Long-tail virtual sample generation method and system based on multi-scale long-tail kernel function

Abstract

The invention provides a long-tail virtual sample generation method and system based on a multi-scale long-tail kernel function, and the method comprises the steps: obtaining an original data set, and recognizing long-tail distribution features in the data set; selecting at least two long-tail kernel functions, and applying constraints to parameter values of the long-tail kernel functions; performing adaptive weighted combination on the plurality of long-tail kernel functions subjected to parameter constraint, and constructing a multi-scale long-tail kernel function; establishing a Gaussian process hidden variable model based on a multi-scale long-tail kernel function, and determining kernel function parameters and weights through optimization; obtaining probability distribution of low-dimensional hidden variables by using a Gaussian process hidden variable model, and sampling from the probability distribution to generate a low-dimensional virtual sample; based on a mapping relationship between the low-dimensional virtual sample and the original data, mapping the low-dimensional virtual sample to an original data space to obtain a high-dimensional virtual sample; according to the method, the combined modeling capability of the local features and the long-tail features can be improved, so that virtual samples with higher quality and more reasonable distribution are generated.

Description

Technical Field

[0001] This invention relates to the field of virtual sample generation technology, and in particular to a method and system for generating long-tailed virtual samples based on a multi-scale long-tailed kernel function. Background Technology

[0002] As machine learning technology is increasingly applied in engineering and industry, model performance largely depends on the scale and quality of training data. However, in practical applications such as battery performance prediction, obtaining high-quality, large-scale datasets is often limited by factors such as cost, time, safety, and experimental conditions, resulting in a limited number of data samples available for modeling. Meanwhile, battery performance is highly sensitive to operating conditions such as temperature and load conditions; batteries from different batches or with different manufacturing processes may exhibit significant differences in key performance indicators, limiting the completeness and representativeness of existing datasets. In cases of insufficient or unevenly distributed data, machine learning models are prone to problems such as insufficient generalization ability and decreased prediction accuracy.

[0003] To mitigate the adverse effects of small sample data on model performance, virtual sample generation technology has gradually become an effective supplementary method. By constructing a reasonable generative model, representative virtual samples can be synthesized in the original sample space or latent variable space, which helps to expand the scale of training data and improve the stability of the model. However, the effectiveness of virtual sample generation largely depends on the generative model's ability to characterize the data distribution characteristics. Many existing virtual sample generation methods based on latent variable models typically use the Gaussian radial basis function (RBF kernel) to measure the similarity between samples. This type of kernel function has good smoothness in the feature space and can effectively describe the overall continuous trend of the data, but its exponential decay characteristic causes the correlation between distant samples to weaken rapidly, which can easily lead to over-smoothing of the model, making it difficult to characterize the local variation features in the data.

[0004] In real-world battery datasets, sample distributions often exhibit significant skewness and imbalance, with some key features displaying pronounced long-tailed distributions and complex nonlinear relationships between features. Traditional virtual sample generation methods based on single-scale Gaussian kernel functions struggle to simultaneously model local details and characterize long-range dependencies in these scenarios, thus affecting the diversity and realism of the virtual samples.

[0005] Therefore, there is an urgent need for a method and system for generating long-tailed virtual samples based on multi-scale long-tailed kernel functions, which can improve the ability to jointly model local features and long-tailed features, thereby generating higher-quality and more reasonably distributed virtual samples, and providing more reliable data support for machine learning modeling under small sample conditions. Summary of the Invention

[0006] To address the aforementioned shortcomings of existing technologies, the present invention aims to provide a method and system for generating long-tailed virtual samples based on multi-scale long-tailed kernel functions. This method aims to solve the technical problem that existing virtual sample generation methods struggle to simultaneously model local features and long-tailed distributions, resulting in insufficient quality of small sample data expansion.

[0007] To achieve the above objectives, in a first aspect, the present invention provides a method for generating long-tailed virtual samples based on a multi-scale long-tailed kernel function, used to generate virtual samples for data with long-tailed distribution characteristics, comprising the following steps: Obtain the original dataset and identify the long-tail distribution characteristics present in the dataset; Select at least two long-tailed kernel functions and impose constraints on the parameter values ​​of the long-tailed kernel functions; An adaptive weighted combination of multiple parameter-constrained long-tailed kernel functions is performed to construct a multi-scale long-tailed kernel function; A Gaussian process latent variable model is established based on the multi-scale long-tailed kernel function, and the kernel function parameters and weights are determined through optimization. The probability distribution of low-dimensional latent variables is obtained using the Gaussian process latent variable model, and low-dimensional virtual samples are generated by sampling from the probability distribution. Based on the mapping relationship between the low-dimensional virtual sample and the original data, the low-dimensional virtual sample is mapped to the original data space to obtain a high-dimensional virtual sample.

[0008] As a further improvement to the above technical solution, the step of identifying long-tail distribution features in the dataset includes: By drawing at least one of histograms, frequency distribution plots, box plots, or violin plots, the data distribution pattern can be analyzed to determine whether long-tailed distribution characteristics exist.

[0009] As a further improvement to the above technical solution, the long-tailed kernel function includes at least one of the following: rational quadratic kernel function, Cauchy kernel function, generalized t-Student kernel function, logarithmic kernel function, heavy-tailed power-law kernel function, or inverse polyquadratic kernel function.

[0010] As a further improvement to the above technical solution, the at least two long-tailed kernel functions belong to the same kernel function type but have different scale parameters, or belong to different types of long-tailed kernel functions, in order to characterize the multi-scale long-tailed characteristics of the data.

[0011] As a further improvement to the above technical solution, the constraints imposed on the parameters of the long-tailed kernel function include: The parameter is limited to a preset value range, which is (0,1).

[0012] As a further improvement to the above technical solution, the parameter constraint is achieved by one of the following methods: applying a Sigmoid function, a Tanh function, or a Softmax function to the kernel function parameters; Introduce a parameter penalty term into the loss function of the Gaussian process latent variable model.

[0013] As a further improvement to the above technical solution, the multi-scale long-tail kernel function is constructed in one of the following ways: A weighted summation of multiple long-tailed kernel functions is performed. Weighted multiplication combination of multiple long-tailed kernel functions; Perform weighted addition and multiplication operations on multiple long-tailed kernel functions.

[0014] As a further improvement to the above technical solution, the weights of each long-tailed kernel function in the multi-scale long-tailed kernel function are obtained by optimizing the loss function of the Gaussian Process Latent Variable Model (GPLVM), and satisfy the constraint that the weights are non-negative and the sum of the weights is 1, as shown in the following formula: ; Among them, w i These are the weighting coefficients. Let m be the loss function of GPLVM, and m be the number of long-tailed kernel functions.

[0015] As a further improvement to the above technical solution, the process of obtaining the probability distribution of low-dimensional latent variables includes: The latent variables were initialized using the principal component analysis results of the original data; The multi-scale long-tailed kernel function is used to characterize the covariance structure among samples; The Gaussian process latent variable model is solved using numerical optimization methods to obtain the probability distribution of low-dimensional latent variables.

[0016] As a further improvement to the above technical solution, when the original data contains tag information, the method further includes: Based on the Gaussian process regression model, the labels corresponding to the generated high-dimensional virtual samples are predicted to obtain the label information of the virtual samples.

[0017] Secondly, the present invention also provides a long-tail sample generation system based on a Gaussian process latent variable model, comprising: The long-tail feature analysis module is used to perform distribution analysis on the input raw dataset to identify long-tail distribution features in the data; The long-tail kernel function construction module is used to select at least two long-tail kernel functions and to constrain the parameter values ​​of the long-tail kernel functions. The multi-scale kernel combination module is used to adaptively weight and combine multiple parameter-constrained long-tailed kernel functions to construct multi-scale long-tailed kernel functions. The latent variable modeling module is used to establish a Gaussian process latent variable model based on the multi-scale long-tailed kernel function, and to determine the kernel function parameters and corresponding weights through the optimization process. The latent variable sampling module is used to obtain the probability distribution of low-dimensional latent variables based on the Gaussian process latent variable model, and to sample from the probability distribution to generate low-dimensional virtual samples; The sample mapping generation module is used to map the low-dimensional virtual sample to the original data space based on the mapping relationship between the low-dimensional virtual sample and the original data, thereby generating a high-dimensional virtual sample.

[0018] Because the present invention adopts the above technical solutions, the beneficial effects of this application are as follows: This invention provides a method for generating long-tailed virtual samples based on a multi-scale long-tailed kernel function. First, a multi-scale long-tailed kernel function is constructed. This multi-scale long-tailed kernel function selects at least two long-tailed kernel functions, which can be of the same type but with different parameter settings, or kernel functions of different categories. These kernel functions are then fused using weighted combination or product methods to form a composite kernel structure with stronger expressive power. Simultaneously, this invention imposes constraints on the parameter range of the kernel function to precisely control its similarity measurement ability in the input space. This structural design enables the multi-scale long-tailed kernel function to effectively capture multi-scale variation information and subtle local features contained in the data, making it particularly suitable for processing complex data structures with skewed or long-tailed distribution characteristics. The Gaussian process latent variable model constructed based on this kernel function significantly improves its fitting ability to complex data distributions, thus laying a solid foundation for the subsequent generation of high-quality virtual samples.

[0019] Secondly, by constructing the aforementioned multi-scale long-tail kernel function, this invention can simultaneously consider both the local detailed features and the global long-tail distribution characteristics of the data. With the support of this kernel function, the Gaussian process latent variable model can more accurately characterize the complex distribution pattern of the original data during the optimization process, especially the data features of the long-tail portion. This fundamentally solves the technical problem of existing virtual sample generation methods struggling to simultaneously consider local features and long-tail distribution modeling, improving the quality of small sample data expansion, and resulting in more representative, diverse, and realistic virtual samples.

[0020] Furthermore, this invention adaptively weights and combines (or multiplies and fuses) multiple long-tailed kernel functions with parameter constraints, enabling the model to dynamically adjust the contribution weights or interaction patterns of each component based on the characteristics of the data itself. This adaptive mechanism endows the model with greater flexibility and adaptability to different types and degrees of long-tailed data. Simultaneously, the inherently good generalization performance of the Gaussian process latent variable model, combined with the accurate modeling of data distribution by multi-scale long-tailed kernel functions, further enhances the generalization ability of the model in generating virtual samples, ensuring that the generated samples better reflect the underlying patterns of the original data.

[0021] Furthermore, this invention employs a strategy of first obtaining the probability distribution of low-dimensional latent variables and sampling them to generate low-dimensional virtual samples, then mapping them back to the original data space to obtain high-dimensional virtual samples. This method decomposes the complex problem of mapping the original high-dimensional space into relatively simple low-dimensional latent space operations and mapping steps, reducing the complexity of direct operations in the high-dimensional space and thus improving the efficiency of virtual sample generation. In addition, since the low-dimensional latent space has relatively less noise and redundant information, the sampling process is more stable, which helps to generate virtual samples with higher quality and better consistency.

[0022] The high-quality virtual samples generated by the method of this invention can effectively alleviate the problem of insufficient training data in long-tail data scenarios, providing more sufficient and representative training data for subsequent machine learning models (such as classifiers and regressors). This helps improve the learning performance, generalization ability, and prediction accuracy of models on long-tail data, providing a more reliable data foundation and technical support for various machine learning tasks on long-tail data. In this invention, the overall parameters of the multi-scale long-tail kernel function and the mixing weights (or fusion coefficients) are adaptively determined by optimizing the objective function. This adaptive optimization process ensures that the generated virtual samples maintain the distribution characteristics of the original data while possessing higher diversity and generalization ability, avoiding the problems of overfitting or monotonous patterns in the generated samples. Attached Figure Description

[0023] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0024] Figure 1 This is a flowchart illustrating a method for generating long-tailed virtual samples based on a multi-scale long-tailed kernel function disclosed in this invention. Figure 2This is a visual diagram illustrating the characteristics of the two kernel functions (RBF kernel function and multi-scale long-tailed RQ kernel function) disclosed in this invention and their corresponding sample generation functions. Figure 3 This is a histogram of the feature distribution of the example dataset disclosed in this invention; Figure 4 This is a two-dimensional visualization diagram of the latent variables disclosed in this invention, wherein... Figure 4 (a) represents the latent variables of the original data. Figure 4 (b) is a two-dimensional visualization diagram of the latent variables of the virtual sample generated by the long-tailed virtual sample generation method based on the multi-scale long-tailed kernel function provided by the present invention. Figure 5 This is a visualization diagram of the distribution of virtual labels generated using the long-tail virtual sample generation method based on a multi-scale long-tail kernel function disclosed in this invention. Figure 5 (a) is a bar chart showing the distribution of virtual labels. Figure 5 (b) is a box plot of the virtual label distribution; Figure 6 This is a scatter plot showing the performance of virtual samples generated by the long-tailed virtual sample generation method based on the multi-scale long-tailed kernel function disclosed in this invention on four learning models. Figure 7 This is a line graph showing the prediction model performance of virtual samples generated by a multi-scale long-tailed kernel function-based method, using a rational quadratic kernel function as an example. The method generates virtual samples based on a mixture of different numbers of long-tailed kernel functions.

[0025] The objectives, features, and advantages of this invention will be further explained in conjunction with the embodiments and with reference to the accompanying drawings. Detailed Implementation

[0026] The technical solutions of 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 a part of the embodiments of the present invention, and not all of them. 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.

[0027] It should be noted that the technical solutions of the various embodiments of the present invention can be combined with each other, but only if they are based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0028] Example 1

[0029] See Figure 1 This invention provides a method for generating long-tailed virtual samples based on a multi-scale long-tailed kernel function, specifically including the following steps: Step S1: Obtain the original dataset and identify the long-tail distribution characteristics present in the dataset.

[0030] First, the original dataset to be processed is obtained. This dataset exhibits a long-tail distribution, meaning that a few categories (head categories) contain a large number of samples, while the majority categories (tail categories) contain only a small number of samples. This step aims to clarify the distribution characteristics of the data, providing a basis for subsequent selection and processing of long-tail kernel functions. By identifying the long-tail distribution characteristics, this invention enables the targeted design of kernel functions, thereby effectively improving the modeling ability of long-tail data structures and laying the foundation for solving the problem of insufficient quality expansion of small sample data.

[0031] Step S2: Select at least two long-tailed kernel functions and impose constraints on the parameter values ​​of the long-tailed kernel functions.

[0032] Based on the long-tail distribution characteristics identified in step S1, at least two long-tail kernel functions K are selected. i (i=1,2,...,m,m≥2). These long-tailed kernel functions can be long-tailed kernel functions belonging to the same category but with different parameter settings, or they can be long-tailed kernel functions of different categories. For example, exponentially decaying long-tailed kernel functions and power-law decaying long-tailed kernel functions can be combined. Then, for each selected long-tailed kernel function K... i The parameter values ​​are constrained. These parameter constraints are used to regulate the similarity measurement ability of the kernel function in the input space, enabling it to better adapt to the characteristics of long-tail data. For example, for the parameter controlling the decay rate of the kernel function, its value range can be set to prevent it from decaying too quickly or too slowly, thus effectively capturing data features in both local and global contexts. This step, through parameter constraints, allows the long-tail kernel function to more accurately characterize the similarity of long-tail data at different scales, providing a basic component for constructing multi-scale long-tail kernel functions and effectively improving the ability to capture multi-scale changes and local features of data.

[0033] Step S3: Adaptively weightedly combine multiple long-tailed kernel functions with parameter constraints to construct a multi-scale long-tailed kernel function.

[0034] For multiple long-tailed kernel functions k after parameter constraints in step S2 i Adaptive weighted combination is performed to construct a multi-scale long-tailed kernel function K. mixture The weighted combination can be a linear weighted combination or a non-linear combination, such as a product. For example, a linear weighted combination can be expressed as... ,in Let be the weight of the i-th long-tail kernel function, and satisfy . , Weight The value can be predetermined or determined adaptively through subsequent optimization processes.

[0035] Through this adaptive weighted combination method, different long-tail kernel functions can contribute different weights based on their advantages in capturing features at specific scales, thus forming a composite kernel structure with stronger expressive power. This multi-scale long-tail kernel function can effectively integrate long-tail feature information at different scales, capturing both local detailed features of the data and grasping the global long-tail distribution trend, thereby significantly improving the modeling ability of complex long-tail data distributions and providing core support for the subsequent generation of high-quality virtual samples.

[0036] Step S4: Establish a Gaussian process latent variable model based on the multi-scale long-tailed kernel function, and determine the kernel function parameters and weights through optimization.

[0037] Based on the multi-scale long-tailed kernel function K constructed in step S3 mixture A Gaussian process latent variable model is established. This model achieves dimensionality reduction and generation by learning the latent low-dimensional representation of the data. In this model, a multi-scale long-tailed kernel function is used to define the covariance relationship between data points, thereby characterizing the data distribution. Subsequently, the kernel function parameters and the weights of the weighted combination involved in step S3 are determined through optimization algorithms (such as maximizing marginal likelihood). This optimization process enables the model to adaptively learn the kernel function structure and parameter configuration best suited to the current data, ensuring the model's fitting accuracy and generalization ability to the original data distribution. By optimizing the determined parameters and weights, the process of generating virtual samples becomes more stable and efficient, and better preserves the distribution characteristics of the original data.

[0038] Step S5: Obtain the probability distribution of low-dimensional latent variables using the Gaussian process latent variable model, and sample from the probability distribution to generate low-dimensional virtual samples.

[0039] Using the Gaussian process latent variable model optimized in step S4, the probability distribution P(Z) of the original data in the low-dimensional latent space can be obtained, where Z is the low-dimensional latent variable. Then, sampling is performed from the probability distribution P(Z) to generate several low-dimensional virtual samples. This step transforms the complex problem of generating original high-dimensional data into a problem of sampling in a low-dimensional latent space. Since the low-dimensional space has lower dimensions, the sampling process is simpler and more stable, which helps to improve the efficiency and quality of virtual sample generation.

[0040] Step S6: Based on the mapping relationship between the low-dimensional virtual sample and the original data, map the low-dimensional virtual sample to the original data space to obtain a high-dimensional virtual sample.

[0041] Finally, based on the mapping relationship between the low-dimensional latent variables learned in the Gaussian process latent variable model and the original high-dimensional data, which is usually a nonlinear mapping, the low-dimensional virtual samples generated in step S5 are... Mapping back to the original data space yields the corresponding high-dimensional virtual sample. This completes the process of generating new high-dimensional virtual samples from the original long-tailed distribution data. These generated virtual samples effectively supplement the original dataset, especially by increasing the number of samples in the tail categories, while maintaining the distribution characteristics of the original data, thereby improving the training effect and generalization ability of machine learning models under small sample conditions.

[0042] In a preferred embodiment, visual charts are used to analyze the data distribution pattern and identify long-tail distribution characteristics in the dataset. Specifically, in this embodiment, statistical analysis is performed on the characteristic variables of the dataset, and corresponding visual charts are drawn to intuitively determine the data distribution pattern, thereby identifying whether long-tail distribution characteristics exist. One or more combinations of the following chart types can be used, but are not limited to: Histogram: A histogram divides feature values ​​into several intervals with a preset width, counts the frequency of samples within each interval, and displays the distribution of frequency as a function of feature values ​​in a bar chart. Histograms can visually reflect the differences in the distribution density of feature values ​​across intervals. By observing whether a significant "tailing" phenomenon appears in the tail region of the histogram—that is, a small number of feature values ​​are far from the main distribution interval but still have a certain frequency—it can be preliminarily determined whether the data exhibits a long-tailed distribution characteristic.

[0043] Frequency distribution plot: Calculate the frequency corresponding to each feature value. The frequency is equal to the frequency count divided by the total number of samples. Present the frequency variation trend with the feature value in the form of a line graph or scatter plot. The frequency distribution plot facilitates observation of the decay rate of high-frequency and low-frequency values. If the low-frequency value shows a slow decreasing trend as the feature value deviates from the main range, it indicates that the data may have a long-tailed distribution characteristic.

[0044] Box plots: Based on the quartiles (Q1, median, Q3) and their upper and lower bounds, box plots are drawn with extended line segments to show the dispersion of the data and the distribution of outliers. The presence of many discrete points far from the main body of the box plot often indicates asymmetrical data distribution and a long tail, thus providing a reference for identifying long-tailed distributions.

[0045] Violin plots combine the structures of kernel density estimation curves and box plots to illustrate the overall distribution shape and probability density variations of the data. Violin plots not only demonstrate the symmetry of the distribution but also visually represent the density decay in the tail region. If the density curve in the tail region decreases slowly and extends far, it helps to more accurately identify long-tailed distribution characteristics.

[0046] After completing the drawing of at least one of the above charts, a comprehensive judgment should be made based on the distribution pattern characteristics presented by the chart: If a clear asymmetric pattern is observed in the data distribution, and a long tail region appears in the direction of large or small feature values, specifically manifested as a trailing histogram, slow decay of low-frequency values ​​in the frequency distribution, many discrete points in the box plot, and a long tail density curve in the violin plot, then it can be determined that the dataset has long-tail distribution characteristics. Conversely, if the data distribution is roughly symmetrical and the tail is short and decays rapidly, it can be determined that there is no significant long-tail distribution characteristic.

[0047] By following the steps above, long-tail distribution characteristics can be identified in a timely manner during the data preprocessing stage, providing a basis for subsequent balancing strategies for long-tail distributions, thereby improving model training effectiveness and overall performance.

[0048] As a preferred embodiment, to further optimize the processing capability of long-tailed distributed data, in this embodiment, the long-tailed kernel function includes at least one of the following: rational quadratic kernel function, Cauchy kernel function, generalized t-Student kernel function, logarithmic kernel function, heavy-tailed power-law kernel function, and inverse multiquadratic kernel function.

[0049] These kernel functions all exhibit slow tail decay in their mathematical form, enabling them to better preserve sparse but important information in long-tail samples within the kernel space. This avoids distortion in similarity measurements due to excessive compression of tail samples. By using these long-tail kernel functions, samples far from the center can be assigned relatively higher weights during kernel matrix construction or similarity calculation. This allows the model to better perceive minority class or extreme value samples in the long-tail distribution during training or prediction, thereby improving its adaptability to long-tail data and overall generalization performance.

[0050] In this embodiment, to more precisely characterize the long-tail features of data at different scales, the long-tail kernel function employs at least two combinations, and the selection principles include: 1. Same kernel function type, but different scale parameters: For example, choose two Cauchy kernel functions with different bandwidth (or length scale) parameters, one to capture long-tail effects within a closer "neighbor" range, and the other to capture long-tail effects within a more "distant" range. By setting different scale parameters within the same kernel function type, long-tail features of both local details and global structure can be considered simultaneously. This allows the kernel function to remain sensitive to nearby long-tail samples within a small range, while maintaining a certain response to distant long-tail samples within a large range, thus improving the modeling ability of multi-scale long-tail structures.

[0051] 2. Different types of long-tail kernel functions: For example, combining heavy-tailed power-law kernel functions with logarithmic kernel functions can emphasize the long-tail decay law under different mathematical forms. Different types of long-tail kernel functions have their own characteristics in terms of tail decay rate and smoothness. By combining them, the long-tail distribution can be characterized from multiple perspectives, making the model more robust in complex long-tail scenarios and reducing the performance loss caused by improper selection of a single kernel function.

[0052] By selecting the long-tail kernel function and combining it with multiple scales as described above, this invention can more fully and accurately characterize the multi-scale long-tail characteristics of data, and effectively improve the processing effect of long-tailed distributed data in kernel methods, similarity measurements and related machine learning tasks.

[0053] As a preferred embodiment, in order to further optimize the constraint method of the long-tail kernel function parameters, so that the parameters can be smoothly and controllably adjusted during the model training process while satisfying the interval restrictions, this embodiment applies constraints to the kernel function parameters through a nonlinear function based on the above technical solution.

[0054] In this embodiment, the parameter constraint can be implemented in one of the following ways: applying a Sigmoid function, a Tanh function, or a Softmax function to the kernel function parameters to map the original parameters to a preset range, thereby achieving flexible constraint.

[0055] In practice, the following steps can be taken: Introducing intermediate parameters: Let the original parameter be θ. Instead of using it directly as the actual parameter of the kernel function, we map it through a differentiable nonlinear function to obtain the restricted actual parameter ϕ.

[0056] The choice and correspondence of mapping functions: When it is necessary to constrain the parameters to the interval (0,1), the Sigmoid function can be used: ; This function maps any real number θ to the open interval (0,1). The Sigmoid function is continuously differentiable in its domain, which can smoothly compress the original parameters to (0,1), avoid abrupt changes at the boundary, facilitate stable gradient propagation, and is suitable for constraining the parameters of a single kernel function.

[0057] If you need to constrain the parameters to (-1, 1) or other ranges that can be transformed into the target interval through linear transformation, you can use the Tanh function: ; Then through linear transformation The range can be converted to (0,1). The Tanh function is approximately linear near the origin and is sensitive to small parameter changes, which is beneficial for fine-tuning when the parameters are close to their optimal values.

[0058] When it is necessary to apply constraints that sum to 1 to multiple related parameters simultaneously, such as in a multi-kernel function weighted fusion scenario, to constrain the weight coefficients of each kernel function, the Softmax function can be used: ; in, , k=1,2...n. The Softmax function can, while ensuring that all parameters are positive and their sum is 1, calculate the maximum value using the original parameters. The relative size of the parameters is used to control the final weight allocation, which facilitates parameter normalization constraints in scenarios such as multi-core fusion.

[0059] Through the aforementioned nonlinear mapping method, this embodiment achieves flexible constraints on kernel function parameters, which avoids the gradient breakage problem caused by hard truncation and ensures that the parameters always fall within the preset reasonable range, thereby improving the stability and convergence effect of model training.

[0060] In this embodiment, a parameter penalty term can also be introduced into the loss function of the Gaussian process latent variable model to further guide the parameters toward the preset interval, thereby automatically meeting the constraint requirements during the optimization process.

[0061] In practice, the following steps can be taken: Construct the loss function: The overall loss function of a Gaussian process latent variable model generally consists of two parts: the first part is the data fitting term (such as marginal likelihood loss), and the second part is the parameter penalty term.

[0062] The overall loss function can be expressed as: ; This represents the fitting error based on observed data and latent variables; For parameters The penalty term is designed; λ is the penalty coefficient, used to adjust the severity of the penalty.

[0063] By introducing a parameter penalty term into the loss function, this embodiment transforms the parameter constraint problem into a differentiable optimization problem, enabling the parameters to automatically and smoothly adjust to a reasonable range during training. This eliminates the need for separate pruning or correction after each parameter update, thereby simplifying the implementation process and improving the efficiency and stability of model training.

[0064] As a preferred embodiment, in order to adapt to the modeling needs of complex data distributions, especially data with long-tail characteristics, this embodiment fuses multiple long-tail kernel functions that have been constrained and selected in a specific way through weighted addition, weighted multiplication or a combination thereof to form a multi-scale kernel function with stronger expressive power.

[0065] Specifically, the construction of the multi-scale long-tail kernel function mainly includes the following combination methods: 1. Weighted addition combination method: First, select one or more basic long-tail kernel functions from a pre-defined set of long-tail kernel functions, such as the Cauchy kernel, the log kernel, or the rational quadratic kernel. Then, assign corresponding weight coefficients to each selected long-tail kernel function and combine them by weighted summation.

[0066] For example, a weighted summation of the Cauchy kernel function, the logarithmic kernel function, and the rational quadratic kernel function can be used to obtain a multi-scale kernel function of the following form: ; in, , , These are the weight coefficients of each long-tail kernel function, and they satisfy... + + (or other normalization conditions, set according to actual needs).

[0067] By combining weighted kernel functions, we can integrate the advantages of different long-tailed kernel functions in capturing features at different scales of data. For example, the Cauchy kernel function typically has strong heavy-tailed characteristics, making it suitable for capturing extreme values; the logarithmic kernel function may perform well when dealing with data with certain specific distributions; and the rational quadratic kernel function has a certain degree of smoothness. Through linear superposition, the combined kernel function can possess multiple characteristics simultaneously, thereby improving the model's adaptability to different types of long-tailed data.

[0068] 2. Weighted multiplication combination method: Similarly, one or more basic long-tail kernel functions are first selected and assigned corresponding weight coefficients, and then combined by weighted product.

[0069] For example, a weighted multiplication of the Cauchy kernel function and the logarithmic kernel function can be used to obtain a multiscale kernel function of the following form: ; in, , These are weighting coefficients used to adjust the relative contribution of each kernel function to the product term.

[0070] Multiplicative combination can generate interactions between kernel functions, resulting in a combined kernel function with a stronger response or weaker influence in certain regions, thus providing a more refined characterization of the local structure of the data. For example, when two kernel functions indicate that two sample points are similar, their product amplifies this similarity; conversely, if one kernel function indicates dissimilarity, the product suppresses overall similarity. This approach helps enhance the model's ability to capture complex dependencies in the data.

[0071] 3. Weighted addition and multiplication combination operation: This method combines the two basic combination methods mentioned above, that is, first performing multiplicative combination of some kernel functions, and then adding the result with other kernel functions, or vice versa.

[0072] For example, the Cauchy kernel function and the logarithmic kernel function can be multiplied together, and then combined with the rational quadratic kernel function by addition, to obtain a multi-scale kernel function of the following form: + ; in, , , These are the corresponding weighting coefficients.

[0073] This hybrid approach combines the advantages of addition and multiplication. It can capture the complex interactions between kernel functions through multiplication terms, while introducing independent features at different scales through addition terms. This allows multi-scale long-tailed kernel functions to adapt more flexibly to the inherent structure of the data, making them particularly suitable for scenarios where data features have complex relationships and multi-scale characteristics.

[0074] Among the various combinations mentioned above, the following measures can be taken to further improve the performance of multi-scale long-tail kernel functions and the stability of parameter optimization: Determining the weight coefficients: Weight coefficients α, β, γ, etc., can be preset based on experience, or optimized through data-driven learning methods such as cross-validation. Appropriate weight allocation allows the model to focus more on kernel function characteristics that are more relevant to the current data distribution, thereby enhancing the model's adaptability.

[0075] Constraints on key parameters: To prevent unreasonable values ​​of kernel function parameters during optimization, such as excessively large or small values ​​leading to numerical instability, nonlinear transformation functions like the Sigmoid function can be used to constrain key parameters in the long-tailed kernel function (such as the scaling parameter of the Cauchy kernel function and the shape parameter of the logarithmic kernel function), limiting their values ​​to a preset reasonable range. For example, suppose the original parameter of a long-tailed kernel function is p, and the parameter after constraint by the Sigmoid function... , where a and b are the lower and upper limits of the allowed range of values ​​for parameter p.

[0076] The multi-scale long-tail kernel function constructed in this invention can be applied to machine learning models such as Support Vector Machines and Gaussian Process Regression for kernel function design and improvement. Furthermore, because the multi-scale long-tail kernel function of this invention is an organic combination of multiple long-tail kernel functions with different characteristics and capturing capabilities, it has stronger expressive power than a single long-tail kernel function, and can better fit complex data patterns, including nonlinear, non-stationary, and mixed distributions, making it particularly suitable for datasets with obvious long-tail characteristics. This construction method significantly enhances the model's ability to capture local features and multi-scale changes. In virtual sample generation tasks, it can generate higher-quality and more realistic virtual samples, thereby improving the model's generalization performance; in other data modeling tasks, it can also improve the model's prediction accuracy and robustness. At the same time, this combination method has good flexibility, and can be freely adjusted and customized according to the distribution characteristics of actual data to achieve the best modeling effect.

[0077] As a preferred embodiment, the loss function of the Gaussian Process Latent Variable Model (GPLVM) is used to determine the weights of each long-tailed kernel function.

[0078] Specifically, during the optimization process, the objective function constructed is to minimize the loss function of GPLVM, i.e. Simultaneously, the constraints are set as follows: the weights are non-negative and the sum of the weights is 1, which can be expressed mathematically as: ; in, The weight coefficients are the values ​​corresponding to the i-th long-tail kernel function. Let m be the loss function of GPLVM, and m be the number of long-tailed kernel functions.

[0079] In practical calculations, mature optimization algorithms, such as gradient descent, can be used to solve the above objective function and constraints. Starting from the initial guessed weight values, the weight values ​​are iteratively updated until the convergence condition is met, i.e., the change in the loss function is less than a preset threshold or the maximum number of iterations is reached.

[0080] This optimization method determines the weights of each long-tailed kernel function. On one hand, by optimizing the GPLVM loss function to determine the weights, the combination of long-tailed kernel functions can better fit the intrinsic structure and features of the data, improving the model's expressive power and prediction accuracy. On the other hand, the constraint that the weights are non-negative and their sum is 1 ensures the rationality and standardization of the weight allocation, avoiding the negative impact of unreasonable weight values ​​on model performance and helping to maintain the model's stability and reliability. Finally, the optimized weights of each long-tailed kernel function are applied to multi-scale long-tailed kernel functions, enabling the method provided by this invention to achieve better results in practical applications, such as exhibiting higher accuracy and better generalization ability in time series prediction and image recognition.

[0081] As a preferred embodiment, the process of obtaining the probability distribution of low-dimensional latent variables is as follows: S51, Implicit variable initialization: First, principal component analysis (PCA) is performed on the original data. PCA can extract the main feature directions of the data, reduce data dimensionality, and retain most of the variation information. The latent variables are then initialized using the PCA results. Specifically, the initial values ​​of the latent variables are set according to the weights or scores of the PCA principal components, so that the latent variables reflect the basic structure of the data in their initial state. The purpose of this step is to provide a reasonable starting point for the subsequent optimization process, accelerate the optimization convergence speed, and improve the stability of the results.

[0082] S52. Using multi-scale long-tailed kernel functions to characterize the covariance structure: In Gaussian process latent variable models, the covariance structure between samples is a core component of the model. This invention employs a multi-scale long-tailed kernel function to characterize the covariance structure between the original high-dimensional samples. Multi-scale long-tailed kernel functions (e.g., three additively weighted rational quadratic kernel functions) exhibit good local smoothness and adaptability. When two sample points are close to each other, the kernel function value approaches 1, indicating a strong correlation between the two samples; as the distance between them increases, the correlation rapidly decreases, demonstrating the local smoothing characteristic of the kernel function. Compared to the traditional RBF kernel function, the multi-scale long-tailed kernel function decreases more slowly with increasing distance, meaning it can capture correlations between samples over a wider range, thus making it more suitable for handling complex and diverse data patterns.

[0083] S53. Solve the model using numerical optimization methods: To obtain the probability distribution of low-dimensional latent variables, the Gaussian process latent variable model needs to be optimized. This invention employs the Scaled Conjugate Gradient Method (SCGP) to optimize this model. SCGP is an efficient numerical optimization algorithm suitable for optimization problems with large-scale data, achieving good convergence within a relatively small number of iterations. This method continuously adjusts the model parameters to maximize the model's likelihood function, thereby obtaining the optimal latent variable configuration.

[0084] S54. Obtain the probability distribution and matrix of low-dimensional latent variables: Based on the optimized Gaussian process latent variable model, the probability distributions and matrices of low-dimensional latent variables corresponding to high-dimensional data can be further obtained. The low-dimensional latent variable probability distributions describe the probabilistic characteristics of each latent variable, while the low-dimensional latent variable matrix contains the representations of all samples in the low-dimensional space. These results provide a foundation for subsequent data analysis, feature extraction, or visualization.

[0085] This invention provides an efficient and adaptable method for solving the probability distribution of low-dimensional latent variables in high-dimensional data by combining principal component analysis, Gaussian process latent variable model and multi-scale long-tailed kernel function, which can effectively cope with complex and diverse data patterns.

[0086] To further illustrate the inventive concept of this invention, a visual analysis of the RBF kernel function and the multi-scale long-tailed RQ kernel function was performed. See details... Figure 2 , Figure 2 Figure (a) illustrates the RBF kernel function. The left-hand plot shows the characteristics of the RBF kernel function, with the horizontal axis representing the independent variable x, ranging from 0 to 10, and the vertical axis representing the value of the kernel function k(x, 1.0), i.e., the value of the kernel function at the independent variable x when the second parameter of the kernel function is fixed at 1.0. This shows how the value of the RBF kernel function at x changes with x. The right-hand plot shows the sample functions generated by the RBF kernel function, with the horizontal axis representing the independent variable x, ranging from 0 to 10, and the vertical axis representing the values ​​of the sample functions generated by the RBF kernel function. This shows multiple sample function curves generated by the RBF kernel function, reflecting the values ​​of the sample functions under different inputs x. These sample functions are sampled from the Gaussian process prior defined by the RBF kernel function.

[0087] Figure 2Figure (b) illustrates the multi-scale long-tailed RQ kernel function. The left-hand plot shows the characteristics of the multi-scale long-tailed RQ kernel function. The horizontal axis represents the independent variable x, ranging from 0 to 10, and the vertical axis represents the value of the kernel function k(x, 1.0), that is, the value of the multi-scale long-tailed rational quadratic (RQ) kernel function at the independent variable x when the second parameter of the kernel function is fixed at 1.0. This shows the change of the value of the multi-scale long-tailed RQ kernel function at x with x. The right-hand plot shows the sample functions generated by the multi-scale long-tailed RQ kernel function. The horizontal axis represents the independent variable x, ranging from 0 to 10, and the vertical axis represents the value of the sample functions generated by the multi-scale long-tailed RQ kernel function. This shows multiple sample function curves generated by the multi-scale long-tailed RQ kernel function, reflecting the value of the sample functions under different inputs x. These sample functions are sampled from the Gaussian process prior defined by the multi-scale long-tailed RQ kernel function.

[0088] As can be seen from the two left-hand plots, when the distances are close, the values ​​of both kernel functions are close to 1, indicating a strong correlation between adjacent samples. As the distance between the distances increases, the correlation decreases rapidly, reflecting the local smoothing characteristic of the kernel function. However, the multi-scale long-tailed RQ kernel function decreases more slowly, meaning it can capture correlations between samples over a larger range.

[0089] The two right-hand plots illustrate the sample function images that Gaussian processes using these two kernel functions might generate when no observation data is available. Both the RBF kernel and the multi-scale long-tailed RQ kernel produce smooth sample functions, but the RBF kernel produces a smoother sample function, while the multi-scale long-tailed RQ kernel produces a more varied and adaptable sample function, suitable for handling complex data patterns. Therefore, the multi-scale long-tailed kernel is more effective and adaptable in modeling high-dimensional long-tail similarity.

[0090] To further illustrate the inventive concept of this invention, this embodiment selects a commercial lithium iron phosphate (LFP) battery as the research object and conducts cycle tests in a temperature-controlled environmental chamber (temperature set at 30°C). The same discharge conditions are used during the experiment, i.e., discharge from 4 C to 2.0 V, where 1 C corresponds to a current of 1.1 A.

[0091] Based on this, different fast charging conditions were adjusted, setting the fast charging rate range to the manufacturer-recommended average charging rate of 3.6 C to 6 C, to simulate and probe the performance of the current generation of power batteries under extreme fast charging conditions (charging completed in 10 minutes). During the experiment, multi-dimensional primary index data were collected, mainly including the following eight aspects: temperature-related parameters, state of charge, voltage and current-related parameters, depth of discharge and charging rate, duty cycle, cycle count and time, energy and power, and internal resistance and resistance; these eight primary indices are further subdivided into a total of 17 secondary indices.

[0092] Exploratory analysis of the collected raw data revealed that most features did not conform to a standard normal distribution, but rather exhibited left-skewed or right-skewed distributions with long-tailed characteristics. The following section combines... Figure 3 The example shown will be explained in detail: Figure 3 (a) is the distribution histogram of the label variable “Discharge_Q” of LFP, which shows the distribution bias of the variable in the numerical range. Figure 3 (b) is the distribution histogram of the feature variable “Slope CCCV-CCCT”. This feature shows a clear left-skewed distribution pattern, that is, the amount of data in the left region is significantly less than the amount of data in the right region.

[0093] Regarding the characteristic "Slope CCCV-CCCT", statistical analysis revealed a distribution with a mean of 18.02 and a standard deviation of 1.08, indicating a relatively concentrated distribution. This suggests that the skewed distribution of the data is not due to outliers, but rather stems from the inherent characteristics of the data acquisition process in the actual experiment. In machine learning modeling scenarios, such skewed data often suffers from bias because some information is present in smaller proportions and is thus "masked" by other, more significant information during model training and inference. Therefore, if this data is to be used to build models for power battery performance evaluation or fast charging strategy optimization, this distribution characteristic needs to be fully considered. Targeted methods such as data transformation, sample weighting, or selecting algorithms more robust to skewed data should be employed to mitigate the negative impact of skewed distribution on model performance, thereby more accurately uncovering the inherent patterns in the data.

[0094] To further illustrate the inventive concept of this invention, based on the long-tailed virtual sample generation method based on the multi-scale long-tailed kernel function provided by this invention, 452 virtual samples were successfully generated. During this process, 126 outlier data points were excluded through specific rules and judgment criteria, ultimately retaining 326 valid virtual samples for subsequent analysis or application.

[0095] To more intuitively demonstrate the differences in the distribution of the original data and the virtual samples generated using the method of this invention in the latent variable space, two-dimensional visualization of the latent variables was performed.

[0096] Visualization of raw data: such as Figure 4 As shown in (a), this figure is a two-dimensional visualization of the latent variables in the original LFP data. It clearly presents the distribution shape of all original samples in the latent variable space in the form of a scatter plot.

[0097] Visualization of generated virtual samples: such as Figure 4 As shown in (b), this figure is a two-dimensional visualization of the latent variables of the virtual samples generated using the long-tailed virtual sample generation method based on the multi-scale long-tailed kernel function provided by this invention. The distribution of the generated virtual samples in the latent variable space is also described in the form of a scatter plot.

[0098] The unique feature of this invention lies in its use of a multi-scale long-tail kernel function. This kernel function possesses powerful feature capture capabilities, accurately capturing long-tail and local features of the data. Due to this characteristic, the generated virtual samples exhibit a non-linear arrangement around the original data. This non-linear arrangement is significant, effectively covering sparse points and outliers in the original data, breaking the limitation of traditional methods where virtual samples are distributed along a linear direction of the original samples. Therefore, the method of this invention can generate diverse virtual samples with more flexible and comprehensive distribution, better reflecting the complex characteristics of the original data and providing richer and more representative data support for related research and applications.

[0099] As a preferred embodiment, when the original dataset contains label information, in order to complete the label of the high-dimensional virtual sample, a Gaussian process regression model is further used to predict the corresponding label of the generated high-dimensional virtual sample to obtain the label information of the virtual sample.

[0100] Based on the existing long-tail virtual sample generation process, that is, after generating high-dimensional virtual samples through a multi-scale long-tail latent variable model, if the original data has a label column, then the "label prediction branch" is activated: First, the original sample features and original labels from the original dataset are used as training data to train a Gaussian Process Regression (GPR) model. GPR is a nonparametric regression method based on a Bayesian framework that can model the distribution of a function and has advantages in fitting and generalizing labels in small-sample, high-dimensional scenarios.

[0101] Secondly, the feature vectors of the generated high-dimensional virtual samples are input into the trained GPR model, and the model outputs the label prediction values ​​of the corresponding virtual samples, thereby assigning label information to the virtual samples.

[0102] To visually demonstrate the universality of this invention in "generating virtual samples from labeled data" and the quality of the virtual labels, this embodiment provides a visual analysis of the label distribution. Figure 5 The diagram shows a visualization of the distribution of virtual labels.

[0103] Figure 5 (a) Bar chart of virtual label distribution: The horizontal axis represents the label value range, and the vertical axis represents the number of samples; the distribution of "original labels" and "virtual labels" is distinguished by different fill styles in the figure. It can be observed that both original labels and virtual labels show a left-skewed distribution pattern; and from the height of the bars, the distribution range of virtual labels overlaps with that of original labels, indicating that the label range of most original samples has been covered by virtual samples.

[0104] Figure 5 (b) Box plot of virtual label distribution: The box plot describes the distribution characteristics using statistics such as median, quartiles, minimum, and maximum. Comparing the box plots of the original labels and the virtual labels, it can be seen that they are highly similar in core statistics such as median and interquartile range (IQR); only slight differences exist in the minimum and some quartile indicators. This high degree of consistency at the statistical level indicates that the virtual samples generated by this invention can inherit the statistical regularity of the original data well in the label dimension, and the label quality of the virtual samples is reliable.

[0105] By introducing Gaussian process regression (GPR) for virtual sample label prediction, building upon the "multi-scale long-tailed latent variable model to generate high-dimensional virtual samples," the generated high-dimensional virtual samples not only possess feature distributions matching the original data but also acquire semantically consistent label information. This provides complete "pseudo-real" data support for subsequent supervised tasks based on virtual samples (such as classification and regression). Leveraging GPR's modeling capabilities for function distributions, combined with visual verification... Figure 5 The consistency between the distribution of bar charts and box plots demonstrates that the virtual labels highly match the original labels in terms of overall distribution and statistical characteristics. This ensures both the representativeness of the virtual samples to the original data and guarantees the generalization ability for downstream tasks. This invention addresses the scenario of "generating virtual samples from labeled data" by supplementing the high-dimensional feature generation with a label prediction mechanism. From technical implementation to effect verification, it reflects the optimization of "virtual sample authenticity and usability," solving the technical problem in existing technologies where virtual samples can only cover the feature dimension and are difficult to maintain consistency across the label dimension.

[0106] To further illustrate the effectiveness of the long-tailed virtual sample generation method (MKLTGP-VSG) based on multi-scale long-tailed kernel functions provided in this invention, the impact of virtual samples on machine learning models in five different scenarios was investigated: "no generative model used", "GMM-VSG (AIC)", "GMM-VSG (BIC)", "VSG-GP", and the MKLTGP-VSG provided in this invention. "No generative model used" means using only the original samples. To evaluate the impact of the algorithm-generated virtual samples, four machine learning models were used for learning and prediction: SVR, Lasso, RF, and XGBoost. This comprehensive analysis aims to understand how introducing virtual samples affects the predictive performance of these machine learning models under different learning algorithms. Specific results are shown in Table 1: ; The GMM algorithm is a small-sample technique that obtains virtual samples from a Gaussian mixture distribution. Table 1 shows the performance of the virtual samples generated by this method on each machine learning algorithm. The results show that in LFP, regardless of whether AIC or BIC is used, the prediction performance of the proposed algorithm is significantly lower than that of the "no generative model" among the four machine learning algorithms. The VSG-GP algorithm uses the RBF kernel function, and on LFP data, except for the RF model, the performance of the learned model is improved after the VSG-GP algorithm. The main reason may be that overfitting occurs in LFP. Using the MKLTGP-VSG algorithm provided in this invention, according to the results shown in Table 1, the Random Forest model has the most accurate prediction in LFP data, achieving R0. 2 = 0.9992, MAE = 0.9086, and MSE = 2.0025. Except for the Lasso model, the MAE, MSE, and R-values ​​for all other models are... 2 All models showed significant improvements compared to learning models using the original samples, with the SVR model showing the most significant improvement. 2 The improvement rate was 10.97%. Figure 6 Plot the true labels of the data on the horizontal axis and the predicted labels of the model on the vertical axis. Figure 6 (a) is a scatter plot of the effect of the SVR model. Figure 6 (b) is a scatter plot of the effects of the Lasso model. Figure 6 (c) is a scatter plot of the RF model's performance. Figure 6 (d) is a scatter plot of the XGBoost model's performance, and the prediction curve is shown in the figure. The model's prediction performance is evaluated by the similarity between the scatter distribution and this curve. Compared with other generation algorithms, the virtual samples generated by MKLTGP-VSG perform the best, demonstrating the high performance of the MKLTGP-VSG algorithm.

[0107] To further explore the impact of the number of multi-scale long-tail kernel functions in the long-tail virtual sample generation method based on multi-scale long-tail kernel functions provided in this invention on the algorithm results, this study aims to ensure improved sample quality without introducing unnecessary computational complexity or overfitting issues. See [link to relevant documentation]. Figure 7 This paper demonstrates the impact of generating virtual samples with different mixture numbers in the XGBoost model on prediction performance. The XGBoost model was chosen as the prediction model because of its scalability and robustness, and its widespread use in machine learning. Variations in the mixture number of the multi-scale long-tailed kernel function cause fluctuations in the MSE metric, affecting MAE and RQ. 2 The impact of the metrics was relatively small. In the seven experiments, the algorithm that mixed three long-tailed kernel functions still performed best. It can be considered that the number of multi-scale long-tailed kernel functions does have a certain impact on the quality of the generated virtual samples. It introduces more kernel parameters to identify data structures and fit data features. However, too many mixing numbers may also lead to overfitting risks, resulting in poor learning model performance.

[0108] Example 2

[0109] This embodiment discloses a long-tail sample generation system based on a Gaussian process latent variable model, including: The long-tail feature analysis module performs distribution analysis on the input raw dataset to identify long-tail distribution features. Specifically, statistical analysis methods (such as calculating frequency histograms for each feature dimension, cumulative distribution functions, etc.) or machine learning-based methods can be used to determine whether the data exhibits significant skewed distribution or tail sparsity across different dimensions. Identifying long-tail features provides a basis for subsequent kernel function selection and parameter constraints, enabling the system to specifically enhance its ability to model the correlation of samples in long-tail regions, avoiding distortion or overfitting that occurs when generating samples in long-tail regions using traditional methods.

[0110] A long-tail kernel function construction module is used to select at least two long-tail kernel functions and constrain the parameter values ​​of these functions. In practice, kernel functions with long-tail characteristics, such as rational quadratic kernel functions, can be selected. Based on the output of the long-tail feature analysis module, reasonable value ranges or prior constraints are set for parameters such as the kernel function's length scale and scale mixing coefficient. For example, for feature dimensions with a significant long tail, the upper limit of the length scale can be appropriately relaxed to expand the effective range of the kernel function. Through parameter constraints, the kernel function can better match the long-tail distribution characteristics of the data, ensuring both the smoothness of the kernel function in local regions and the preservation of correlation between samples over a larger distance, thereby improving the accuracy of characterizing the sample structure in long-tail regions.

[0111] The multi-scale kernel combination module adaptively weights and combines multiple parameter-constrained long-tail kernel functions to construct a multi-scale long-tail kernel function. In practice, a weighted summation method can be used to linearly combine multiple long-tail kernel functions according to preset or learned weights, forming a composite kernel function with multi-scale characteristics. The weights can be determined based on the results of the long-tail feature analysis module; for example, higher weights can be assigned to feature dimensions with significant long-tail features, or the weights can be dynamically adjusted through an optimization process, ensuring that the combined kernel function can effectively capture the correlation between samples at different scales. Multi-scale kernel combination can simultaneously consider local details and global structure, enhancing the kernel function's adaptability to data patterns at different scales, and is particularly suitable for complex scenarios in long-tail data where dense regions and sparse tails coexist.

[0112] The latent variable modeling module is used to establish a Gaussian Process Latent Variable Model (GPLVM) based on the multi-scale long-tailed kernel function, and to determine the kernel function parameters and corresponding weights through an optimization process.

[0113] In practice, a multi-scale long-tailed kernel function is embedded into the covariance function of GPLVM to construct a likelihood function that includes the relationship between latent variables and observed data. Subsequently, numerical optimization methods (such as scaled conjugate gradient method, L-BFGS, etc.) are used to train the model, with the optimization objective being to maximize the data likelihood or minimize the reconstruction error, thereby determining the optimal parameters and combined weights of the kernel function. Through this optimization process, the model can automatically learn the nonlinear mapping relationship between data and latent variables, and fully utilize the advantages of the multi-scale long-tailed kernel function to improve its modeling ability for high-dimensional long-tailed data structures.

[0114] The latent variable sampling module is used to obtain the probability distribution of low-dimensional latent variables based on the Gaussian process latent variable model, and to sample from the probability distribution to generate low-dimensional virtual samples.

[0115] In practice, after model training is complete, a set of low-dimensional virtual sample points can be obtained by sampling the prior distribution of the low-dimensional latent variable space or by conditional sampling based on the posterior distribution. Sampling in the latent space can effectively utilize the learned low-dimensional structure, and the generated low-dimensional virtual samples are consistent with the distribution of the real data, providing a reliable foundation for subsequent mapping to the high-dimensional space.

[0116] The sample mapping generation module maps low-dimensional virtual samples back to the original data space based on the mapping relationship between low-dimensional virtual samples and the original data, generating high-dimensional virtual samples. Specifically, it utilizes the mapping function learned in the latent variable modeling module, which maps the low-dimensional virtual samples point-by-point back to the high-dimensional space, obtaining virtual samples of the same dimension as the original data. Through this mapping process, the generated virtual samples maintain the distribution characteristics of the original data while supplementing the number of samples in the long-tail region, alleviating the data imbalance problem caused by the long-tail distribution, and providing more comprehensive sample support for subsequent data analysis, model training, and other tasks.

[0117] After the raw data is input, the long-tail feature analysis module first identifies the long-tail distribution characteristics; the long-tail kernel function construction module selects and constrains the parameters of the long-tail kernel function accordingly; the multi-scale kernel combination module adaptively weights and combines multiple constrained kernel functions into a multi-scale long-tail kernel function; the latent variable modeling module builds GPLVM based on this kernel function and optimizes the parameters; the latent variable sampling module extracts the low-dimensional latent variable probability distribution from the optimized model and samples it; finally, the sample mapping generation module maps the low-dimensional virtual samples to a high-dimensional space, completing the generation of long-tail virtual samples. Through the collaborative work of the above modules, this system can effectively generate virtual samples covering the long-tail region in high-dimensional long-tail data scenarios, improving data utilization efficiency and the performance of downstream tasks.

[0118] The above are merely preferred embodiments of the present invention and do not limit the patent scope of the present invention. All equivalent structural transformations made using the contents of the present invention's specification and drawings under the inventive concept of the present invention, or direct / indirect applications in other related technical fields, are included within the patent protection scope of the present invention.

Claims

1. A long-tail virtual sample generation method based on a multi-scale long-tail kernel function, characterized by, A method for generating virtual samples for data with long-tail distribution characteristics, comprising the following steps: Obtaining an original data set and identifying the long-tail distribution characteristics existing in the data set; Selecting at least two long-tail kernel functions and imposing constraints on the parameter values of the long-tail kernel functions; Adaptively combining the multiple long-tail kernel functions with parameter constraints to construct a multi-scale long-tail kernel function; Based on the multi-scale long-tail kernel function, a Gaussian process latent variable model is established, and the kernel function parameters and weights are determined by optimization; Using the Gaussian process latent variable model to obtain the probability distribution of low-dimensional latent variables, and sampling from the probability distribution to generate low-dimensional virtual samples; Based on the mapping relationship between the low-dimensional virtual samples and the original data, the low-dimensional virtual samples are mapped to the original data space to obtain high-dimensional virtual samples.

2. The long-tail virtual sample generation method based on the multi-scale long-tail kernel function according to claim 1, characterized in that, The step of identifying the long-tail distribution characteristics existing in the data set comprises: By drawing at least one of the histogram, frequency distribution chart, box plot or violin plot, the data distribution pattern is analyzed to determine whether there is a long-tail distribution characteristic.

3. The long-tail virtual sample generation method based on the multi-scale long-tail kernel function according to claim 1, characterized in that, The long-tail kernel function at least includes one of the following: rational quadratic kernel function, Cauchy kernel function, generalized t-Student kernel function, logarithmic kernel function, heavy-tailed power law kernel function or inverse multi-quadratic kernel function.

4. The long-tail virtual sample generation method based on the multi-scale long-tail kernel function according to claim 1, characterized in that, The at least two long-tail kernel functions belong to the same kernel function type but have different scale parameters, or belong to different types of long-tail kernel functions to represent the multi-scale long-tail characteristics of the data.

5. The long-tail virtual sample generation method based on the multi-scale long-tail kernel function according to claim 1, characterized in that, The constraints imposed on the parameters of the long-tail kernel function include: Limiting the parameters within a preset numerical interval, the preset numerical interval is (0, 1).

6. The long-tail virtual sample generation method based on the multi-scale long-tail kernel function according to claim 5, characterized in that, The parameter constraint is realized by one of the following ways: imposing Sigmoid function, Tanh function or Softmax function on the kernel function parameters; Introducing a parameter penalty term in the loss function of the Gaussian process latent variable model.

7. The long-tail virtual sample generation method based on the multi-scale long-tail kernel function according to claim 1, characterized in that, The multi-scale long-tail kernel function is constructed by one of the following ways: Weighted additive combination of multiple long-tail kernel functions; Weighted multiplication combination of multiple long-tail kernel functions; Combination operation of weighted addition and multiplication of multiple long-tail kernel functions.

8. The long-tail virtual sample generation method based on the multi-scale long-tail kernel function according to claim 1, characterized in that, The weights of each long-tail kernel function in the multi-scale long-tail kernel function are obtained by optimizing the loss function of the Gaussian process latent variable model, and satisfy the constraint conditions that the weights are non-negative and the sum of the weights is 1.

9. The long-tail virtual sample generation method based on the multi-scale long-tail kernel function according to claim 1, characterized in that, The process of obtaining the probability distribution of low-dimensional latent variables includes: Using the principal component analysis result of the original data to initialize the latent variables; Using the multi-scale long-tail kernel function to describe the covariance structure between samples; Solving the Gaussian process latent variable model by numerical optimization method to obtain the probability distribution of low-dimensional latent variables.

10. A long-tail sample generation system based on a Gaussian process latent variable model, characterized by, It includes: A long-tail feature analysis module for analyzing the distribution of input original data set to identify the long-tail distribution characteristics existing in the data; A long-tail kernel function construction module for selecting at least two long-tail kernel functions and constraining the parameter values of the long-tail kernel functions; A multi-scale kernel combination module for adaptively combining multiple long-tail kernel functions with parameter constraints to construct a multi-scale long-tail kernel function; a latent variable modeling module configured to establish a Gaussian process latent variable model based on the multi-scale long-tailed kernel function, and determine kernel function parameters and corresponding weights through optimization; a latent variable sampling module configured to obtain a probability distribution of a low-dimensional latent variable based on the Gaussian process latent variable model, and sample from the probability distribution to generate a low-dimensional virtual sample; a sample mapping generation module configured to map the low-dimensional virtual sample to an original data space based on a mapping relationship between the low-dimensional virtual sample and the original data, and generate a high-dimensional virtual sample.

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