Multiquery quasi-interpolation Boosting method based on residual error
By introducing residual-based Boosting method and dynamic shape parameter adjustment in the multivariate quasi-interpolation algorithm, the existing algorithms have solved the problem of insufficient accuracy and poor flexibility in complex function approximation, and efficient and accurate function approximation effect is achieved.
Patent Information
- Application Number
- CN202510028089.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-01-08
- Publication Date
- 2025-06-06
AI Technical Summary
When handling complex and variable functions, the existing multivariate quasi-interpolation algorithms lack accuracy, are difficult to capture high-frequency oscillation or steep changes, and lack optimization mechanisms, so they cannot fully utilize the intrinsic relationships between data points.
The multiquadric quasi-interpolation Boosting method based on residuals is used to finely adjust the shape parameters dynamically and optimize the approximation effect through initial smooth fitting and residual-based boosting technology.
It significantly improves the approximation accuracy of complex functions, enhances the flexibility and generalization capabilities of the model, can effectively handle high-frequency oscillations or steep changes, reduces computational costs and improves the efficiency of processing large-scale data sets.
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Figure CN120104913A_ABST
Abstract
Description
Technical Field
[0001] The invention belongs to the technical field of numerical analysis and function approximation, and specifically relates to a multiquadric quasi-interpolation Boosting method based on residual. Background Art
[0002] In the technical field of numerical analysis and function approximation, interpolation algorithms occupy an important position. Traditional interpolation algorithms, such as polynomial interpolation and radial basis functions (RBFs) interpolation, rely on constructing an interpolation function that can accurately pass through all given data points. This approach ensures the high accuracy of the function value at the data point and provides an accurate basis for subsequent numerical calculations or data processing. However, traditional interpolation algorithms have the following significant disadvantages: (1) high computational complexity, (2) sensitivity to data: In contrast, the multiquadric quasi-interpolation algorithm has obvious advantages. It does not need to perform complex equation-solving operations, but directly constructs an approximate function based on given data points using specific basis functions and relatively simple calculation rules, thereby quickly and conveniently obtaining the function value of the function point. This makes the multivariate quadratic quasi-interpolation algorithm perform well in processing large-scale data sets or in application scenarios with high real-time requirements, effectively reducing computational costs and improving efficiency.
[0003] However, the existing multivariate quadratic quasi-interpolation algorithm also has limitations:
[0004] (1) Insufficient precision: Due to the use of relatively fixed basis function forms and shape parameter settings, it has poor flexibility when facing complex and changeable function shapes, and it is difficult to capture high-frequency oscillations or steep change characteristics, resulting in a large deviation between the function value at the test point and the true value.
[0005] (2) Lack of optimization mechanism: It fails to fully explore the intrinsic relationship between data points, especially in the use of residual information, and lacks an effective mechanism to flexibly adjust the shape parameters according to the function characteristics, which limits its performance in complex function approximation. Summary of the invention
[0006] The purpose of the present invention is to provide a residual-based multiquadric quasi-interpolation Boosting method in order to solve the above-mentioned problem.
[0007] The technical solution adopted by the present invention is as follows: a residual-based multiquadric quasi-interpolation Boosting method, the method comprising the following steps:
[0008] S1: Initialization and parameter setting: define the number of nodes, x-axis range, data point spacing and shape parameter coefficients
[0009] S2: Perform boundary processing: construct boundary polynomials to adapt to the data edges and ensure that boundary conditions are correctly handled during calculations
[0010] S3: Perform smooth fitting: Apply multivariate quadratic function for initial smoothing to obtain preliminary fitting results
[0011] S4: Perform fine-tuning based on residual boosting techniques: Analyze the residuals after the first fit and use a smaller shape parameter coefficient for a second fit to correct the error and optimize the approximation effect
[0012] S5: Display the results: draw a comparison chart of the original data and the fitting curve to evaluate the algorithm performance.
[0013] In a preferred embodiment, in step S1, first, global variables such as the number of nodes n, the minimum x_min and maximum x_max of the x-axis are set, and the fixed spacing h between data points is calculated based on these parameters. Then, two shape parameter coefficients of different sizes are set based on h: coeff_initial is used for initial fitting, and coeff_residual is used for subsequent fine adjustment to ensure the smoothness and smooth transition of the image.
[0014] In a preferred embodiment, in step S2, data points near the boundary are selected to construct a high-order polynomial model for approximating the function behavior near the boundary to ensure that the boundary information is properly processed. Specifically, data points near the boundary are first selected from the data set, usually including the first and last data points of the data sequence, to fully capture the information of the boundary area.
[0015] In a preferred embodiment, in step S2, the selected data points are fitted using the np.polyfit() function to obtain the coefficients of the polynomial and generate a high-order polynomial expression p(x) that can describe the boundary characteristics. This step not only provides a transformed data basis for subsequent numerical approximation, but also provides a mathematical basis for the processing of boundary conditions.
[0016] In a preferred embodiment, in step S3, the constructed polynomial is applied to the data points in the boundary area to ensure that the boundary conditions can be accurately reflected in the subsequent numerical approximation process, effectively avoiding the boundary effect problem common in traditional interpolation algorithms, and improving the overall approximation quality and stability. Through this method, the present invention significantly enhances the performance of the approximation algorithm in the boundary area and improves the overall approximation accuracy.
[0017] In a preferred embodiment, in step S3, a multiquadric basis function is used. As the kernel function, its good locality and positive definiteness characteristics are used for initial smoothing. For non-boundary data, the weight function of the data point is calculated:
[0018]
[0019] in
[0020] Then, the weight function is multiplied and accumulated with the transformed data after removing the boundary influence, and the contribution of the boundary polynomial is added to finally obtain the result of the initial fitting.
[0021] In a preferred embodiment, in step S4, after the initial fitting, we calculate the residuals on all data points and apply the multiquadric pseudo-interpolation algorithm to fit them again. This time, a smaller shape parameter coefficient is used to better capture the detail features. In this step, the residual-based boosting technology uses the residual after the first fitting as the new objective function for secondary fitting, gradually reducing the error and achieving a more accurate approximation effect.
[0022] In a preferred embodiment, in step S5, a chart is drawn by using the Matplotlib library of Python, so that the difference between the original function and the fitting results under two different strategies can be intuitively compared.
[0023] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:
[0024] 1. In the present invention, relying on a unique algorithm framework, the goal of efficient approximation of complex functions is successfully achieved, especially showing significant advantages in dealing with boundary effects. This method not only significantly improves the accuracy of approximation, but also greatly enhances the flexibility and generalization ability of the model, providing a new perspective and technical solution for solving similar problems. By introducing innovative boundary polynomial processing technology and a residual-based boosting mechanism, the present invention has made substantial improvements to the multiquadric quasi-interpolation algorithm, which not only broadens its scope of application, but also greatly improves the accuracy of the algorithm, effectively overcomes the shortcomings of traditional methods, and brings important innovative results to the field of numerical analysis and function approximation.
[0025] 2. In the present invention, by introducing the residual-based boosting technology, the approximation accuracy of complex functions is significantly improved, especially when dealing with high-frequency oscillations or steep change features, the deviation between the function value at the test point and the true value can be effectively reduced. The multiquadric quasi-interpolation algorithm inherits the advantages of no need to solve equations and high computational efficiency, and at the same time, through algorithm optimization, the efficiency of processing large-scale data sets is further improved and the computational cost is reduced. The present invention dynamically adjusts the shape parameters so that the algorithm can flexibly optimize the quasi-interpolation process according to the different characteristics of the function, thereby enhancing the model's adaptability to complex and variable function forms.
[0026] 3. In the present invention, the present invention adopts high-order polynomials to process boundary conditions, which effectively avoids the boundary effect problem in traditional interpolation algorithms and improves the approximation quality and stability. The technical solution of the present invention not only improves the approximation accuracy, but also enhances the generalization ability of the model, so that it can be widely used in various types of function approximation problems. By drawing a comparison chart of the original data and the fitting curve, the present invention provides an intuitive way to evaluate the performance of the algorithm, so that the user can clearly see the approximation effect and error distribution. Relying on a unique algorithm framework, the present invention successfully realizes efficient approximation of complex functions, especially showing significant advantages in dealing with boundary effects. In summary, the technical solution of the present invention has significant advantages in improving approximation accuracy, optimizing calculation efficiency, enhancing model flexibility, and improving boundary processing, and has brought important innovative results to the field of numerical analysis and function approximation. BRIEF DESCRIPTION OF THE DRAWINGS
[0027] Figure 1 is a flow chart of the overall method of the present invention;
[0028] Figure 2 It is a schematic diagram of the error distribution generated by the multiquadric quasi-interpolation algorithm and its residual-based boosting improvement technology in the present invention. DETAILED DESCRIPTION
[0029] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.
[0030] Embodiment:
[0031] Reference Figure 1-2 , a multiquadric quasi-interpolation Boosting method based on residuals. It includes the following steps:
[0032] (1) Initialization and parameter setting: define the number of nodes, x-axis range, data point spacing, shape parameter coefficients, etc.
[0033] (2) Boundary processing: Construct boundary polynomials to adapt to the data edges and ensure that boundary conditions are correctly handled during the calculation process.
[0034] (3) Smooth fitting: Apply multivariate quadratic function to perform initial smoothing to obtain preliminary fitting results.
[0035] (4) Fine-tuning based on residual boosting technology: Analyze the residuals after the first fitting and use a smaller shape parameter coefficient for the second fitting to correct the error and optimize the approximation effect.
[0036] (5) Result presentation: Plot a comparison chart of the original data and the fitted curve to evaluate the algorithm performance.
[0037] Among them: initialization and parameter setting include:
[0038] First, set global variables such as the number of nodes n, the minimum x_min and maximum x_max of the x-axis, and calculate the fixed spacing h between data points based on these parameters. Then, set two shape parameter coefficients of different sizes based on h - coeff_initial for initial fitting and coeff_residual for subsequent fine-tuning to ensure the smoothness and smooth transition of the image.
[0039] Boundary processing includes:
[0040] In order to effectively handle boundary conditions, the present invention selects data points near the boundary to construct a high-order polynomial model, which is used to approximate the function behavior near the boundary and ensure that the boundary information is properly handled. Specifically, firstly, data points located near the boundary are selected from the data set, usually including the first and last data points of the data sequence, to fully capture the information of the boundary area. Then, the selected data points are fitted using the np.polyfit() function to obtain the coefficients of the polynomial and generate a high-order polynomial expression p(x) that can describe the boundary characteristics. This step not only provides a transformed data basis for subsequent numerical approximation, but also provides a mathematical basis for the processing of boundary conditions. Finally, the constructed polynomial is applied to the data points in the boundary area to ensure that the boundary conditions can be accurately reflected in the subsequent numerical approximation process, effectively avoiding the common boundary effect problems in traditional interpolation algorithms, and improving the overall approximation quality and stability. Through this method, the present invention significantly enhances the performance of the approximation algorithm in the boundary area and improves the overall approximation accuracy.
[0041] Smooth fitting includes:
[0042] At this stage, the multiquadric basis function is used As the kernel function, its good locality and positive definiteness characteristics are used for initial smoothing. For non-boundary data, the weight function of the data point is calculated:
[0043]
[0044] ],That
[0045] Then, the weight function is multiplied and accumulated with the transformed data after removing the boundary influence, and the contribution of the boundary polynomial is added to finally obtain the result of the initial fitting.
[0046] Residual-based boosting techniques for fine-tuning include:
[0047] After the initial fitting, we calculate the residuals on all data points and apply the multiquadric quasi-interpolation algorithm to fit them again. This time, a smaller shape parameter coefficient is used to better capture the detailed features. In this step, the residual-based boosting technique uses the residuals after the first fitting as the new objective function for secondary fitting, gradually reducing the error and achieving a more accurate approximation effect. This method can gradually optimize the approximation results and enhance the approximation ability of the original function. Specifically, the residual-based boosting technique includes the following steps:
[0048] (1) Calculate the residual after the initial fitting;
[0049] (2) Use a smaller shape parameter coefficient to perform a quadratic fit on the residuals;
[0050] (3) Add the initial fitting result and the secondary fitting result to obtain the final approximation result;
[0051] (4) Insert the true values of the boundary points to ensure the accuracy of the boundary conditions.
[0052] Results include:
[0053] Finally, by drawing a chart using Python's Matplotlib library, we can visually compare the differences between the original function and the fitting results under the two different strategies. Figure 2 The error distribution of the multiquadric quasi-interpolation algorithm and its residual-based boosting improvement technique is demonstrated to verify the effectiveness of the proposed method.
[0054] like Figure 2 As shown, the error of the original multiquadric quasi-interpolation algorithm is on the order of 10 -5 After the improvement based on residual boosting, the error magnitude is significantly reduced to 10 -9This change is very significant. In the field of scientific computing and data analysis, the reduction in the order of magnitude of error means that the accuracy of the results has been greatly improved. For example, in physical simulation, such an increase in accuracy may mean a more accurate description and prediction of physical phenomena; in engineering design, it may lead to more reliable and optimized design solutions.
[0055] This improved technique can control the error within a smaller range by effectively processing the residual, reflecting the powerful ability of this method in improving the accuracy of the multiquadric quasi-interpolation algorithm. In practical application scenarios, whether in data mining, model fitting in machine learning, or numerical calculation in scientific research, this high-precision algorithm has broad application prospects.
[0056] In summary, the present invention relies on a unique algorithmic framework to successfully achieve the goal of efficient approximation of complex functions, especially showing significant advantages in dealing with boundary effects. This method not only significantly improves the accuracy of the approximation, but also greatly enhances the flexibility and generalization ability of the model, providing a new perspective and technical solution for solving similar problems. By introducing innovative boundary polynomial processing technology and a residual-based boosting mechanism, the present invention has made substantial improvements to the multiquadric quasi-interpolation algorithm, which not only broadens its scope of application, but also greatly improves the accuracy of the algorithm, effectively overcoming the shortcomings of traditional methods, and bringing important innovative results to the fields of numerical analysis and function approximation.
[0057] Compared with the existing technical solutions, the technical solution of the present invention has the following advantages or beneficial effects:
[0058] Improve approximation accuracy: The present invention significantly improves the approximation accuracy of complex functions by introducing residual-based boosting technology, especially when dealing with high-frequency oscillations or steep change features, and can effectively reduce the deviation between the function value at the test point and the true value.
[0059] Optimize computational efficiency: This invention inherits the advantages of the multiquadric quasi-interpolation algorithm, which does not require solving equations and is computationally efficient. At the same time, through algorithm optimization, it further improves the efficiency of processing large-scale data sets and reduces computational costs.
[0060] Enhanced model flexibility: The present invention dynamically adjusts shape parameters so that the algorithm can flexibly optimize the quasi-interpolation process according to the different characteristics of the function, thereby enhancing the model's adaptability to complex and variable function forms.
[0061] Improved boundary processing: The present invention uses high-order polynomials to process boundary conditions, effectively avoiding the boundary effect problem in traditional interpolation algorithms and improving the approximation quality and stability.
[0062] Enhanced generalization capability: The technical solution of the present invention not only improves the approximation accuracy, but also enhances the generalization capability of the model, so that it can be widely applied to various types of function approximation problems.
[0063] Intuitive result display: By plotting the comparison chart of the original data and the fitting curve, the present invention provides an intuitive way to evaluate the algorithm performance, so that the user can clearly see the approximation effect and error distribution.
[0064] Innovative algorithm framework: Relying on a unique algorithm framework, the present invention successfully achieves efficient approximation of complex functions, especially showing significant advantages in dealing with boundary effects.
[0065] In summary, the technical solution of the present invention has significant advantages in improving approximation accuracy, optimizing computational efficiency, enhancing model flexibility, and improving boundary processing, and has brought important innovative results to the field of numerical analysis and function approximation.
[0066] The present invention combines the two stages of smooth fitting and fine adjustment to form a unique function approximation boosting method. In the smooth fitting stage, the characteristics of the multiquadric basis function are used for the initial smoothing process, and the fine adjustment stage uses the residual boosting technology to perform a secondary fit on the residual of the initial fitting. This two-stage architecture is the key to achieving high-precision approximation.
[0067] For example, the way of calculating the weight function of data points in smooth fitting and the strategy of using residuals to construct a new objective function in fine-tuning work together to enable the algorithm to effectively handle complex function forms and gradually optimize the approximation results.
[0068] Boundary Processing Technology
[0069] The important innovation of the present invention is to select data points close to the boundary to construct a high-order polynomial to handle the boundary conditions. By accurately selecting boundary data points, using the np.polyfit() function to fit the polynomial coefficients, generating a boundary polynomial expression, and applying it to the boundary area data points, the boundary effect problem of the traditional interpolation algorithm is effectively avoided. This technology ensures the approximation accuracy and stability in the boundary area, plays a key role in improving the overall approximation quality, and is particularly important for processing function data with irregular boundaries.
[0070] Residual Utilization Mechanism
[0071] One of the core means of improving the accuracy of the present invention is to fully exploit the residual information after the initial fitting and use it as the objective function of the secondary fitting. By calculating the residual, using a smaller shape parameter coefficient to perform a secondary fitting on the residual, and reasonably combining the two fitting results, the gradual correction of the error and the optimization of the approximation effect are achieved.
[0072] This residual-based boosting technology enables the algorithm to better capture the detailed features of the function, adapt to functions of different complexities, and enhance the adaptability and accuracy of the algorithm.
[0073] Parameter setting and adjustment strategy
[0074] The key technical details of the present invention are to set the shape parameter coefficients of different sizes (coeff_initial is used for initial fitting, coeff_residual is used for fine adjustment) according to factors such as the data point spacing, and flexibly use these parameters based on the function characteristics. Reasonable parameter settings ensure the smoothness and smooth transition of the image, and play different roles in different fitting stages, which helps to improve the approximation accuracy and calculation efficiency.
[0075] It should be noted that, in this article, relational terms such as first and second, etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the term "comprise" or any other variant thereof is intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device. In the absence of further restrictions, the elements defined by the sentence "comprise a ..." do not exclude the existence of other identical elements in the process, method, article or device including the elements.
[0076] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that the technical solutions described in the aforementioned embodiments may still be modified, or some of the technical features may be replaced by equivalents. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.
Claims
1. Residual-based multiquadric quasi-interpolation Boosting method, characterized by: The method comprises the following steps: S1: Initialization and parameter setting: define the number of nodes, x-axis range, data point spacing and shape parameter coefficients S2: Perform boundary processing: construct boundary polynomials to adapt to the data edges and ensure that boundary conditions are correctly handled during calculations S3: Perform smooth fitting: Apply multivariate quadratic function for initial smoothing to obtain preliminary fitting results S4: Perform residual-based boosting techniques for fine-tuning: Analyze the residuals after the first fit and use a smaller shape parameter coefficient for a second fit to correct the error and optimize the approximation effect S5: Display the results: draw a comparison chart of the original data and the fitting curve to evaluate the algorithm performance.
2. The residual-based multiquadric pseudo-interpolation Boosting method according to claim 1, characterized in that: In step S1, first, global variables such as the number of nodes n, the minimum x_min and maximum x_max of the x-axis are set, and the fixed spacing h between data points is calculated based on these parameters; then, two shape parameter coefficients of different sizes are set based on h - coeff_initial is used for initial fitting, and coeff_residual is used for subsequent fine adjustment to ensure the smoothness and smooth transition of the image.
3. The residual-based multiquadric pseudo-interpolation Boosting method according to claim 1, characterized in that: In step S2, data points close to the boundary are selected to construct a high-order polynomial model for approximating the function behavior near the boundary to ensure that the boundary information is properly processed; specifically, data points near the boundary are first selected from the data set, usually including the first and last data points of the data sequence, to fully capture the information of the boundary area.
4. The residual-based multiquadric pseudo-interpolation Boosting method according to claim 1, characterized in that: In step S2, the np.polyfit() function is used to fit the selected data points, obtain the coefficients of the polynomial, and generate a high-order polynomial expression p(x) that can describe the boundary characteristics; this step not only provides a transformed data basis for subsequent numerical approximation, but also provides a mathematical basis for the processing of boundary conditions.
5. The residual-based multiquadric pseudo-interpolation Boosting method according to claim 1, characterized in that: In step S3, the constructed polynomial is applied to the data points in the boundary area to ensure that the boundary conditions can be accurately reflected in the subsequent numerical approximation process, effectively avoiding the common boundary effect problems in traditional interpolation algorithms and improving the overall approximation quality and stability; through this method, the present invention significantly enhances the performance of the approximation algorithm in the boundary area and improves the overall approximation accuracy.
6. The residual-based multiquadric pseudo-interpolation Boosting method according to claim 1, characterized in that: In step S3, a multiquadric basis function is used. As the kernel function, its good locality and positive definiteness characteristics are used for initial smoothing; for non-boundary data, the weight function of the data point is calculated: in Then, the weight function is multiplied and accumulated with the transformed data after removing the boundary influence, and the contribution of the boundary polynomial is added to finally obtain the result of the initial fitting.
7. The residual-based multiquadric pseudo-interpolation Boosting method according to claim 1, characterized in that: In step S4, after the initial fitting, we calculate the residuals on all data points and apply the multiquadric quasi-interpolation algorithm to fit them again; this time, a smaller shape parameter coefficient is used in order to better capture the detail features; In this step, the residual-based boosting technique uses the residual after the first fitting as the new objective function for secondary fitting, gradually reducing the error and achieving a more accurate approximation effect.
8. The residual-based multiquadric pseudo-interpolation Boosting method according to claim 1, characterized in that: In step S5, a chart is drawn by using the Matplotlib library of Python, so that the difference between the original function and the fitting results under the two different strategies can be intuitively compared.