A double-layer mixed gaussian process functional regression time series prediction method and system

Abstract

The application discloses a double-layer mixed Gaussian process functional regression time series prediction method and system, and belongs to the technical field of time series prediction and machine learning. The method comprises the following steps: periodically cutting and time aligning an original non-stationary time series to construct a functional period data set; constructing a double-layer probability model architecture, clustering the period characteristics by using a K-Means algorithm to divide dynamic modes, and constructing a low-layer sparse Gaussian process functional regression expert model comprising a B-spline mean function and a stable period kernel function; performing sparse processing by using an induced point selection strategy based on extreme point reservation and clustering centers; alternately optimizing the posterior probability of the period belonging to a category and the hyperparameters of each expert model by using a variational expectation maximization algorithm; and fusing the outputs of each expert based on a mixed weighting mechanism to realize smooth prediction of a future period, and solving the problems that a traditional Gaussian process has high computational complexity in long sequence modeling and is difficult to capture quasi-periodic and non-stationary characteristics.

Description

Technical Field

[0001] This invention belongs to the field of machine learning and data mining technology, specifically relating to a two-layer mixture Gaussian process functional regression time series prediction method and system. Background Technology

[0002] Time series forecasting is of great significance in fields such as astronomical observation, traffic management, and weather forecasting. However, many real-world time series (such as sunspot activity cycles and traffic flow) exhibit significant quasi-periodicity and non-stationarity. That is, the amplitude, period length, and waveform shape of the data change dynamically over time. Currently, commonly used forecasting methods mainly include: Traditional statistical models (such as ARIMA) struggle to capture complex nonlinear characteristics.

[0003] Standard Gaussian process regression (GPR): While it can provide uncertainty estimates, its computational complexity is O(n log n). It is difficult to handle long sequence data; and the standard GPR usually assumes that the data is stationary, making it difficult to fit the variable periodic pattern using a single kernel function.

[0004] Neural networks (such as LSTM / Transformer) require massive amounts of data for training, lack probabilistic interpretability, and are difficult to provide confidence intervals.

[0005] Referring to the existing patent CN118503621A, a method for optimizing a GPR model based on the Grey Wolf Optimization Algorithm is disclosed. Although this method optimizes hyperparameters through swarm intelligence algorithms, it is essentially still a single-layer GPR model, failing to solve the problem of high computational complexity and unable to effectively handle the multimodal characteristics of the data. Patent CN112070157A proposes a multi-limit learning machine model. Although it adopts a hybrid model approach, it lacks utilization of the "functional periodic" structure of time series data and ignores the geometric features within the period.

[0006] To address the aforementioned issues, this invention proposes a two-layer mixture Gaussian process functional regression (SV-TMGPFR) architecture based on the sparse variational EM algorithm. Summary of the Invention

[0007] To address the problems of high computational cost and difficulty in handling non-stationary quasi-periodic data in existing Gaussian process models, this invention provides a two-layer mixture Gaussian process functional regression time series prediction method. This method treats the time series as a collection of "functional curves," captures different dynamic patterns through a two-layer structure (high-level clustering + low-level experts), and introduces sparse variational inference technology, which significantly reduces computational costs while ensuring prediction accuracy.

[0008] To achieve the above objectives, in a first aspect, the present invention provides a method for predicting time series data using a two-layer mixture Gaussian process functional regression, comprising the following steps: S1. Obtain the original univariate time series data, perform denoising and smoothing processing on it, and divide the continuous time series into several independent functionalized periodic segments based on the extreme value detection algorithm to construct a functional periodic dataset. S2, extract the statistical features of each functionalized period, and use a clustering algorithm to divide all functionalized periods into K initial clusters as the dynamic mode of the system; S3. For each initial cluster, a sparse Gaussian process functional regression expert model is constructed. The expert model uses a mean function based on B-splines to fit the non-zero mean trend and a mixed kernel function to capture local correlation. S4. For each expert model, a hybrid sampling strategy based on extreme value preservation and cluster center is used to select the induction point; S5, the variational expectation-maximization algorithm is used to iteratively train the model; in the E-step, the posterior responsibility value belonging to each expert model in each cycle is calculated; in the M-step, the hyperparameters and variational parameters of each expert model are updated using the variational lower bound (ELBO) maximization strategy. S6. For the future time period to be predicted, calculate the weight of each expert model, and perform weighted fusion by combining the prediction distributions of each expert to obtain the final time series prediction result.

[0009] Furthermore, step S1 specifically includes the following steps: S101 uses a Savitzky-Golay filter to smooth the original time series in order to eliminate the influence of high-frequency noise on extreme value detection. S102, identify the local minimum points of the smoothed sequence as period splitting points, and extract the data between two adjacent splitting points as a complete physical period; S103, Define relative time

[0010] Align the start time of all cycles to zero and perform Z-score normalization on the amplitude to form a functional sample set.

[0011] Furthermore, in step S3, constructing the mean function based on B-splines specifically includes the following steps: S301, in the defined domain Generate a uniformly distributed node vector and fill the nodes at both ends to satisfy the boundary conditions; S302, based on node vectors, uses recursive formulas to calculate the projection of input time t onto each B-spline basis function, and constructs the basis function matrix. ; S303, the mean function is constructed through a linear combination of basis functions, and its expression is as follows:

[0012] in, The output is the mean function at time point t; M is the number of basis functions; d is the order of the B-spline. For the corresponding weight parameters; Let be the j-th d-th B-spline basis function.

[0013] Furthermore, in step S3, constructing the hybrid kernel function specifically includes the following steps: S311, establish a stable periodic kernel to characterize the main periodic oscillation characteristics of the sequence, and set interval constraints on the period length parameter; S312, establish a long-scale radial basis function (RBF) kernel to describe the slow change of periodic amplitude over time; S313, establish a short-scale RBF kernel to capture local non-periodic random fluctuations; S314, the final covariance function is formed by combining the above kernel function through addition and multiplication, as shown in the following equation:

[0014] Where k(t,t') represents the covariance between time points t and t'; The periodic term represents the signal variance; p represents the fixed period length. This represents the length scale of the periodic kernel; This refers to the RBF kernel used for amplitude modulation; This represents the RBF kernel used for local fluctuations.

[0015] Furthermore, the induction point selection strategy in step S4 specifically includes the following steps: S401: Traverse all training samples within the cluster and extract peak and valley points as first-class induction points to preserve key morphological features of the sequence; S402: K-Means clustering is applied to non-extreme points, and the cluster centers are selected as second-type induction points to cover the overall distribution structure; S403: Combine the above two types of induction points to form an induction point set:

[0016] And it is used as a variational parameter for optimization in subsequent training.

[0017] Furthermore, in step S5, the calculation of the posterior responsibility value in step E specifically includes: Calculate the marginal likelihood of the i-th period sample under the k-th expert model based on the current model parameters; The posterior probability (responsibility value) is calculated based on Bayes' theorem using the mixing coefficients. Its expression is as follows:

[0018] in, Represents the mixing coefficients of the k-th expert model; Represents the Gaussian probability density function; This represents the vector of observations in the i-th period; and These represent the inputs of the k-th sparse Gaussian process expert model. The predicted mean vector and predicted covariance matrix at the given location.

[0019] Furthermore, in step S5, updating the model parameters in step M specifically includes: Responsibility value calculated by fixed E-step ; Construct a variational lower bound (ELBO) objective function consisting of a log-likelihood expectation term and a KL divergence regularization term; Stochastic gradient descent is used to maximize the objective function in order to update the kernel function hyperparameters, mean function weights, and induced point positions. The objective function of the k-th expert model is as follows:

[0020] in, This represents the variational lower bound of the k-th expert model; Represents the variational distribution Expectations; Represents the log-likelihood for a given latent function value; This represents the Kullback-Leibler divergence, used to constrain the variational posterior distribution. Without deviating from the prior distribution Too far.

[0021] In a second aspect, the present invention provides a two-layer mixture Gaussian process functional regression time series prediction system, including a data processing module, a clustering module, a model building and training module, and a prediction module; The data processing module is used to acquire raw univariate time series data, perform denoising and smoothing processing on it, and divide the continuous time series into several independent functionalized periodic segments based on the extreme value detection algorithm to construct a functional periodic dataset. The clustering module is used to extract the statistical features of each functionalized period and uses a clustering algorithm to divide all functionalized periods into K initial clusters as the dynamic pattern of the system. The model building and training module is used to construct a sparse Gaussian process functional regression expert model for each initial cluster. The expert model uses a B-spline-based mean function to fit the non-zero mean trend and a hybrid kernel function to capture local correlations. For each expert model, a hybrid sampling strategy based on extremum preservation and cluster centers is used to select induction points. The variational expectation-maximization algorithm is used to iteratively train the model. In the E-step, the posterior responsibility value belonging to each expert model in each period is calculated. In the M-step, the hyperparameters and variational parameters of each expert model are updated using the variational lower bound (ELBO) maximization strategy. The prediction module is used to calculate the weights of the future time period to be predicted belonging to each expert model, and to perform weighted fusion by combining the prediction distributions of each expert to obtain the final time series prediction result.

[0022] Thirdly, the present invention can also provide a computer device, including a processor and a memory, wherein the memory is used to store a computer executable program, the processor reads the computer executable program from the memory and executes it, and the processor can realize the two-layer mixed Gaussian process functional regression time series prediction method of the present invention when executing the executable program.

[0023] Simultaneously, a computer-readable storage medium is provided, in which a computer program is stored. When the computer program is executed by a processor, it can implement the two-layer mixed Gaussian process functional regression time series prediction method described in this invention.

[0024] Compared with the prior art, the present invention has at least the following beneficial effects: By using B-spline mean and a two-layer expert mechanism, the amplitude variation and asymmetric waveform of non-stationary sequences can be accurately fitted. By introducing a sparse variational framework, the computational complexity is reduced from cubic to linear, enabling the processing of long sequence data containing thousands of cycles. The designed induction point strategy ensures the capture of key information such as peak values, avoiding the problem of underfitting extreme values ​​in traditional sparse GP. It can provide a complete confidence interval for prediction, providing a quantitative basis for decision-making regarding uncertainty. Attached Figure Description

[0025] Figure 1This is an overall flowchart of a two-layer mixture Gaussian process functional regression (SV-TMGPFR) time series prediction method based on the sparse variational EM algorithm proposed in this invention.

[0026] Figure 2 This is a schematic diagram of the two-layer hybrid architecture of the multi-task transfer learning architecture of the SV-TMGPFR model in this invention.

[0027] Figure 3 This is a schematic diagram illustrating the logical interaction between model training and parameter optimization in an embodiment of the present invention.

[0028] Figure 4 This is a visual comparison of the prediction results of the present invention on the Sunspots dataset.

[0029] Figure 5 This is a visualization comparison of the prediction results of the present invention on the Synthetic Cycles dataset.

[0030] Figure 6 This is a visual comparison chart of the prediction results on the Metro Traffic Volume dataset in the embodiments of the present invention. Detailed Implementation

[0031] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are for illustration and explanation only and are not intended to limit the scope of protection of the present invention.

[0032] refer to Figure 1 This invention provides a two-layer mixture Gaussian process functional regression (SV-TMGPFR) time series prediction method based on the sparse variational EM algorithm, comprising the following steps: S1. Obtain the original univariate time series data, perform denoising and smoothing processing on it, and divide the continuous time series into several independent functionalized periodic segments based on the extreme value detection algorithm to construct a functional periodic dataset. S2, extract the statistical features of each functionalized period, and use a clustering algorithm to divide all functionalized periods into K initial clusters as the dynamic mode of the system; S3. For each initial cluster, a sparse Gaussian process functional regression expert model is constructed. The expert model uses a mean function based on B-splines to fit the non-zero mean trend and a mixed kernel function to capture local correlation. S4. For each expert model, a hybrid sampling strategy based on extreme value preservation and cluster center is used to select the induction point; S5, the variational expectation-maximization algorithm is used to iteratively train the model; in the E-step, the posterior responsibility value belonging to each expert model in each cycle is calculated; in the M-step, the hyperparameters and variational parameters of each expert model are updated using the variational lower bound (ELBO) maximization strategy. S6. For the future time period to be predicted, calculate the weight of each expert model, and perform weighted fusion by combining the prediction distributions of each expert to obtain the final time series prediction result.

[0033] refer to Figure 2 The functional data perspective and two-layer architecture of this invention differ from traditional methods that treat time series as discrete points. This invention preprocesses the series into independent functional periods. Based on this, a two-layer model is established: Layer 2 (higher layer) is responsible for capturing long-term macroscopic patterns (Regime) by clustering periodic features (such as amplitude and duration) to divide the data into different clusters; Layer 1 (lower layer) trains an expert model—a Gaussian Process Functional Regression model (GPFR)—within each cluster to fit the specific waveform details of the dynamic pattern corresponding to that cluster.

[0034] To address the high computational complexity of Gaussian processes, this invention introduces a Sparse Gaussian process (SGP). Unlike randomly selecting induced points, this invention designs an extremum preservation and K-Means induced point selection strategy. By preserving the peaks and troughs of key feature points and covering the regions outside the key feature points with cluster centers, it can approximate the entire dataset with high accuracy using a very small number of induced points (e.g., 100). Combined with variational inference, it achieves rapid training on large-scale datasets by maximizing ELBO (Evidence Lower Bound) instead of the traditional log-likelihood maximization.

[0035] To accommodate the complex asymmetric periodic shapes of data such as sunspots, this invention introduces a B-Spline (B-spline) mean function into the Gaussian General Process (GP) model. Compared to zero-mean or constant-mean GP, ​​the B-spline mean can explicitly learn the skeletal shape of the period, allowing the Gaussian process to simply fit the residuals, significantly reducing the learning difficulty. Simultaneously, a constrained, stable periodic kernel function is designed to prevent the model from degenerating into a non-periodic model due to parameter drift during training.

[0036] refer to Figure 3This invention uses the Expectation-Maximization (EM) algorithm to couple cluster assignment with model parameter learning. In the E-step, the probability of belonging to each expert in each cycle is dynamically updated, and in the M-step, each expert model is specifically optimized, achieving the co-evolution of dynamic pattern recognition and waveform fitting.

[0037] The present invention also provides a two-layer mixture Gaussian process functional regression time series prediction system, including a data processing module, a clustering module, a model building and training module, and a prediction module; The data processing module is used to acquire raw univariate time series data, perform denoising and smoothing processing on it, and divide the continuous time series into several independent functionalized periodic segments based on the extreme value detection algorithm to construct a functional periodic dataset. The clustering module is used to extract the statistical features of each functionalized period and uses a clustering algorithm to divide all functionalized periods into K initial clusters as the dynamic pattern of the system. The model building and training module is used to construct a sparse Gaussian process functional regression expert model for each initial cluster. The expert model uses a B-spline-based mean function to fit the non-zero mean trend and a hybrid kernel function to capture local correlations. For each expert model, a hybrid sampling strategy based on extremum preservation and cluster centers is used to select induction points. The variational expectation-maximization algorithm is used to iteratively train the model. In the E-step, the posterior responsibility value belonging to each expert model in each period is calculated. In the M-step, the hyperparameters and variational parameters of each expert model are updated using the variational lower bound (ELBO) maximization strategy. The prediction module is used to calculate the weights of the future time period to be predicted belonging to each expert model, and to perform weighted fusion by combining the prediction distributions of each expert to obtain the final time series prediction result.

[0038] To verify the effectiveness and advancement of the proposed two-layer mixture Gaussian process functional regression (SV-TMGPFR) time series prediction method based on the sparse variational EM algorithm in handling different types of non-stationary time series, this invention selects three representative datasets for experimental verification: Sunspots dataset: Represents data on natural physical phenomena with quasi-periodic characteristics.

[0039] Synthetic Cycles dataset: used to validate the theoretical performance of the model under nonlinear trends and complex fluctuations.

[0040] Metro Traffic Volume dataset: Represents real-world engineering data with strong noise, large numerical values, and obvious daily / weekly periodicity.

[0041] This invention selects several mainstream methods, including Naive Prediction, Support Vector Regression (SVR), Random Forest, Long Short-Term Memory Network (LSTM), and Traditional Single-Task Gaussian Process (Single GP), as benchmark models for comparison; the evaluation metrics include Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Coefficient of Determination (R²). 2 ) and Predicted Interval Coverage Probability (PICP).

[0042] Example 1: Prediction of Quasi-Periodic Sunspot Sequences We selected an internationally recognized monthly sunspot observation dataset, which exhibits a quasi-periodicity of approximately 11 years, although the amplitude and length of each cycle dynamically change over time. The experiment was divided into 24 solar cycles, and the model needed to learn and predict future activity intensity within a finite historical period. The last two complete cycles were selected as the test set. Comparison models included Naive Baseline, Support Vector Regression (SVR), Random Forest, Long Short-Term Memory (LSTM) networks, and the traditional Single Layer Sparse Gaussian Process (Single GP).

[0043] Results analysis: The performance comparison results of the model of this invention and various benchmark models on the sunspot test set are shown in Table 1.

[0044] Table 1. Performance comparison of different models on the sunspot dataset

[0045] As shown in Table 1, the method of the present invention (TMGPFR) is significantly better than the comparative model on all error metrics.

[0046] Improved prediction accuracy: Compared to the better-performing Single GP and SVR (R 2 (All are approximately 0.68), R of the model of this invention 2 The value was increased to 0.793, and the RMSE decreased by approximately 18.8%. This indicates that the present invention, through a transfer learning mechanism, more effectively captures the phase changes and amplitude fluctuations of the solar cycle.

[0047] Uncertainty Quantification Optimization: It is worth noting that Single GP achieves a PICP of 100%, but its RMSE is high, indicating that it covers the true value by outputting an overly wide confidence interval, which lacks practicality. In contrast, this invention significantly reduces prediction error while stabilizing the PICP at 96.77%, achieving the optimal balance between prediction interval tightness and coverage.

[0048] Visualization of prediction results: Figure 4 This is a visualization of the prediction results on a sunspot dataset in this embodiment. As shown, the present invention can accurately capture the peaks and troughs of the solar activity cycle. The black solid line represents the actual observed values. The prediction curve (blue line) of the present invention not only closely follows the cyclical fluctuations of the actual values, but also significantly outperforms the comparative model, which can only fit the average trend, in peak prediction. A magnified view shows that in areas of drastic data fluctuation, Single GP often exhibits underfitting, while the prediction curve of the present invention closely follows the actual values, and the confidence interval (light shading) effectively covers the range of data fluctuations.

[0049] Example 2: Validation of Non-stationary Multimodal Synthesis Data To verify the theoretical advantages of this invention in handling the Regime Switching problem, a synthetic dataset containing two distinct dynamic modes was constructed: Mode A (quiet period): low amplitude, long period, symmetrical waveform; Mode B (active period): high amplitude, short period, severely skewed waveform. A generated synthetic time series of 28 periods was used, and nonlinear trend changes were introduced into the data to test the model's rapid adaptability with small samples (24 periods in the training set). The scaling factor was set to 0.35 to simulate observation noise. The data randomly switched between the two modes on the time axis.

[0050] 1. Analysis of experimental results: The test results of each model on the synthetic dataset are shown in Table 2.

[0051] Table 2 Performance comparison of different models on multimodal synthetic datasets

[0052] 2. Discussion of Results: The data in Table 2 demonstrate that this invention possesses a significant leading advantage in capturing complex nonlinear periodic trends. Single GP's R 2 A negative value (-0.111) indicates that the single-layer Gaussian process attempts to fit two drastically different modes using a single set of kernel function parameters, resulting in a severe averaging effect and the model completely failing.

[0053] Effectiveness of the invention: R of TMGPFR 2 The RMSE is as high as 0.948, which is about 78% lower than Single GP. This is due to the two-layer architecture of this invention: the high-level clustering network successfully identifies the current dynamic pattern and activates the corresponding low-level expert model, demonstrating the powerful representation ability of the multi-output Gaussian process flow regression structure in processing complex manifold data.

[0054] 3. Visualization of prediction results Visual verification: such as Figure 5 As shown, at the mode switching point, from the active period to the calm period, the comparison model exhibited severe overshoot oscillations, while the model of this invention quickly converged to the new waveform, demonstrating strong adaptive capability. The figure clearly shows that the model of this invention almost perfectly reproduced the data fluctuation trajectory, while other models could only fit an approximate straight line average.

[0055] Example 3: Multi-modal Traffic Volume Prediction Application This embodiment selects the Metro Interstate Traffic Volume dataset, which exhibits significant differences between "weekday (bi-peak)" and "weekend (uni-peak)" patterns and contains numerous sudden fluctuations, making it ideal data for evaluating model performance in complex engineering application scenarios. The continuous time series is divided into functional periods based on days (24 hours). An ensemble learning method, Random Forest, is added to the comparison model.

[0056] Analysis of experimental results: Table 3 presents detailed evaluation metrics for each model on the traffic flow test set. As shown in Table 3, the system maintains extremely high performance even with massive datasets of thousands of units. Firstly, there is a significant improvement in accuracy; the RMSE of the model in this invention is 522.03, a 44.2% reduction compared to Single GP (936.08) and a 58.1% reduction compared to Random Forest (1245.66). (The coefficient of determination R...) 2 The accuracy reached 0.930, with MAPE controlled at 15.21%, far exceeding other models. In the transportation sector, this improvement in accuracy has significant practical value for alleviating congestion and optimizing traffic light timing, demonstrating that this invention can perfectly decouple weekday and weekend traffic patterns.

[0057] Table 3 Performance comparison of different models on traffic flow datasets

[0058] Visualization of prediction results: Figure 6The visualization of the experimental results on the traffic flow dataset shows that, during a magnified weekend period, Single GP (dashed line) still predicted a weekday morning rush hour, leading to significant errors. In contrast, the model of this invention (solid line) accurately predicted the smooth, low-flow curve of the weekend, fully demonstrating the crucial role of the Layer2 clustering mechanism in practical applications. The model of this invention can accurately capture abrupt changes in traffic flow during morning and evening rush hours, and the confidence interval band (shaded area) tightly covers the actual observations, demonstrating high robustness.

[0059] In summary, the SV-TMGPFR method proposed in this invention has significant accuracy advantages and robustness compared with existing technologies when processing time series with quasi-periodic, multimodal switching and non-stationary characteristics. Furthermore, the sparse variational framework ensures computational efficiency, making it highly valuable for engineering applications.

[0060] The present invention can also provide a computer device, including a processor and a memory, wherein the memory is used to store a computer executable program, the processor reads the computer executable program from the memory and executes it, and the processor can implement the two-layer mixed Gaussian process functional regression time series prediction method and system described in the present invention when executing the computer executable program.

[0061] This invention provides a computer-readable storage medium storing a computer program. When the computer program is executed by a processor, it can implement the two-layer mixture Gaussian process functional regression time series prediction method and system described in this invention.

[0062] The computer device may be a laptop, a desktop computer, or a workstation.

[0063] The processor can be a central processing unit (CPU), a digital signal processor (DSP), an application-specific integrated circuit (ASIC), or an off-the-shelf programmable gate array (FPGA).

[0064] The memory described in this invention can be an internal storage unit of a laptop, desktop computer, or workstation, such as memory or hard disk; or it can be an external storage unit, such as a portable hard disk or flash memory card.

[0065] Computer-readable storage media can include computer storage media and communication media. Computer storage media includes volatile and non-volatile, removable and non-removable media implemented using any method or technology for storing information such as computer-readable instructions, data structures, program modules, or other data. Computer-readable storage media can include: read-only memory (ROM), random access memory (RAM), solid-state drives (SSDs), or optical discs, etc. Random access memory can include resistive random access memory (ReRAM) and dynamic random access memory (DRAM).

Claims

1. A method for predicting time series data using functional regression of a two-layer mixture Gaussian process, characterized in that, Includes the following steps: S1. Obtain the original univariate time series data, perform denoising and smoothing processing on it, and divide the continuous time series into several independent functionalized periodic segments based on the extreme value detection algorithm to construct a functional periodic dataset. S2, extract the statistical features of each functionalized period, and use a clustering algorithm to divide all functionalized periods into K initial clusters as the dynamic mode of the system; S3. For each initial cluster, a sparse Gaussian process functional regression expert model is constructed. The expert model uses a mean function based on B-splines to fit the non-zero mean trend and a mixed kernel function to capture local correlation. S4. For each expert model, a hybrid sampling strategy based on extreme value preservation and cluster center is used to select the induction point; S5, the variational expectation-maximization algorithm is used to iteratively train the model; in the E-step, the posterior responsibility value belonging to each expert model in each period is calculated; in the M-step, the hyperparameters and variational parameters of each expert model are updated using the variational lower bound maximization strategy. S6. For the future time period to be predicted, calculate the weight of each expert model, and perform weighted fusion by combining the prediction distributions of each expert to obtain the final time series prediction result.

2. The method for predicting time series using a two-layer mixture Gaussian process functional regression according to claim 1, characterized in that, Step S1 specifically includes the following steps: S101 uses a Savitzky-Golay filter to smooth the original time series in order to eliminate the influence of high-frequency noise on extreme value detection. S102, identify the local minimum points of the smoothed sequence as period splitting points, and extract the data between two adjacent splitting points as a complete physical period; S103, Define relative time Align the start time of all cycles to zero and perform Z-score normalization on the amplitude to form a functional sample set.

3. The method for predicting time series using functional regression of a two-layer mixture Gaussian process according to claim 1, characterized in that, Step S3, constructing the mean function based on B-splines, specifically includes the following steps: S301, in the defined domain Generate a uniformly distributed node vector and fill the nodes at both ends to satisfy the boundary conditions; S302, based on node vectors, uses recursive formulas to calculate the projection of input time t onto each B-spline basis function, and constructs the basis function matrix. ; S303, the mean function is constructed through a linear combination of basis functions, and its expression is as follows: in, The output is the mean function at time point t; M is the number of basis functions; d is the order of the B-spline. For the corresponding weight parameters; Let be the j-th d-th B-spline basis function.

4. The method for predicting time series using a two-layer mixture Gaussian process functional regression according to claim 1, characterized in that, Step S3, constructing the hybrid kernel function specifically includes the following steps: S311, establish a stable periodic kernel to characterize the main periodic oscillation characteristics of the sequence, and set interval constraints on the period length parameter; S312, establish a long-scale radial basis function kernel to describe the slow change of periodic amplitude over time; S313, establish a short-scale RBF kernel to capture local non-periodic random fluctuations; S314, the final covariance function is formed by combining the above kernel function through addition and multiplication, as shown in the following equation: Where k(t,t') represents the covariance between time points t and t'; The periodic term represents the signal variance; p represents a fixed period length. This represents the length scale of the periodic nucleus; This refers to the RBF core used for amplitude modulation; This represents the RBF kernel used for local fluctuations.

5. The method for predicting time series using a two-layer mixture Gaussian process functional regression according to claim 1, characterized in that, The induction point selection strategy in step S4 specifically includes the following steps: S401: Traverse all training samples within the cluster and extract peak and valley points as first-class induction points to preserve key morphological features of the sequence; S402: K-Means clustering is applied to non-extreme points, and the cluster centers are selected as second-type induction points to cover the overall distribution structure; S403: Combine the above two types of induction points to form an induction point set: And it is used as a variational parameter for optimization in subsequent training.

6. The method for predicting time series using functional regression of a two-layer mixture Gaussian process according to claim 1, characterized in that, In step S5, the calculation of the posterior responsibility value in step E specifically includes: Calculate the marginal likelihood of the i-th period sample under the k-th expert model based on the current model parameters; The posterior probability is calculated using Bayes' theorem based on the mixing coefficients. Its expression is as follows: in, Represents the mixing coefficients of the k-th expert model; Represents the Gaussian probability density function; This represents the vector of observations in the i-th period; and These represent the inputs of the k-th sparse Gaussian process expert model. The predicted mean vector and predicted covariance matrix at the given location.

7. The method for predicting time series using functional regression of a two-layer mixture Gaussian process according to claim 1, characterized in that, In step S5, updating the model parameters in step M specifically includes: Responsibility value calculated by fixed E-step ; Construct a variational lower bound objective function consisting of a log-likelihood expectation term and a KL divergence regularization term; Stochastic gradient descent is used to maximize the objective function in order to update the kernel function hyperparameters, mean function weights, and induced point positions. The objective function of the k-th expert model is as follows: in, This represents the variational lower bound of the k-th expert model; Represents the variational distribution Expectations; Represents the log-likelihood for a given latent function value; This represents the Kullback-Leibler divergence, used to constrain the variational posterior distribution. Without deviating from the prior distribution Too far.

8. A two-layer mixture Gaussian process functional regression time series prediction system, characterized in that, It includes a data processing module, a clustering module, a model building and training module, and a prediction module; The data processing module is used to acquire raw univariate time series data, perform denoising and smoothing processing on it, and divide the continuous time series into several independent functionalized periodic segments based on the extreme value detection algorithm to construct a functional periodic dataset. The clustering module is used to extract the statistical features of each functionalized period and uses a clustering algorithm to divide all functionalized periods into K initial clusters as the dynamic pattern of the system. The model building and training module is used to build a sparse Gaussian process functional regression expert model for each initial cluster; the expert model uses a mean function based on B-splines to fit the non-zero mean trend and a mixed kernel function to capture local correlations; For each expert model, a hybrid sampling strategy based on extreme value preservation and cluster center is used to select induction points; the variational expectation maximization algorithm is used to iteratively train the model; in the E-step, the posterior responsibility value belonging to each expert model in each period is calculated; in the M-step, the hyperparameters and variational parameters of each expert model are updated using the variational lower bound maximization strategy. The prediction module is used to calculate the weights of the future time period to be predicted belonging to each expert model, and to perform weighted fusion by combining the prediction distributions of each expert to obtain the final time series prediction result.

9. A computer device, characterized in that, It includes a processor and a memory, the memory being used to store a computer-executable program, the processor reading part or all of the computer-executable program from the memory and executing it, and the processor executing part or all of the computed executable program is able to implement the two-layer mixture Gaussian process functional regression time series prediction method according to any one of claims 1-6.

10. A computer-readable storage medium, characterized in that, Used to store computer-readable programs or instructions, which, when executed by a processor, are capable of implementing the two-layer mixture Gaussian process functional regression time series prediction method as described in any one of claims 1-6.

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