OWA operator-based time series hybrid model and method

Abstract

The invention provides a time series hybrid model and method based on an OWA operator, and the method comprises the steps: carrying out the weighting processing of time series data through employing an ordered weighted average operator, calculating an OWA-based autocorrelation coefficient and an OWA-based partial autocorrelation coefficient, obtaining ACF-OWA and PACF-OWA, and carrying out the calculation of the time series hybrid model based on an ARIMA model, according to the self-correlation coefficient of the OWA and the partial self-correlation coefficient based on the OWA, an ARIMA-OWA model is constructed and obtained, and compared with an ARIMA model, the obtained ARIMA-OWA model has better stability, and prediction of linear part data can be better achieved. Moreover, the invention provides a mixed time series prediction method on the basis of the ARIMA-OWA model, and compared with an existing time series prediction method, the mixed time series prediction method has better prediction performance.

Description

Technical Field

[0001] The present invention relates to the field of time series analysis, and in particular to a time series hybrid model and method based on OWA operator. Background Art

[0002] The ARIMA (Autoregressive integrated moving average) model has become the most commonly used model tool in the field of time series analysis and has been widely used in the fields of environment, health care, finance, etc. In order to improve the accuracy of prediction, researchers have proposed a large number of hybrid time prediction methods in recent years. One of the representative ones is the hybrid model of ARIMA and neural network (ANN), which improves the accuracy of ARIMA prediction.

[0003] At present, the time series hybrid forecasting model based on OWA operator has been widely used in data fusion, risk assessment, and intelligent decision-making. The hybrid model uses the mean, variance, and covariance calculation methods based on OWA to calculate the coefficients in the forecasting model equation. However, the accuracy of ARIMA model prediction still needs to be improved, but there is a lack of effective methods, so it is an urgent problem to be solved. Summary of the invention

[0004] In view of the above situation, the main purpose of the present invention is to propose a time series hybrid model and method based on OWA operator to solve the above technical problems.

[0005] The present invention proposes a time series hybrid model based on the OWA operator, which uses the OWA operator to perform weighted processing on the time series data and calculates the autocorrelation coefficient based on the OWA; Given the autoregressive order and the moving average order, and obtaining the corresponding number of OWA-based autocorrelation coefficients according to the autoregressive order and the moving average order, the moment estimation method is used for parameter estimation to obtain the OWA-based autoregressive parameters and the OWA-based moving average parameters; Based on the ARIMA model, the ARIMA model parameters are replaced with the autoregressive parameters based on OWA and the moving average parameters based on OWA to obtain the ARIMA-OWA model; The maximum lag order is set, and the final autoregressive order, moving average order and difference order of the ARIMA-OWA model are determined by the cyclic traversal method. In each cyclic traversal process, the autoregressive parameters based on OWA and the moving average parameters based on OWA are dynamically adjusted according to the autoregressive order and the moving average order. After the cyclic traversal is completed, the final ARIMA-OWA model is obtained.

[0006] The present invention also proposes a hybrid time series prediction method based on the OWA operator, the method adopts the above-mentioned time series hybrid model based on the OWA operator, and the method comprises the following steps: Step 1: Obtain time series data and decompose the time series data using an MA filter to obtain a linear component and a nonlinear residual sequence; Step 2, preprocessing the linear component to generate a linear component sequence, and defining the historical sequence in the linear component sequence as the OWA mean of the current sequence in the linear component sequence; Step 3: transform the ARIMA-OWA model according to the core assumptions of the ARIMA model to obtain a transformed model; initialize the historical error of the transformed model, and then substitute the linear component sequence to calculate the historical random error sequence; Step 4: Substitute the historical linear component series and the historical random error series into the ARIMA-OWA model to obtain the predicted linear variables; Step 5: Select a number of data from the time series data as the first time series; Selecting a number of data from the nonlinear residual sequence as the second time series; The first time series, the second time series and the predicted linear variable are sent to the ANN network to obtain the mixed prediction results of the time series.

[0007] Compared with the prior art, the present invention has the following beneficial effects: 1. Inspired by variance, covariance, autocorrelation coefficient, and partial correlation coefficient, ACF-OWA and PACF-OWA coefficients are proposed. The proposed ACF-OWA and PACF-OWA can better capture and extract the complex characteristics of time series data, and obtain a more stable time series data that is more in line with the ARIMA model assumptions.

[0008] 2. Based on ACF-OWA and PACF-OWA, the ARIMA model was reconstructed to obtain the ARIMA-OWA model, and the performance of the constructed ARIMA-OWA model is better than that of the ARIMA model. ARIMA-OWA enhances the dependence of time series data at different time points and improves the model's ability to handle data stability, so that the model can better capture the long-term trend changes and linear relationships between data, thereby improving the accuracy and stability of model predictions.

[0009] 3. Based on ARIMA-OWA and inspired by ARIMA-ANN, a time series forecasting method is proposed. This invention fully considers the stationarity in the data and the strong dependence between the data, and solves these problems through innovative methods, creating a model with high versatility and flexibility, and has obvious advantages over other methods in this field.

[0010] Additional aspects and advantages of the present invention will be given in part in the following description and in part will be obvious from the following description or learned through embodiments of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] Figure 1 A flow chart for constructing a time series hybrid model based on the OWA operator proposed in the present invention; Figure 2 The algorithm flow chart of the hybrid time series prediction method based on OWA operator proposed by the present invention; Figure 3 Schematic diagram of autocorrelation and partial autocorrelation coefficients of each data set; Figure 4 This is a time series line graph of the sunspot dataset; Figure 5 Comparison chart of actual and predicted values ​​of 8 time series prediction methods on the sunspot dataset; Figure 6 This is a time series line graph of the lynx dataset; Figure 7 This is a comparison chart of the actual and predicted values ​​of 8 time series prediction methods on the lynx dataset; Figure 8 A time series line graph for the aviation dataset; Fig. 9 Comparison chart of actual and predicted values ​​of 8 time series prediction methods on aviation data set; Fig.10 This is a time series line graph of the star brightness dataset; Fig.11 Comparison chart of actual and predicted values ​​of 8 time series prediction methods on star brightness dataset; Fig.12 A time series line graph for the exchange rate dataset; Fig.13 Comparison chart of actual and predicted values ​​of 8 time series prediction methods on random time series data sets; Fig.14 A bar chart showing the average values ​​of MAE, MSE, and MASE for 8 time series prediction methods on 5 data sets. DETAILED DESCRIPTION

[0012] Embodiments of the present invention are described in detail below, examples of which are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and cannot be understood as limiting the present invention.

[0013] These and other aspects of the embodiments of the present invention will be apparent with reference to the following description and accompanying drawings. In these descriptions and accompanying drawings, some specific implementations of the embodiments of the present invention are specifically disclosed to represent some ways of implementing the principles of the embodiments of the present invention, but it should be understood that the scope of the embodiments of the present invention is not limited thereto.

[0014] See also Figure 1 , this embodiment provides a time series mixed model based on the OWA (ordered weighted average) operator: Step 1: Use the OWA operator to perform weighted processing on the time series data and calculate the autocorrelation coefficient based on OWA; Step 2: Given the autoregressive order and the moving average order, and obtaining the corresponding number of OWA-based autocorrelation coefficients according to the autoregressive order and the moving average order, the moment estimation method is used to perform parameter estimation to obtain the OWA-based autoregressive parameters and the OWA-based moving average parameters; Step 3: Based on the ARIMA model, replace the ARIMA model parameters with the autoregressive parameters based on OWA and the moving average parameters based on OWA to obtain the ARIMA-OWA model; Step 4: Set the maximum lag order, and use the loop traversal method to determine the final autoregressive order, moving average order, and difference order of the ARIMA-OWA model. In each loop traversal process, dynamically adjust the OWA-based autoregressive parameters and OWA-based moving average parameters according to the autoregressive order and moving average order. After the loop traversal is completed, the final ARIMA-OWA model is obtained.

[0015] In the preferred implementation manner provided by the present invention, the OWA operator is used to perform weighted processing on the time series data, and the process corresponding to the calculation of the autocorrelation coefficient based on OWA has the following relationship: ; Formula (14) in, represents the autocorrelation coefficient based on OWA, Indicates the time series data t The weight vector corresponding to a large number, Represents the first t elements, Represents the number of elements in the time series, Indicates the time interval, represents the covariance based on OWA, represents the variance based on OWA, represents the OWA mean of the time series data, denote the autoregressive order and the moving average order respectively; The calculation process corresponding to the mean of OWA has the following relationship: ; in, Represents the first n elements, represents the OWA operator, Indicates the time series data i The weight vector corresponding to a large number, Indicates the time series data i a large number; Time series data i The weight vector calculation process corresponding to a large number has the following relationship: ; in, They represent the mean and standard deviation of time series data respectively. The calculation process of the mean and standard deviation of time series data has the following relationship: .

[0016] In the preferred embodiment provided by the present invention, given the autoregressive order and the moving average order, and obtaining a corresponding number of autocorrelation coefficients based on OWA according to the autoregressive order and the moving average order, using the moment estimation method to perform parameter estimation, obtaining the autoregressive parameters based on OWA and the moving average parameters based on OWA specifically includes the following steps: Given the autoregressive order and the moving average order, the corresponding number of autocorrelation coefficients based on OWA are obtained according to the autoregressive order and the moving average order to construct the Yule-Walker equation. The autoregressive parameters based on OWA are estimated using the Yule-Walker equation. The corresponding process has the following relationship: ; Formula (15) in, represents the pth OWA-based autoregressive coefficient, represents the q+p+2th OWA-based autocorrelation coefficient; The moving average parameters based on OWA are calculated based on the obtained autoregressive parameters based on OWA and the moving average order; Among them, when the moving average order is 0, the moving average parameter based on OWA is calculated using the first relationship, which is as follows: ; in, Indicates that the moving average coefficient based on OWA is 0; When the moving average order is greater than or equal to 1, the moving average parameter based on OWA is calculated using the second relational expression, which is as follows: ; Formula (16) in, Indicates i Based on the OWA moving average coefficient, represents the estimate of the covariance based on OWA, represents the OWA covariance of the error vector, Represents the variance of the error. When the autoregressive order is 0, the variance of the error is calculated as follows: ; in, represents the estimate of the covariance based on OWA, ; The OWA covariance calculation process of the error vector has the following relationship: ; Formula (17) in, Represents random error.

[0017] In a preferred embodiment of the present invention, the second relationship is solved by a linear iteration method, which specifically includes the following steps: Given and A set of initial values , ; Sets the iteration conditions for the OWA-based moving average parameters and the variance of the error. The iteration conditions include the first precision of positive real numbers. and second precision of positive real numbers ; The initial value , Substitute the second relation and solve it iteratively until it satisfies At the same time, the iteration is terminated. is the number of iterations, and the final output result is and The solution.

[0018] In the preferred implementation provided by the present invention, based on the ARIMA model, the ARIMA model parameters are replaced with the autoregressive parameters based on OWA and the moving average parameters based on OWA to obtain the ARIMA-OWA model, and the corresponding process has the following relationship: ; in, represents the first OWA-based autoregressive coefficient, Indicates hysteresis p Time series data of order, Indicates hysteresis q The order error value, Indicates q Based on the OWA moving average coefficient.

[0019] See also Figure 2 This embodiment provides a hybrid time series prediction method based on the OWA operator. The method adopts the above-mentioned time series hybrid model based on the OWA operator. The method includes the following steps: Step 1: Obtain time series data and decompose the time series data using an MA filter to obtain a linear component and a nonlinear residual sequence; Step 2, preprocessing the linear component to generate a linear component sequence, and defining the historical sequence in the linear component sequence as the OWA mean of the current sequence in the linear component sequence; Step 3: transform the ARIMA-OWA model according to the core assumptions of the ARIMA model to obtain a transformed model; initialize the historical error of the transformed model, and then substitute the linear component sequence to calculate the historical random error sequence; Step 4: Substitute the historical linear component series and the historical random error series into the ARIMA-OWA model to obtain the predicted linear variables; Step 5: Select a number of data from the time series data as the first time series; Selecting a number of data from the nonlinear residual sequence as the second time series; The first time series, the second time series and the predicted linear variable are sent to the ANN network to obtain the mixed prediction results of the time series.

[0020] In the preferred embodiment provided by the present invention, the time series data is decomposed by using an MA filter, and the process corresponding to the linear component and the nonlinear residual sequence is obtained, and the following relationship exists: ; ; in, Indicates tThe linear component of the elements, represents the filter length, represents the nonlinear residual sequence, Indicates the time series data i elements, Indicates t elements.

[0021] In the preferred embodiment provided by the present invention, the ARIMA-OWA model is transformed according to the core assumption of the ARIMA model to obtain a transformed model; the historical error of the transformed model is initialized, and then the linear component sequence is substituted to calculate the process corresponding to the historical random error sequence. The following relationship exists: ; in, represents the normalized lag p The linear component of order, It means the normalized t The linear component of the elements.

[0022] In the preferred embodiment provided by the present invention, the historical linear component sequence and the historical random error sequence are substituted into the ARIMA-OWA model to obtain the process corresponding to the predicted linear variable, and the following relationship exists: ; in, represents the predictor linear variable, Represents the ARIMA-OWA model.

[0023] In the preferred embodiment provided by the present invention, the first time series, the second time series and the predicted linear variable are sent to the ANN network, and the process corresponding to the hybrid prediction result of the time series is obtained, and the following relationship exists: ; in, Respectively represent the length of the selected data in the time series data and the nonlinear residual sequence, represents the first time series, represents the second time series, represents the nonlinear function of the neural network, Indicates Time series data, Indicates A nonlinear residual sequence.

[0024] The model of Example 1 is recorded as ARIMA-OWA, and the method of Example 2 is recorded as ARIMA-OWA-ANN. In order to verify the performance of the present invention, the mean absolute error (MAE), root mean square error (MSE) and mean absolute proportional error (MASE) indicators are used to evaluate the advantages and disadvantages of different models. The evaluation indicators are shown in the following formula: ; ; ; in, represents the mean absolute error, represents the root mean square error, represents the mean absolute proportional error.

[0025] Five time series datasets will be used as the basis for the experiment. Among them, the Wolf Sunspot Dataset and the Canadian Lynx Dataset have been widely used in statistics and neural network literature. The other three are: Airline Passengers Dataset, Stellar Brightness Dataset, and USD / MXN exchange rate dataset. These three time series datasets come from different application fields and are also often used as basic datasets for time series prediction. Here is a brief overview of these five datasets: (1) The Wolf Sunspot Dataset records the number of sunspots observed each month from 1700 to 1987, with a total of 288 observations (e.g. Figure 4 ), researchers often use sunspot datasets to study the relationship between solar activity and various phenomena on Earth, such as climate relationships, geodetic disturbances, and radio wave propagation.

[0026] (2) The Canadian lynx dataset contains the number of lynx captured each year in the Mackenzie River region of northern Canada. The dataset records observations from 1821 to 1934, with a period of about 10 years and a total of 114 observations (e.g. Figure 6 as shown).

[0027] (3) The airline passenger data set corresponds to the number of international flights per month (in thousands), including data from January 1949 to December 1960 (e.g. Figure 8 As shown in the figure, the data size is 144 months.

[0028] (4) Star brightness datasets (such as Fig.10 The 1920 dataset (shown) contains daily star brightness observations at the same location and time (probably the same time every day), totaling 600 observation points.

[0029] (5) The exchange rate dataset contains daily fixed exchange rate data of the U.S. dollar against the Mexican peso (USD / MXN) from August 2017 to October 2019, and is presented monthly in the form of arithmetic means. A total of 27 months of monthly arithmetic means are used as observations (e.g. Fig.12 as shown).

[0030] During the experiment, in order to evaluate the performance of different time series forecasting models, each data set was divided into a training set and a test set. The training data was used to train the model, while the test data was used to evaluate the performance of the trained model. In order to adjust the hyperparameters, 20% of the training data was selected as the validation set so that real-time adjustment and evaluation could be performed during the model training process.

[0031] The ACF-OWA and PACF-OWA coefficients of the five data sets are given, such as Figure 3 As shown, from Figure 3 It can be seen that after adding the OWA operator, most of the positive values ​​in the autocorrelation and partial autocorrelation coefficients of each data set are larger, and the positive correlation is enhanced, while the negative values ​​are smaller, and the negative correlation is weakened. Compared with the ARIMA model, the ARIMA-OWA model has a stronger dependence between different time points on these five time series data sets.

[0032] The variance of the error terms of ARIMA and ARIMA-OWA on the five data sets can be calculated, and the calculation results are shown in Table 1.

[0033] Table 1. Error variance table of each data set

[0034] From the comparison in Table 1, it can be seen that the error variance of the ARIMA-OWA method is smaller than that of the ARIMA method on the five given data sets, which indicates that the results obtained by using ARIMA-OWA to calculate time series data are relatively more stable than those obtained by ARIMA, so that a prediction more favorable to the linear part of the data can be obtained. The ARIMA-OWA-ANN proposed in the present invention uses the ARIMA-OWA method to obtain the predicted data. Compared with the ARIMA-ANN prediction model proposed by Buyuksahin and Ertekin, the ARIMA-OWA-ANN proposed in the present invention improves the stability of ARIMA.

[0035] On the five data sets given above, experiments were conducted using ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy, Buyuksahin-Ertekin, and ARIMA-OWA-ANN time series prediction models. During the experiment, when solving the most suitable ARIMA model parameters, the maximum lag order was set to 20, and then the simplest ARIMA model parameters were solved by looping. In addition, since there is no recognized method for selecting ANN network parameters, this embodiment uses an empirical method to adjust the ANN network parameters

[10] . The Adam optimization algorithm is then used to train the neural network and initialize a suitable learning rate (lr), which determines the update speed when the model is initialized. During the ANN learning process, the weights are modified during the iteration process until the value of the L1 loss function is minimized. In order to prevent the iteration from entering an infinite loop and to achieve acceptable time efficiency, the number of iterations is set to 500 during the implementation process, that is, the number of epochs is 500. The transfer function used in the hidden layer of the ANN network is LeakReLU. During the implementation, their MAE, MSE, and MASE performances were compared and analyzed. From Table 2, it can be seen that the MAE, MSE, and MASE performances of ARIMA-OWA are better than ARIMA. At the same time, the MAE, MSE, and MASE performances of ARIMA-OWA-ANN are better than ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy, and Buyuksahin-Ertekin. Therefore, the ARIMA-OWA-ANN proposed in the present invention has the best performance. The performance comparison of all data sets is shown in Table 2.

[0036] Table 2. Performance comparison of all datasets

[0037] The sunspot dataset has 288 observations (such as Figure 4 As shown in Figure 2, it is considered to be nonlinear and non-Gaussian and is often used to evaluate many linear and nonlinear models. In order to evaluate the prediction performance of the proposed hybrid model, the sunspot dataset is divided into two samples: a training set and a test set. The training dataset, 221 observations from 1700 to 1920, is specifically used to build and train the model, and then the test dataset, the last 67 observations from 1921 to 1987, is used to evaluate the performance of the model.

[0038] In the sunspot dataset, similar to other studies, when the model is a single ARIMA, the present invention sets the ARIMA model parameters to 9 (AR(9)); when the model is a single ANN, a 4x4x1 neural network architecture is used, consisting of 4 input nodes, 4 hidden nodes, and 1 output node.

[0039] In the method proposed by the present invention, the length of the MA filter is 15, and the linear component can be extracted. At this time, the ADF test of the time series data of sunspots can be solved to be 0.003. Because 0.003 is much smaller than the threshold value of 0.05, the time series of sunspots is stable. When the ARIMA-OWA model is used, the same order 9 (AR (9)) as the ARIMA model is used. During the experiment, when the network architecture of the hybrid method is 8x6x1, the best fitting neural network model is obtained. The neural network model has 8 nodes in the input layer, 4 of which are past observations, 1 node is used for the prediction results of the linear component, and 3 are residual sequences. The network architecture of other hybrid methods corresponds to the Buyuksahin-Ertekin method.

[0040] The performance indicators of the proposed sunspot dataset model are shown in Table 2. It can be seen from Table 2 that: (1) ARIMA and ANN have the lowest performance compared with other hybrid methods, which shows that the use of a single model cannot fully handle all the features in the time series; (2) The ARIMA-OWA method significantly improves the performance of the ARIMA model, which shows that the linear model has a better ability to handle linear components; (3) The hybrid methods of Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin do produce better results. However, the prediction performance they produce is lower than that of the ARIMA-OWA-ANN method. Specifically, in terms of MAE indicators, ARIMA-OWA-ANN improves by 25.6%, 18.4%, 16.6%, 22.0%, 4.1%, 11.7% and 2.8% compared with ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin methods, respectively. In terms of MSE, ARIMA-OWA-ANN improves by 40.3%, 36.7%, 33.4%, 30.3%, 7.8%, 19.8% and 3.9% compared with ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin methods, respectively. In terms of MASE, ARIMA-OWA-ANN improves by 24.8%, 18.4%, 16.6%, 22.0%, 4.1%, 11.7% and 2.8% compared with ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin methods, respectively. Figure 5 A comparison of the actual and predicted values ​​of eight time series prediction methods on the sunspot dataset is given.

[0041] The lynx dataset has 114 observations (e.g. Figure 6 As shown in Figure 1, the first 100 observations from 1821 to 1920 were used to train the model as the training set, and the last 14 observations from 1921 to 1934 were used to evaluate the performance of the model as the test set. As in other studies, the logarithm of the data (base 10) was used in the analysis. When the ARIMA model is used as a single model, the present invention also sets the ARIMA model parameters to 12 (AR(12)) like Khashei-Bijari and Khashei-Bijari. When the ANN model is used alone, a 7x5x1 neural network architecture is used.

[0042] In the method proposed in the present invention, when the length of the fixed MA filter is 5, the linear component is extracted. At this time, the ADF test value of the lynx time series data can be solved to be 0.0019, which is much smaller than the threshold value of 0.05, indicating that the data set is stable. The ARIMA-OWA model has an order of 12 (AR(12)), and the architecture adopted by the ANN is 9x9x1, specifically: the input layer consists of 9 nodes, 5 of which are past observations, 1 node is used for the prediction results of the linear component, and 3 are residual sequences.

[0043] The performance indicators of the Lynx dataset model are shown in Table 2. As can be seen from Table 2: (1) The performance of the ARIMA-ANN hybrid method is higher than that of the single model ARIMA and ANN; (2) The ARIMA-OWA method significantly improves the performance of the traditional ARIMA model; (3) The performance indicators of the ARIMA-OWA-ANN method are the best. In terms of MAE indicators, ARIMA-OWA-ANN improves by 27.2%, 24.4%, 15.1%, 16.9%, 12.5%, 16.3% and 6.3% compared with ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin methods respectively. In terms of MSE, ARIMA-OWA-ANN improves by 34.3%, 33.2%, 21.9%, 13.5%, 9.7%, 15.0%, and 5.4% compared with ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy, and Buyuksahin-Ertekin methods, respectively. In terms of MASE, ARIMA-OWA-ANN improves by 27.2%, 24.4%, 15.1%, 16.9%, 12.5%, 16.3%, and 6.3% compared with ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy, and Buyuksahin-Ertekin methods, respectively. Figure 7 A comparison between the actual and predicted values ​​of eight time series forecasting methods on the lynx dataset is given.

[0044] The size of the aviation dataset is 144 months of observations (such as Figure 8 The first 114 months of data from January 1948 to July 1958 were used as training data, and the last 29 months from August 1958 to December 1960 were used as test data to test the performance of the model. The data were processed using natural logarithms when analyzed.

[0045] When the linear model is a separate ARIMA model, the experiment found that the order AR(13) is the simplest of all ARIMA models. At the same time, after repeated experimental tuning, when the nonlinear model is a separate ANN, the ANN in ARIMA-ANN uses 40x20x1, which is the most suitable neural network architecture.

[0046] In the method proposed by the present invention, when the length of the fixed MA filter is 27, the linear component is extracted. The ADF test value of the aviation data set is 0.002, which is much less than 0.05, and the data set is stationary. The order of the ARIMA-OWA model is 13 (AR(13)). According to the experiment: when the network architecture in the ARIMA-OWA-ANN model is 40x40x1, the best fitting neural network model is obtained. The neural network model consists of 40 nodes in the input layer, 39 of which are past observations, and 1 node is used for the prediction results of the linear component. The residual sequence becomes 0 during the tuning process. In addition, the network system and component allocation of the Khashei-Bijari and Buyuksahin-Ertekin methods are the same as those of the ARIMA-OWA-ANN method, the network structure of the Zhang method is 40x20x1, and the architecture of the Babu-Reddy method is 18x18x1.

[0047] The performance indicators of the proposed models for the aviation dataset are shown in Table 2. It can be seen from Table 2 that: (1) In this dataset, among the methods used alone, ANN has better accuracy than ARIMA compared to the sunspot dataset and lynx dataset. This may be because the aviation dataset is more stable than the sunspot dataset and lynx dataset; (2) The performance of the ARIMA-ANN hybrid method is lower than that of the ANN model, for example, the Zhang and Babu-Reddy methods; (3) The ARIMA-OWA method improves the performance of the traditional ARIMA model; (4) The ARIMA-OWA-ANN performs best, as shown in the following: In terms of MAE, ARIMA-OWA-ANN improves by 26.8%, 65.5%, 43.1%, 33.3%, 6.7%, 59.3% and 5.9% compared to ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin methods, respectively. In terms of MSE, ARIMA-OWA-ANN improves by 42.3%, 87.3%, 64.2%, 56.5%, 12.4%, 83.5% and 10.0% compared with ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin methods, respectively. In terms of MASE, ARIMA-OWA-ANN improves by 26.8%, 65.5%, 43.1%, 33.3%, 6.7%, 59.3% and 5.9% compared with ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin methods, respectively. Fig. 9 The comparison between the actual and predicted values ​​of 8 decision methods on aviation data sets is given.

[0048] The star brightness dataset has 600 observations (such as Fig.10 The first 480 data are used as training data, and the last 120 data are used as test data to test the performance of the model. When the linear model is a separate ARIMA model, the experiment found that the order AR(7) is the simplest of all ARIMA models. At the same time, after repeated experimental tuning, when the nonlinear model is a separate ANN, the 8x15x1 neural network architecture is the most suitable.

[0049] In the method proposed in the present invention, the linear component is extracted when the length of the MA filter is fixed to 5. The ADF test value of the given star brightness data set is , much less than 0.05, the data set is stable. The order of ARIMA-OWA is 7 (AR (7)). According to the experiment, when the network architecture of ARIMA-OWA-ANN is 19x19x1, the best fitting neural network model is obtained. The input layer consists of 19 nodes, 18 of which are past observations, and 1 node is used for the prediction results of the linear component. The residual sequence becomes 0 during the tuning process. In addition, the network system and component allocation of the Khashei-Bijari and Buyuksahin-Ertekin methods are the same as those of ARIMA-OWA-ANN. The ANN network structure of the Zhang method is 8x15x1, and the ANN network architecture of the Babu-Reddy method is 4x7x1.

[0050] The performance indicators of the stellar brightness data set are shown in Table 2. It can be seen from Table 2 that: (1) except for the Babu-Reddy hybrid method, the performance of the hybrid method is better than that of the single ARIMA and ANN methods; (2) the ARIMA-OWA method improves the performance of the traditional ARIMA model; (3) ARIMA-OWA-ANN has the best performance. Specifically, in terms of MAE indicators, ARIMA-OWA-ANN improves by 27.1%, 25.2% and 23.0%, 19.1%, 3.3%, 37.5% and 2.6% compared with ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin methods, respectively. In terms of MSE, ARIMA-OWA-ANN improves by 47.5%, 48.2%, 43.3%, 40.8%, 6.2%, 59.2% and 3.4% compared with ANN, ARIMA and ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin methods, respectively. In terms of MASE, ARIMA-OWA-ANN improves by 27.1%, 25.1% and 23.0%, 19.1%, 3.2%, 37.5% and 2.6% compared with ANN, ARIMA and ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin methods, respectively. Fig.11 A comparative analysis of the actual and predicted values ​​of 8 methods on the stellar brightness dataset is given.

[0051] The exchange rate dataset has a total of 27 months of monthly arithmetic means as observations (such as Fig.12 As shown in the figure, the observations from August 2017 to July 2019 were used to train the model as the training set, and the observations from the last 3 months were used to evaluate the performance of the model as the test set.

[0052] When the linear model is a separate ARIMA model, experiments show that the order AR(4) is the best among all ARIMA models. At the same time, when the nonlinear model is a separate ANN, the 10x2x1 neural network architecture is the most suitable.

[0053] In the method proposed in the present invention, when the length of the MA filter is 10, the linear component is extracted. The ADF test of the given exchange rate data set is 0.042, which is much less than 0.05, and the data is stationary. The ARIMA-OWA order is: 4 (AR(4)). According to the experiment, when the network architecture of ARIMA-OWA-ANN is 7x2x1, the best fitting neural network model is obtained, and the input layer consists of 7 nodes, 4 of which are past observations, 1 node is used for the prediction results of the linear component, and 2 are residual sequences. In addition, the network system and component allocation of the Khashei-Bijari and Buyuksahin-Ertekin methods are the same as those of the ARIMA-OWA-ANN method, the network structure of the Zhang method is 10x2x1, and the architecture of the Babu-Reddy method is 4x6x1.

[0054] The performance indicators of the proposed exchange rate dataset model are shown in Table 2. It can be seen from Table 2 that: (1) the Babu-Reddy method has limited model performance for this dataset, and the performance is lower than that of the single model, but the performance of other hybrid methods is still higher than that of the ARIMA or ANN model; (2) the ARIMA-OWA method also improves the performance of the traditional ARIMA model; (3) the performance indicators of ARIMA-OWA-ANN are better than those of GARCH-OWA. The best MAE and MSE results of the GARCH-OWA model are 0.0149 and 0.0006 respectively. The performance of ARIMA-OWA-ANN method is much higher than that of GARCH-OWA; (4) ARIMA-OWA-ANN method has certain advantages over existing time series prediction methods, which are reflected in the following aspects: In terms of MAE index, ARIMA-OWA-ANN improves by 30.0%, 35.7%, 29.2%, 21.7%, 70.6%, and 6.2% compared with ANN, ARIMA, ARIMA-OWA, Zhang, Babu-Reddy, and Buyuksahin-Ertekin methods, respectively. In terms of MSE index, ARIMA improves by 61.2%, 51.7%, 61.4%, 47.6%, 18.3%, 91.7%, and 6.1% compared with ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy, and Buyuksahin-Ertekin methods, respectively. On MASE, ARIMA-OWA-ANN improves by 30.0%, 35.7%, 29.2%, 21.7%, 70.6%, and 6.2% compared to ANN, ARIMA and ARIMA-OWA, Zhang, Babu-Reddy, and Buyuksahin-Ertekin methods, respectively.

[0055] For these index evaluation methods, the present invention will use the random (function) in the Python programming language to randomly generate 100 data to construct a data set with a value range of 0-1. In the case of one-step prediction, the prediction range is 10. According to the experiment, the order of the ARIMA model is set to AR (3, 0, 0). In addition, the network structure of the nonlinear model ANN is 3x3x1, and the ANN network structure of the hybrid model is set to 4x9x1. The obtained data are modeled using ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy and Buyuksahin-Ertekin methods, and the proposed ARIMA-OWA-ANN method, and their MAE, MSE and MASE evaluation values ​​are calculated respectively. During the experiment, the filter length was fixed to 16, and the stationarity test (ADF) result of the modified data set was 0.0015, which is much less than the threshold value of 0.05. For the Khashei-Bijari method, Buyuksahin-Ertekin method and the proposed ARIMA-OWA-ANN method, the number of input nodes is 4, of which 3 are original data values ​​and one is the prediction result of the linear component. The number of nonlinear residual sequences becomes 0 during the tuning process. The performance measurement results obtained with this data are shown in Table 3. The one-step prediction results are shown in Fig.13 shown.

[0056] Table 3. Performance comparison of different methods for randomly generated data

[0057] From Table 3 and Fig.13 It can be seen that the MAE, MSE, and MASE performance of ARIMA-OWA are better than ARIMA. The MAE, MSE, and MASE performance of ARIMA-OWA-ANN are better than those of ANN, ARIMA, ARIMA-OWA, Zhang, Khashei-Bijari, Babu-Reddy, and Buyuksahin-Ertekin methods. Therefore, the ARIMA-OWA-ANN hybrid model developed by the present invention can achieve better performance on random time series.

[0058] Several important results were obtained in the experiments of the present invention: (1) When using a single model, the prediction performance of the ARIMA model is better than that of the ANN model in the sunspot dataset and the lynx dataset. However, the opposite is true for the aviation dataset and the star brightness dataset. (2) In all datasets, the ARIMA-OWA model proposed in the present invention significantly improves the prediction performance of the ARIMA model (e.g. Fig.14(as shown), because the model enhances the dependence of time series data between different time points and improves the model's ability to handle data stability, so that the model can better capture the long-term trend changes and linear relationships between data, thereby improving the accuracy and stability of model predictions; (3) When data fluctuations are large, hybrid methods may have better performance than single methods. When special data values ​​appear in the data, the assumptions made by other hybrid methods will reduce the prediction performance. For example, the Zhang and Babu-Reddy method assumes that the relationship between linear components and nonlinear components is additive, which reduces the prediction performance compared with the ANN model in the aviation data set and the star brightness data set. In addition, the Buyuksahin-Ertekin method combines the Khashei-Bijari and Babu-Reddy methods. Although it overcomes the risk of reduced performance caused by some assumptions, it will be more or less restricted by some situations in the time series data, such as data stability, dependence between data, etc., thus affecting the prediction results; (4) The ARIMA-OWA-ANN time series prediction method has the best performance. To this end, the present invention fully considers the problems of stationarity and dependence in the data, and solves these problems through innovative methods, thereby creating a model with high versatility and flexibility, and has obvious advantages over other methods in this field.

[0059] In order to further analyze the ARIMA-OWA-ANN hybrid time series prediction method, the average values ​​of MAE, MSE, and MASE of the 8 time series prediction methods in Table 2 on the 5 data sets given above were statistically analyzed. The results are as follows: Fig.14 shown.

[0060] from Fig.14 It can be seen that the average MAE, MSE, and MASE of the ARIMA-OWA-ANN model on the five data sets are all the smallest, and therefore the best. At the same time, the average MAE, MSE, and MASE of the ARIMA-OWA model on the five data sets are smaller than those of ARIMA, so ARIMA-OWA is better than ARIMA. At the same time, compared with ARIMA-OWA, it can be seen that the data sequence processed by the OWA operator is relatively stable and more conducive to the establishment of a mixed model. The ARIMA model has good adaptability to fixed-order and high-frequency data, but poor adaptability to non-stationary and sensitive data. The ARIMA-OWA-ANN model proposed in the present invention eliminates these limitations, uses OWA to process linear components, improves the stability of time series data, and the use of ANN improves the adaptability of the model to the data and the accuracy of prediction.

[0061] It should be understood that, although each step in the flow chart of each embodiment of the present invention is shown in sequence according to the indication of the arrow, these steps are not necessarily performed in sequence according to the order indicated by the arrow. Unless there is a clear explanation in the present invention, the execution of these steps does not have strict order restrictions, and these steps can be performed in other orders. Moreover, at least a portion of the steps in each embodiment may include a plurality of sub-steps or a plurality of stages, and these sub-steps or stages are not necessarily performed at the same time, but can be performed at different times, and the execution order of these sub-steps or stages is not necessarily performed in sequence, but can be performed in turn or alternately with at least a portion of other steps or sub-steps or stages of other steps.

[0062] It should be understood that the various parts of the present invention can be implemented by hardware, software, firmware or a combination thereof. In the above-mentioned embodiments, multiple steps or methods can be implemented by software or firmware stored in a memory and executed by a suitable instruction execution system. For example, if implemented by hardware, as in another embodiment, it can be implemented by any one of the following technologies known in the art or their combination: a discrete logic circuit having a logic gate circuit for implementing a logic function for a data signal, a dedicated integrated circuit having a suitable combination of logic gate circuits, a programmable gate array (PGA), a field programmable gate array (FPGA), etc.

[0063] In the description of this specification, the description with reference to the terms "one embodiment", "some embodiments", "examples", "specific examples", or "some examples" means that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representation of the above terms does not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described may be combined in any one or more embodiments or examples in a suitable manner.

[0064] The above-mentioned embodiments only express several implementation methods of the present invention, and the description thereof is relatively specific and detailed, but it cannot be understood as limiting the scope of the patent of the present invention. It should be pointed out that, for ordinary technicians in this field, several variations and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention. Therefore, the protection scope of the patent of the present invention shall be subject to the attached claims.

Claims

1. A time series mixed model based on OWA operator, characterized in that: The OWA operator is used to perform weighted processing on the time series data and calculate the autocorrelation coefficient based on OWA; Given the autoregressive order and the moving average order, and obtaining the corresponding number of OWA-based autocorrelation coefficients according to the autoregressive order and the moving average order, the moment estimation method is used for parameter estimation to obtain the OWA-based autoregressive parameters and the OWA-based moving average parameters; Based on the ARIMA model, the ARIMA model parameters are replaced with the autoregressive parameters based on OWA and the moving average parameters based on OWA to obtain the ARIMA-OWA model; The maximum lag order is set, and the final autoregressive order, moving average order and difference order of the ARIMA-OWA model are determined by the cyclic traversal method. In each cyclic traversal process, the autoregressive parameters based on OWA and the moving average parameters based on OWA are dynamically adjusted according to the autoregressive order and the moving average order. After the cyclic traversal is completed, the final ARIMA-OWA model is obtained.

2. A time series hybrid model based on OWA operator according to claim 1, characterized in that: The OWA operator is used to perform weighted processing on time series data. The process of calculating the autocorrelation coefficient based on OWA has the following relationship: ； in, represents the autocorrelation coefficient based on OWA, Indicates the time series data t The weight vector corresponding to a large number, Represents the first t elements, Represents the number of elements in the time series, Indicates the time interval, represents the covariance based on OWA, represents the variance based on OWA, represents the OWA mean of the time series data, denote the autoregressive order and the moving average order respectively; The calculation process corresponding to the mean of OWA has the following relationship: ； in, Represents the first n elements, represents the OWA operator, Indicates the time series data i The weight vector corresponding to a large number, Indicates the time series data i a large number; Time series data i The weight vector calculation process corresponding to a large number has the following relationship: ； in, Respectively represent the mean and standard deviation of time series data. The calculation process corresponding to the mean and standard deviation of time series data has the following relationship: 。 3. A time series hybrid model based on OWA operator according to claim 2, characterized in that: Given an autoregressive order and a moving average order, and obtaining a corresponding number of autocorrelation coefficients based on OWA according to the autoregressive order and the moving average order, the moment estimation method is used to perform parameter estimation, and the autoregressive parameters based on OWA and the moving average parameters based on OWA are obtained. Specifically, the steps include: Given the autoregressive order and the moving average order, the corresponding number of autocorrelation coefficients based on OWA are obtained according to the autoregressive order and the moving average order to construct the Yule-Walker equation. The autoregressive parameters based on OWA are estimated using the Yule-Walker equation. The corresponding process has the following relationship: ； in, Indicates p The autoregressive coefficients based on OWA, represents the q+p+2th OWA-based autocorrelation coefficient; The moving average parameters based on OWA are calculated based on the obtained autoregressive parameters based on OWA and the moving average order; Among them, when the moving average order is 0, the moving average parameter based on OWA is calculated using the first relationship, which is as follows: ； in, Indicates that the moving average coefficient based on OWA is 0; When the moving average order is greater than or equal to 1, the moving average parameter based on OWA is calculated using the second relational expression, which is as follows: ； in, Indicates i Based on the OWA moving average coefficient, represents the estimate of the covariance based on OWA, represents the OWA covariance of the error vector, Represents the variance of the error. When the autoregressive order is 0, the variance of the error is calculated as follows: ； in, represents the estimate of the covariance based on OWA, ; The OWA covariance calculation process of the error vector has the following relationship: ； in, Represents random error.

4. The time series hybrid model based on OWA operator according to claim 3 is characterized in that: The second relationship is solved by linear iteration method, which specifically includes the following steps: Given and A set of initial values , ; Sets the iteration conditions for the OWA-based moving average parameters and the variance of the error. The iteration conditions include the first precision of positive real numbers. and second precision of positive real numbers ; The initial value , Substitute the second relation and solve it iteratively until it satisfies At the same time, the iteration is terminated. is the number of iterations, and the final output result is and The solution.

5. The time series hybrid model based on OWA operator according to claim 4 is characterized in that: Based on the ARIMA model, the ARIMA model parameters are replaced with the autoregressive parameters based on OWA and the moving average parameters based on OWA to obtain the ARIMA-OWA model. The corresponding process has the following relationship: ； in, represents the first OWA-based autoregressive coefficient, Indicates hysteresis p Time series data of order, Indicates hysteresis q The order error value, Indicates q Based on the OWA moving average coefficient.

6. A hybrid time series prediction method based on OWA operator, the method adopts the time series hybrid model based on OWA operator according to any one of claims 1 to 5, characterized in that: The method comprises the following steps: Step 1: Obtain time series data and decompose the time series data using an MA filter to obtain a linear component and a nonlinear residual sequence; Step 2, preprocessing the linear component to generate a linear component sequence, and defining the historical sequence in the linear component sequence as the OWA mean of the current sequence in the linear component sequence; Step 3: transform the ARIMA-OWA model according to the core assumptions of the ARIMA model to obtain a transformed model; initialize the historical error of the transformed model, and then substitute the linear component sequence to calculate the historical random error sequence; Step 4: Substitute the historical linear component series and the historical random error series into the ARIMA-OWA model to obtain the predicted linear variables; Step 5: Select a number of data from the time series data as the first time series; Selecting a number of data from the nonlinear residual sequence as the second time series; The first time series, the second time series and the predicted linear variable are sent to the ANN network to obtain the mixed prediction results of the time series.

7. A hybrid time series prediction method based on OWA operator according to claim 6, characterized in that: The time series data is decomposed using an MA filter to obtain linear components and nonlinear residual sequences. The corresponding process has the following relationship: ； ； in, Indicates t The linear component of the elements, represents the filter length, represents the nonlinear residual sequence, Represents time series data i elements, Indicates t elements.

8. The hybrid time series prediction method based on OWA operator according to claim 7 is characterized in that: According to the core assumptions of the ARIMA model, the ARIMA-OWA model is transformed to obtain the transformed model; the historical error of the transformed model is initialized, and then the linear component sequence is substituted to calculate the process corresponding to the historical random error sequence. The following relationship exists: ； in, represents the normalized lag p The linear component of order, It means the normalized t The linear component of the elements.

9. The hybrid time series prediction method based on OWA operator according to claim 8, characterized in that: Substituting the historical linear component series and the historical random error series into the ARIMA-OWA model to obtain the corresponding process of predicting linear variables has the following relationship: ； in, represents the predictor linear variable, Represents the ARIMA-OWA model.

10. A hybrid time series prediction method based on OWA operator according to claim 9, characterized in that: The first time series, the second time series and the predicted linear variable are sent into the ANN network to obtain the mixed prediction results of the time series. The corresponding process has the following relationship: ； in, Respectively represent the length of the selected data in the time series data and the nonlinear residual sequence, represents the first time series, represents the second time series, represents the nonlinear function of the neural network, Indicates Time series data, Indicates A nonlinear residual sequence.

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