44.11  39.66  39.57  38.43  39.52 
We have now performed a seasonal ARIMA fitting process in order to get the best forecasting model. The SARIMA(2, 0, 2) (1, 1, 1)_{7} model had minimal BIC value.
In forecasting, the municipal traffic flow for 11 samples from the test set with 95% confidence limits, using SARIMA(2, 0, 2) (1, 1, 1)_{7} model, values were obtained for MAE, RMSE, MSE, and MAPE, and maximal error of estimate were 28.27, 35.13, 1234.13, 42.38, and 85.18, respectively. Corresponding performance measures in the case of linearly combined best ANN topologies and their extensions with linear regression model did not outperform the traditional SARIMA modelling, considering different types of forecasting errors. This is systematized in Table
Electric load consumption forecasting performances of different approaches.
Method  Error type  

MAE  RMSE  MSE  MAPE  Maximal error  
Single ANN  34.42  42.31  1789.95  39.00  93.45 


Combined ANN  
Averaged  30.67  38.18  1457.48  39.66  73.15 
MAE weighted  30.55  38.05  1448.13  39.57  72.93 
RMSE weighted  30.53  38.00  1443.75  38.43  72.85 
MSE weighted  29.72  37.28  1389.59  39.52  72.49 


Combined ANNLIN.REG.  
MAE weighted  35.49  41.59  1729.46  50.78  87.10 
RMSE weighted  35.83  41.94  1759.12  51.42  87.45 
MSE weighted  37.58  43.49  1897.06  54.26  87.63 


SARIMA  28.27  35.13  1234.13  42.38  85.18 
Combined ANN and linear regression models for predictions of daily electric load consumption: true, MAEweighted, RMSEweighted, and MSEweighted values.
Results obtained using neural networks in predicting the GNI for the Republic of Serbia have multiple qualities. Specifically, the use of mathematical methods and models in the prediction of future economic trends in the Balkan region at the Southeast Euro zone, encompassing the Republic of Serbia, is very ungrateful. GNI is particularly vulnerable to turbulent changes and numerous of noneconomic factors to a great extent. Observing different types of forecasting performance measures in suggested ANN methodology just confirm the quality of the achieved results.
The results in predicting the movement of the GNI are of extreme importance for candidates for full membership in the European Union, such as Serbia. The size of the GNI determines the obligations of membership in this integration as well as the benefits that can be granted from numerous funds. The EU budget is financed by its own system of resources whose amount is limited to 1.23% of EU GNI (for the period 2014–2020). These funds are to be filled from the budged of each member country with the amount of 0.73% from its own GNI value. With the amount of about 80 billion Euros per year, these resources represent the largest source of budget revenue, reaching 69% of all revenues. On the other hand, each member can expect from the EU budget the maximal amount of 4% of its national GNI. In some situations, a small difference in the amount of the GNI can significantly change the ratio of liabilities and benefits of new members. Future GNI values are also important for creating a budget revenues and expenditures, due to restrictions in monetary and fiscal policy. Finally, the design of GNI forecasting system is crucial for making decisions on large investment projects because one gets a realistic picture of the capacity of the national economy. This reduces risks of illiquidity and even insolvency.
On the other hand, with the intention to preserve sustainable future, the importance of prediction of local traffic in large cities comes in for many reasons such as environmental and pollution monitoring; fuel usage reduction; journey planning; traffic control; urban planning; realtime route guidance; and ITS (intelligent transport system). In this analysis, ANNbased forecasting systems were developed enabling prediction of travel times, travel speeds, and traffic volumes on transportation networks using historic and realtime data.
At the end, we can also conclude that electric power load forecasting is the foundation of planning, development, and the assurance of operation efficiency and reliability of electric power systems. Because of the inherent characteristics of uncertainty, randomness, and nonlinearity, the load forecast has always been a forefront and hot issue. In the case of this forecasting task, we have achieved results using different types of forecasting performance measures in treated ANN models in order to confirm their quality.
In this paper, a novel methodology for increasing the predictions accuracy of different ANNbased systems has been suggested. Throughout analysis of three different time series of important everyday parameters, we have introduced some efficient improvements for prediction of short time series. The proposed method has been verified on GNI forecasting at national economy level, municipal traffic volume forecasting, and suburban daily electric load consumption forecasting. ANNbased models have been trained, and the performance of the models has been analyzed by applying various performance evaluation criteria and statistical tests that included MAE, RMSE, MSE, MAPE, and maximal error of estimate. Based on their accuracy, best performing ANN topologies, considering number of the neurons in the hidden layer, have been selected and implemented into a new forecasting system that linearly combines the forecasts of most accurate individual networks. More accurate one has been assigned a greater weight value in the linear combination. The same forecasting accuracy tests have been repeated for a test set data in order to conclude which model is superior. We have concluded that results of these three case studies reveal that linear combination of three most accurate ANN forecasts could predict trend of the future changes more accurately and with more confidence and that in most cases outperforms individual ANN forecasts, ARIMA forecasts, and hybrid ANNlinear regression forecasts. Moreover, we have determined the that accuracy improvement in these three particular cases ranges from 13% in the case of the electric load prediction and up to 65% in the case of Serbian GNI prediction. Based on the analysis presented in the paper, we can anticipate that the applicability of the method can be extended to other AI forecasting and modelling methods, as well as different types of ANN topologies. Our future research would also be oriented toward further error reductions and the discussion on the smallest number of the neurons in ANN layers in order to achieve this goal.
Data representing the Serbian GNI time series used to support this study are obtained from World Bank National Accounts data and OECD National Accounts data files. Data sets representing daily electric load consumption are obtained from the EUNITE 2001 competition file and are cited at relevant places within the text as references [
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was partly funded by the Ministry of Education and Science of Republic of Serbia under contract no. TR32004.