Carbon emission transaction price prediction method based on MPA-VMD and QRMGM-KDE

Abstract

The invention discloses a carbon emission transaction price prediction method based on MPA-VMD and QRMGM-KDE, and the method comprises the steps: carrying out the decomposition of an original carbon price time sequence through MPA-VMD, and obtaining an optimal parameter and a sub-sequence; predicting a conditional quantile of each sub-sequence by utilizing a method of combining a minimum gated memory network and quantile regression; performing linear superposition on each sub-sequence according to a decomposition integration strategy to obtain an interval prediction result; a probability density function of the carbon price is estimated according to the conditional quantile of the carbon price by using a kernel density estimation method; and finally, performance verification is performed on five aspects of superiority of a decomposition method, point prediction precision, interval prediction performance, probability prediction comprehensive performance and training time of the proposed models through seven reference models. According to the method, a high-accuracy point prediction result, a reasonable prediction interval and a good probability distribution result can be obtained in a short time.

Description

Technical Field

[0001] The present invention belongs to the technical field of carbon price measurement, and specifically relates to a carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE. Background Art

[0002] Carbon emissions trading, referred to as carbon trading, is a market mechanism used to promote global greenhouse gas emissions reduction and reduce global carbon dioxide emissions. With the low-carbon transformation of energy, power system dispatching must meet the requirements of carbon emission reduction. If the price of carbon trading is too high, it will excessively increase the operating costs of the energy system and affect the enthusiasm of power generation companies; if the price of carbon trading is too low, the actual carbon emissions will not meet the corresponding carbon emission control requirements. Therefore, the carbon trading price is predicted when the power system is dispatched.

[0003] Existing carbon price forecasting methods are mainly divided into two categories, one is the traditional econometric model, and the other is the machine learning model. The econometric models mainly include the generalized autoregressive conditional heteroskedasticity model (GARCH), the autoregressive integrated moving average model (ARIMA) and exponential smoothing (ES). Econometric models have a traditional statistical basis and rely on a complete statistical theoretical basis, which has a positive significance for carbon price forecasting. However, these models rely heavily on assumptions about time series, such as normal distribution. Therefore, these models are difficult to handle complex and nonlinear time series data. In addition, the application scenarios of these models are mainly limited to specific economic management issues. Due to the limitations of econometric models, researchers have begun to turn to machine learning models to solve the problems of large volatility, high complexity and low prediction accuracy in carbon price forecasting. Due to the advantages of high prediction accuracy, good generalization performance and strong fitting ability of machine learning models, a large number of studies have shown that machine learning models have better performance in the field of carbon price forecasting than traditional econometric models. However, when the carbon price time series is highly random and nonlinear, the prediction ability of these models will perform poorly on the test set.

[0004] In recent years, with the rapid development of deep learning, many machine learning methods are not as good as deep learning performance. Recurrent neural networks (RNNs) are used to solve time series prediction problems because they can consider time series information. However, when encountering long sequences, RNNs cannot overcome the dependency problem of long sequences. In order to solve the above problems, researchers began to use long short-term memory networks (LSTMs) for carbon price prediction. Through experiments on different data sets, they found that LSTMs are reliable and stable in carbon price prediction. Today, LSTMs have multiple variants, among which the gated recurrent unit (GRU) simplifies the network structure and shortens the training time. The minimum gated memory network (MGM) has a simpler network unit than GRU, and has a shorter training time than LSTM and GRU without changing the prediction accuracy and reliability.

[0005] However, the above methods are all point prediction results, which are inevitably limited in real-world scenarios. It is difficult to describe the specific fluctuation range of the real carbon market and cannot reflect the uncertainty of the carbon market. If the point prediction misjudges the future market trend, it may cause huge economic losses. Therefore, interval prediction came into being, which can provide more valuable information, reduce uncertainty, and help market decision makers. Interval prediction is widely used. In addition to wind speed prediction and load prediction, it also has a good application effect in carbon price prediction. Probabilistic prediction can describe the uncertainty of the carbon market and is widely used in the field of wind power prediction. However, most decomposition and integration prediction frameworks often ignore interval prediction, and even fewer researchers pay attention to the probabilistic prediction of carbon prices. Probabilistic prediction can describe the fluctuation of carbon prices under uncertain environments, and can give financial investors more information to make multiple decisions and avoid the risks of single-point prediction results. However, the prediction results obtained by quantile regression are conditional quantiles of a series of carbon prices, which need to be estimated by probability density method to obtain the probability density function (PDF). Among them, the probability estimation method can be divided into parameter estimation method and non-parametric estimation method. Kernel density estimation (KDE) is a typical nonparametric estimation method. Compared with the parameter estimation method, it does not require prior assumptions when estimating the original data distribution.

[0006] Decomposition technology has gradually attracted the attention of researchers because of its significant prediction accuracy in other fields, and has been widely used in the study of complex carbon price time series. Decomposition technology decomposes the original time series into several subsequences with low complexity, which can remove the noise of the original time series, and then predicts each subsequence separately, and superimposes the prediction results to obtain the final prediction result. There are currently several widely used decomposition methods, including empirical mode decomposition (EMD), ensemble empirical mode decomposition (EEMD), adaptive noise ensemble empirical mode decomposition (CEEMDAN) and variational mode decomposition (VMD). Some literature combines complementary ensemble empirical mode decomposition (CEEMD), empirical wavelet transform (EWT) and machine learning algorithm model extreme learning machine (ELM) to predict carbon price data and obtain better prediction accuracy than a single model, proving that the combination of decomposition method and machine learning method can improve the prediction ability of carbon trading prices.

[0007] After research, it was found that these decomposition methods have certain limitations. EMD is too sensitive to the frequency of the decomposed data. EEMD attempts to solve the problem of modal aliasing by adding Gaussian white noise during the decomposition process, but there is no way to remove the added white noise. VMD shows strong robustness in frequency domain feature extraction and noise reduction, and can solve the problems caused by modal aliasing and endpoint effects. Some literature has proved through experimental research that VMD has good frequency resolution and strong noise resistance, and is more suitable for time series processing with large volatility. However, VMD requires preset hyperparameters. Some literature searches for the K value in VMD with the goal of reducing the reset error; some literature optimizes VMD through PSO, with the goal of reducing reconstruction error and complexity, and determines the values ​​of K and α to further improve the performance of VMD. Summary of the invention

[0008] The main purpose of the present invention is to overcome the shortcomings and deficiencies of the prior art and to provide a carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE. By utilizing MPA-VMD to decompose the original carbon price time series and proposing a method combining a minimum gated memory network and quantile regression, a highly accurate point prediction result, a reasonable prediction interval and a good probability distribution result can be obtained in a relatively short time.

[0009] In order to achieve the above object, the present invention adopts the following technical solutions:

[0010] In a first aspect, the present invention provides a method for predicting carbon emission trading prices based on MPA-VMD and QRMGM-KDE, comprising the following steps:

[0011] Initialize MPA parameters, use MPA to optimize VMD parameters, and obtain optimal VMD parameters; construct an MPA-VMD model based on the optimal VMD parameters, and use the MPA-VMD model to decompose the original carbon price time series to obtain multiple subsequences;

[0012] Construct a QR-MGM model and set relevant parameters. The QR-MGM model includes a minimum gated memory network and quantile regression. The minimum gated memory network includes multiple output nodes, each of which corresponds to a quantile. Use the QR-MGM model to divide multiple subsequences into intervals according to quantiles. Each interval includes several continuous time steps. Use the output nodes to predict the subsequences of each interval according to the time steps to obtain carbon price prediction samples at different quantile levels.

[0013] Linearly superimpose the subsequences in the same interval to obtain the carbon price interval prediction results;

[0014] The optimal bandwidth is determined, and kernel density estimation is performed on the carbon price prediction samples at different quantile levels to obtain the probability density of carbon prices.

[0015] As a preferred technical solution, the optimization of VMD parameters by using MPA includes:

[0016] Set the size of VMD parameters, which include the number of decomposition modes K and the quadratic penalty term α;

[0017] MPA is used to allocate the search space to predators and preys, and the calculation is iteratively updated in the search space. If the fitness of the prey is greater than that of the predator at the corresponding position, the prey will replace the corresponding predator until the preset termination condition is reached to obtain the optimal VMD parameters.

[0018] As a preferred technical solution, the MPA is used to allocate the search space to the predator and the prey, as shown in the following formula:

[0019]

[0020] Elite Matrix:

[0021] Prey Matrix:

[0022] Among them, Y i (t) represents the position vector in iteration number t, J i represents the objective function, D and N represent the total number of dimensions and variables respectively, and Y i,j ′ represents the best predator vector Y i ′jth dimension, Y i ′ is replicated N times to obtain the elite matrix.

[0023] As a preferred technical solution, the MPA-VMD model is constructed according to the optimal VMD parameters, comprising the following steps:

[0024] S21. Initialize the modal function of VMD using the MPA algorithm

[0025] S22, decomposing the signal using a VMD algorithm to obtain multiple intrinsic mode functions;

[0026] S23, performing denoising processing on each intrinsic mode function;

[0027] S24. The denoised modal function is embedded into the ADMM algorithm, and the complete VMD is obtained by optimizing in the Fourier domain.

[0028] As a preferred technical solution, the MPA-VMD model is used to decompose the time series of the original carbon price, including:

[0029] The MPA-VMD model is used to decompose the original carbon price time series into multiple components with different frequencies;

[0030] Introduce the quadratic penalty term α and the Lagrange multiplier λ for each component to obtain the modal component As follows:

[0031]

[0032] Where f(ω) represents the real signal with the center frequency, represents the modal component of the center frequency, χ(ω) represents the instantaneous amplitude of the center frequency, η represents the phase function, ω l represents the lth modal component The center frequency of

[0033] Each modal component is transformed using wavelet transform Perform denoising processing to obtain multiple subsequences.

[0034] As a preferred technical solution, the minimum gated memory network includes:

[0035] The forget gate and input gate are as follows:

[0036] Forget Gate t :f t =σ(net t )=σ(w h h t-1 +w x x t )

[0037] Input Gate I t :It =1-f t

[0038] where h t-1 Indicates the cell state at the previous moment, x t represents the input at the current moment, w h and w x are the corresponding weight parameters respectively;

[0039] The cell state at the current moment is as follows:

[0040] a t =tanh(net t )=tanh(w h h t-1 +w x x t )

[0041] Among them, tanh represents the activation function;

[0042] The output of the hidden layer at the current moment is as follows:

[0043] h t =f t h t-1 +a t I t

[0044] The output of the output layer at the current moment is as follows:

[0045] y t =σ(w y h t )

[0046] where y t is the output of the output layer, w y represents the weight of the output layer.

[0047] As a preferred technical solution, the quantile regression is as follows:

[0048]

[0049] in, is the value of the dependent variable at the τ quantile, and τ∈(0,1), x t Represents the input at the current moment, is the estimated value of the regression coefficient

[0050] The estimated values ​​of the regression coefficients It is obtained by minimizing the loss function L, as follows:

[0051]

[0052] Symmetric Function The formula is:

[0053]

[0054] Among them, y t represents the output of the output layer of the minimum gated memory network, and b represents the independent variable.

[0055] As a preferred technical solution, the optimal bandwidth is determined by grid search and cross-validation methods.

[0056] As a preferred technical solution, the kernel density estimation is performed on the carbon price range prediction results at different quantile levels, as shown in the following formula:

[0057]

[0058] Where B represents the bandwidth and is greater than zero, n is the total number of samples, and Z t,i Represents the prediction result Z of the tth carbon price interval t The ith carbon price prediction sample in , x represents the independent variable, and K(·) is the kernel function.

[0059] As a preferred technical solution, the kernel of the kernel function is an Epanechnikov function, as shown in the following formula:

[0060]

[0061] Among them, α represents the independent variable.

[0062] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0063] (1) The present invention decomposes the time series of the original carbon price by combining the variational mode decomposition (VMD) and the marine predator optimization algorithm (MPA). On the one hand, the VMD algorithm shows strong robustness in frequency domain feature extraction and noise reduction, and can solve the problems caused by modal aliasing and endpoint effects. On the other hand, the use of MPA and VMA can achieve complementary advantages, that is, the global search capability of the MPA algorithm can assist the VMD algorithm to better find the optimal solution, while the local search capability of the VMD algorithm can refine the search results of the MPA algorithm, thereby further improving the efficiency and accuracy of signal processing.

[0064] (2) The present invention proposes a non-parametric interval prediction model QR-MGM for predicting carbon prices. The model is based on the minimum gated memory network MGM and combined with quantile regression QR to achieve non-parametric interval prediction of carbon prices. At the same time, the present invention proposes a minimum gated memory network to simplify the network structure and reduce the training time without reducing the prediction accuracy.

[0065] (3) The kernel density estimation (KDE) method is introduced to generate the probability prediction results of carbon prices. The optimal bandwidth of KDE is determined through grid search and cross-validation methods, and Epanechnikov is used as the kernel function to achieve the probability prediction of carbon prices. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the drawings required for use in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work.

[0067] Figure 1 This is a flow chart of a method for predicting carbon emission trading prices based on MPA-VMD and QRMGM-KDE according to an embodiment of the present invention;

[0068] Figure 2 A schematic diagram of IMF components after decomposition according to an embodiment of the present invention;

[0069] Figure 3 Schematic diagram of training and prediction of the QR-MGM model according to an embodiment of the present invention;

[0070] Figure 4 A schematic diagram showing the comparison between the prediction results of each model and the true value in the embodiment of the present invention;

[0071] Figure 5 This is a point prediction comparison diagram of the prediction model of the embodiment of the present invention;

[0072] Figure 6 This is a schematic diagram of interval prediction results according to an embodiment of the present invention;

[0073] Figure 7 These are nine probability density curves obtained by equidistant sampling in the Hubei carbon price market using the MPA-VMD-QRMGM model of an embodiment of the present invention. DETAILED DESCRIPTION

[0074] In order to enable those skilled in the art to better understand the present application, the technical solutions in the embodiments of the present application will be clearly and completely described below in conjunction with the drawings in the embodiments of the present application. Obviously, the described embodiments are only part of the embodiments of the present application, rather than all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without making creative work are within the scope of protection of the present application.

[0075] Reference to "embodiments" in this application means that a particular feature, structure, or characteristic described in conjunction with the embodiments may be included in at least one embodiment of the present application. The appearance of the phrase in various locations in the specification does not necessarily refer to the same embodiment, nor is it an independent or alternative embodiment that is mutually exclusive with other embodiments. It is explicitly and implicitly understood by those skilled in the art that the embodiments described in this application may be combined with other embodiments.

[0076] Marine Predators Optimization Algorithm, commonly referred to as Marine Predators Algorithm (MPA), is a new meta-heuristic optimization algorithm.

[0077] Minimal Gated Memory Network (MGM) is a neural network architecture that has certain advantages in processing sequence data and performing time series prediction tasks.

[0078] Quantile regression (QR) is a statistical modeling technique used to estimate the linear relationship between a set of regressors X and the quantiles of the explained variable Y.

[0079] Example 1

[0080] See also Figure 1 This embodiment provides a method for predicting carbon emission trading prices based on MPA-VMD and QRMGM-KDE, comprising the following steps:

[0081] S1. Data preprocessing: The first step is to input the original carbon price data into VMD. The second step is to initialize the MPA parameters, set the number of decomposition modes K and the size of the quadratic penalty α. The third step is to optimize the parameters K and α according to (1.1)-(1.3) to determine whether the iteration termination condition Max_iteration is reached. The fourth step is to obtain the optimal solution if the conditions of the third step are met. Otherwise, execute the next iteration until the termination condition is met. Through the above four steps, the optimal parameters K and α of VMD are obtained, and the original carbon price time series is decomposed into multiple subsequences through MPA-VMD, namely IMF1, IMF2, ..., IMF K ,like Figure 2 shown.

[0082] In particular, in the marine predator optimization method, the search space is randomly divided into predators and preys, and its mathematical formula is expressed as follows:

[0083]

[0084] The elite matrix E is a matrix composed of the best predators, and the prey matrix is ​​represented by P. Their formulas are as follows:

[0085] Elite Matrix:

[0086] Prey Matrix:

[0087] Among them, Y i (t) represents the position vector in iteration number t, J i represents the objective function, D and N represent the total number of dimensions and variables respectively, and Y i,j ′ represents the best predator vector Y i ′jth dimension, Y i ′ is replicated N times to obtain the elite matrix.

[0088] In particular, this embodiment adopts a new non-recursive signal decomposition method in the process of decomposing subsequences. VMD can decompose the original carbon price time series signal into multiple smooth components of different frequencies, reducing the complexity of the time series. In order to minimize the sum of the bandwidth of each decomposed subsequence, VMD decomposes the original signal into several different modes. VMD can be described by the following formula:

[0089]

[0090] in, and ωl are the lth modal component and its center frequency, respectively, γ(t) represents the Dirac function, t represents time, θ t represents partial derivative, represents the modal function, and f represents the augmented Lagrangian function. In order to make the above constraint problem unconstrained, the quadratic penalty term α and the Lagrangian multiplier λ are introduced, and the generalized Lagrangian is formed by combining the two. Its mathematical formula is expressed as follows:

[0091]

[0092] Where f(ω) represents the real signal with the center frequency, represents the modal component of the center frequency, χ(ω) represents the instantaneous amplitude of the center frequency, and η represents the phase function.

[0093] It should be explained that the Lagrange multiplier method refers to a method for finding the extreme value of a multivariate function whose variables are subject to one or more conditions. This method converts an optimization problem with n variables and k constraints into an extreme value problem of a system of equations with n+k variables, whose variables are not subject to any constraints. In variational mode decomposition (VMD), the Lagrange multiplier method is used when it is necessary to solve a minimization problem under certain conditions. By introducing Lagrange multipliers, an optimization problem with constraints can be converted into an unconstrained optimization problem, thereby simplifying the solution process.

[0094] In this way, the solution of the original constrained variational problem is transformed into a saddle point that gradually enhances the Lagrangian in a series of iterative optimization processes. The process of gradually alternating to find a new Lagrangian saddle point decomposes the original signal into multiple modal components. The acquisition is as follows:

[0095]

[0096] The above optimization algorithm is embedded into the ADMM algorithm, and the complete VMD is obtained by optimizing in the Fourier domain.

[0097] The combination of MPA algorithm and VMD algorithm can achieve complementary advantages. The global search capability of MPA algorithm can assist VMD algorithm to better find the optimal solution, while the local search capability of VMD algorithm can refine the search results of MPA algorithm. This combination can further improve the efficiency and accuracy of signal processing.

[0098] Specifically, the steps for constructing the MPA-VMD model according to the optimal VMD parameters are as follows:

[0099] 1. Initialize the mode function of VMD using the MPA algorithm.

[0100] 2. Use the VMD algorithm to decompose the signal into the sum of multiple intrinsic mode functions.

[0101] 3. Perform denoising on each intrinsic mode function, for example, using methods such as wavelet transform.

[0102] 4. The denoised intrinsic mode functions are reassembled into signals, that is, the optimization algorithm is embedded in the ADMM algorithm, and the complete VMD is obtained by optimizing in the Fourier domain.

[0103] 5. Repeat steps 2-4 until the stop condition is reached.

[0104] S2, such as Figure 3 As shown in the figure, the training and prediction of the QR-MGM model: construct the QR-MGM model and set the relevant parameters, train and predict the multiple subsequences after decomposition, obtain the prediction results at different quantile levels, and thus obtain the point prediction results.

[0105] Specifically, it is necessary to first build a QR-MGM model and set relevant parameters, wherein the QR-MGM model includes a minimum gated memory network and quantile regression, wherein the minimum gated memory network includes multiple output nodes, each output node corresponding to a quantile. This neural network structure is used to perform regression predictions on different quantiles of the data, rather than just predicting the mean. By training the QR-MGM model, prediction results at different quantiles can be obtained, thereby better understanding the distribution of the data.

[0106] In this embodiment, the mathematical calculation process of the unit structure of the minimum gated memory network MGM is as follows:

[0107] (1) The formulas for the forget gate and input gate are and

[0108] f t =σ(net t )=σ(w h h t-1 +w x x t )(0.1)

[0109] I t =1-f t (0.2)

[0110] where h t-1 Indicates the cell state at the previous moment, x t represents the input at the current moment, w h and w x are the corresponding weight parameters respectively.

[0111] (2) The cell state at the current moment can be expressed by the following formula:

[0112] a t =tanh(net t )=tanh(w h h t-1 +w x x t )(0.3)

[0113] Tanh represents the activation function.

[0114] (3) The output calculation formula of the hidden layer at the current moment is:

[0115] h t =f t h t-1 +a t I t (0.4)

[0116] (4) The output calculation formula of the output layer at the current moment is:

[0117] y t =σ(w y h t )(0.5)

[0118] where y t is the output of the output layer, w y It represents the weight of the output layer.

[0119] The linear regression QR model is expressed as follows:

[0120]

[0121] is the value of the dependent variable at the τ quantile, and τ∈(0,1), β(τ) is the regression coefficient. The estimated value of β(τ) It can be obtained by minimizing the loss function L:

[0122]

[0123] Symmetric Function The formula is:

[0124]

[0125] Here, b represents the independent variable.

[0126] Finally, the estimated value of the dependent variable at the τ quantile is obtained through the QR model:

[0127]

[0128] Next, the QR-MGM model is used to divide multiple subsequences into intervals according to quantiles. Each interval includes several continuous time steps. The output nodes are used to predict the subsequences in each interval according to the time steps to obtain carbon price prediction samples at different quantile levels.

[0129] For step S2, QR and MGM are combined to predict each component to obtain prediction results under different quantile conditions. Specifically, the QR-MGM model divides the time series into different intervals, each of which contains several continuous time steps. In order to predict the values ​​of different quantile intervals, the model introduces multiple output nodes in the output layer of MGM, each node corresponds to a quantile, so that the values ​​of different quantiles can be predicted at the same time. In general, the output prediction result is the mean prediction value, that is, the prediction value at the 0.5 quantile level. The quantile can be set as needed. In general, a 90% or 95% confidence interval is selected, that is, an interval with an upper bound of 0.95 and a lower bound of 0.05 quantiles.

[0130] S3. Linearly superimpose the subsequences in the same interval to obtain the carbon price interval prediction result.

[0131] In this embodiment, according to the interval divided in step S2, the subsequences in the interval are linearly superimposed, and then the same operation is performed on each interval, so that the carbon price interval prediction results at different quantile levels can be obtained. For example, the prediction results of the 8 subsequences obtained at the 0.95 quantile level are superimposed, and the same operation is performed at each quantile.

[0132] S4. Determine the optimal bandwidth, perform kernel density estimation on the carbon price prediction samples at different quantile levels, and obtain the probability density of carbon prices.

[0133] Bandwidth is very important in KDE estimation. Too small bandwidth will generate excessive noise and affect accuracy, while too large bandwidth will lead to estimation deviation. Therefore, in order to obtain the optimal bandwidth, this embodiment adopts a grid search algorithm and cross-validation.

[0134] After obtaining the optimal bandwidth, the QR model can only obtain the predicted values ​​at different quantile levels. By introducing kernel density estimation, the probability density function (PDF) can be obtained. Kernel density estimation is a classic non-parametric estimation method that does not require prior assumptions. The predicted values ​​at different conditional quantiles obtained by QRMGM are: By using kernel density estimation, Z t The probability density function of is:

[0135]

[0136] Where B represents the bandwidth and is greater than zero, n is the total number of samples, As a kernel function, this embodiment uses Epanechnikov as the kernel function kernel, and the calculation formula is:

[0137]

[0138] Kernel density estimation deals with point prediction results. Different quantiles are not considered here. It is to use kernel density estimation to obtain probability density functions for different subsequences, and then add the probability density functions of these 8 subsequences to obtain the probability density function of the final prediction result.

[0139] Example 2

[0140] Data source: The data comes from Hubei Carbon Emission Trading Center (https: / / www.hbets.cn / .html). From it, a total of 1,626 trading days of data from Hubei carbon market from April 5, 2017 to March 18, 2024 were selected for analysis, of which 80% were training sets and 20% were validation sets.

[0141] The data preprocessing based on MPA-VMD adopts: MPA-VMD processes the carbon price series, decomposes it into multiple subsequences, mines the main features and eliminates the noise in the original sequence. First, set the population size to 10, the maximum number of iterations to 10, and use the minimum value of the envelope entropy as the fitness function to find the optimal solution of K and α. As the number of iterations increases, the envelope entropy gradually decreases. The 9th iteration reaches the minimum value and stabilizes at 2.8925. At this time, the intrinsic mode component has more characteristic information and less noise. Through a large number of experiments, the optimal parameter combination is obtained: K=8, α=100. The carbon price series is decomposed to obtain 8 IMF components, and the noise is extracted from the remaining components, such as Figure 2 .

[0142] Comparative experiment of decomposition methods: In this experiment, based on the model of QRMGM framework, different decomposition methods are combined for comparative experiments. Different decomposition methods include EMD, EEMD, CEEMDAN, VMD and MPA-VMD. This experiment is mainly to verify the superiority of MPA-VMD decomposition strategy. The experimental results are as follows Figure 4, the figure shows the comparison between the prediction results of each model and the true value, where the black curve is the true carbon price series, the blue curve is the QRMGM model without the decomposition method, the orange curve is the EMD-QRMGM model, the pink curve is the EEMD-QRMGM model, the yellow curve is the CEEMDAN-QRMGM model, the purple curve is the VMD-QRMGM model, and the green curve is the MPA-VMD-QRMGM model. It can be seen that the curve of the EEMD-QRMGM model deviates the most from the true value curve at most time points, and the CEEMDAN-QRMGM model deviates the most from the true value. The prediction results of other models are close to the true value, but the prediction results of the MPA-VMD-QRMGM model are basically consistent with the true carbon price series. The figure also shows the comparison of the two error evaluation indicators of the model. It can be seen that the RMSE and MAPE of the EEMD-QRMGM model are the largest, followed by the CEEMDAN-QRMGM model. Their error values ​​even exceed the QRMGM model without the decomposition technology. In addition, the MAPE of the EMD-QRMGM model is slightly higher than that of the QRMGM model. This is because EMD has the problem of modal aliasing. Although EEMD solves the problem of modal aliasing by adding white noise, it cannot eliminate white noise during the decomposition process, so it cannot avoid being affected by white noise, which reduces the prediction accuracy. CEEMDAN is sensitive to noise other than Gaussian white noise during the decomposition process, and the decomposition results are easily affected. There will be modal aliasing problems, and different modes will affect each other, which will reduce the prediction effect. The error values ​​of the VMD-QRMGM model and the MPA-VMD-QRMGM model are smaller, and the MPA-VMD-QRMGM model has the smallest error value. Therefore, the carbon price prediction results using the VMD model are more accurate. Compared with EMD, EEMD, CEEMDAN, and VMD, using MPA-VMD to decompose carbon prices can obtain more accurate prediction results and improve the prediction performance of the model.

[0143] Point prediction comparison experiment of prediction models: This experiment compares the MPA-VMD-QRMGM model with 6 different models for point prediction to further verify the prediction performance based on the MGM hybrid model. These 6 benchmark models include not only SVR, GPR, QR, but also QRLSTM, QRGRU and QRNN that combine MPA-VMD decomposition technology with quantile regression. The point prediction evaluation results of the 6 models are shown in Table 1.

[0144] Table 1

[0145]

[0146] The best results are filled in gray, and the following conclusions can be drawn from the table:

[0147] ⑴The RMSE values ​​of MPA-VMD-QRMGM, MPA-VMD-QRLSTM, MPA-VMD-QRGRU and MPA-VMD-QRNN are 0.3269 yuan / ton, 0.3355 yuan, 0.3761 yuan and 0.4146 yuan respectively, while the RMSE values ​​of SVR, GPR and QR are 1.0074 yuan, 1.0514 yuan and 1.0960 yuan respectively. It can be seen that the prediction accuracy of the four models (MPA-VMD-QRMGM, MPA-VMD-QRLSTM, MPA-VMD-QRGRU and MPA-VMD-QRNN) combined with MPA-VMD decomposition technology is significantly higher than that of the other three models (SVR, GPR and QR). Carbon price is a time series, and the decomposition technology combined with the optimization algorithm can effectively remove noise. In addition, the first four models can capture the information of the time series while the last three models cannot consider the information of the time series. Therefore, the prediction accuracy of the four models combined with the MPA-VMD decomposition technology is better than that of the last three models.

[0148] ⑵By comparing the four models (MPA-VMD-QRMGM, MPA-VMD-QRLSTM, MPA-VMD-QRGRU and MPA-VMD-QRNN) that combined the MPA-VMD decomposition technology, the RMSE and MAPE values ​​of MPA-VMD-QRMGM were 0.3269 yuan and 0.9368% respectively, which is the model with the highest prediction accuracy among all six models. Compared with the QR model with the worst prediction accuracy, the RMSE value of MPA-VMD-QRMGM decreased by 70% and the MAPE value decreased by 45%, indicating that MPA-VMD-QRMGM can obtain competitive prediction accuracy in carbon price forecasting.

[0149] ⑶ By comparing the training time of the four neural network models combined with MPA-VMD decomposition technology, the training time of MPA-VMD-QRMGM, MPA-VMD-QRLSTM, MPA-VMD-QRGRU and MPA-VMD-QRNN are 190s, 286s, 303s and 246s respectively. Obviously, the training time of MPA-VMD-QRMGM is the shortest. Because the number of gates in the gate structure unit in the MPA-VMD-QRMGM neural network is reduced to 1, the MPA-VMD-QRMGM model can reduce the training time without affecting the prediction accuracy. The experimental results are as follows Figure 5 .

[0150] In summary, MPA-VMD-QRMGM is a lightweight prediction framework that can reduce training time without reducing prediction accuracy.

[0151] Comparative experiment of interval prediction of prediction models: In order to judge the rationality of the prediction interval, the interval coverage and average bandwidth of the interval are verified through the interval prediction result evaluation. The evaluation index results of the interval prediction of the seven models in the Hubei carbon price market are shown in Table 2.

[0152] Table 2

[0153]

[0154] The best results are filled in gray, and the following conclusions can be drawn from the table:

[0155] ⑴ In the Hubei carbon price market data, the 95% CP values ​​of MPA-VMD-QRMGM, MPA-VMD-QRLSTM, MPA-VMD-QRGRU, MPA-VMD-QRNN, SVR, GPR and QR are 0.9454, 0.9438, 0.9490, 0.9562, 0.9354, 0.9323 and 0.9231, respectively. The coverage of the seven models is close to 95%, indicating that the prediction intervals of these seven models are reasonable.

[0156] (2) In the Hubei carbon price market data, the 95% MWP values ​​of MPA-VMD-QRMGM, MPA-VMD-QRLSTM, MPA-VMD-QRGRU, MPA-VMD-QRNN, SVR, GPR, and QR are 0.1680, 0.1781, 0.1980, 0.2047, 0.2265, 0.2329, and 0.2525, respectively. Obviously, the 95% MWP values ​​of the first four neural network models combined with the optimization algorithm decomposition technology are smaller than those of the last three single models, because the prediction accuracy of SVR, GPR, and QR is lower than that of the neural network model, so these three models can only improve the coverage by increasing the interval bandwidth.

[0157] ⑶ In the Hubei carbon price market data, the 95% MC values ​​of MPA-VMD-QRMGM, MPA-VMD-QRLSTM, MPA-VMD-QRGRU, MPA-VMD-QRNN, SVR, GPR, and QR are 0.1777, 0.1887, 0.2087, 0.2141, 0.2422, 0.2498, and 0.2735, respectively. The 95% MC value of MPA-VMD-QRMGM is the best, indicating that the model covers as many observations as possible with the smallest possible interval bandwidth.

[0158] In conclusion, the prediction interval obtained by MPA-VMD-QRMGM is the most appropriate. The interval prediction results of Hubei carbon price market are as follows: Figure 6As shown in the figure, it can be seen that most of the observations fall within the prediction interval, and the prediction interval bandwidth is small enough, indicating that the prediction interval of MPA-VMD-QRMGM is the most appropriate.

[0159] Probabilistic prediction comparison experiment of prediction model: The comparison experiment of probability prediction results is to verify the comprehensive performance of probability prediction. In the Hubei carbon price market data, the CRPS values ​​of MPA-VMD-QRMGM, MPA-VMD-QRLSTM, MPA-VMD-QRGRU, MPA-VMD-QRNN, SVR, GPR and QR are 0.1060, 0.1085, 0.1101, 0.1301, 0.2388, 0.2537 and 0.4021 respectively, among which the CRPS value of MPA-VMD-QRMGM is the smallest, indicating that the probability prediction performance of the model is the best, which is consistent with the results of point prediction and interval prediction. Figure 7 These are the 9 probability density curves obtained by the MPA-VMD-QRMGM model in the Hubei carbon price market by equidistant sampling. The peaks of the 9 probability density curves are not too high or too low, and the curve shape is not too bloated or thin. Overall, they are very full, indicating that the probability density curve is very suitable. In periods 1, 41, 82, 122, 163, 204 and 244, the observed values ​​are almost in the center of the curve. In period 285, the observed values ​​are also very close to the center of the curve, indicating that the prediction accuracy of these periods is very high and the prediction effect is good. In period 325, the observed values ​​are a certain distance away from the center of the curve, indicating that the prediction error in this period is high. By analyzing the probability prediction results of the validation set, it is found that the observed values ​​are both close to the center of the curve and far away from the center, but in most cases they are close to the center, indicating that the probability prediction results are reliable. If all the observed values ​​are in the center of the curve, it means that the experimental results are not credible.

[0160] It should be noted that, for the sake of convenience, the aforementioned method embodiments are all expressed as a series of action combinations, but those skilled in the art should know that the present invention is not limited to the described order of actions, because according to the present invention, certain steps can be performed in other orders or simultaneously.

[0161] The technical features of the above embodiments may be combined arbitrarily. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0162] The above embodiments are preferred implementation modes of the present invention, but the implementation modes of the present invention are not limited to the above embodiments. Any other changes, modifications, substitutions, combinations, and simplifications that do not deviate from the spirit and principles of the present invention should be equivalent replacement methods and are included in the protection scope of the present invention.

Claims

1. A carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE, characterized in that: The steps include: Initialize MPA parameters, use MPA to optimize VMD parameters, and obtain optimal VMD parameters; construct an MPA-VMD model based on the optimal VMD parameters, and use the MPA-VMD model to decompose the original carbon price time series to obtain multiple subsequences; Construct a QR-MGM model and set relevant parameters. The QR-MGM model includes a minimum gated memory network and quantile regression. The minimum gated memory network includes multiple output nodes, each of which corresponds to a quantile. Use the QR-MGM model to divide multiple subsequences into intervals according to quantiles. Each interval includes several continuous time steps. Use the output nodes to predict the subsequences of each interval according to the time steps to obtain carbon price prediction samples at different quantile levels. Linearly superimpose the subsequences in the same interval to obtain the carbon price interval prediction results; The optimal bandwidth is determined, and kernel density estimation is performed on the carbon price prediction samples at different quantile levels to obtain the probability density of carbon prices.

2. The carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE according to claim 1 is characterized in that: The method of optimizing VMD parameters by using MPA comprises: Set the size of VMD parameters, which include the number of decomposition modes K and the quadratic penalty term α; MPA is used to allocate the search space to predators and preys, and the calculation is iteratively updated in the search space. If the fitness of the prey is greater than that of the predator at the corresponding position, the prey will replace the corresponding predator until the preset termination condition is reached to obtain the optimal VMD parameters.

3. The carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE according to claim 2 is characterized in that: The MPA is used to allocate the search space to the predator and the prey as follows: Elite Matrix: Prey Matrix: Among them, Y i (t) represents the position vector in iteration number t, J i represents the objective function, D and N represent the total number of dimensions and variables respectively, and Y i,j ′ represents the best predator vector Y i ′jth dimension, Y i ′ is replicated N times to obtain the elite matrix.

4. The carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE according to claim 1 is characterized in that: The MPA-VMD model is constructed according to the optimal VMD parameters, comprising the following steps: S21. Initialize the modal function of VMD using the MPA algorithm S22, decomposing the signal using a VMD algorithm to obtain multiple intrinsic mode functions; S23, performing denoising processing on each intrinsic mode function; S24. The denoised intrinsic mode function is embedded into the ADMM algorithm, and the complete VMD is obtained by optimizing in the Fourier domain.

5. The carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE according to claim 1 is characterized in that: The MPA-VMD model is used to decompose the time series of the original carbon price, including: The MPA-VMD model is used to decompose the original carbon price time series into multiple components with different frequencies; Introduce the quadratic penalty term α and the Lagrange multiplier λ for each component to obtain the modal component As follows: Where f(ω) represents the real signal with the center frequency, represents the modal component of the center frequency, χ(ω) represents the instantaneous amplitude of the center frequency, η represents the phase function, ω l represents the lth modal component The center frequency of Each modal component is transformed using wavelet transform Perform denoising processing to obtain multiple subsequences.

6. The carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE according to claim 1 is characterized in that: The minimum gated memory network comprises: The forget gate and input gate are as follows: Forget Gate t :f t =σ(net t )=σ(w h h t-1 +w x x t ) Input Gate I t :I t =1-f t where h t-1 Indicates the cell state at the previous moment, x t represents the input at the current moment, w h and w x are the corresponding weight parameters respectively; The cell state at the current moment is as follows: a t =tanh(net t )=tanh(w h h t-1 +w x x t ) Among them, tanh represents the activation function; The output of the hidden layer at the current moment is as follows: h t =f t h t-1 +a t I t The output of the output layer at the current moment is as follows: y t =σ(w y h t ) where y t is the output of the output layer, w y represents the weight of the output layer.

7. The carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE according to claim 1 is characterized in that: The quantile regression is as follows: Among them, Q yt (τ|x t ) is the value of the dependent variable at the τ quantile, and τ∈(0,1), x t Represents the input at the current moment, is the estimated value of the regression coefficient The estimated values ​​of the regression coefficients It is obtained by minimizing the loss function L, as follows: Symmetric Function The formula is: Among them, y t represents the output of the output layer of the minimum gated memory network, and b represents the independent variable.

8. The carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE according to claim 1 is characterized in that: The optimal bandwidth was determined by grid search and cross-validation methods.

9. The carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE according to claim 1 is characterized in that: The kernel density estimation of the carbon price range prediction results at different quantile levels is as follows: Where B represents the bandwidth and is greater than zero, n is the total number of samples, and Z t，i Represents the prediction result Z of the tth carbon price interval t The ith carbon price prediction sample in , x represents the independent variable, and K(·) is the kernel function.

10. The carbon emission trading price prediction method based on MPA-VMD and QRMGM-KDE according to claim 9 is characterized in that: The kernel of the kernel function is the Epanechnikov function, as shown below: Among them, α represents the independent variable.

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