A method for processing heterogeneous multi-scale data by Gaussian process regression

Abstract

This application relates to the field of data processing technology and discloses a method for processing heterogeneous multi-scale data in Gaussian process regression. The method first acquires the raw data and performs feature transformation, standardization, and dimensionality reduction preprocessing to construct a multi-level validation dataset. Second, it reconstructs the Gaussian process regression kernel function, introducing a learnable Minkowski distance parameter to replace the traditional fixed Euclidean distance metric. Next, it uses a tree-structured Parzen estimator strategy to perform global iterative optimization on the validation set with the goal of minimizing prediction error, determining the optimal Minkowski distance parameter. Subsequently, it trains the model based on the optimal parameter and an automatic correlation determination mechanism, jointly optimizing hyperparameters such as signal variance and length scale through maximum likelihood estimation. Finally, it performs prediction and outputs the mean and variance. This invention, through adaptive parameterization of the kernel function metric, effectively overcomes the shortcomings of traditional models in adapting to non-spherical distributions and multi-dimensional heterogeneous features, significantly improving prediction accuracy and generalization performance.

Description

Technical Field

[0001] This invention relates to the field of data processing technology, specifically to a method for processing heterogeneous multi-scale data using Gaussian process regression. Background Technology

[0002] Gaussian process regression, as a nonparametric Bayesian modeling method, can utilize kernel functions to measure the similarity between samples and construct a mapping relationship between input and output, thus possessing application value in small-sample learning and the quantification of prediction uncertainty. In practical engineering applications, the key to building high-precision models lies in the construction of kernel functions and the effective estimation of hyperparameters.

[0003] Existing Gaussian process regression models typically employ kernel functions based on fixed Euclidean distance when dealing with data exhibiting complex distribution characteristics. This approach presupposes that the data possesses an isotropic spherical distribution in the feature space, while actual industrial or meteorological data often exhibit non-spherical decay characteristics. When the true correlation structure of the data does not match the geometric structure presupposed by the Euclidean distance, the model struggles to accurately capture the topological relationships between samples, thus limiting the improvement of prediction accuracy.

[0004] To address the aforementioned geometric mismatch problem, some studies have attempted to introduce variable distance parameters. However, at the parameter optimization level, existing techniques typically optimize the distance parameters controlling the geometric topology and the physical hyperparameters controlling the signal strength on the same level. Since changes in the distance parameters alter the metric of the feature space, leading to non-convex and multimodal characteristics in the objective function, while physical hyperparameters usually correspond to relatively smooth function surfaces, mixing these two types of parameters with significantly different mathematical properties can easily lead to numerical computational instability. Specifically, during iteration, the kernel matrix condition number is prone to deterioration, resulting in matrix factorization failure, or the optimization algorithm may converge to local extrema rather than the global optimum, making the model training process lack robustness.

[0005] Actual collected data typically contains features across multiple physical dimensions, which vary significantly in scale and weight in influencing the results. Conventional data processing workflows often lack systematic preprocessing and weight adaptation mechanisms for heterogeneous, multi-scale data. This can lead to large-scale features masking key information from small-scale features, or noisy features interfering with the model's learning direction, further impacting the model's generalization ability under complex conditions. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention provides a method for processing heterogeneous multi-scale data in Gaussian process regression. This method solves the technical problems of existing Gaussian process regression models relying on fixed Euclidean distance metrics, making it difficult to adapt to non-spherical data distributions and multidimensional heterogeneous feature structures, and lacking global adaptability in hyperparameter optimization, resulting in low prediction accuracy and weak generalization ability under complex nonlinear conditions.

[0007] To achieve the above objectives, the present invention provides the following technical solution: The first aspect of this invention provides a method for processing heterogeneous multi-scale data in Gaussian process regression, comprising the following steps: acquiring raw heterogeneous multi-scale data, performing feature preprocessing on it to construct a standardized input space, and dividing the preprocessed dataset into a training set and an independent test set, further dividing the training set into a sub-training set and a validation set; constructing a Gaussian process regression kernel function based on Minkowski distance, wherein the kernel function uses an adjustable Minkowski distance parameter instead of a fixed Euclidean distance metric; using a tree-structured Parzen estimator strategy, iteratively optimizing the Minkowski distance parameter on the validation set to determine the optimal Minkowski distance parameter with the goal of minimizing the prediction error; based on the metric form of the kernel function with the optimal Minkowski distance parameter, training a Gaussian process regression model using the training set, and jointly estimating the signal variance, noise variance, and feature length scale other than the Minkowski distance parameter by maximizing the log marginal likelihood; inputting the data to be predicted into the trained Gaussian process regression model, and outputting the prediction result.

[0008] Furthermore, to address the heterogeneity and multi-scale characteristics present in the original data, this invention employs a feature preprocessing strategy to eliminate skewness and dimensional differences in the data distribution. Specifically, this strategy includes: detecting and removing samples containing missing values; performing mathematical transformations on the distribution patterns of different feature dimensions, applying logarithmic transformations to data with right-skewed distributions, power transformations to data with left-skewed distributions, and square root transformations to data with moderately skewed distributions; performing standardization processing on dimensional differences in different feature dimensions, applying Z-score standardization to features with approximately normal distributions, applying max-min normalization to bounded distributions, and applying robust scaling to features containing outliers; and performing principal component analysis for dimensionality reduction when the number of feature dimensions exceeds a preset threshold, retaining principal components whose cumulative variance contribution rate meets a preset standard.

[0009] Furthermore, the Gaussian process regression kernel function based on Minkowski distance specifically replaces the distance term used to measure the similarity between samples in the basic kernel function with the Minkowski distance. The calculation logic of the Minkowski distance is as follows: calculate the absolute value of the difference between the feature values ​​of two samples in each dimension raised to the power of p, divide by the corresponding dimension's feature length scale raised to the power of p, sum the results of each dimension, and finally take the p-th root of the sum. Here, p is the Minkowski distance parameter, and its value is a real number greater than zero. By adjusting the value of the parameter p, this kernel function can represent multiple geometric metric spaces from Manhattan distance to Chebyshev distance, thus adapting to the true distribution structure of the data.

[0010] Furthermore, when the exponential kernel function is selected as the basic kernel function, the calculation logic of the improved Gaussian process regression kernel function is as follows: the exponent of the natural constant e is calculated using the amplitude of the signal variance control function, and the exponent term of this exponent is the negative value of the aforementioned Minkowski distance.

[0011] Furthermore, to adapt to the feature correlation of different types of data, this invention configures a constraint mode for the feature length scale. When configured in isotropic mode, the feature length scale of all feature dimensions is constrained to be equal, and the model shares a unique feature length scale parameter; when configured in anisotropic mode, each feature dimension is allowed to have an independent feature length scale, and the model automatically identifies the weights of different feature dimensions through training.

[0012] Furthermore, this invention employs a tree-structured Parzen estimator strategy to optimize the Minkowski distance parameters. This strategy sets the objective function to minimize the root mean square error on the validation set and performs iterations within a defined parameter search space. In each iteration, based on the root mean square error distribution of historical observation points, a good-value probability density function and a non-good-value probability density function are constructed, and new candidate Minkowski distance parameters are sampled according to the maximization of expected improvement criterion.

[0013] Furthermore, the tree-structured Parzen estimator strategy includes an anomaly handling mechanism. During the iterative optimization process, a temporary model is constructed using the sub-training set under the current candidate Minkowski distance parameters. If the kernel matrix exhibits non-positive definite numerical anomalies due to the candidate Minkowski distance parameters, the model evaluation stage is skipped, and a preset maximum error value is fed back to the strategy. This mechanism guides the algorithm to reduce the sampling probability of invalid parameter regions in subsequent iterations.

[0014] Furthermore, after determining the optimal geometric metric, this invention employs maximum likelihood estimation for the joint estimation of the remaining hyperparameters of the model. Specifically, this involves constructing the logarithmic marginal likelihood function of the training set and maximizing this function using a numerical optimization algorithm, thereby obtaining the optimal estimates of the signal variance, noise variance, and feature length scale.

[0015] Furthermore, the prediction results output by this invention include deterministic and uncertain components. Specifically, by calculating the posterior prediction distribution of the Gaussian process regression model, the mean of the posterior prediction distribution is output as the predicted value; the diagonal elements of the covariance matrix of the posterior prediction distribution are extracted as the variance of the prediction result, which is used to quantify the uncertainty of the prediction.

[0016] This invention provides a method for processing heterogeneous multi-scale data using Gaussian process regression. It offers the following advantages: 1. This invention enhances the model's adaptability to non-spherical data distributions by extending the distance metric in the Gaussian process regression kernel function from a fixed Euclidean distance to a parameterized Minkowski distance, overcoming the limitation of traditional models that can only construct correlations with fixed geometric shapes. By introducing a learnable Minkowski distance parameter, the model can adaptively adjust the metric form between Manhattan distance and Chebyshev distance according to the true distribution characteristics of the data, thereby more accurately capturing the sample correlation structure within complex nonlinear data (such as critical heat flux density or meteorological data) and significantly improving prediction accuracy.

[0017] 2. This invention automates and globally optimizes kernel function metric parameters by employing a tree-structured Parzen estimator (TPE) strategy to replace traditional grid search or manual trial-and-error methods for finding the optimal Minkowski distance parameter. Based on Bayesian optimization principles, this strategy utilizes historical observations to construct a probabilistic model, efficiently locating the parameter value that minimizes the validation set error in the continuous parameter space. This not only avoids subjective biases from manual parameter setting but also significantly reduces the time cost of model tuning, ensuring the stability of model performance.

[0018] 3. This invention enhances the robustness of the model in handling multidimensional heterogeneous features. It combines targeted feature preprocessing strategies (such as distribution transformation, standardization, and dimensionality reduction) with an Automatic Correlation Detection (ARD) mechanism, effectively solving the common problems of large feature scale differences and skewed distribution in engineering data. The ARD mechanism allows the model to independently learn length scales for features of different physical dimensions, thereby automatically identifying and retaining key feature information, suppressing interference from irrelevant features or noise, and improving the model's generalization ability under complex multivariate conditions. Attached Figure Description

[0019] Figure 1 This is a flowchart of the method of the present invention; Figure 2 Different Minkowski distance parameters of the present invention And a comparison chart of the predictive performance of the GPR model under ARD settings; Figure 3 This is a comparison chart of the predicted and actual critical heat flux densities under different p values ​​and ARD parameters according to the present invention; Figure 4 This is a kernel density distribution map of meteorological data (temperature) for this invention; Figure 5 This is a distribution diagram after feature transformation according to the present invention; Figure 6 This is a comparison chart of the prediction performance of the GPR model under different Minkowski distance parameters p and ARD settings according to the present invention; Figure 7 This is a comparison chart of the predicted and actual temperature values ​​under different p values ​​and ARD parameters according to the present invention. Detailed Implementation

[0020] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0021] Please see the appendix Figure 1 - Appendix Figure 7 This invention provides a method for processing heterogeneous multi-scale data in Gaussian process regression, including: S100, data preprocessing and multi-level dataset partitioning; S200, construction of an improved kernel function based on Minkowski distance; S300, adaptive optimization of Minkowski distance parameters based on a two-layer decoupling strategy; S400, model training and joint estimation of hyperparameters; and S500, prediction execution and performance evaluation.

[0022] Step S100: Data preprocessing and multi-level dataset partitioning To address the issues of inconsistent dimensions, large differences in feature scales, and uneven distribution that often exist in raw data collected in industrial or meteorological scenarios, step S100 cleans and standardizes the data to ensure the effectiveness of subsequent distance measurements.

[0023] Sub-step S110: Data cleaning and feature engineering First, the original dataset is cleaned to remove missing values ​​due to sensor malfunctions or transmission errors, ensuring the integrity of the input data. Then, considering the non-normal distribution of the original data, to improve the fitting effect of the Gaussian process regression model, step S110 abandons the general unified transformation method and instead performs targeted morphological correction based on feature skewness: For features with significant right-skewed distribution characteristics (such as long-tailed data like wind speed and air pressure), log smoothing is performed to compress large numerical intervals to shorten sample distances and correct data asymmetry. For features with left-skewed distribution characteristics (such as relative humidity), perform a power-law expansion transformation to correct the tailing phenomenon of the data distribution. For features exhibiting mild or moderate skewness (such as visibility), a square root transformation is used for gentle distribution adjustment.

[0024] The above transformation aims to map the probability density function of the original features to a form close to a Gaussian distribution, thereby conforming to the theoretical assumption of Gaussian process regression model regarding the stationarity of the input space.

[0025] To address the numerical differences caused by different physical units, step S110 executes a context-aware standardization strategy: For features that approximately follow a normal distribution, Z-Score standardization based on the mean and standard deviation is used to reconstruct the feature space into a distribution with zero mean and unit variance. For features with strict physical boundaries (such as parameters in percentage form), the interval linear mapping technique (Min-MaxScaling) is used to constrain the values ​​within a fixed closed interval to prevent extrapolation from exceeding the limit. For features containing abrupt noise or outliers, we abandon the standard method that is sensitive to the mean and instead adopt a robust scaling strategy based on the median and interquartile range to suppress the pulling effect of outliers on the overall distribution parameters.

[0026] For cases where the feature dimension is too high, step S110 performs principal component analysis (…). ), by using orthogonal transformation, high-dimensional correlated variables are converted into linearly independent low-dimensional features, thereby achieving dimensionality reduction while preserving the variance information of the original data to the maximum extent.

[0027] Sub-step S120: Constructing a multi-level validation system. To achieve unbiased parameter decoupling optimization and objective evaluation of model performance, step S120 divides the dataset into three parts. First, the dataset is divided into a training set and an independent test set. Then, within the training set, a sub-training set and a validation set are further divided. The sub-training set is used for the posterior calculation of the inner-layer physical parameters of the Gaussian process model, the validation set is used to evaluate the selection of outer-layer geometric parameters, and the independent test set is used to validate the generalization performance of the final model.

[0028] Step S200: Construction of the improved kernel function based on Minkowski distance Step S200 reconstructs the kernel function of the Gaussian process regression, the core of which is to construct a mathematical model that decouples the geometric topology from the physical correlation strength.

[0029] Sub-step S210: Define a parameterized Minkowski distance metric. Select a base kernel function based on distance construction and utilize the generalized Minkowski distance (…). This replaces the traditional Euclidean distance metric. It introduces an adjustable geometric control parameter. , used to define the topological structure of the sample feature space. When the parameter When different values ​​are taken, the "equidistant lines" between samples change from a rhombus shape in Manhattan space ( A smooth transition to a circle in Euclidean space ( ), and even the square shape of Chebyshev space ( This parameterized metric allows the model to adaptively capture non-spherical data distribution structures.

[0030] Sub-step S220: Construct the improved kernel function The defined Minkowski distance is incorporated into a basic kernel function (such as an exponential kernel function). The improved kernel function controls the overall magnitude of the signal variance control function and uses the Minkowski distance to quantify the rate of similarity decay between samples.

[0031] Sub-step S230: Configure Feature Correlation Mode (ARD) At the level of physical hyperparameters, the model introduces automatic correlation determination ( The mechanism assigns an independent feature length scale parameter to each input feature dimension. During training, the length scale participates in the distance calculation as a denominator, and its value directly determines the weight contribution of the corresponding feature dimension to the prediction result: the shorter the length scale, the more significant the impact of changes in that dimension's features on the output; conversely, when the length scale approaches infinity, that dimension's features are automatically suppressed by the model, thus achieving soft filtering of features.

[0032] Step S300: Adaptive optimization of Minkowski distance parameters based on a two-layer decoupling strategy Given the Minkowski distance parameter The resulting loss function surface is usually non-convex, multi-peaked and non-differentiable, while hyperparameters such as length scale have good analytical properties. Step S300 abandons the traditional hybrid joint optimization method and instead adopts a two-layer decoupling strategy of "outer layer geometric search - inner layer physical fitting".

[0033] Sub-step S310: Construct the outer geometry search loop Setting parameters The search space is utilized using a tree-structured Parzen estimator ( ) as an outer optimizer. The algorithm does not rely on backpropagation of gradients, but instead constructs "good value probability density" and "non-good value probability density" based on historical observation points, and improves the expected value by maximizing the expected value. The criterion is used to sample the next candidate that is most likely to reduce the validation set error. value.

[0034] Sub-step S320: Perform inner-layer physical fitting and numerical stability protection For each candidate passed in from the outer layer In step S320, the geometry of the kernel function is first fixed, and then the inner optimization loop is entered. In the inner loop, a quasi-Newton method (such as...) is used... Maximizing the log-marginal likelihood function based on a sub-training set ( Thus, the optimal analytical solution for signal variance and feature length scale is obtained.

[0035] In this process, a numerical stability monitoring mechanism (i.e., anomaly detection mechanism) is incorporated: during the... Decomposition and calculation of the kernel matrix ( Inverse matrix ( When monitoring the eigenvalue spectrum distribution of the matrix in real time ( ), If the current candidate The value leads to the condition number of the kernel matrix ( If the value exceeds the preset safety threshold (i.e., a non-positive definite value or a singular value appears), the circuit breaker mechanism is immediately triggered, skipping subsequent calculations in the inner layer and directly proceeding to the outer layer. The optimizer feeds back a preset maximum penalty error value. This mechanism forces the optimization to... The algorithm's sampling focus is deviated from those parameter regions that are theoretically feasible but numerically unstable, ensuring the robustness of the optimization process.

[0036] Sub-step S330: Determine the optimal parameters After the iteration is complete, the parameter that minimizes the error on the validation set is output as the optimal Minkowski distance parameter. This parameter represents the geometric metric that best fits the current data manifold structure.

[0037] Step S400: Model Training and Joint Hyperparameter Estimation Numerically stable optimal geometric parameters locked in step S300 The geometric metric form of the kernel function is fixed. Then, the final hyperparameter fitting is performed using the full quantum training set. The optimization algorithm maximizes the log-marginal likelihood function (...). For physical hyperparameters other than the Minkowski distance parameter (including signal variance) Noise variance and the length scale of each feature dimension The final joint estimation is then performed to complete the construction of the Gaussian process regression model.

[0038] Step S500: Predictive Execution and Performance Evaluation The sample to be predicted is input into the trained model, and by utilizing the posterior distribution characteristics of the Gaussian process, not only is the predicted mean output ( ) as a point estimate, and simultaneously extract the posterior covariance matrix ( The main diagonal elements () ) as prediction variance ( This provides a quantitative confidence interval for each prediction result, thereby enabling the quantification of uncertainty in heterogeneous multi-scale data.

[0039] This embodiment details the specific processing logic for heterogeneous, multi-scale, and high-dimensional feature data in step S100. Step S100 is used to eliminate the dimensional differences and distribution skewness of the original data and construct a standard input space adapted to the Minkowski distance metric. Step S100 is further decomposed into data cleaning and feature engineering step S110 and multi-level dataset partitioning step S120.

[0040] Step S110: Data Cleaning and Feature Engineering Step S110 addresses the issues of missing values, skewed distribution, and inconsistencies in dimensions in the original collected data (such as nuclear engineering critical heat flux density data or meteorological observation data) by performing structured processing. Step S110 specifically includes sub-steps S111 to S114.

[0041] Sub-step S111: Missing value determination and processing For the input dataset, the proportion of missing values ​​for each feature dimension is calculated. When the proportion of missing values ​​in a sample is lower than a preset threshold (e.g., 1%), a direct deletion method is used to remove sample rows containing missing values ​​to ensure the numerical stability of subsequent matrix operations. For features with a proportion of missing values ​​higher than the preset threshold, mean imputation or interpolation methods are used to complete the data. The specific imputation algorithms for missing values ​​are well-known to those skilled in the art and will not be elaborated upon here.

[0042] Sub-step S112: Transformation of non-spherical distribution characteristics In order to improve the fitting effect of the Gaussian process regression model, step S112 performs a mathematical transformation on the original data to improve the distribution shape, given that the original data exhibits a non-normal distribution.

[0043] For data with a right-skewed distribution (such as wind speed and air pressure in meteorological data), a natural logarithmic transformation is used. The transformation formula is as follows: In the formula, These are the original eigenvalues. These are the transformed eigenvalues. To prevent the logarithm from being a constant with negative infinity.

[0044] For data with left-skewed distribution characteristics (such as relative humidity), power transformation (such as square transformation) is used for processing: For features with moderately skewed distributions (such as visibility), the square root transformation is used: Through the above transformation, the distribution of the original features is close to a normal distribution, thus satisfying the Gaussian process regression's data assumption of stationarity in the input space.

[0045] Sub-step S113: Standardization of multi-scale features To eliminate the order-of-magnitude differences between different physical units (such as temperature in degrees Celsius and pressure in Pascals) and to prevent large-scale features from masking the contribution of small-scale features to the distance metric, step S113 selects the corresponding standardization strategy based on the specific statistical characteristics of the features.

[0046] For features that approximately follow a normal distribution (such as perceived temperature), Z-Score standardization is used. Z-Score standardization transforms the feature into a standard normal distribution with a mean of 0 and a variance of 1 by subtracting the mean and dividing by the standard deviation. The calculation formula is as follows: In the formula, These are the original eigenvalues. The sample mean of the current feature. The sample standard deviation of the current feature. This is the standardized value.

[0047] For features with a fixed value range and no obvious outliers (such as humidity in percentage form), max-min normalization is used to linearly map the feature values ​​to the [0,1] interval. The calculation formula is as follows: In the formula, and These are the minimum and maximum values ​​of the current feature in the dataset, respectively.

[0048] Robust scaling is employed for features containing outliers or anomalies (such as abrupt changes in wind speed data). Robust scaling utilizes the median and interquartile range for scaling, effectively mitigating the impact of outliers. The calculation formula is as follows: In the formula, The median of the current feature. It is the 25th percentile (lower quartile). It is the 75th percentile (upper quartile).

[0049] Sub-step S114: Dimensionality reduction of high-dimensional features Principal component analysis is performed when the feature dimensions of the input data exceed a preset threshold (e.g., 20 dimensions) or when significant multicollinearity exists. The eigenvalues ​​and eigenvectors of the covariance matrix are calculated, the eigenvalues ​​are sorted from largest to smallest, and the cumulative variance contribution rate is calculated. The top eigenvalues ​​with a cumulative variance contribution rate of 95% are selected. Each principal component is used as a new input feature matrix to reduce computational complexity while retaining most of the information in the original data.

[0050] Step S120: Construct a multi-level dataset To support subsequent adaptive optimization of the Minkowski distance parameter and evaluation of the model's generalization ability, step S120 performs a two-level data partitioning to establish a complete validation system that includes a training set, a validation set, and a test set.

[0051] Sub-step S121: First-level split (training-test split) The complete dataset processed in step S110 is randomly divided into a "total training set" and an "independent test set" according to a preset ratio (e.g., 9:1). The data in the independent test set is completely invisible in the subsequent parameter optimization and model training process, and is only used in step S500 to finally evaluate the prediction accuracy of the model, so as to objectively reflect the generalization ability of the model.

[0052] Sub-step S122: Secondary partitioning (training-validation partitioning) Within the "total training set", it is further divided into "sub-training set" and "validation set" according to a preset ratio (e.g., 9:1).

[0053] The “sub-training set” is used to calculate the kernel matrix and posterior distribution parameters in a Gaussian process regression model.

[0054] The "validation set" serves as the evaluation benchmark for the objective function of the Tree Structure Parzen Estimator (TPE) optimization algorithm. During the iterative optimization process in step S300, the Minkowski distance parameter is sampled at each step. Both methods train the model on a "sub-training set" and calculate the prediction error on a "validation set." The TPE algorithm updates its parameter search strategy based on the error feedback from the validation set.

[0055] The aforementioned two-level partitioning mechanism effectively isolates the parameter optimization process from the final performance evaluation process, avoiding model overfitting issues caused by data leakage and ensuring the accuracy of the Minkowski distance parameter. The global optimization results have real statistical significance.

[0056] This embodiment details the specific implementation of kernel function reconstruction in step S200. Step S200 extends the Euclidean distance metric in Gaussian process regression to a general Minkowski distance metric, and incorporates the Minkowski distance parameters... The hyperparameter system is incorporated for optimization. Step S200 includes defining the general Minkowski distance formula (step S210), constructing the improved kernel function (step S220), and configuring the feature length scale constraint mode (step S230).

[0057] Step S210: Define the general Minkowski distance formula Step S210 selects a distance-based kernel function as the basis and defines a general Minkowski distance calculation method to replace the fixed Euclidean distance in the traditional kernel function.

[0058] Regarding the parameter definition of Minkowski distance, let... For the input feature dimension, and The first two samples are respectively the first two samples. 3D eigenvalues For the first The length scale of the dimensional feature These are the parameters for the Minkowski distance.

[0059] The Minkowski distance constructed in step S210 The calculation formula is as follows: Among them, parameters This is an adjustable parameter. Step S210 involves introducing the parameter. This allows the similarity measurement between samples to no longer be limited to Euclidean space, but to be based on the data distribution characteristics in Manhattan distance (…). Distance from Chebyshev ( Adjust between ) .

[0060] Step S220: Construct the improved kernel function Step S220 will use the Minkowski distance defined in step S210. An improved kernel function is constructed by integrating it into the basic kernel function.

[0061] Taking the improved exponential kernel function as an example, the above Minkowski distance is... Substituting into the mathematical model of the exponential kernel function, the improved kernel function form is obtained as follows: in, The standard deviation of the signal is used to control the amplitude of the kernel function. Improved kernel functions utilize parameters... Capturing nonlinear correlations between samples, when Value and When the value is determined, the kernel function uniquely determines the covariance structure between samples.

[0062] Step S230: Configure feature length scale constraint mode Step S230 controls the feature length scale based on the feature dimension properties of the data through the Automatic Relevance Determination (ARD) parameter. Learning constraint methods.

[0063] Sub-step S231: Isotropic mode configuration When the ARD parameter is set to False, step S231 enters the isotropic mode. In this mode, step S231 constrains all feature dimensions to share the same length scale, i.e., the condition is satisfied. In this case, the model assumes that all features have the same rate of change in space, which is suitable for data scenarios with highly homogeneous features.

[0064] Sub-step S232: Anisotropic mode configuration When the ARD parameter is set to True, step S231 enters the anisotropic mode. In this mode, step S232 independently assigns a length scale parameter to each feature dimension, i.e., the feature length scale set is... The model learns each element independently during training. The value of . When the length scale of a certain dimension. When a feature is learned to have a larger value, the weight of that feature in the Minkowski distance formula decreases, thereby suppressing irrelevant features and filtering key features.

[0065] This embodiment details the specific implementation of step S300. Step S300 utilizes the Tree Structure Parzen Estimator (TPE) algorithm to evaluate the Minkowski distance parameter on the validation set. A global optimization is performed to determine the optimal kernel function metric. Step S300 specifically includes step S310 of constructing the optimization objective and configuring the search environment, step S320 of performing TPE iterative sampling, step S330 of candidate parameter evaluation and anomaly handling, and step S340 of determining the optimal parameters.

[0066] Step S310: Construct optimization objectives and configure the search environment Step S310 configures the parameter search task based on the Optuna hyperparameter optimization framework.

[0067] Step S310 defines the optimization objective as minimizing the root mean square error (RMSE) on the validation set. This is applied to the Minkowski distance parameters sampled in each iteration. , The formula for calculating the loss function corresponding to the parameters is as follows: In the formula: The total number of samples in the validation set; For the verification set The true target value of each sample; For the model in parameters Under the action of the first The predicted value for each sample.

[0068] Step S310: Set the Minkowski distance parameters The search space is Or a continuous interval adjusted according to the actual data distribution. Meanwhile, the core parameters for configuration optimization include the total number of iterations ( ) and the number of parallel tasks ( ), to control the computational scale of the optimization process.

[0069] Step S320: Perform TPE iterative sampling In step S320, new candidate parameters are generated using the TPE algorithm during each iteration. The TPE algorithm models historical observation data in a non-parametric manner, and its specific execution logic is as follows: Step S320: Based on the parameters generated from the historical iterations The historical observation points are divided into two groups based on the corresponding validation set RMSE values: one group is the "good value distribution" with smaller errors (usually the points corresponding to the top 20% quantiles with the smallest RMSE in the historical observations), and the other group is the "non-good value distribution" with larger errors (the remaining 80% of the points).

[0070] Step S320: Construct two probability density functions respectively. and ,in A good fit distribution represents the probability density of parameter values ​​leading to low error; a bad fit distribution represents the probability density of parameter values ​​leading to high error.

[0071] Step S320 selects the next sampling point based on the criterion of maximizing the expected improvement (EI). Specifically, the algorithm searches for a ratio that... The largest The value is used as the current candidate parameter. This sampling strategy ensures that the search process mainly focuses on the parameter region that may produce a lower RMSE, while retaining some exploration capability to avoid getting trapped in local optima.

[0072] Step S330: Candidate Parameter Evaluation and Anomaly Handling Step S330 applies to each candidate parameter generated in step S320 Execution model construction and performance evaluation.

[0073] Step S330 constructs a model incorporating Minkowski distance parameters on the sub-training set. An improved Gaussian process regression model was developed. During model construction, the L-BFGS-B optimization algorithm was used to maximize the logarithmic marginal likelihood function, optimizing the model for all Gaussian processes except... The remaining hyperparameters, including signal variance, noise variance, and feature length scale, are estimated.

[0074] Step S330 applies the trained model to the validation set, calculates the RMSE value according to the formula in step S310, and feeds the calculated RMSE value back to the Optuna framework to update the probability model of the TPE algorithm.

[0075] During execution, if due to parameters Extreme values ​​or data characteristics can cause the kernel matrix to be non-positive definite, resulting in the model being unable to perform Cholesky decomposition or failing to train. Step S330 triggers the anomaly capture mechanism.

[0076] Step S330 skips the model prediction stage for anomalous experiments and returns a preset maximum penalty value to the optimization framework. This anomaly feedback mechanism enables the TPE algorithm to identify parameter regions causing numerical instability as "non-optimal regions," thereby automatically reducing the sampling probability of these regions in subsequent iterations and ensuring the continuity and stability of the optimization process.

[0077] Step S340: Determine the optimal parameters When the preset is completed After the first iteration, step S340 iterates through all historical iteration records.

[0078] Step S340 selects the iteration with the smallest RMSE value on the corresponding validation set and records the Minkowski distance parameter used in that iteration as the optimal parameter. .

[0079] Step S340 outputs the optimal parameters This is used for training the final model in the subsequent step S400.

[0080] This embodiment details the specific implementation of steps S400 and S500. After determining the optimal Minkowski distance parameters through the aforementioned steps... Next, step S400 performs parameter fitting for the final model, and step S500 uses the trained model to predict unseen samples and evaluate the model's generalization performance. Step S400 includes constructing the final kernel function and estimating hyperparameters in step S410; step S500 includes calculating the posterior prediction distribution in step S510 and evaluating multidimensional performance metrics in step S520.

[0081] Step S410: Construct the final kernel function and estimate hyperparameters Step S410 utilizes the optimal Minkowski distance parameters output in step S340. Determine the geometric metric structure of the kernel function.

[0082] Step S410 directly selects the "sub-training set" divided in step S120 as training data. Let the input data of the sub-training set and the corresponding observations constitute training sample pairs.

[0083] Step S410 constructs the final covariance function (i.e., the improved kernel function). In the improved kernel function, the Minkowski distance parameter is fixed at... It no longer participates in subsequent gradient optimization. At this point, the hyperparameters to be estimated in the model only include signal variance, noise variance, and feature length scale (in anisotropic mode, they are in vector form).

[0084] Step S410 employs the maximum likelihood estimation method to solve for the hyperparameters to be estimated. Specifically, step S410 constructs a logarithmic marginal likelihood function based on the well-known principles of Gaussian process regression. This function measures the probability of observed data occurring under the current combination of hyperparameters.

[0085] Step S410 uses the L-BFGS-B numerical optimization algorithm to maximize (or minimize) the logarithmic marginal likelihood function until convergence, thereby determining the optimal estimates of the signal variance, noise variance, and feature length scale, and completing model training. The specific construction of the logarithmic marginal likelihood function and the solution process of the L-BFGS-B algorithm are well-known techniques to those skilled in the art and will not be elaborated upon here.

[0086] Step S510: Calculate the posterior prediction distribution Step S510 inputs the "independent test set" reserved in step S121 into the trained Gaussian process regression model.

[0087] According to the Bayesian inference principle of Gaussian process regression, the output of the test set samples follows a Gaussian posterior distribution. Step S510 calculates the mean vector and covariance matrix of this posterior distribution based on the trained kernel matrix and the observed data.

[0088] Step S510 uses the mean vector of the posterior distribution as the model's final predicted output value for the test sample.

[0089] Step S510 extracts the diagonal elements of the posterior distribution covariance matrix as the prediction variance, which is used to quantify the uncertainty of the prediction result (i.e., the confidence interval). The specific matrix calculation process for the posterior mean and variance of the Gaussian process is also a well-known technique in this field. This embodiment directly uses the prediction interface in standard algorithm libraries (such as GPy or Scikit-learn) for implementation.

[0090] Step S520: Multidimensional performance index evaluation Step S520 compares the predicted mean output in step S510 with the actual observed values ​​in the test set and calculates multiple statistical indicators to evaluate the model performance.

[0091] The metrics calculated in step S520 include root mean square error (RMSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and coefficient of determination (R-Squared).

[0092] The root mean square error is calculated in the same way as defined in step S310 above.

[0093] Mean absolute percentage error (MASE) measures the proportion of prediction deviation relative to the true value. It can eliminate the influence of dimensions and objectively reflect the accuracy of prediction.

[0094] The coefficient of determination is used to measure the model’s ability to explain the variability of data; the closer its value is to 1, the better the model fits the data.

[0095] Step S520 compares the results based on the optimal Minkowski parameters. The model and the model based on traditional Euclidean distance ( The above evaluation metrics of the model verify the accuracy improvement effect of the method in this embodiment under complex non-spherical data distribution. If the coefficient of determination is close to 1 and the RMSE and MAPE are significantly reduced compared with the traditional model, it indicates that the model is successfully constructed and can output prediction results for engineering applications.

[0096] This embodiment applies the improved Gaussian process regression method based on the Minkowski distance framework described in Embodiments 1 to 5 to the critical heat flux density prediction scenario in the field of nuclear engineering.

[0097] This embodiment utilizes experimental data on critical heat flux density to construct and validate a prediction model following the procedures outlined in steps S100 to S500. The specific implementation process is as follows: I. Data Acquisition and Preprocessing (corresponding step S100) This embodiment selects experimental data on boiling flow in a vertical circular tube under low-pressure, low-mass-flux conditions as the basis for modeling. The critical heat flux density dataset contains 512 valid samples, covering 6 key input features describing the thermal-hydraulic state.

[0098] The input features specifically include: pipe diameter (D), heating length (L), system pressure (P), mass flow rate (G), and inlet subcooling (…). ) and export vapor content (X).

[0099] The output target is the critical heat flux density (CHF).

[0100] Statistical analysis shows that the critical heat flux density data is concentrated in a system with a pressure range of 106-951 kPa and a mass flow rate range of 20-277 kg / m³. 2 There are significant scale differences among different features.

[0101] This embodiment follows the partitioning strategy described in Embodiment 2, dividing the 512 samples into a training set (462 sets) and an independent test set (50 sets). This is done to determine the Minkowski distance parameters. To optimize the training set, this embodiment further divides the training set into a sub-training set (416 sets) and a validation set (46 sets).

[0102] This embodiment performs standardization processing on the input features to eliminate the significant order-of-magnitude difference between the pipe diameter and the system pressure, and constructs a standardized input matrix.

[0103] II. Parameter Optimization and Model Construction (corresponding steps S200-S400) This embodiment constructs a Gaussian process regression model based on the improved exponential kernel function described in Embodiment 3, and executes the TPE strategy described in Embodiment 4 to adjust the Minkowski distance parameter. Perform global optimization.

[0104] In this embodiment, the optimization objective is set as minimizing the root mean square error on the validation set (46 samples). The TPE algorithm performs iterative sampling within a preset search space.

[0105] Through iterative search, the TPE algorithm determined the optimal Minkowski distance parameter for the critical heat flux density dataset. It is 1.56.

[0106] The optimal parameter of 1.56 indicates that, for the critical heat flux density prediction task, the similarity measure between samples differs from the standard Euclidean geometric space. ).

[0107] Based on the defined parameters, this embodiment constructs four comparative models: Model A: Traditional Euclidean distance isotropic model ( ,ARD=False); Model B: Improved Minkowski distance isotropic model ( ,ARD=False); Model C: Traditional Euclidean distance anisotropy model ( ,ARD=True); Model D: Improved Minkowski distance anisotropy model ( ,ARD=True).

[0108] III. Predictive Performance Verification Analysis (corresponding to step S500) This embodiment evaluates four comparative models on an independent test set (50 samples) and calculates the root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE).

[0109] 1. Comparison results of isotropic models Model A (Traditional) The RMSE of Model B was 40.519, and the MAPE was 3.437%; Model B (improved) The RMSE decreased to 36.111, and the MAPE decreased to 3.143%. Data comparison shows that, without distinguishing feature weights, adjusting the distance metric in the kernel function from Euclidean distance to Minkowski parameters... This reduces the model's prediction error.

[0110] 2. Comparison Results of Anisotropic Models Model C (traditional) The RMSE of model D (improved) is 23.706. The RMSE further decreased to 22.091, the MAPE reached its lowest value of 1.683%, and the coefficient of determination increased to 0.994712.

[0111] Model D performed best across all evaluation metrics. This result demonstrates that the combined effect of the "Minkowski distance parameterization" and "anisotropic feature weighting" mechanisms improved prediction accuracy. The anisotropic mechanism identifies key influencing factors such as mass flow rate and system pressure by independently learning feature length scales. The Minkowski distance, on the other hand, fits the influence of input features on CHF in geometric space.

[0112] Comparison chart of predicted and actual values ​​( Figure 3 In the diagram, the red pentagram represents the actual value, and the blue curve represents the predicted value. The predicted value curve of model D (p=1.56, ARD=True) highly overlaps with the actual value data points, and the deviation is significantly smaller than that of models A and C. Model D exhibits good following performance in the high heat flux density region, without obvious saturation or drift.

[0113] In summary, this embodiment verifies the applicability of the improved Gaussian process regression method based on the Minkowski distance framework in the prediction of key nuclear safety parameters. This method improves the prediction accuracy of the model by adaptively matching the data distribution structure.

[0114] This embodiment applies the improved Gaussian process regression method based on the Minkowski distance framework described in Embodiments 1 to 5 to a complex meteorological temperature prediction scenario with high dimensionality and massive sample characteristics.

[0115] This embodiment utilizes historical meteorological datasets from the Szeged region of Hungary, constructing a prediction model following steps S100 to S500. This embodiment verifies the adaptability of the method of this invention to non-spherical distributed data in a multidimensional feature space after dimensionality reduction via Principal Component Analysis (PCA). The specific implementation process is as follows: I. Data Acquisition and Preprocessing (corresponding step S100) This embodiment uses hourly meteorological observation data from 2006 to 2016 as the basis for modeling. The original data contains 96,453 records, covering 12 fields including temperature, perceived temperature, humidity, wind speed, wind direction, visibility, air pressure, precipitation type, and weather description. In this embodiment, "temperature" is used as the target variable to be predicted.

[0116] In view of the complex characteristics of meteorological data distribution, this embodiment performs the following feature engineering operations: Feature Distribution Transformation: Statistical analysis shows that the original features do not satisfy the Gaussian process regression assumption of input stationarity. Therefore, this embodiment performs a logarithmic transformation on the right-skewed "wind speed" and "air pressure" features; a square transformation on the left-skewed "humidity" feature; and a square root transformation on the moderately skewed "visibility" feature. These feature transformations make the probability density functions of each feature dimension approximate a normal distribution, optimizing the distance calculation environment for the kernel function.

[0117] Encoding and Standardization: One-hot encoding is used for discrete features such as "precipitation type" to convert text categories into numerical vectors. To address the dimensional differences of different features, this embodiment uses Z-Score standardization for normally distributed features such as "feeling temperature," Min-Max normalization for bounded features such as "humidity," and robust scaling for features containing outliers such as "wind speed."

[0118] Principal Component Analysis (PCA) Dimensionality Reduction: After encoding, the input feature dimension is expanded to 33. To avoid the curse of dimensionality and improve computational efficiency, PCA is performed on the training data in this embodiment. The calculation results show that the cumulative variance contribution rate of the first 7 principal components reaches 96.24%. Therefore, this embodiment extracts the first 7 principal components to construct a new low-dimensional input matrix for subsequent model training.

[0119] According to the strategy described in Example 2, this example performs random sampling on the preprocessed data to balance the computational load, and divides the sampled dataset into a training set (including a sub-training set and a validation set) and an independent test set.

[0120] II. Parameter Optimization and Model Construction (corresponding steps S200-S400) This embodiment constructs a Gaussian process regression model based on an improved exponential kernel function, and utilizes the TPE strategy to refine the Minkowski distance parameter on the validation set. To find the best option.

[0121] The TPE algorithm iteratively samples within a predefined search space to ultimately determine the optimal Minkowski distance parameter for the meteorological dataset. It is 1.198.

[0122] The optimal parameter of 1.198 indicates that, after PCA dimensionality reduction and mixed distribution transformation, the correlation structure of the meteorological data's feature space conforms to the Minkowski distance metric (i.e., close to the Manhattan distance property) with a parameter close to 1. This result demonstrates that, in the multidimensional meteorological feature space, the similarity decay between samples exhibits a non-isotropic distribution.

[0123] Based on optimal parameters In this embodiment, an improved Gaussian process regression model was constructed, and the L-BFGS-B algorithm was used to jointly estimate the signal variance, noise variance, and feature length scale (in anisotropic mode) on the total training set.

[0124] III. Predictive Performance Verification Analysis (corresponding to step S500) This embodiment tests the optimal Minkowski distance on an independent test set. The improved model and the model based on traditional Euclidean distance ( A comparative test was conducted on the baseline model of ).

[0125] 1. Performance improvements in isotropic mode Under isotropic settings (i.e., forcing all feature lengths to be equal): Traditional Euclidean distance model ( The root mean square error (RMSE) was 0.392 and the mean absolute error (MAE) was 0.239.

[0126] Improved Minkowski distance model ( The RMSE decreased to 0.343 and the MAE decreased to 0.197.

[0127] Data comparison results show that the RMSE index decreased by approximately 12.5%, and the MAE index decreased by approximately 17.6%. These results demonstrate that, when the model is limited to a single length scale, optimizing the distance metric parameters can improve performance. This improves the model's ability to represent complex data distributions and enhances prediction accuracy.

[0128] 2. Performance under anisotropic mode Under anisotropic settings (i.e., allowing independent learning of feature length scales): Improved Minkowski distance model ( The coefficient of determination () The value reached 0.999038, RMSE was 0.297, and MAE was 0.151.

[0129] Although the traditional Euclidean distance anisotropy model performs slightly better (RMSE 0.285) in the PCA dimensionality reduction feature space of this specific dataset, the improved model ( The performance of (ARD=True) is still better than that of the isotropic model with the same parameters, and it maintains a high level of prediction. ).

[0130] Comparison chart of predicted and actual values ​​( Figure 7 In the diagram, red stars represent actual values, and blue curves represent predicted values. The predicted curves of the improved model closely match the actual values, indicating that the improved model can capture the nonlinear patterns of temperature changes over time and with meteorological factors.

[0131] In summary, this embodiment verifies the effectiveness of the improved Gaussian process regression method based on the Minkowski distance framework in the field of weather forecasting. Especially in scenarios with limited computational resources or requiring simplified model structures (using isotropic kernels), the introduction of learnable Minkowski distance parameters... The method in this embodiment improves the prediction accuracy of the model.

[0132] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method for processing heterogeneous multi-scale data in Gaussian process regression, characterized in that, Includes the following steps: Step S100: Obtain the original heterogeneous multi-scale data, perform feature preprocessing on it, and divide the preprocessed dataset into a training set and an independent test set. The training set is then further divided into a sub-training set and a validation set. Step S200: Construct a Gaussian process regression kernel function based on Minkowski distance. The kernel function decouples the geometric metric parameters of the sample feature space from the physical hyperparameters of the kernel function. The geometric metric parameters are adjustable Minkowski distance parameters, and the physical hyperparameters include signal variance and feature length scale. Step S300: Execute a two-level iterative optimization strategy based on numerical stability constraints to determine the optimal Minkowski distance parameter on the validation set; The two-layer iterative optimization strategy specifically includes: in the outer loop, a tree-structured Parzen estimator strategy is used to sample candidate Minkowski distance parameters in a preset geometric space; in the inner loop, the kernel function metric is fixed for the currently sampled candidate Minkowski distance parameters, and the physical hyperparameters are analytically or solved by maximizing the logarithmic marginal likelihood function using the sub-training set; at the same time, an anomaly capture mechanism is run in the iteration: the positive definiteness of the kernel matrix constructed by the current candidate Minkowski distance parameters is detected in real time, and if the kernel matrix is ​​non-positive definite or the condition number is abnormal, the inner calculation is blocked and a preset penalty error value is fed back to the tree-structured Parzen estimator to guide the sampling to avoid numerically unstable regions; Step S400: Based on the numerically stable optimal Minkowski distance parameter determined in step S300, fix the geometric metric form of the kernel function, train the final Gaussian process regression model using the sub-training set, and obtain the final joint estimate of the physical hyperparameters. Step S500: Input the data to be predicted into the trained Gaussian process regression model and output the prediction results.

2. The method for processing heterogeneous multi-scale data using Gaussian process regression according to claim 1, characterized in that, In step S100, feature preprocessing is performed on the original heterogeneous multiscale data. Specifically, this includes detecting and removing samples containing missing values, and performing feature transformations for different feature dimensions: performing logarithmic transformation on features exhibiting a right-skewed distribution; performing power transformation on features exhibiting a left-skewed distribution; and performing square root transformation on features exhibiting a moderately skewed distribution.

3. The method for processing heterogeneous multi-scale data using Gaussian process regression according to claim 1, characterized in that, The feature preprocessing also includes handling the dimensional differences and high-dimensional redundancy of the feature dimensions: standardization is performed for the dimensional differences of different feature dimensions; Z-Score standardization is performed for features with approximately normal distribution; max-min normalization is performed for bounded distribution features; and robust scaling is performed for features containing outliers. When the number of feature dimensions exceeds a preset threshold, principal component analysis is performed to reduce dimensionality, and principal components whose cumulative variance contribution rate meets the preset standard are retained as input features.

4. The method for processing heterogeneous multi-scale data using Gaussian process regression according to claim 1, characterized in that, The step S200 of constructing the Gaussian process regression kernel function based on Minkowski distance specifically refers to: Select a basic kernel function and replace the distance term used to measure the similarity between samples in the basic kernel function with Minkowski distance; The calculation logic of the Minkowski distance is as follows: calculate the absolute value of the difference between the feature values ​​of two samples in each dimension raised to the power of p, divide it by the corresponding dimension feature length scale raised to the power of p, add the calculation results of each dimension to obtain the sum, and finally take the p-th root of the sum; where p is the Minkowski distance parameter.

5. The method for processing heterogeneous multi-scale data using Gaussian process regression according to claim 4, characterized in that, The basic kernel function is an exponential kernel function. The calculation logic of the improved Gaussian process regression kernel function is as follows: the natural constant e is calculated using the amplitude of the signal variance control function, and the exponent term of the exponent is the negative value of the Minkowski distance.

6. The method for processing heterogeneous multi-scale data using Gaussian process regression according to claim 4, characterized in that, Step S200 further includes configuring the constraint mode for the feature length scale: When configured in isotropic mode, the feature length scale of all feature dimensions is constrained to be equal, and the model shares a unique feature length scale parameter. When configured in anisotropic mode, each feature dimension is allowed to have an independent feature length scale, and the model automatically identifies the weights of different feature dimensions during training.

7. The method for processing heterogeneous multi-scale data using Gaussian process regression according to claim 1, characterized in that, The iterative optimization using the tree-structured Parzen estimator strategy in step S300 specifically includes: The objective function is set to minimize the root mean square error on the validation set. Define the search space range for the Minkowski distance parameters; in each iteration, construct the good value probability density function and the non-good value probability density function based on the root mean square error distribution of historical observation points; and sample new candidate Minkowski distance parameters within the search space range according to the maximization expectation improvement criterion.

8. The method for processing heterogeneous multi-scale data using Gaussian process regression according to claim 1, characterized in that, In step S400, the joint estimation of model hyperparameters other than the Minkowski distance parameter specifically refers to: The model hyperparameters include signal variance, noise variance, and feature length scale; a log-marginal likelihood function of the training set is constructed; and a numerical optimization algorithm is used to maximize the log-marginal likelihood function to obtain the optimal estimate of the model hyperparameters.

9. The method for processing heterogeneous multi-scale data using Gaussian process regression according to claim 1, characterized in that, The output of the prediction result in step S500 specifically includes: Calculate the posterior prediction distribution of the Gaussian process regression model, and output the mean of the posterior prediction distribution as the predicted value; extract the diagonal elements of the covariance matrix of the posterior prediction distribution as the variance of the prediction result, which is used to quantify the uncertainty of the prediction.

10. The method for processing heterogeneous multi-scale data using Gaussian process regression according to claim 1, characterized in that, The step S500 is followed by a model performance verification step: The trained Gaussian process regression model was tested using the independent test set, and the statistical evaluation index between the predicted value and the actual value was calculated. The statistical evaluation indicators include root mean square error, mean absolute error, mean absolute percentage error, and coefficient of determination. The differences between the model using the optimal Minkowski distance parameter and the benchmark model using the fixed Euclidean distance parameter on the statistical evaluation index were compared to verify the model's generalization ability under non-spherical data distribution.

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