In this paper the intelligent methods for solving the problem of short- and middle-term forecasting electricity sales to network for short- and middle-term forecasting was performed. Experimental studies of hybrid Problems of forecasting non-stationary time series and market indexes at stock exchanges pay great attention of managers of enterprises and various scientific researchers. For its solution were developed and applied for a long time powerful statistical methods, first of all ARIMA [1, 2]. In recent years, various intelligent methods, and technologies, such as fuzzy logic systems and neural networks, have also been proposed and widely used for forecasting in economics and technology.
hybrid
ARIMA for short- and middle-term forecasting have been conducted. The accuracy of the obtained forecasts was compared. The expediency of applying the researched methods of artificial intelligence for the considered intervals is substantiated.
2023 Copyright for this paper by its authors. overall training time. Based on the studied two classes of hybrid DL networks, their effectiveness for forecasting in the financial sphere was compared.
In the previous papers these methods were applied and investigated in the problem of forecasting in financial sphere for market indices Dow Jones industrial average and NASDAQ. That is why it is interesting to investigate the effectiveness of ARIMA, GMDH and hybrid DL networks in forecasting in other areas, such as technology and production, specifically in short- and middle-term forecasting tasks. The goal of this paper is to investigate the accuracy of intelligent methods – hybrid DL networks, GMDH and ARIMA at the problem of forecasting Electricity Sales to Ultimate Customers, Residential (USA), June 7, 2023 – at the different forecasting intervals (short-term and middle-term), compare their efficiency and to determine which computational intelligence methods are the most perspective for forecasting in the economy and technology.
The architecture of the evolving hybrid DL-network is shown in Fig. 1. The input of the system accepts an ( × 1)-dimensional vector of signals that are considered input. Then the first hidden layer receives this signal. At this level there are 1 = 2 nodes, each of which has strictly two inputs.
Outputs [
It selects among the output signals ̂[
[
As a result, due to the GMDH algorithm, it is possible to obtain a well-trained network with an
optimal structure that was synthesized automatically. High-dimensionality problems, as well as
vanishing or exploding gradients, are avoided because the learning is sequential layer-by-layer.
[
In Fig. 2 shows the architecture of the node selected as the quality for the proposed GMDH
system. This is a neo-fuzzy neuron (NFN) proposed by Takeshi Yamakawa et al. in [
In the case of a predefined dataset, the training process can be performed in a single epoch in batch
mode. For this purpose, the conventional least squares method is used [
The standard local quadratic error function is used as the goal function (i.e., the learning criterion):
( )=
( ( )− ̂( )) = ( )2 =
( ( )− ∑
∑ ( ( )))
non-linear system that has one output and several inputs. In the proposed GMDH system,
neo-fuzzy-neurons with only two inputs are used, which implements the following mapping:
of
where ̂ is the output of the system, is the input signal ( = 1,2, … , ). The nonlinear synapses
are the building blocks of a neo-fuzzy neuron. Their task is to convert the input signal in the form
[
∑ [
1 2 =1 2 1 2 1 2 + =1 2 ℎ =1 =1
2 =1 (1) (2) (3) (4) 3. Data set , (5) ( − 1)( ( )− ( ( − 1)) ( ( ))) ( ( ))
1 + ( ( ( ))) ( − 1) ( ( )) ( − 1) ( ( ))( ( ( ))) ( − 1)
1 + ( ( ( ))) ( − 1) ( ( ))
.
With the sequential receipt of training observations, i.e., in the online mode, the recurrent form of the ANN can be represented as
As the data set for forecasting were taken monthly Electricity Sales to Ultimate Customers, Residential (USA) since 01-2002 till 01-2023 taken. The whole sample consisted of 251 instances. The sample was divided into training and test subsamples. The dynamics of monthly energy power supply to Ultimate Customers, Residential (USA) is shown in the Fig. 3. preceding and conceding values and the process is periodical with period 6 months. Autocorrelation function (ACF) was determined for this process of power supply which is shown in the Fig. 5. The check for stationarity of this process was performed using Dickey-Fuller test.
P-value: 0.5117527467140699 > 0.05. As it follows from this test the initial time series is not stationary. Using differencing this time series was transformed to the stationary one that’s confirmed by Dickey-Fuller test with P-value: 1.3594288749888985e-14 < 0.05.
In the investigations was explored the forecasting accuracy of hybrid DL neo-fuzzy networks at various forecasting intervals: short-term forecasting with intervals 1, 3 5 and 7 days and middle term forecasting with intervals 10 and 20 days. At the first step the variable experimental parameters of hybrid network were chosen which are presented in the Table 1.
The optimization of these parameters was performed, in result the following optimal values were determined inputs: 3; linguistic variables: 3; ratio: 0,7.
After that the structure optimization of hybrid DL neo-fuzzy network was constructed using GMDH method. The process of structure generation is presented in the Table 2.
In result the optimal structure of three layers was determined: at the first layer 3 inputs, second layer – two neurons, third layer – one output neuron.
Further the training of the best hybrid network was carried out using method SGD (stochastic gradient descent with variable step. Flow chart of forecasting results for interval 1day in presented in the Fig. 6. The values of MSE and MAPE for this experiment are shown in the Table 3.
In the Fig. 6. flow chart of MAPE values for the best model of hybrid network is shown.
Further the similar experiments of hybrid network were performed with forecasting interval 3, 5, 10 and 20 days. After optimization the parameters and structure of hybrid network it was trained using training subsample. The forecasting accuracy at the test sample for interval 3 days is presented
SB 1 0.138 0.23 0.116 at the Table 4. In the succeeding experiments forecasting accuracy of Hybrid neo-fuzzy network was investigated with forecasting intervals 5, 10 and 20 days.
In the Table 5 accuracy of forecasting of the hybrid NFN optimal structure is presented with forecasting interval 10 days and in the table 6 with forecasting interval 20 days.
For estimating forecasting accuracy of hybrid DL network, it was compared with alternative methods: ARIMA and GMDH. The forecasting accuracy of GMDH for interval 1 day is shown in the Table 7 and for 5 days in the Table 8. The flowchart of forecasting results for the interval 5 days is shown in the Fig. 7 and for 20 days in the Fig. 8.
In the next experiments forecasting efficiency of method ARIMA was investigated and analyzed. After the preliminary investigations the optimal parameters for ARIMA were found which were used in the following experiments. The forecasting accuracy of ARIMA for interval 1 day is presented in the Table 9 and for interval 5 days in the Table 10. The flowchart of real and forecasting results for ARIMA with interval 20 days is shown in the Fig. 9.
The comparative experiments were performed in which the accuracy of forecasting by hybrid DL network, GMDH and ARIMA at the different forecasting intervals was estimated and compared. The corresponding results are presented in the Tables 11, 12 and Fig. 10, 11.
Analyzing the presented results in the Fig. 10 and 11 one may conclude that GMDH method appears to be the best at short term forecasting 1, 3 days which complies the theory.
Hybrid deep learning neo-fuzzy networks are the best at middle-term forecasting 5, 7, 10, 20 days. ARIMA appeared to be the worst by accuracy as compared with intelligent methods – hybrid DL networks and GMDH.
In this paper the investigations of artificial intelligence methods: hybrid Deep learning networks and GMDH and ARIMA were carried out in the problem of forecasting Electricity Sales to Ultimate Customers, Residential (USA) since 01-2002 till 01-2023.
During the experiments the optimal structure and optimal parameters: number of inputs, number of linguistic values, ratio training/test samples of hybrid neo-fuzzy networks were determined.
After optimization of hybrid neo-fuzzy networks and parameters of GMDH method the experiments on forecasting Electricity Sales to Ultimate Customers, were performed at different intervals: 1, 3, 5, 7 (short term forecast) and 10, 20 days (middle term forecast).
The accuracy of forecasting by Hybrid DL networks was compared with alternative methods – GMDH and ARIMA.
The analysis of obtained results have shown that GMDH method is the best at short term forecasting 1, 3 days while hybrid deep learning neo-fuzzy networks are the best at middle-term forecasting 7, 10, 20 days. Method ARIMA appeared to be the worst by accuracy as compared with intelligent method – hybrid DL networks and GMDH.