Mathematical Problems in Engineering
Volume 2021 (2021), Article ID 5573170, 16 pages
https://doi.org/10.1155/2021/5573170

Research Article

An Adaptive Control Combination Forecasting Method for Time Series Data

Hongyan Jiang,1,2 Dianjun Fang,1,3 and Xinyan Zhang¹

¹School of Mechanical Engineering, Tongji University, Shanghai 201804, China
²School of Mechatronic and Power Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China
³Sino-German Institute of Intelligent Technologies, Qingdao 266000, China

Correspondence should be addressed to Dianjun Fang; 1733316@tongji.edu.cn

Received 19 January 2021; Revised 6 May 2021; Accepted 19 May 2021; Published 3 June 2021

Academic Editor: Samuel N. Jator

Copyright © 2021 Hongyan Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

According to the individual forecasting methods, an adaptive control combination forecasting (ACCF) method with adaptive weighting coefficients was proposed for short-term prediction of the time series data. The US population dataset, the American electric power dataset, and the vibration signal dataset in a hydraulic test rig were separately tested by using ACCF method, and then, the accuracy analysis of ACCF method was carried out in the study. The results showed that, in contrast to individual methods or combination methods, the proposed ACCF method was adaptive to adopt one or some of prediction methods and showed satisfactory forecasting results due to flexible adaptability and a high accuracy. It was also concluded that the higher the noise ratio of the tested datasets, the lower the prediction accuracy of the ACCF method; the ACCF method demonstrated a better prediction trend with good volatility and following quality under noisy data, as compared with other methods.

1. Introduction

A time series is a set of statistics and usually collected at regular intervals. Time series data occur naturally in many application areas, such as economics, medicine, weather data, ocean engineering, finance, and engineering control. Time series data are obtained by the sensors, and they refer to the large, diverse datasets of information that cannot be easily processed by using standard computers. Based on the past performance, time series forecasting is an analysis used to forecast future value, which is still a challenging research topic nowadays [1–3].

The three common methods for time series forecasting included physical, statistical, and artificial intelligence. In physical methods, effective forecasting results rely on physical information [4, 5], but it was not proficient in dealing with short-term series with complex calculation process. Statistical models, including Period-Sequential Index (PSI) [6], moving average (MA) [7], autoregressive integrated moving average (ARIMA) [8], exponential smoothing [9], Kalman filter [10], and grey forecasting [11], effectively tackled linear features but gave larger error for a fluctuant, seasonal one [12], noise, or instability [13]. Artificial intelligence models, subsuming BP neural network (BP-NET) [14, 15], support vector machines (SVM) [16], fuzzy logic models [17], and least square support vector machine (LSSVM) [18], have exhibited significant advantages in dealing with nonlinear problems. These artificial intelligence models offered higher forecasting accuracy than physical or statistical models, but their prediction was mostly relying on training datasets, and they are easy to get stuck or suffer from overfitting in the local optima [19, 20].

Because of the inherent disadvantages of each model, nowadays, the effective information of multiple models has been used to predict time series, and weight problem of combination model is becoming the research focus. Weights could be allocated to the various forecasts produced by individual models, so as to achieve a combined forecast [21, 22]. For example, Clark et al. [23] derived mean square error-minimizing weights for combining the restricted and unrestricted forecasts and assigned more weights on the restricted model and less weights on the unrestricted model. However, Clark’s method supported the conventional wisdom that simple averages were hard to beat and applied only to averaging to nested models. Hao et al. [24] introduced entropy weight method into the combination prediction model. Xie et al. [25] combined linear regression prediction model and grey model to get a various weight combination model. The two methods have better forecasting accuracy in only Dam’s settlement or landslip, but weights were assigned to all participating single models, and forecast accuracy in other time series cases was unwarrantable or unknown. Gao et al. [26] established five kinds of combination forecasting models, including suboptimal weight, optimal weight, grey comprehensive correlation degree weight, entropy weight, and neural network. In Gao’s study, the weighted constraint criterion was given particular attention; nevertheless, negative weight of single item prediction model might occur in the combination model. Song and Fu [27] have also found out that some models failed to combine the advantages of single models, such as the combination of the autoregressive integrated moving average model and neural network model, or the combination of neural network and other forecasting models.

In my opinion, the main problem of the above combination methods is that the statistical distribution information of the forecasting errors with the historical time is not paid more attention or is ignored, leading to unreasonable weight distribution and even negative weights. Therefore, existing combined forecast models were still lacking the predicted reliability, particularly under a condition of noise. In the study, weighting coefficient for each model was adaptively determined based on their own statistical forecasting performance for historical data. The rest of the paper is organized as follows. Section 2 contains the methodology of combination forecasting method. Section 3 contains the steps of computation. Section 4 contains the results and discussion of short-term prediction cases. Section 5 contains the conclusions.

2. Methodology

2.1. The Forecasting Methods

2.1.1. Individual Methods

(1)Period-Sequential Index Method The Period-Sequential Index (PSI) method [6] had been studied by the author in the previous work. The PSI model introduced the period index (PI) and sequential index (SI) to describe the dataset structure information in vertical and horizontal dimensions, respectively. Figure 1 shows the schematic diagram of the PSI algorithm. H₋₂, H₋₁ denote reference historical periods, i.e., the year before last year and last year. H₀ represents the forecasting period. The period for H₋₂, H₋₁ and H₀ is uniform, defined as T. When time ti is upcoming, the forecasting value at time of ti based on PSI method is mainly dependent on the PI and the SI. The PSI method is described as follows:

 where ti (independent variable) is a forecasting time, Y₁(ti) (dependent variable) is a forecasting value at time of ti, K₀ is the reference coefficient for period index, y(ti−1) is the observed value at historical time of ti−1, α is the optimized weighing factor of PSI method, PI(ti) is the forecasting period index (dependent variable) at time of ti and SI(ti−1) is the forecasting sequential index (dependent variable) at time of ti−1.

 where y(ti−2T) and y(ti−T) describe the reference historical data at time of ti−2T and ti−T, respectively. K₋₂ and K₋₁ are reference functions of period index. A standard period average is originally set to be a reference function of period index, where it is defined as a constant. The more detailed derivation can be found in Ref. [6].(2)Exponential Smoothing Method The Exponential Smoothing (ES) method [9] is often used in practice to forecast time series. Suppose that the observed values for time series are y(t₁), y(t₂),…, y(ti−1) at time of t1,t2,…, ti−1, respectively. For ES method, the forecast value at time of ti is dependent on the observed value at time of ti−1 and the forecasting value at time of ti−1. The ES method is defined as

 where Y₂(ti) and Y₂(ti−1) represent the forecasting values at time of ti and ti−1 by using ES method, y(ti−1) is the observed value at historical time of ti−1, and β is the smoothing parameter, which can be adjusted between 0 and 1. Higher β will produce a forecast, which is more responsive to recent changes in the data, whilst also being less robust to any errors that could occur.(3)Moving Average Method The moving average (MA) method [7] is simple and widely used, and it performs well in forecasting competitions against more sophisticated approaches. A simple Moving Average is a common average of the previous n data points in time series data, and each point in the time series data is equally weighted. The MA method can be described as follows:

 where Y₃(ti) represents the forecasting values at time of ti by using MA method, n is the number of data points used in the calculation, and y(ti−n) is the observed data point value at time of ti−n.(4)Autoregressive Integrated Moving Average Method Autoregressive integrated moving average (ARIMA) [8] is one of the most popular statistical linear models for forecasting time series data. It is a combination of autoregression AR(p) (an additive linear function of p past observations), moving average MA(q) (q random errors), and d which is an integer making a series to be stationary. The general form of the forecast equation for ARIMA (p, q, d) model can be written as follows:

 where Y₄(ti) is a forecasting value at time of ti by using ARIMA method, c is the constant representing the intercept, φj and y(ti−j) are the parameters and regressors for AR part of the model, respectively, θk and ε(ti−k) are the parameters and regressors of the MA part of the model, respectively, and ε(ti) is the white noise at time ti.(5)BP Neural Network Method BP neural network (BP-NET) method [14, 15] can realize self-learning and memory functions of machine. Figure 2 gives the simple structure of BP-NET. It can be seen that BP-NET is composed of input layer, output layer, and hidden layer. Using this three-layer structure, BP can simulate any complex nonlinear relationship through nonlinear elements. The basic calculation principle is divided into three steps: forward calculation (calculate the output of each node in turn based on the input); error backpropagation (calculate the gradient of each node according to the loss function); weight update. Because this method has excellent data processing and relationship building capabilities, it has been widely used in forecasting.(6)Grey Forecasting Method Grey forecasting method (GM) [11] indicates one variable and one-order grey forecasting model. This grey differential equation is formed by an original time series y(ti) using accumulated generating operation (AGO) technique. It is denoted as follows:

 where Y₆(ti) is a forecasting value at time of ti by using GM method, a is a developing coefficient, and b is a control variable. a and b are denoted as

 where

Figure 1: The schematic diagram of the PSI algorithm.

Figure 2: The simple structure of BP-NET.

2.1.2. Combination Forecasting Methods

Because it is too risky to rely on the forecasts produced by an individual method, the combination forecasting method was widely used in the study. There is one historical piece of data y(t₁), y(t₂),..., y(tm) occurring at the corresponding time t₁, t₂,..., tm. For the same forecasting problem at time of ti, n kinds of single forecasting model can give forecasts: Y₁(ti), Y₂(ti), …, Yn(ti). The forecasting value of the j-th model (j = 1,2, …, n) and the corresponding weight coefficient are Yj(ti) and respectively, at time ti. The linear combination is generally calculated according to [28]

Considering the actual situation and the calculation complexity, three common weight methods [27], including inverse variance (IV) method, mean square error inverse (MSEI) method, and simple weighted average (SWA) method, were used to compute weight coefficients in the study. The methods are based on the sum of squared errors.

The weight coefficients of the IV method were computed in equation (10). In a case of the larger sum of squared errors in a single method, this method is assigned a smaller weight. On the contrary, a larger weight is assigned to the smaller squared errors in a single model.

where n indicates the number of single models, and ej is the j-th single model.

Similarly, the weight coefficients of the MSEI method were computed as follows:

For the SWA method, the sum of squared errors of each model was ranked by a descending ranking order. Then, a new array of jr(=1,2, …, n) could be defined and represent a ranking order of single model. This meant that one individual model with a higher value of jr would have a lower forecasting error. The weight coefficients of the SWA method were further given in equation (13). It is seen that the weight coefficient is larger in a condition of higher jr in order to minimize the sum of squared errors.

2.2. The Adaptive Control Combination Forecasting Method

In the following work, one total tested data set occurred at time t₁, t₂, …, tN−1, tN, and is defined as

y(t₁), y(t₂), …, y(ti−1), y(ti), (ti+1), …, y(tN−1), y(tN).

This total tested data set is a time series based on an equal interval of time (dt) and a fixed time period (T). For example, while dt is month, day, hour, minute, or second, the corresponding T of the tested data are year, month, day, and minute, respectively. Now, the time ti (3TtitN) is assumed to be upcoming, so the tested data set during three time periods (3T) before ti and y(ti) is extracted from the total tested data set. It is defined that the extracted tested data set contains s + 1 (s = 3T/dt) data in number and is given in detail as follows: y(ti−s), y(ti−s−1), y(ti−s−2), …, y(ti−2), y(ti−1), and y(ti). Using this extracted tested data set, each individual method can give forecasts at time of ti based on that mentioned in Section 2.1.1 and get the modeling data set: Y₁(ti), Y₂(ti), …, Yn−1(ti), Yn(ti).

Similarly, as the forecasting time ti moves back to ti, ti−1, ti−2, ……, ti−s−2, ti−s−1, and ti−s in sequence, the all extracted tested datasets can be generated, and they are listed in Table 1.

Table 1: The all extracted tested datasets.

Number
Extracted tested datasets

1
y(ti−s), y(ti−s−1), y(ti−s−2), …, y(ti−2), y(ti−1), and y(ti).

2
y(ti−s−1), y(ti−s), y(ti−s+1), …, y(ti−3), y(ti−2), and y(ti−1).

3
y(ti−s−2), y(ti−s−1), y(ti−s), …, y(ti−4), y(ti−3), and y(ti−2).

……
……

s+1
y(ti−2s), y(ti−2s+1), y(ti−2s+2), …, y(ti−s−2), y(ti−s−1), and y(ti−s).

Using each individual method, corresponding forecasts at time of ti, ti−1, ti−2, …, ti−s−1, and ti−s, are defined as the first modeling datasets, which are listed in Table 2.

Table 2: The first modeling datasets.

Number
n forecasting datasets

1
Y₁(ti), Y₁(ti−1), Y₁(ti−2),..., Y₁(ti−s−1), Y₁(ti−s).

2
Y₂(ti), Y₂(ti−1), Y₂(ti−2),..., Y₂(ti−s−1), Y₂(ti−s).

……
……

N
Yn(ti), Yn(ti−1), Yn(ti−2),..., Yn(ti−s−1), Yn(ti−s).

In fact, the first modeling datasets are n forecasting datasets corresponding to every individual forecasting model, respectively. Then, the mean absolute percentage errors (MAPE) [29, 30] are computed to obtain the datasets MAPE₁, MAPE₂, …, and MAPEn, given in equation (14). The mean values and the standard deviations for MAPE₁, MAPE₂, ……, and MAPEn can be further given in equation (14). They are defined as the second modeling datasets, which are listed in Table 3.

Table 3: The second modeling datasets.

Number
Error datasets

1
, ,..., ,.

2
, , …, ,.

3
σ₁, σ₂ , …, σn−1, σn

Based on the previous preparation work of equation (14), our present work is to produce a forecast F(ti) for the upcoming time ti (3siN). Consequently, an adaptive control combination forecasting (ACCF) method is given in equation (14).

where , and are the weighting coefficients, which are dependent variables with the change of the upcoming time ti. Based on the performance-based approach [31], an adaptive weight for each model is determined based on their own forecasting performance and can be defined as follows:

where kj is either 0 or 1, and it is computed as follows:

In the equations (16) and (17), there are the third modeling datasets, shown in Table 4.

Table 4: The third modeling datasets.

Number
Datasets for solving weights

1
.

2
k₁, k₂, …, kn−1, kn.

3
,, …, , .

Clearly, using this ACCF approach, models producing smaller values of (j = 1,2, …, n) will be assigned larger weights in comparison to models with higher ones. The smallest mean value and the corresponding standard deviation are defined as and σm, respectively. Thus, the smallest MAPE range is defined between and . When statistical arrays range from to produced by models that overlap partially or fully with the smallest MAPE range (from to ), where kj is 1; otherwise, kj is 0.

In the ACCF model, the modeling datasets, including , and kj must be corrected on every upcoming time ti according to the solution of equations (14), (16), and (17). Finally, key parameters are finally taken into equation (15) to make a prediction on time ti. Because j in equation (16) must be updated and revised on every upcoming time ti so as to ensure prediction accuracy with high robustness, the new developed model is propitious to short-term prediction. In the condition of long-term prediction, however, this new model cannot update the value of the weights in time and may be result in a large percentage forecast error.

2.3. Evaluation Index

The MAPE during time period (T) is further measured to evaluate the obtained results. Because there are s/3 data in number every time period, the error MAPEk (k = 1, 2, …, (N−3s)/(s/3)) during the kth time period can be calculated by equation (17). For total tested data set, the number of predicting time periods is equal to (N−3s)/(s/3), and calculation equations of total error indicators MAPEall are given in the below formula (18). To verify the superiority of the ACCF approach, statistical count lower than MAPEall (defined Count) and percentage of Count in the sum (defined Per) are also calculated by equation (18) for this new array MAPEk.

where y(ti) is the measured value at time of ti; F(ti) is the predicting value at time of ti.

After obtaining the predicting error of MAPEall, the forecasting accuracy (FAall) can be calculated by using equation (19). Thus, the forecasting accuracy of the model will be better when the FA is approaching 100%.

3. Steps of Computation

In the study, six individual forecasting methods, including PSI, ES, MA, ARIMA, GM, and BP-NET, are used to construct ACCF method described in Section 2.2. The flow chart of the proposed ACCF model for short-term prediction is summarized in Figure 3. It gives a rolling forecast process, and the detailed steps are as follows:

 Step 1: The total tested data set (y(t₁), y(t₂), …… y(ti−1)) is initialized at time of ti−1. Step 2: The all tested datasets in Table 1 are extracted from total tested data set. Step 3: By solving equations (1) and (3)–(6), ..., the first modeling datasets in Table 2 are predicted by using each individual model, such as PSI method, ES method, ARIMA method, MA method, GM method, and BP-NET method. Step 4: Equation (14) is solved, and the second modeling datasets in Table 3 are got; Step 5: Equations (16) and (17) are solved, and the third modeling datasets in Table 4 are further obtained; Step 6: Using Y₁(ti), Y₂(ti), …, Y₆(ti) (in Table 2) and , , …, (in Table 4), equation (15) is calculated to give a new combination forecast F(ti) at time step of ti. Step 7: If the time steps of the stop condition (ti ≥ tN) are satisfied, the search stops, as well as output parameters of MAPEall, Count, Per, and FAall by solving equations (18) and (19); Otherwise, the time step is added, new generated data y(ti) is added into the total tested data set, and then the procedure returns to step 2.

Figure 3: The flow chart of the ACCF method for short-term prediction.

4. Results and Discussion

Table 5 shows three groups of tested time series datasets. As shown in Table 5, the first dataset (named USP), with total samples of 492, is US population between January 1979 and December 2019, which is from the US Census Bureau hosted by the Federal Reserve Economic Database (FRED) [32]. FRED has a data platform found US population data and updated population information of every month. The second dataset (named AEP), with total samples of 3432, is the American hourly electric power consumption data between March 13, 0 : 00, and August 2, 23 : 00, in 2018, which comes from PJM’s website and is in megawatts (MW) [33]. PJM is a regional transmission organization (RTO) in the United States, and part of the Eastern Interconnection grid operating an electric transmission system. The third dataset (named VS) was experimentally obtained with a hydraulic test rig. This test rig consists of a primary working, a test system, and a secondary cooling-filtration circuit, which are connected via the oil tank [34, 35]. The system cyclically repeats constant load cycles (duration 60 seconds). The test system is equipped with several sensors measuring process values such as vibration, with standard industrial 20 mA current loop interfaces connected to a data acquisition system. In the study, these vibration signals in hydraulic test rig with a sampling frequency of 1 Hz during 8580 seconds [34] were measured and used as the third dataset.

Table 5: Time series datasets used in the experiments.

Dataset name
Time for historical data
Total samples/time interval/time period

U.S. population (USP) dataset [32]
January 1979–December 2019
492/month/year

American electric power (AEP) dataset [33]
March 13, 2018 0 : 00–August 2, 2018 23 : 00
3,432/hour/day

Vibration signal (VS) dataset [34] in a hydraulic test rig
0 minute 0 second–142 minutes 59 seconds
8,580/second/minute

By using the developed ACCF method, a comparison analysis between the real value and the forecasting value was implemented and showed a direct observation of the prediction, so as to evaluate the confidence of the ACCF method.

4.1. Periodic Recognition and Prediction on USP Dataset

The USP datasets were used as a tested dataset to show periodic detection and prediction results. In order to calculate second modeling datasets ( produced by six individual models, respectively) in equation (14), monthly population data of USP during contiguous 36 months are in turn trained to forecast the population data during next month. Figure 4 shows evolution of the mean values of MAPE with year when predicting by using the PSI, ES, ARIMA, MA, BP-NET, and GM methods. It can be seen that the ARIMA method gives a much better prediction accuracy.

Figure 4: Evolution of the MAPE mean values with time on USP dataset.

Based on the ACCF method, the weighting coefficients can be given by solving equations (16) and (17):  = 0 for PSI,  = 0 for ES,  = 1 for ARIMA,  = 0 for MA,  = 0 for BP-NET and  = 0 for GM. It also shows that the ACCF method can adaptively seek the prediction methods with much higher accuracy and abandon other prediction methods with poor accuracy. Then, the USP dataset from January 1988 to December 2019 is circularly predicted by solving equation (15), as shown in Figure 5. It demonstrates a good prediction trend with good volatility and following quality by using the ACCF method.

Figure 5: Time series pattern and prediction on USP dataset.

4.2. Periodic Recognition and Prediction on AEP Dataset

The AEP datasets were used as a tested data set to show periodic detection and prediction results. Figure 6 shows evolution of the mean values of MAPE with time when predicting by using the PSI, ES, ARIMA, MA, BP-NET, and GM methods. By solving equations (16) and (17), the weighting coefficients of models at different forecasting times are shown in Table 6. It can be seen that the sum of six weight coefficients is always equal to 1; , and most of are equal to zero; the values of are always less than those of and . Thus, because of the worst performance of MA method and GM method, their weights are not assigned. But PSI method and ARIMA method play more important role than other individual methods in the prediction of AEP dataset. Then, the AEP dataset from March 22, 0 : 00, to August 2, 23 : 00, in 2018 performed a rolling prediction by solving equation (15), as shown in Figure 7. It demonstrates that the ACCF method shows a better prediction trend with good volatility and following quality.

Figure 6: Evolution of the MAPE mean values with time on AEP dataset.

Table 6: The weighting coefficients of models at different forecasting time on AEP dataset.

Forecasting time/seconds
PSI weight
ES weight
ARIMA weight
MA weight
BP-NET weight
GM weight

217
0.4878
0.0000
0.5122
0.0000
0.0000
0.0000

432
0.5859
0.0000
0.4141
0.0000
0.0000
0.0000

648
0.3875
0.0000
0.3929
0.0000
0.2196
0.0000

864
0.4762
0.0000
0.5238
0.0000
0.0000
0.0000

1080
0.2967
0.2414
0.2566
0.0000
0.2052
0.0000

1296
0.4296
0.0000
0.3370
0.0000
0.2334
0.0000

1512
0.5832
0.0000
0.4168
0.0000
0.0000
0.0000

1728
0.3642
0.0000
0.3795
0.0000
0.2563
0.0000

1944
0.3324
0.0000
0.3977
0.0000
0.2699
0.0000

2160
0.3998
0.0000
0.3525
0.0000
0.2477
0.0000

2376
0.3449
0.0000
0.3589
0.0000
0.2962
0.0000

2592
0.3369
0.0000
0.3623
0.0000
0.3008
0.0000

2808
0.4346
0.0000
0.5654
0.0000
0.0000
0.0000

3024
0.3791
0.0000
0.3557
0.0000
0.2652
0.0000

3240
0.3900
0.0000
0.3844
0.0000
0.2257
0.0000

3432
0.4782
0.0000
0.5218
0.0000
0.0000
0.0000

(a)

(b)

Figure 7: Time series pattern and prediction on AEP dataset. (a) Forecasts from the 216th hour to the 1728th hour. (b) Forecasts from the 1729th hour to the 3432th hour.

4.3. Periodic Recognition and Prediction on VS Dataset

The VS datasets were used as a tested dataset to show periodic detection and prediction results. Figure 8 shows evolution of the mean values of MAPE with time when predicting by using the PSI, ES, ARIMA, MA, BP-NET, and GM methods. Based on the ACCF method, the weighting coefficients by each base model can be computed by solving equations (16) and (17). Table 7 shows the weighting coefficients of models at different forecasting times. It can be seen that the sum of six weight coefficients is equal to 1, which is similar to the result in Section 4.2, but only and are equal to zero. PSI method, ES method, ARIMA method, and BP-NET method all play important role in the prediction of VS dataset. Then, the VS dataset from the 541th second to the 8580th second is further predicted by using the ACCF method, as shown in Figure 9. Thus, once again, the ACCF method demonstrates a better prediction trend with good volatility and following quality.

Figure 8: Evolution of the MAPE mean values with time on VS dataset.

Table 7: The weighting coefficients of models at different forecasting time on VS dataset.

Forecasting time/seconds
PSI weight
ES weight
ARIMA weight
MA weight
BP-NET weight
GM weight

541
0.2713
0.2498
0.2428
0.0000
0.2361
0.0000

1080
0.2575
0.2571
0.2482
0.0000
0.2372
0.0000

1620
0.5295
0.0000
0.4705
0.0000
0.0000
0.0000

2160
0.2621
0.2517
0.2463
0.0000
0.2400
0.0000

2700
0.2682
0.2467
0.2465
0.0000
0.2386
0.0000

3240
0.2744
0.2431
0.2451
0.0000
0.2374
0.0000

3780
0.3634
0.3163
0.3203
0.0000
0.0000
0.0000

4320
0.2710
0.2498
0.2454
0.0000
0.2338
0.0000

4860
0.3547
0.3216
0.3238
0.0000
0.0000
0.0000

5400
0.2635
0.2520
0.2544
0.0000
0.2301
0.0000

5940
0.2683
0.2441
0.2565
0.0000
0.2310
0.0000

6480
0.3760
0.3081
0.3159
0.0000
0.0000
0.0000

7020
0.3700
0.3112
0.3188
0.0000
0.0000
0.0000

7560
1.0000
0.0000
0.0000
0.0000
0.0000
0.0000

8100
0.5281
0.0000
0.4719
0.0000
0.0000
0.0000

8580
1.0000
0.0000
0.0000
0.0000
0.0000
0.0000

(a)

(b)

Figure 9: Time series pattern and prediction on VS dataset. (a) Forecasts from the 541th second to the 4320th second. (b) Forecasts from the 4321th second to the 8580th second.

4.4. Accuracy Analysis of ACCF Method

4.4.1. Comparison with Other Forecasting Models

Based on each prediction model, MAPEk (k = 1, 2, …, (N−3s)/(s/3)), MAPEall, Count, and Per are calculated by equation (18). Then, the statistical test of MAPEk is shown in Table 8 on USP dataset, in Table 9 on AEP dataset, and in Table 10 on VS dataset. Max value of MAPEk, standard deviation of MAPEk can also be got and listed in Tables 8–10. It can be seen from Tables 8 and 9 that MAPEall, max value of MAPEk and standard deviation of MAPEk are smaller by ACCF method, as compared to those by other methods. Meanwhile, greater values of Count and Per can be got by ACCF method than other methods. Furthermore, in Table 10, max value and standard deviation of MAPEk by ACCF method are slightly greater than that of IV method and MSEI method, but ACCF method has the smaller value of MAPEall and the greater values of Count and Per over all methods. A possible reason is that statistical distribution law of historical forecasting errors was delved deeper by using ACCF method, and weighting coefficient for each model was modified more reasonably, leading to smaller MAPEall as well as greater statistical Count. Therefore, the ACCF method has the highest incidence of delivering the best predictions over all compared forecasting methods on USP dataset, AEP dataset, and VS dataset.

Table 8: Statistical test of MAPEk (k = 1,2, …, 32) on USP dataset.

Models
Max value of MAPEk (%)
Standard deviation of MAPEk (%)
MAPEall (%)
Count {MAPEk0.006}
Per (=Count/32) (%)

PSI
0.112
0.022
0.061
0
0

ES
0.112
0.019
0.079
0
0

ARIMA
0.033
0.006
0.006
21
66

MA
0.168
0.028
0.119
0
0

BP-NET
0.126
0.026
0.048
0
0

GM
6.437
1.013
4.688
0
0

IV
0.033
0.007
0.007
19
59

MSEI
0.054
0.011
0.021
0
0

SWA
1.003
0.158
0.726
0
0

ACCF
0.033
0.006
0.006
22
69

Table 9: Statistical test of MAPEk (k = 1,2, …, 134) on AEP dataset.

Models
Max value of MAPEk (%)
Standard deviation of MAPEk (%)
MAPEall (%)
Count {MAPEk1.094}
Per (=Count/134) (%)

PSI
2.390
0.401
1.270
46
34

ES
4.519
0.697
3.127
0
0

ARIMA
3.018
0.358
1.314
35
26

MA
6.416
0.937
4.617
0
0

BP-NET
4.589
0.689
1.959
10
7

GM
66.269
10.719
22.116
0
0

IV
2.891
0.319
1.141
67
50

MSEI
3.130
0.392
1.406
24
18

SWA
9.842
1.464
2.892
0
0

ACCF
2.270
0.279
1.094
76
57

Table 10: Statistical test of MAPEk (k = 1,2, …, 134) on VS dataset.

Models
Max value of MAPEk (%)
Standard deviation of MAPEk (%)
MAPEall (%)
Count {MAPEk1.189}
Per (=Count/134) (%)

PSI
1.669
0.140
1.237
50
37

ES
1.746
0.139
1.376
12
9

ARIMA
1.933
0.169
1.367
22
16

MA
2.574
0.209
1.964
0
0

BP-NET
1.913
0.169
1.454
7
5

GM
8.304
1.013
4.705
0
0

IV
1.593
0.136
1.220
56
42

MSEI
1.571
0.129
1.274
35
26

SWA
1.953
0.161
1.266
44
33

ACCF
1.618
0.138
1.189
68
51

Table 11 presents a more visual view of prediction accuracy of each prediction model. It can be noticed that most of FAall values for individual methods are all more than 95%, and the four combination forecasting methods have higher FAall values of more than 97%. Obviously, the combination forecasting effect is most desirable and superior to the individual methods. It is interesting to know that the ACCF method is observed to be the best for the prediction of the USP AEP and VS datasets, due to higher forecasting accuracy. When judging by the FAall values of the four combination methods, ACCF method and IV method are superior to MSEI method and SWA method. In general, the developed ACCF algorithm is adaptive to adopt one (for USP) or some (for AEP and VS) among individual prediction methods, achieving satisfactory accuracy (FAall > 98.8%) in time series prediction, and it is suitable to the prediction of three datasets used in this study due to higher values of FAall.

Table 11: Accuracy indicators of each prediction model.

Models
Accuracy indicators on USP dataset
Accuracy indicators on AEP dataset
Accuracy indicators on VS dataset

FAall (%)
FAall (%)
FAall (%)

PSI
99.939
98.730
98.763

ES
99.921
96.873
98.624

ARIMA
99.994
98.686
98.633

MA
99.881
95.383
98.036

BP-NET
99.952
98.041
98.546

GM
95.312
77.884
95.295

IV
99.993
98.859
98.780

MSEI
99.979
98.594
98.726

SWA
99.274
97.108
98.734

ACCF
99.994
98.906
98.811

4.4.2. Impact of Noise Ratio

In order to test robustness of the ACCF algorithm, the noisy data were further added to the AEP data set and the VS data set. The ratio of the standard deviation (STD) of added noise to the STD of original dataset is in a range from 0.00 to 0.50 in the study. The prediction accuracy of the ACCF method under noisy data was computed and compared with that of other forecasting methods, as shown in Tables 12 and 13.

Table 12: Prediction accuracy of comparison algorithms on AEP dataset.

Models
FAall (%) under different STD

0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50

PSI
98.073
97.252
96.324
95.348
94.381
93.466
92.587
91.716
90.824
89.881

ES
96.725
96.346
95.792
95.115
94.370
93.603
92.827
92.050
91.278
90.517

ARIMA
97.853
96.997
96.126
95.244
94.358
93.475
92.603
91.753
90.933
90.154

MA
95.304
95.047
94.632
94.080
93.409
92.642
91.816
90.967
90.135
89.357

BP-NET
96.986
95.825
94.608
93.386
92.209
91.118
90.106
89.156
88.252
87.377

GM
77.871
77.785
77.649
77.489
77.332
77.192
77.047
76.865
76.612
76.257

IV
98.120
97.344
96.542
95.724
94.900
94.080
93.266
92.461
91.666
90.882

MSEI
97.905
97.164
96.386
95.583
94.771
93.960
93.155
92.359
91.573
90.801

SWA
97.048
96.689
96.110
95.390
94.610
93.835
93.078
92.336
91.608
90.892

ACCF
98.184
97.399
96.576
95.741
94.918
94.127
93.354
92.581
91.788
90.955

Max values
98.184
97.399
96.576
95.741
94.918
94.127
93.354
92.581
91.788
90.955

Table 13: Prediction accuracy of comparison algorithms on VS dataset.

Models
FAall (%) under different STD

0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50

PSI
98.725
98.637
98.510
98.354
98.180
97.996
97.805
97.609
97.409
97.205

ES
98.600
98.521
98.402
98.255
98.095
97.933
97.768
97.599
97.425
97.245

ARIMA
98.596
98.506
98.377
98.223
98.055
97.886
97.716
97.542
97.365
97.181

MA
98.022
97.963
97.867
97.746
97.609
97.464
97.312
97.151
96.977
96.789

BP-NET
98.503
98.418
98.301
98.161
98.005
97.841
97.671
97.495
97.316
97.132

GM
95.290
95.281
95.267
95.242
95.203
95.149
95.082
95.008
94.931
94.856

IV
98.742
98.663
98.552
98.416
98.266
98.108
97.945
97.779
97.611
97.443

MSEI
98.693
98.619
98.513
98.382
98.236
98.081
97.921
97.757
97.590
97.423

SWA
98.702
98.632
98.530
98.405
98.263
98.111
97.932
97.758
97.521
97.414

ACCF
98.770
98.686
98.570
98.431
98.276
98.115
97.949
97.782
97.616
97.452

Max values
98.770
98.686
98.570
98.431
98.276
98.115
97.949
97.782
97.616
97.452

As can be observed from Tables 12 and 13, when the proportion of noisy data increases, the FAall value of each algorithm decreases in all cases. Considering the different natural periodicity of each time series dataset, comparison methods can obtain different accuracy on different datasets. The FAall values of forecasting methods are between 76.257% and 98.184% for the AEP dataset and are in a range of 94.856%–98.770% for the VS dataset. In addition, for AEP dataset or VS dataset with noise ratio of 0.00, the ACCF method is superior to other methods, and the IV method is in the second place. With the increasing noise ratios, however, the ACCF method almost keeps the highest FAall value in two cases against other comparison algorithms (including not only individual methods, but also combination methods). For example, as the noise ratio changes from 0.0 to 0.5, the FAall of IV method decreases from 98.120% to 90.882% for AEP dataset, and from 98.742% to 97.443% for VS dataset, while the FAall of ACCF method decreases only from 98.184% to 90.955% for AEP dataset, and from 98.770% to 97.452% for VS dataset. This might be due to the high stochasticity of the tested data set under high noise ratio, and thus the comparison models could not capture the actual trend of historical forecasting errors. Especially for IV, MSEI, and SWA models, the statistical distribution information of the forecasting errors with the historical time is not considered, so they show lower robustness than developed ACCF model. On the contrary, because the statistical forecasting errors are used to correct weights in real time, the robustness of ACCF method is better than other comparison methods for noisy data.

Therefore, it is concluded that the proposed ACCF algorithm obtains higher prediction accuracy on time series datasets and is more robust to noisy data than other individual methods, as well as combination methods.

5. Conclusions

(i)According to the individual forecasting methods, such as PSI, ES, ARIMA, MA, and BP-NET methods, an ACCF method with adaptive weighting coefficients is proposed for short-term prediction of the time-series data.(ii)The combination forecasting methods are most desirable and superior to the individual methods. In contrast to other forecasting methods, the proposed ACCF method is adaptive to adopt one or some of prediction methods and shows satisfactory forecasting quality due to its flexible adaptability and high forecasting accuracy. The ACCF method is extremely suitable for short-term prediction of time series datasets.(iii)The higher the noise ratio of the tested datasets, the lower the prediction accuracy of the ACCF method. But the proposed ACCF methods can still achieve significant advantages compared with other forecasting methods in terms of forecasting accuracy. The ACCF method demonstrates a better prediction trend with good volatility and following quality.

Abbreviations

dt: | Equal interval of time
ej: | The sum of squared errors of j-th single model
FA: | Forecasting accuracy
kj: | Either 0 or 1 (j = 1,2, …, n)
K₀: | Correction coefficient for period index
MAPE: | Mean absolute percentage error
MAPE₁: | New historical arrays given in equation (14) by PSI method
MAPE₂: | New historical arrays given in equation (14) by ES method
MAPE₃: | New historical arrays given in equation (14) by ARIMA method
MAPE₄: | New historical arrays given in equation (14) by MA method
MAPE₅: | New historical arrays given in equation (14) by BP-NET method
MAPE₆: | New historical arrays given in equation (14) by GM method
N: | Number of forecasting methods
PI(ti): | Period index at time of ti
SI(ti): | Sequential index at time of ti
ti: | Time (ti = t₁ , t₂, …, tN)
T: | Fixed time period
y(ti),: | Observed value at time of ti
Y₁(ti): | Forecasting value at time of ti by PSI method
Y₂(ti): | Forecasting value at time of ti by ES method
Y₃(ti): | Forecasting value at time of ti by ARIMA method
Y₄(ti): | Forecasting value at time of ti by MA method
Y₅(ti): | Forecasting value at time of ti by BP-NET method
Y₆(ti): | Forecasting value at time of ti by GM method
: | Weighting coefficients by ACCF method (j = 1,2, …, n)
: | Mean values for historical arrays MAPEj (j = 1,2, …, n)
: | The smallest values in (j = 1,2, …, n)
σj: | Standard deviations for historical arrays MAPEj (j = 1,2, …, n)
σm: | Standard deviations for historical arrays MAPEm
F(ti): | Forecasting value at time of ti by ACCF method
Α: | Optimized weighing factor of PSI method
β: | Smoothing parameter.

Data Availability

The data presented in this study are available upon request from the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

Conceptualization was performed by H. J., D. F.; formal analysis was performed by H.J. , D. F.; investigation was performed by H. J., D. F.; original draft was written by H. J., X. Z.; reviewing and editing were performed by H. J., D. F.

Acknowledgments

This study was funded by the Key Projects for International Cooperation in Scientific and Technological Innovation between Governments (grant no. 2017YFE0101600). The authors would like to express their sincere thanks to technical support from Sino–German Institute for Intelligent Technologies.

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