Attribute solving method for solving multi-attribute linear system based on single attribute difference proportion method

Abstract

This invention discloses a method for solving the properties of multi-attribute linear systems based on the single-attribute difference ratio method. The method includes: calculating the difference in known attribute values ​​between any two points in the multi-attribute linear system; selecting the largest attribute value difference; determining whether the largest attribute value difference is less than a predetermined threshold; if so, determining that the multi-attribute linear system is ill-conditioned and ending the solution process; otherwise, using the two points corresponding to the largest attribute value difference as the benchmark point B and reference point R. By selecting the point pair and attribute with the largest attribute difference when determining the calculation benchmark, the method essentially selects the calculation path that maximizes the denominator. This fundamentally avoids small or near-zero denominators, significantly suppressing the risk of division by zero and the problem of floating-point error amplification caused by excessively small denominators. The method can maintain stable output even when the data has minor perturbations or redundancy, solving the technical problems of error amplification and even division by zero errors leading to poor stability in existing property solution methods.

Description

Technical Field

[0001] This invention belongs to the field of numerical computation and linear system solution, and relates to property solution methods, specifically a property solution method for solving multi-attribute linear systems based on the single-attribute difference ratio method. Background Technology

[0002] Interpolation and solving of linear relationships are the cornerstone of scientific and engineering computing. In many applications, such as predicting the properties (coordinates, temperature, physical quantities, etc.) of unknown points based on known points, the essence can be abstracted as solving a multi-attribute linear system.

[0003] In the existing technology, the mainstream methods for addressing this type of problem include:

[0004] 1. Distance Proportion Method: This method uses weighted interpolation based on geometric distance. While intuitive, it involves squaring and square root operations, resulting in high computational complexity. Furthermore, when the target point lies on the extension of the line connecting known points, additional handling of the direction sign is required, making the logic complex.

[0005] 2. Parametric Curve Method: This method expresses the linear system as parametric equations and then performs interpolation. However, it requires prior establishment of the parametric equations, which is not straightforward, and the parametric process itself can become cumbersome in high-dimensional or complex situations.

[0006] 3. Projection / Elimination Methods: These methods solve problems through projection or elimination and matrix operations in linear algebra. While these methods are generally applicable, they may suffer from numerical stability issues (such as the risk of division by zero and error amplification) when dealing with "incomplete rank" systems with specific structures. Furthermore, the computational process is highly holistic and difficult to optimize using the specific distribution of known information within the system.

[0007] In particular, existing methods exhibit significant limitations when dealing with the following two common but challenging types of systems: Complementary boundary value systems: These are systems that are solvable as a whole, but the known information at a single point is insufficient, requiring the cross-complementation of information from multiple points to complete the solution. Existing methods lack a systematic iterative mechanism to fully utilize this dispersed information.

[0008] Multi-attribute boundary value systems: When the attribute dimension increases, existing methods usually perform high-dimensional operations directly, failing to effectively utilize the inherent linear correlation of the system (i.e., the "incomplete rank" characteristic) for "dimensionality reduction", which leads to challenges in computational efficiency and numerical stability.

[0009] In summary, existing technologies lack a unified, efficient, and stable method for handling linear systems ranging from one-dimensional natural boundary values ​​to complementary boundary values ​​and high-dimensional multi-attribute boundary value systems. In particular, when dealing with incomplete rank systems with dispersed information or high dimensionality, there are shortcomings in terms of computational efficiency, numerical robustness, and implementation simplicity. Summary of the Invention

[0010] To address the shortcomings of existing technologies, the present invention aims to provide a property solution method for solving multi-attribute linear systems based on the single-attribute difference ratio method, thereby solving the technical problems of poor stability caused by error amplification or even division by zero errors in existing property solution methods.

[0011] To solve the above-mentioned technical problems, the present invention adopts the following technical solution: A property solution method for solving multi-attribute linear systems based on the single-attribute difference ratio method includes the following steps: Step 1: Calculate the difference in attribute values ​​of known attributes between any two points in the multi-attribute linear system. Select the largest attribute value difference and determine whether the largest attribute value difference is less than a predetermined threshold. If it is, the multi-attribute linear system is determined to be ill-conditioned and the solution process ends. If not, the two points corresponding to the largest attribute value difference are taken as the reference point B and the reference point R, and the known attribute corresponding to the largest attribute value difference is taken as the reference attribute b. The known attributes are pressure, coordinates, or elevation; Step 2: Identify points in the multi-attribute linear system whose values ​​of the baseline attribute b are known, excluding the baseline point B and the reference point R. Use these points as target points to obtain the target point set. Step 3: Calculate the scaling factor based on the benchmark point, reference point, and each target point in the target point set. ;

[0012] in: It is the first The value of the baseline attribute b for each target point; It is the value of the reference attribute b of the reference point; It is the value of the reference attribute b of the reference point; Step 4, based on the common scaling factor obtained in Step 3 Determine the linear relationship of a multi-attribute linear system and solve for the th [missing information] in the target point set. Unknown attributes of each target point value ;

[0013] The known properties of the reference point The value; The known properties of the reference point The value; Step 5: Solve for the unknown attributes of the target point obtained in Step 4. value As new known information, it is entered into the multi-attribute linear system. The difference of the attribute values ​​of the current known attributes of any two points in the multi-attribute linear system is calculated. The maximum attribute value difference is selected. The two points corresponding to the maximum attribute value difference are used as the reference point B and the reference point R. The attribute corresponding to the maximum attribute value difference is used as the reference attribute b. Then, return to step 2 until it is impossible to continue solving or all unknown attributes have been solved. The criterion for determining that the solution cannot continue is: for the unknown attributes of the target point, there are no two points in the multi-attribute linear system that are known.

[0014] This invention also includes the following technical features: In step 1, the attribute value difference is the absolute value of the difference between a certain attribute value between two points.

[0015] In step 2, if no target point is identified, the current maximum attribute difference is discarded and the second largest attribute difference is selected.

[0016] Before step 1, the process also includes: data normalization, which involves scaling and measuring the values ​​of known attributes in a multi-attribute linear system and recording the scaling factor.

[0017] Between steps 4 and 5, the following is also included: applying a scaling factor to the unknown properties of the target point. value Perform denormalization.

[0018] Compared with the prior art, the beneficial technical effects of this invention are: (I) This invention selects the point pair and attribute with the largest attribute difference when determining the calculation benchmark, which essentially selects the calculation path that maximizes the denominator. This fundamentally avoids small or near-zero denominators, significantly suppresses the risk of division by zero and the problem of floating-point error amplification caused by excessively small denominators, and enables the method to maintain stable output even when there are slight disturbances or redundancies in the data. It solves the technical problem of poor stability caused by error amplification or even division by zero errors in the existing attribute solving methods.

[0019] (II) The entire solution process of this invention involves only basic addition, subtraction, multiplication, division, and comparison operations, completely avoiding complex operations such as squaring and square root extraction in the distance ratio method. While achieving similar high accuracy, its computational efficiency is significantly higher than that of the projection method and the distance ratio method. The algorithm steps are fixed, the logic is clear, and it is easy to implement efficiently on various hardware and software platforms, making it particularly suitable for embedded systems or real-time computing scenarios.

[0020] (III) The scaling factor in this invention is determined by the data itself, and can be positive or negative, supporting interpolation and extrapolation of linear relationships without the need for additional direction judgment logic. Furthermore, through continuous iteration, it can effectively solve complementary boundary value systems where known information is scattered across different elements; in redundant boundary value systems with multiple sets of known information, the method's benchmark selection mechanism can automatically select the most stable solution path, achieving a leapfrog improvement in accuracy and possessing good fault tolerance. Detailed Implementation

[0021] It should be noted that, unless otherwise specified, all components in this invention are those known in the art.

[0022] The following are specific embodiments of the present invention. It should be noted that the present invention is not limited to the following specific embodiments. All equivalent modifications made based on the technical solutions of this application fall within the protection scope of the present invention.

[0023] This invention discloses a property solution method for solving multi-attribute linear systems based on the single-attribute difference ratio method, comprising the following steps: Step 1: Calculate the difference in attribute values ​​of known attributes between any two points in the multi-attribute linear system. Select the largest attribute value difference and determine whether the largest attribute value difference is less than a predetermined threshold. If it is, the multi-attribute linear system is determined to be ill-conditioned and the solution process ends. If not, the two points corresponding to the largest attribute value difference are taken as the reference point B and the reference point R, and the known attribute corresponding to the largest attribute value difference is taken as the reference attribute b. The known attributes are pressure, coordinates, or elevation; Step 2: Identify points in the multi-attribute linear system whose values ​​of the baseline attribute b are known, excluding the baseline point B and the reference point R. Use these points as target points to obtain the target point set. Step 3: Calculate the scaling factor based on the benchmark point, reference point, and each target point in the target point set. ;

[0024] in: It is the first The value of the baseline attribute b for each target point; It is the value of the reference attribute b of the reference point; It is the value of the reference attribute b of the reference point; Step 4, based on the common scaling factor obtained in Step 3 Determine the linear relationship of a multi-attribute linear system and solve for the th [missing information] in the target point set. Unknown attributes of each target point value ;

[0025] The known properties of the reference point The value; The known properties of the reference point The value; Step 5: Solve for the unknown attributes of the target point obtained in Step 4. value As new known information, it is entered into the multi-attribute linear system. The difference of the attribute values ​​of the current known attributes of any two points in the multi-attribute linear system is calculated. The maximum attribute value difference is selected. The two points corresponding to the maximum attribute value difference are used as the reference point B and the reference point R. The attribute corresponding to the maximum attribute value difference is used as the reference attribute b. Then, return to step 2 until it is impossible to continue solving or all unknown attributes have been solved. The criterion for determining when a solution cannot be continued is: for the unknown attributes of the target point, in a multi-attribute linear system, there are no two points that are known.

[0026] A multi-attribute linear system refers to a system containing two or more attributes, where any attribute can be represented as a linear function of any set of linearly independent attributes. In the one-dimensional case, any attribute is a linear function of another attribute (e.g., y = ax + b). In the two-dimensional case, if x and y are linearly independent, then any other attribute z can be represented as z = ax + by + c.

[0027] In step 1 above, by selecting the pair of points and attributes with the largest attribute difference when determining the calculation benchmark, the calculation path that maximizes the denominator is essentially chosen. This fundamentally avoids small or near-zero denominators, significantly suppresses the risk of division by zero and the problem of floating-point error amplification caused by excessively small denominators, and enables the method to maintain stable output even when there are minor perturbations or redundancies in the data. This solves the technical problem of poor stability caused by error amplification or even division by zero errors in existing attribute solving methods.

[0028] Furthermore, the entire solution process involves only basic addition, subtraction, multiplication, division, and comparison operations, completely avoiding complex calculations such as squares and square roots found in the distance scaling method. While achieving similar high accuracy, its computational efficiency is significantly higher than that of the projection method and the distance scaling method. The algorithm steps are fixed, the logic is clear, and it is easy to implement efficiently on various hardware and software platforms, making it particularly suitable for embedded systems or real-time computing scenarios.

[0029] The value of the scaling factor is determined by the data itself, and can be positive or negative. It supports interpolation and extrapolation of linear relationships without requiring additional direction judgment logic. Furthermore, through continuous iteration, it can effectively solve complementary boundary value systems where known information is distributed across different elements. In redundant boundary value systems with multiple sets of known information, the method's benchmark selection mechanism can automatically select the most stable solution path, achieving a significant improvement in accuracy and exhibiting good fault tolerance.

[0030] In step 1, the attribute value difference is the absolute value of the difference between two points for a certain attribute value.

[0031] In step 1, if no target point is identified, the current maximum attribute difference is discarded and the second largest attribute difference is selected.

[0032] Before step 1, the process also includes: data normalization, which involves scaling and measuring the values ​​of known attributes in a multi-attribute linear system and recording the scaling factors.

[0033] Between steps 4 and 5, the following is also included: applying a scaling factor to the unknown properties of the target point. value Perform denormalization.

[0034] I. The feasibility and solution accuracy of this solution in different scenarios are verified through the following four examples: Example 1: This embodiment verifies the elevation of the intersection point between an unfinished tunnel and a completed tunnel within the context of tunnel construction, comparing four methods: parametric curve method, projection method, distance ratio method, and the method of this invention. The aim is to demonstrate the accuracy and reliability of each method when rounding errors exist in the recorded values ​​(i.e., the target attribute t is the elevation z). 1.1 Scenarios and Data; The plane coordinates and elevations of the two endpoints B and R of an unfinished tunnel have been obtained through high-precision measurement (0.001m accuracy), and their true values ​​satisfy the strict collinearity relationship of the three points. However, due to the limitation of recording accuracy, the actual recording is rounded to 0.01m. A completed tunnel intersects the unfinished tunnel at point T, and its plane position has been determined. The elevation of the unfinished tunnel at point T needs to be interpolated to verify the elevation matching between the two tunnels. The known true values ​​(0.001m accuracy) and recorded values ​​(0.01m accuracy) are as follows: Reference point B: True value (0.000, 0.000, 0.000), Recorded value (0.00, 0.00, 0.00) Reference point R: True value (0.026, 799.994, 200.000), Recorded value (0.03, 799.99, 200.00) Target point T: True value (0.013, 399.997, 100.000), Recorded value (0.01, 400.00, ?) 1.2 Method Comparison and Calculation Results; Based on the above recorded values, the estimated elevation of point T on the unfinished tunnel was calculated using four different methods and compared with the true value of 100m.

[0035] (1) Parametric curve method; The classic parametric curve method will randomly or by default select the x attribute to calculate the interpolation parameter r.

[0036]

[0037]

[0038] Absolute error:

[0039] (2) Projection method; The projection method calculates the scaling factor based on the vector dot product, using all the information of x and y.

[0040]

[0041]

[0042] Absolute error:

[0043] (3) Distance ratio method; The distance scaling method calculates the scale based on Euclidean distance.

[0044]

[0045]

[0046] Absolute error:

[0047] (4) The method of the present invention; The method of this invention first analyzes the data and selects... (far greater than) The y-direction is used as the calculation reference.

[0048]

[0049]

[0050] Absolute error:

[0051] 1.3 Results and Comparison The same recording error can result in different computational loads and final calculation results due to different calculation methods. The table below compares the efficiency and effectiveness of four methods. The relative error is the ratio of the absolute error to the true value; the computational complexity is the number of multiplication and division operations.

[0052] The table above shows that when the parametric curve method selects the x-axis with minimal changes in direction sensitivity, the elevation calculation error reaches as high as 33m, with a relative error of 33%, due to the amplification effect of recording errors. Selecting the x-axis with greater changes results in the same error as in this invention. The projection method and the distance scaling method offer high accuracy but have high computational complexity, and the distance scaling method involves square root operations, which are time-consuming. This invention achieves the same high accuracy as the projection method and the distance scaling method by automatically selecting the coordinate axis with the greatest changes, but with significantly lower computational complexity.

[0053] Example 2: This embodiment compares four methods—parametric curve method, projection method, distance scaling method, and the method of this invention—in the context of long-distance extrapolation. It demonstrates the accuracy and computational efficiency of each method under the ideal condition where all relevant attributes of the target point are known.

[0054] 2.1 Scenarios and Data; Using the linear relationship established between two nearby benchmark points R and B, the elevation of point T 5000m away is predicted (i.e., the value of the target attribute t is the elevation z).

[0055] The planar coordinates (x, y) of point T are known (i.e., they meet the data requirements of both the projection method and the distance scaling method). The true value is measured with an accuracy of 0.001, recorded with an accuracy of 0.01, and the unit is meters (m). Reference point B: True value (0.00, 0.00, 100.00), Recorded value (0.00, 0.00, 100.00) Reference point R: True value (0.020, 80.000, 300.000), Recorded value (0.02, 80, 300.00); Target point T: True value: (1.255, 5000.005, ?), Recorded value: (1.26, 5000.01, ?); Calculate the target value T: theoretical true value 12600.000.

[0056] Then the target point T has the following true values: (1.255, 5000.005, 12600.000) and recorded values: (1.26, 5000.01, ?). 2.2 Method Comparison and Calculation Results; Based on the above recorded values, the elevation estimates of point T were calculated using four different methods and compared with the true values.

[0057] (1) Classical parametric curve method; The classic parametric curve method will randomly or by default select the x attribute to calculate the interpolation parameters.

[0058]

[0059]

[0060] Absolute error:

[0061] (2) Projection method; The projection method calculates the scaling factor based on the vector dot product, using all the information of x and y.

[0062]

[0063]

[0064] Absolute error:

[0065] (3) Distance ratio method; The distance scaling method calculates the scale based on Euclidean distance.

[0066]

[0067]

[0068] Absolute error:

[0069] (4) The method of the present invention; The method of this invention first analyzes the data and selects... (far greater than) The y-direction is used as the calculation reference.

[0070]

[0071]

[0072] Absolute error:

[0073] 2.3 Performance Comparison and Preliminary Analysis The same recording error can result in different computational loads and final calculation results due to different calculation methods. The table below compares the efficiency and effectiveness of four methods. The relative error is the ratio of the absolute error to the true value; the computational complexity is the number of multiplication and division operations.

[0074] The table above shows that this example once again verifies conclusion 1.3: however, this time the method of this invention and the distance scaling method have the highest accuracy, followed by the projection method. The parameterization method and the method of this invention have the highest efficiency, followed by the projection method, and the distance scaling method has the lowest efficiency. While maintaining the same high computational efficiency as the parameterized curve method, the method of this invention ensures that each calculation automatically approaches the optimal accuracy through intelligent selection.

[0075] Example 3: This embodiment demonstrates that the present invention naturally supports reverse extension interpolation (i.e., the target point is located on the reverse extension line of the line connecting the reference point and the benchmark point), while the traditional distance ratio method requires more complex calculations in principle because the distance ratio is always positive.

[0076] 3.1 Scenarios and Data; Consider a three-dimensional straight line. Given the coordinates of two points B and R on the line, and the planar coordinates of target points T1, T2, and T3 on the line, we need to interpolate its elevation z. The actual values ​​satisfy strict collinearity, but the recorded values ​​have rounding errors.

[0077] Reference point B: True value (0.00, 0.00, 100.00), Recorded value (0.00, 0.00, 100.00) Reference point R: True value (0.028, 79.990, 300.000), Recorded value (0.03, 79.99, 300.00); Target point T1: True value: (-0.014, -39.995, ?), Recorded value: (-0.01, -40.00, ?); Target point T2: True value: (0.014, 39.995, ?), Recorded value: (0.01, 40.00, ?); Target point T3: True value: (0.042, 119.985, ?), Recorded value: (0.04, 119.99, ?); 3.2 Calculation results; The table below shows the results and absolute errors calculated using four methods, all of which use formulas:

[0078] Calculate the z-coordinates of T1, T2, and T3.

[0079]

[0080] The table above shows a serious error in calculating T1 using the distance ratio method. This is a key flaw of the distance ratio method that hasn't been explained before: it fails when the relative positions of the three points cannot be guaranteed. To consistently use the distance ratio method correctly, the relationship between the three points should be determined first, and it should be used when T lies within RB and its extension. Otherwise use calculate.

[0081] 3.3 Key findings; The limitation of the distance ratio method lies in its high computational complexity (involving square root operations) and the need to pre-determine the positional relationship of the three points, resulting in the lowest efficiency. This invention utilizes a scaling factor... It can be positive or negative, and naturally supports bidirectional interpolation. This example further demonstrates that, based on the functionality of supporting backward extension, this invention, through a denominator maximization strategy, can ensure that even in backward interpolation, the most stable attribute is automatically selected for calculation, thus achieving superior functionality and robustness.

[0082] Example 4: This embodiment demonstrates how, in a complementary boundary value system where information is strictly dispersed and there is no direct solution path, the method of the present invention can automatically connect solution information, construct an effective solution path, and ultimately correctly solve for the target point.

[0083] 4.1 Scenarios and Data Given a one-dimensional linear field, four collinear points A, B, C, and D satisfy a linear relationship. The following information leads to the classic problem of complementary boundary values ​​with eight known parameters:

[0084] When it is necessary to solve for the attribute p value at point C or D, the system represented in the table above has a core contradiction: the attribute p is not simultaneously known at all three points. Therefore, the unknown attribute p at any point cannot be directly solved. It is necessary to iterate and first solve for the unknown attribute of some intermediate points to create conditions for subsequent calculations.

[0085] 4.2 The process of the method of the present invention The algorithm runs completely autonomously, requiring no global model or manual intervention. Its steps are as follows: First iteration: Find the optimal path; Within the base point-reference point-target point combination, two paths that can be executed immediately were identified: L1: The baseline point is C(4.0, 2.0, ?), the reference point is D(6.0, 3.0, ?), and the target point is A(0.0, ?, 10.0). (Find the value of the target point) The baseline attribute is x, and the attribute difference is... =6.0 - 4.0 = 2.0 (can be calculated) ).

[0086] L2: Base point is C(4.0, 2.0, ?), reference point is D(6.0, 3.0, ?), target point is B(?, 1.0, 20.0) (Calculate) The baseline attribute is y, and the attribute difference is... =3.0 - 2.0 = 1.0 (can be calculated) ).

[0087] Intelligent Optimal Selection: Based on the principle of maximizing the denominator, the algorithm automatically selects the value with the largest absolute value. Subsequent calculations will be performed. For this embodiment, There is only one; there is no room for choice.

[0088] Perform the calculation: scaling factor .

[0089] Solve .

[0090] State update: Point A becomes the omniscient point (0.0,0.0,10.0) and is removed from the unsolved set; L1 is deleted.

[0091] At this point, only one candidate path remains: L2: Base point is C(4.0, 2.0, ?), reference point is D(6.0, 3.0, ?), target point is B(?, 1.0, 20.0) (Calculate) The baseline attribute is y, and the attribute difference is... =3.0 - 2.0 = 1.0 (can be calculated) ).

[0092] Intelligent Optimal Selection: Based on the principle of maximizing the denominator, the algorithm automatically selects the value with the largest absolute value. Further calculations will be performed. At this point, only... And there is only one, with no room for selection.

[0093] Perform the calculation: scaling factor .

[0094] Solve .

[0095] State update: Point B becomes the omniscient point (2.0, 1.0, 20.0) and is removed from the unsolved set; L2 is deleted.

[0096] The information table then becomes:

[0097] Second iteration: Building the optimal bridge At this point, both A and B are omniscient points and can form a stable baseline. There are 4 candidate paths: L3: Base point is A(0.0,0.0,10.0), reference point is B(2.0,1.0,20), target point is C(4.0,2.0,?) (Calculate) The baseline attribute is x, and the attribute difference is... =2.0 - 0.0 = 2.0, (can be calculated) ).

[0098] L4: Base point is A(0.0,0.0,10.0), reference point is B(2.0,1.0,20), target point is D(6.0,3.0,?) (Find...) The baseline attribute is x, and the attribute difference is... =2.0 - 0.0 = 2.0, (can be calculated) ).

[0099] L5: Base point is A(0.0,0.0,10.0), reference point is B(2.0,1.0,20), target point is C(4.0,2.0,?) (Calculate) The baseline attribute is y, and the attribute difference is... =1.0 - 0.0 = 1.0, (can be calculated) ).

[0100] L6: Base point is A(0.0,0.0,10.0), reference point is B(2.0,1.0,20), target point is D(6.0,3.0,?) (Find...) The baseline attribute is y, and the attribute difference is... =1.0 - 0.0 = 1.0, (can be calculated) ).

[0101] Intelligent Optimal Selection: Based on the principle of maximizing the denominator, the algorithm automatically selects the value with the largest absolute value. That is, path L3 will be used for subsequent calculations.

[0102] Perform the calculation: scaling factor .

[0103] Solve .

[0104] State update: Point C becomes the omniscient point (4.0, 2.0, 30.0) and is removed from the unsolved set; L3 and L5 are deleted.

[0105] At this point, two candidate paths remain: L4: Base point is A(0.0,0.0,10.0), reference point is B(2.0,1.0,20), target point is D(6.0,3.0,?) (Find...) The baseline attribute is x, and the attribute difference is... =2.0 - 0.0 = 2.0, (can be calculated) ).

[0106] L6: Base point is A(0.0,0.0,10.0), reference point is B(2.0,1.0,20), target point is D(6.0,3.0,?) (Find...) The baseline attribute is y, and the attribute difference is... =1.0 - 0.0 = 1.0, (can be calculated) ).

[0107] Intelligent Optimal Selection: Based on the principle of maximizing the denominator, the algorithm automatically selects the value with the largest absolute value. That is, path L4 will be used for subsequent calculations.

[0108] Perform the calculation: scaling factor .

[0109] Solve .

[0110] State update: Point D becomes omniscient point (6.0, 3.0, 40.0) and is removed from the unsolved set; L4 and L6 are deleted.

[0111] Completed: No unknown attributes, the algorithm terminated successfully.

[0112] Final result:

[0113] 4.3 Extension under compatibility redundancy constraints; Consider information-distributed systems under compatible redundancy constraints (such as 10 nodes in a sensor network collecting the same temperature attribute, with 5 nodes providing identical measurements; or 3 data sources providing repetitive pressure data in multi-source data fusion). Such systems require solving the complementary definite solution problem of selecting the optimal solution from redundant paths. The challenge lies in the fact that compatible redundancy increases path complexity (e.g., 45 pairwise paths for 10 sensors). Traditional methods are prone to large redundancy errors due to overall optimization, while greedy strategies tend to select shortest paths and ignore stable information within redundancy. Both necessitate more intelligent optimization mechanisms.

[0114] This invention addresses the above challenges by using the denominator maximization criterion (selecting the path with the largest absolute value of attribute difference at each step): automatically scanning all redundant paths and selecting the most stable path with the largest attribute difference (e.g., selecting the two nodes with the largest temperature difference as initial calibration points in a sensor network); fully checking the redundant correlations of intermediate points (e.g., the cross-influence of temperature and humidity), and retaining the most relevant points as needed; verifying stability at each step (e.g., whether the deviation after calibration is <1%), and gradually constructing a globally robust solution path.

[0115] In this extended scenario, the core value of this invention is the selection of the most stable path in redundancy, that is, using limited computing costs (compute more intermediate points) to exchange for a coordinated improvement in accuracy and reliability, solving the pain point of redundant information overload in information-distributed systems, and is suitable for high-reliability engineering such as sensor network calibration and multi-source data fusion.

[0116] 4.4 Comparative Analysis with Traditional Methods and Shortest Path Strategies This invention achieves two major breakthrough advantages through the denominator maximization criterion (selecting the path with the largest absolute value of attribute difference at each step, i.e., with the core logic of amplifying the effective signal and suppressing error propagation, see the table below): Stability transition: Significantly reduces the sensitivity to the solution error of the final goal, compared with the overall data dependence of traditional methods and the lack of active control of greedy strategies; Accuracy-cost balance: Achieving a decisive improvement in accuracy by incurring a small additional cost for calculating intermediate points, especially suitable for real-world scenarios where errors are unavoidable, such as sensor calibration and multi-source data fusion.

[0117]

[0118] Based on the dimensional differences in the table above, the essential differences between the three methods can be summarized as follows: Traditional global equation methods are superior to theoretical research, but their stability is constrained by data quality and their practicality is poor. The shortest path greedy strategy is superior to speed, but at the cost of stability (short paths may introduce implicit error chains). This invention achieves global stability through the optimal path, thereby improving both accuracy and reliability with limited computational cost.

[0119] II. The reliability and solution stability of this invention will be analyzed in the following two scenarios: (I) The following analysis examines the fundamental differences between the present invention and the classical parametric curve method in terms of algorithm design, expected computational efficiency, and reliability of results, in order to illustrate the efficiency and reliability of the present invention.

[0120] (1.1) Scenarios and Data; Construct an extreme scenario where the attribute change is zero. In the measurement data, the benchmark point and the reference point are perfectly aligned on a certain coordinate axis, and the attribute difference in that direction is zero.

[0121] Reference point B: (1000.000, 2000.000, 100.0) Reference point R: (1000.000, 2000.100, 150.0) Target point T (located at the midpoint of BR): (1000.000, 2000.050, 125.0) (1.2) Two possible paths and their probabilities in the classical parametric curve method; The parametric curve method randomly selects either X or Y attributes as parameters, and the costs of the two paths differ: Path A (incorrect x-direction selection): because If the value is 0, a division by zero error will be triggered immediately. The algorithm must catch the exception and roll back to path B to recalculate. Total computation cost = cost of attempting path A + cost of exception handling + cost of computation for path B.

[0122] Path B (correctly selected y-direction): Calculation successful directly. Total computation cost = cost of a single computation.

[0123] Without any prior information, the probability of choosing either direction is 50%. Therefore, the expected computational cost is a weighted average of the two cases, and includes additional anomaly handling overhead.

[0124] (1.3) The deterministic path of the method of the present invention; This invention first calculates and compares | | and |Δ The algorithm will always and uniquely choose the y-direction as the calculation reference, and the calculation will succeed on the first attempt. Total computational cost = comparison cost + single calculation cost.

[0125] (1.4) Calculation steps and cost analysis (based on typical floating-point operations) Time consumption for each operation: Comparison / branch (C) ≈ Absolute value (A) ≈ Addition / subtraction (SA) < Multiplication (M) < Division (D) < Exception handling (E).

[0126]

[0127] (1.6) Conclusion; Only when the parametric curve method is extremely lucky enough to select the correct path on the first try, and the risk and cost of its incorrect selection are completely ignored, the method of this invention may have negligible additional overhead (approximately 2-3 clock cycles) in a single run due to the addition of two subtractions and one comparison. In real unbiased systems, the parametric curve method has a 50% probability of getting stuck on a high-overhead error path. Considering the costs of exception handling and process jumps, its expected runtime far exceeds that of the method of this invention. This invention completely eliminates performance fluctuations and operational risks with constant and slightly increased basic overhead, achieving predictable high performance.

[0128] (ii) The following demonstrates that in a redundant boundary value system with a large number of attributes, when the vast majority of attribute differences are zero or extremely small, the classical parametric curve method suffers a sharp decline in efficiency due to blindly selecting attributes for trial calculations; while the method of this invention can immediately lock the most stable (attribute difference is the largest) calculation path through a systematic comparative analysis, demonstrating deterministic efficiency and robustness advantages.

[0129] (2.1) Scenarios and Data; In a system containing numerous monitoring indicators (attributes), such as an environmental monitoring station (attributes include PM2.5, temperature, humidity, wind speed, air pressure, etc.) or financial assets (attributes include price, trading volume, volatility, etc.), two states, A and B, are known to have the same values ​​for most indicators, differing only in a few. The task is to predict a certain unknown indicator for a third state, C.

[0130] Baseline state A: (a1=0, a2=0, a3=0, ..., an=20.0) Reference state B: (b1=0, b2=10.0, b3=0, ..., bn=30.0), with only attribute 2 showing a significant difference.

[0131] Attribute differences: Δ1=0, Δ2=10.0, Δ3=0, ..., Δn=10.0.

[0132] Given conditions: The first n-1 attribute values ​​of state C are known, forming a highly redundant definite solution system.

[0133] (2.2) Classical parametric curve method; This method typically selects one known attribute from among many attributes, either sequentially or randomly, to calculate the parameter t.

[0134] An attempt to select Δ1: t=(c1-a1) / Δ1=0 / 0 triggered a division by zero error, and the calculation failed.

[0135] Trying to select Δ3: also failed.

[0136] After trying several zero-difference properties, the result was successful when Δ2 was selected: t=(C2-A2) / Δ2=5.0 / 10.0=0.5.

[0137] Performance Analysis: When the number of non-zero attribute differences is small (one in this example), in the worst case (e.g., random order and always selecting a zero-difference attribute), the algorithm requires n-1 failed attempts, each involving a division operation and an exception handling operation. In the expected case, it requires approximately n / 2 attempts on average to succeed. As the number of system attributes n increases, the computational overhead becomes proportional to n, accompanied by a significant exception handling burden, resulting in extremely low efficiency.

[0138] (2.3) The method of the present invention; The method of this invention first performs a traversal analysis on the differences of all n known attributes.

[0139] Analysis phase (involving only lightweight operations): Calculate |Δ1|, |Δ2|, |Δ3|, ... sequentially, and find the one with the largest absolute value by comparison. This process involves only n-1 subtractions, taking the absolute value, and comparison operations, completely avoiding complex division operations and exception handling.

[0140] Calculation phase (one core operation): After locking the maximum of |Δ2|, use this property to perform a unique division operation to complete the solution.

[0141] Performance Analysis: Regardless of the increase in the number of attributes n, the method of this invention performs only one deterministic, exception-free traversal analysis and one successful division calculation. Its computational complexity is a lightweight operation of O(n), which is far lower than the O(n) division attempts and exception handling of the parameterized curve method.

[0142] (2.4) Quantitative comparison; The system has k=50 monitored attributes. The classic parametric curve method uses random or sequential selection for blind calculation. In the best case, the target attribute is selected on the first try, requiring one division, one multiplication, and a few additions and subtractions to successfully calculate the target attribute value. In the worst case, the first 49 attempts are all incorrect, and the target attribute is finally selected on the 50th attempt, requiring 49 exception handling steps, 50 divisions, one multiplication, and a few additions and subtractions to successfully calculate the target attribute value. On average, it takes about 25 attempts, or approximately 24 exception handling steps, 25 divisions, one multiplication, and a few additions and subtractions to successfully calculate the target attribute value. The calculation process is uncertain, resulting in extremely large performance fluctuations. Frequent divisions and exception handling consume a large amount of CPU resources, leading to a high system resource load.

[0143] The method of this invention employs intelligent traversal selection, ensuring that the target attribute value is successfully calculated in 50 comparisons + 1 division + 1 multiplication + numerous additions and subtractions under any circumstances. The calculation process is deterministic, and the performance is stable and controllable. The system resource load is very low.

[0144] In fact, after calculating the ratio, the computational cost of the two methods is the same. The huge difference lies in the calculation of the ratio t. The table below shows the comparison of the computational costs of the two methods when calculating the ratio t.

[0145]

[0146] (2.5) Conclusion The advantages of this invention amplify with the increase in the number of system attributes: the lower the proportion of non-zero attribute differences in the system (the higher the sparsity), the lower the probability of traditional methods guessing the correct path, and the trial-and-error cost increases linearly with n; while the cost of scanning and locking the correct path in this invention increases slowly and is completely controllable. This proves that the method of this invention has irreplaceable value in dealing with large-scale, sparse attribute linear problems. It clearly demonstrates that this invention is applicable to various systems with a large number of attributes but sparse variation characteristics, such as sensor networks, multi-factor financial models, and hyperspectral image analysis, providing empirical evidence for their wide application.

Claims

1. A property solution method for multi-attribute linear systems based on the single-attribute difference ratio method, characterized in that, Includes the following steps: Step 1: Calculate the difference in attribute values ​​of known attributes between any two points in the multi-attribute linear system. Select the largest attribute value difference and determine whether the largest attribute value difference is less than a predetermined threshold. If it is, the multi-attribute linear system is determined to be ill-conditioned and the solution process ends. If not, the two points corresponding to the largest attribute value difference are taken as the reference point B and the reference point R, and the known attribute corresponding to the largest attribute value difference is taken as the reference attribute b. The known attributes are pressure, coordinates, or elevation; Step 2: Identify points in the multi-attribute linear system whose values ​​of the baseline attribute b are known, excluding the baseline point B and the reference point R. Use these points as target points to obtain the target point set. Step 3: Calculate the scaling factor based on the benchmark point, reference point, and each target point in the target point set. ; in: It is the first The value of the baseline attribute b for each target point; It is the value of the reference attribute b of the reference point; It is the value of the reference attribute b of the reference point; Step 4, based on the common scaling factor obtained in Step 3 Determine the linear relationship of a multi-attribute linear system and solve for the th [missing information] in the target point set. Unknown attributes of each target point value ; The known properties of the reference point The value; The known properties of the reference point The value; Step 5: Solve for the unknown attributes of the target point obtained in Step 4. value As new known information, it is entered into the multi-attribute linear system. The difference of the attribute values ​​of the current known attributes of any two points in the multi-attribute linear system is calculated. The maximum attribute value difference is selected. The two points corresponding to the maximum attribute value difference are used as the reference point B and the reference point R. The attribute corresponding to the maximum attribute value difference is used as the reference attribute b. Then, return to step 2 until it is impossible to continue solving or all unknown attributes have been solved. The criterion for determining that the solution cannot continue is: for the unknown attributes of the target point, there are no two points in the multi-attribute linear system that are known.

2. The attribute solution method for solving multi-attribute linear systems based on the single-attribute difference ratio method as described in claim 1, characterized in that, In step 1, the attribute value difference is the absolute value of the difference between a certain attribute value between two points.

3. The attribute solution method for solving multi-attribute linear systems based on the single-attribute difference ratio method as described in claim 1, characterized in that, In step 1, if no target point is identified, the current maximum attribute difference is discarded and the second largest attribute difference is selected.

4. The attribute solution method for solving multi-attribute linear systems based on the single-attribute difference ratio method as described in claim 1, characterized in that, Before step 1, the process also includes: data normalization, which involves scaling and measuring the values ​​of known attributes in a multi-attribute linear system and recording the scaling factor.

5. The attribute solution method for solving multi-attribute linear systems based on the single-attribute difference ratio method as described in claim 4, characterized in that, Between steps 4 and 5, the following is also included: applying a scaling factor to the unknown properties of the target point. value Perform denormalization.

6. An electronic device comprising a memory, a processor, and a computer program stored in the memory, characterized in that, When the processor executes the program, it implements the attribute solution method for solving multi-attribute linear systems based on the single-attribute difference ratio method as described in any one of claims 1-5.

7. A computer-readable storage medium having a computer program stored thereon, characterized in that, When executed by the processor, the program implements the attribute solution method for solving multi-attribute linear systems based on the single-attribute difference ratio method as described in any one of claims 1-5.

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