Apparatus and method for improving monte carlo particle transport simulation with probability density distribution sampling

Abstract

The application provides a device and method for improving Monte Carlo particle transport simulation of probability density distribution sampling, specifically: introducing particle transport data of an environment to be analyzed; determining a probability density distribution type of the particle transport data, and pre-processing the particle transport data according to the probability density distribution type; optimizing a Monte Carlo particle transport algorithm based on a weight-based alias sampling algorithm, and calculating the pre-processed particle transport data through the optimized Monte Carlo particle transport algorithm to obtain a radiation field calculation result of the environment. The probability density distribution type of different particle transport data can be pre-processed, the Monte Carlo particle transport algorithm can be optimized based on the weight-based alias sampling algorithm, the time complexity of the probability density distribution sampling can be greatly reduced, the efficiency of the Monte Carlo particle transport calculation can be improved, and the radiation field calculation result of the environment to be analyzed can be quickly obtained.

Description

Technical Field

[0001] This invention mainly relates to the field of radiation dose calculation technology, specifically to an apparatus and method for improving the sampling of probability density distribution to simulate Monte Carlo particle transport. Background Technology

[0002] Currently, particle radiation technology has become a powerful tool for humankind to transform the world, helping to improve our productivity and quality of life. This is inseparable from the strong support of the Monte Carlo method and its related software. In nuclear medicine, gamma rays, electrons, neutrons, protons, and heavy ions are used to treat tumors. The Monte Carlo particle transport method is used to accurately calculate and invert radiation doses under complex human body shapes. In non-destructive testing, gamma rays and neutrons are used to obtain information about the internal structure, processing defects, and thickness of objects. The Monte Carlo particle transport calculation method is used to efficiently analyze the reaction modes and intensity attenuation of particles with different substances. In the field of irradiation processing, particles are used to irradiate insulating materials, neutrons are used to create radioactive nuclei, radiation is used to induce gene mutations and control germination and maturation time, and irradiation is used to degrade organic waste and treat sewage. The Monte Carlo particle transport calculation method is used to design the required radiation intensity and duration for different particles. In the field of safety and environmental protection, irradiation is used to eliminate static electricity, prevent fires or explosions, control pests, and manage environmental pollution. The Monte Carlo particle transport method is used to quickly calculate the dose requirements for different types of irradiation.

[0003] However, Monte Carlo particle transport calculations also suffer from slow convergence speeds and difficulty in handling deep penetration of radiation shielding. Therefore, improving the efficiency of Monte Carlo particle transport calculations is of great significance. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to provide an apparatus and method for improving the sampling of probability density distribution for Monte Carlo particle transport simulation, which addresses the shortcomings of the prior art.

[0005] The technical solution of this invention to solve the above-mentioned technical problems is as follows: An apparatus for improving the sampling of probability density distribution in Monte Carlo particle transport simulation, comprising:

[0006] The import module is used to import particle transport data from the environment to be analyzed.

[0007] A data preprocessing module is used to determine the probability density distribution type of the particle transport data and preprocess the particle transport data according to the probability density distribution type.

[0008] The sampling module is used to optimize the Monte Carlo particle transport algorithm based on the weighted aliasing algorithm. The optimized Monte Carlo particle transport algorithm is used to calculate the preprocessed particle transport data to obtain the radiation field calculation results of the environment to be analyzed.

[0009] Another technical solution of the present invention to solve the above-mentioned technical problems is as follows: A method for improving the sampling of probability density distribution in Monte Carlo particle transport simulation, comprising the following steps:

[0010] Import particle transport data from the environment to be analyzed;

[0011] Determine the probability density distribution type of the particle transport data, and preprocess the particle transport data according to the probability density distribution type;

[0012] The Monte Carlo particle transport algorithm is optimized by using a weighted aliasing sampling algorithm. The preprocessed particle transport data is then calculated using the optimized Monte Carlo particle transport algorithm to obtain the radiation field calculation results of the environment to be analyzed.

[0013] Another technical solution of the present invention to solve the above-mentioned technical problems is as follows: an apparatus for improving the sampling of probability density distribution to simulate the transport of Monte Carlo particles, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the method for improving the sampling of probability density distribution to simulate the transport of Monte Carlo particles as described above.

[0014] Another technical solution of the present invention to solve the above-mentioned technical problems is as follows: a computer-readable storage medium storing a computer program, which, when executed by a processor, implements the method for simulating the Monte Carlo particle transport by improved probability density distribution sampling as described above.

[0015] The beneficial effects of this invention are: it can preprocess according to the probability density distribution type of different particle transport data, optimize the Monte Carlo particle transport algorithm with a weighted alias sampling algorithm, which can significantly reduce the time complexity of probability density distribution sampling, improve the efficiency of Monte Carlo particle transport calculation, and quickly obtain the radiation field calculation results of the environment to be analyzed. Attached Figure Description

[0016] Figure 1 A block diagram of a device for simulating Monte Carlo particle transport using improved probability density distribution sampling, provided in an embodiment of the present invention;

[0017] Figure 2 This is a flowchart illustrating an improved method for Monte Carlo particle transport simulation based on probability density distribution sampling, provided in an embodiment of the present invention. Detailed Implementation

[0018] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0019] Example 1:

[0020] like Figure 1 As shown, an apparatus for improving the sampling of probability density distributions in Monte Carlo particle transport simulation includes:

[0021] The import module is used to import particle transport data from the environment to be analyzed.

[0022] A data preprocessing module is used to determine the probability density distribution type of the particle transport data and preprocess the particle transport data according to the probability density distribution type.

[0023] The sampling module is used to optimize the Monte Carlo particle transport algorithm based on the weighted aliasing algorithm. The optimized Monte Carlo particle transport algorithm is used to calculate the preprocessed particle transport data to obtain the radiation field calculation results of the environment to be analyzed.

[0024] In the above embodiments, preprocessing can be performed according to the probability density distribution type of different particle transport data, and the Monte Carlo particle transport algorithm can be optimized by weighted alias sampling algorithm. This can significantly reduce the time complexity of probability density distribution sampling, improve the efficiency of Monte Carlo particle transport calculation, and quickly obtain the radiation field calculation results of the environment to be analyzed.

[0025] Specifically, the probability density distribution types include discrete distribution types, continuous distribution types, and functional distribution types.

[0026] Because the Monte Carlo method requires generating a large number of random numbers to simulate particle transport processes, and then sampling from these random numbers across various probability density distributions, the efficiency of this sampling directly impacts the efficiency of the Monte Carlo particle transport simulation. Traditional sampling methods generate a random number between 0 and 1, and then compare this random number with the critical point values ​​of the probabilities of each discrete variable to determine which interval it falls into. This algorithm has a time complexity of O(logN), resulting in high time complexity and low efficiency.

[0027] Sampling of probability density distributions is involved in various processes, including source particle sampling, transport length sampling, reaction nuclide and reaction type sampling, post-reaction particle state sampling, and secondary particle state sampling. A weighted alias sampling algorithm improves the sampling methods for different probability density distribution types, thereby enhancing the overall efficiency of Monte Carlo particle transport simulations.

[0028] The following section details the process of optimizing traditional Monte Carlo particle transport for various probability density distribution types.

[0029] The data preprocessing module is specifically used to, when the probability density distribution type is determined to be a discrete distribution type...

[0030] The discrete distribution type is represented as {L1,L2,…,L…} N |P1,P2,…,P N}, where L i P represents the value of a discrete variable. i This represents the probability of a discrete variable taking a value, and

[0031] Multiplying each probability by the number of discrete distribution values ​​N, the processed discrete distribution probabilities are {NP1, NP2, ..., NP...} N};

[0032] The discrete distribution probabilities are assembled into an equally probable binomial distribution. Specifically, if the discrete distribution probability is less than 1.0, the difference in probability is subtracted from the discrete variable values ​​whose discrete distribution probabilities are greater than 1.0 to supplement them to an equally probable 1.0. This process continues until the discrete distribution probabilities corresponding to all discrete variable values ​​are filled to an equally probable 1.0, thus obtaining a new discrete distribution probability array {P′1, P′2, ..., P′}. N} and the corresponding array of filler item positions {t1,t2,…,t N}, where t i =-1 indicates that discrete variable values ​​with an original probability greater than or equal to 1.0 do not need to be filled.

[0033] It should be understood that for each column distribution formed by each discrete term, there are at most two cases that conform to the binomial distribution.

[0034] When the probability density distribution type is a discrete distribution type, the sampling module is specifically used to optimize the sampling method in the Monte Carlo particle transport algorithm based on a weighted aliasing sampling algorithm, specifically as follows:

[0035] Generate a random number ξ1 between 0 and 1, and calculate the corresponding discrete sequence. And round down;

[0036] Generate another random number ξ2 between 0 and 1. When ξ2 < P′ m When, the variable takes the value L m Otherwise, the variable takes the value Then, the radiation field of the environment to be analyzed is calculated.

[0037] Specifically, the data preprocessing module is used to, when the probability density distribution type is determined to be a continuous distribution type...

[0038] The continuous distribution type is represented as {C0, C1, ..., C...} N |0,P1,…,P N}, where C i P represents the boundary value of a continuous variable. i Represents the range of variables (C) i-1 C i The probability of ] taking the value, and

[0039] Multiplying the probability of each value by the number of continuous distribution values ​​N, the processed continuous distribution probability is {NP1, NP2, ..., NP...} N};

[0040] The continuous distribution values ​​are combined into equal-probability binomial distributions. Specifically, if the continuous distribution value probability is less than 1.0, the difference in probability is deducted from the boundary values ​​of continuous variables with a continuous distribution value probability greater than 1.0, and then supplemented to equal probability 1.0. The above operation is repeated until the continuous distribution value probabilities corresponding to all continuous variable boundary values ​​are filled to equal probability 1.0.

[0041] Record the new variable critical point value, specifically:

[0042] Assuming the original probability of a value in column i is greater than 1.0, it is moved to column j, where the original probability is less than 1.0, and column k is used to fill the gaps in column i to bring the probability of a value in column i to 1.0. Then the critical value of the variable in column i is...

[0043]

[0044] This yields a new continuous distribution probability array {P′1, P′2, ..., P′}. N} and the binomial distribution of each column corresponds to the new variable critical point value {t}. 11 ,t 12 ,t 13 ,t 14 ,t 21 ,t 22 ,t 23 ,t 24 …,t N1 ,t N2 ,t N3 ,t N4}, where t i1 ,t i2 ,t i3 ,t i4t represents the upper and lower critical values ​​of the binomial distribution in the i-th column. If the current column has only one term, then t i3 ,t i4 All are -1.

[0045] During this process, the source of the supplementary probability is also recorded.

[0046] When the probability density distribution type is determined to be a continuous distribution type, the sampling module is specifically used to optimize the sampling method in the Monte Carlo particle transport algorithm based on a weighted aliasing sampling algorithm, specifically as follows:

[0047] Generate a random number ξ1 between 0 and 1, and calculate the corresponding column. And round down;

[0048] Generate another random number ξ2 between 0 and 1. When ξ2 < P′ m When, the variable takes the value Otherwise, the variable takes the value Then, the radiation field of the environment to be analyzed is calculated.

[0049] When the probability density distribution type is determined to be a function distribution type, the data preprocessing module is specifically used for:

[0050] Divide the function distribution type into several uniformly distributed continuous spaces within the range of values ​​of the function's independent variable to obtain the distribution function P(x). The distribution function P(x) is equivalent to {C0, C1, ..., C...} N |0,P1,…,P N},in, Represents the range of variables (C) i-1 C i The probability of ] taking the value, and

[0051] When the probability density distribution type is determined to be a function distribution type, the sampling module is specifically used to optimize the sampling method in the Monte Carlo particle transport algorithm based on a weighted aliasing sampling algorithm, specifically as follows:

[0052] Generate a random number ξ1 between 0 and 1, and calculate the corresponding column. And round down;

[0053] Generate another random number ξ2 between 0 and 1. When ξ2 < P′ m When, the variable takes the value Otherwise, the variable takes the value Then, the radiation field of the environment to be analyzed is calculated.

[0054] In the above embodiments, the weighted alias sampling algorithm can significantly reduce the time complexity of probability density distribution sampling and improve the efficiency of Monte Carlo particle transport calculation.

[0055] It should be understood that the particle transport data of the environment to be analyzed is particle transport data in the field of radiation protection or particle transport data in the field of irradiation processing.

[0056] Example 2:

[0057] like Figure 2 As shown, a method for improving the sampling of probability density distribution in Monte Carlo particle transport simulation includes the following steps:

[0058] S1: Import particle transport data of the environment to be analyzed;

[0059] S2: Determine the probability density distribution type of the particle transport data, and preprocess the particle transport data according to the probability density distribution type;

[0060] S3: The weighted aliasing algorithm optimizes the Monte Carlo particle transport algorithm. The preprocessed particle transport data is calculated using the optimized Monte Carlo particle transport algorithm to obtain the radiation field calculation results of the environment to be analyzed.

[0061] Example 3:

[0062] An apparatus for simulating Monte Carlo particle transport using improved probability density distribution sampling includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements a method for simulating Monte Carlo particle transport using improved probability density distribution sampling as described above.

[0063] Example 4:

[0064] A computer-readable storage medium storing a computer program that, when executed by a processor, implements a method for simulating Monte Carlo particle transport using an improved probability density distribution sampling method as described above.

[0065] It should be noted that, in this document, relational terms such as "first" and "second" are used only to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such process, method, article, or apparatus.

[0066] Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the specific working process of the above-described apparatus and unit can be referred to the corresponding process in the foregoing method embodiments, and will not be repeated here.

[0067] In the several embodiments provided in this application, it should be understood that the disclosed apparatus and methods can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative. For instance, the division of units is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed.

[0068] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of the embodiments of the present invention, depending on actual needs.

[0069] Furthermore, the functional units in the various embodiments of the present invention can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit.

[0070] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this invention, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods of the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0071] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An apparatus for simulating Monte Carlo particle transport using improved probability density distribution sampling, characterized in that, include: The import module is used to import particle transport data of the environment to be analyzed, wherein the particle transport data of the environment to be analyzed is particle transport data in the field of radiation protection or particle transport data in the field of irradiation processing. A data preprocessing module is used to determine the probability density distribution type of the particle transport data and preprocess the particle transport data according to the probability density distribution type. The sampling module is used to optimize the Monte Carlo particle transport algorithm using a weighted aliasing sampling algorithm. The Monte Carlo particle transport algorithm involves sampling of probability density distributions in the source particle sampling, transport length sampling, reaction nuclide and reaction type sampling, post-reaction particle state sampling, and secondary particle state sampling processes. The weighted aliasing sampling algorithm is used to optimize the probability density distribution sampling in each process. The preprocessed particle transport data is calculated using the optimized Monte Carlo particle transport algorithm to obtain the radiation field calculation results of the environment to be analyzed. These radiation field calculation results are used for radiation dose control, irradiation process optimization, or emergency rescue decision-making. The data preprocessing module is specifically used to, when the probability density distribution type is determined to be a discrete distribution type... The discrete distribution type is represented as ,in, Represents the values ​​of discrete variables. This represents the probability of a discrete variable taking a value, and ; Multiply the probability of each value by the number of discrete distribution values. N Then the probability of the processed discrete distribution is ; The discrete distribution probabilities are assembled into an equally probable binomial distribution. Specifically, if the discrete distribution probability is less than 1.0, the difference in probability is subtracted from the discrete variable values ​​whose discrete distribution probabilities are greater than 1.0 to supplement them to an equally probable 1.0, until the discrete distribution probabilities corresponding to all discrete variable values ​​are filled to an equally probable 1.0, thus obtaining a new array of discrete distribution probabilities. and the corresponding filler item position array ,in, This indicates that discrete variable values ​​whose original probability is greater than or equal to 1.0 do not need to be filled. The data preprocessing module is further specifically used to, when determining that the probability density distribution type is a continuous distribution type... The continuous distribution type is represented as ,in, Represents the boundary values ​​of continuous variables. Represents the range of variables The probability of taking the value, and ; a. Multiply the probability of each value by the number of continuous distribution values. N That is, the probability of the continuous distribution after processing is ; b. Combine the continuous distribution probabilities into equal-probability binomial distributions. Specifically, if the continuous distribution probability is less than 1.0, subtract the difference in probability from the boundary values ​​of continuous variables with a continuous distribution probability greater than 1.0, and supplement them to equal probability 1.

0. Repeat steps a to b above until the continuous distribution probabilities corresponding to all continuous variable boundary values ​​are filled to equal probability 1.

0. Record the new variable critical point value, specifically: Assume the first i If the original probability of a column's value is greater than 1.0, move it to the column whose original probability is less than 1.

0. j Column, and by the first k Complete the column i If the probability of a column taking a value is 1.0, then the [number]th [column name]... i The critical value of the column variable is ， This yields a new continuous distribution probability array. And the binomial distribution of each column corresponds to the new variable critical point value. ,in, Representing the i The upper and lower limits of the binomial distribution are defined. If the current column contains only one term, then... All are -1, where, i Indicates the index of the column currently being processed. j The column index represents the probability of receiving additional data. k Auxiliary column index representing supplementary probability, Indicates the first i The left and right boundary values ​​of the column. Indicates the first k The left and right boundary values ​​of the column. Indicates the first i The probability values ​​after column preprocessing Indicates the first j The probability values ​​after column preprocessing Indicates the first k The probability values ​​after column preprocessing; The data preprocessing module is further specifically used to, when determining that the probability density distribution type is a function distribution type... Dividing the function distribution type into several uniformly distributed continuous spaces within the range of values ​​of the function's independent variable yields the distribution function. , distribution function Equivalent to ,in, Represents the range of variables The probability of taking the value of represents the distribution function P(x) within the range of the variable. Performing a definite integral within the variable gives the probability that the variable falls within the range of that variable. ,and .

2. The apparatus for simulating Monte Carlo particle transport using improved probability density distribution sampling according to claim 1, characterized in that, The probability density distribution types include discrete distribution types, continuous distribution types, and functional distribution types.

3. The apparatus for simulating Monte Carlo particle transport using improved probability density distribution sampling according to claim 1, characterized in that, When the probability density distribution type is a discrete distribution type, the sampling module is specifically used to optimize the sampling method in the Monte Carlo particle transport algorithm based on a weighted aliasing sampling algorithm, specifically as follows: Generate a random number between 0 and 1 The corresponding discrete columns are calculated as follows: and round down. Indicates a discrete column index. This represents the floor operation; Generate another random number between 0 and 1. ,when When, the variable takes the value Otherwise, the variable takes the value Then, the radiation field calculation results of the environment to be analyzed are obtained. Indicates the number of times after preprocessing. The percentage of equal probability intervals in the column.

4. The apparatus for simulating Monte Carlo particle transport using improved probability density distribution sampling according to claim 1, characterized in that, When the probability density distribution type is determined to be a continuous distribution type, the sampling module is specifically used to optimize the sampling method in the Monte Carlo particle transport algorithm based on a weighted aliasing sampling algorithm, specifically as follows: Generate a random number between 0 and 1 The corresponding column is calculated as follows And round down; Generate another random number between 0 and 1. ,when When, the variable takes the value Otherwise, the variable takes the value Then, the radiation field calculation results of the environment to be analyzed are obtained. Indicates the first List the start and end points of the main interval of the binomial distribution. Indicates the first List the start and end points of the alias interval of the binomial distribution. This represents the sampled value within the alias interval.

5. The apparatus for simulating Monte Carlo particle transport using improved probability density distribution sampling according to claim 1, characterized in that, When the probability density distribution type is determined to be a function distribution type, the sampling module is specifically used to optimize the sampling method in the Monte Carlo particle transport algorithm based on a weighted aliasing sampling algorithm, specifically as follows: Generate a random number between 0 and 1 The corresponding column is calculated as follows And round down; Generate another random number between 0 and 1. ,when When, the variable takes the value Otherwise, the variable takes the value Then, the radiation field of the environment to be analyzed is calculated. Indicates the first List the start and end points of the main interval of the binomial distribution. Indicates the first List the start and end points of the alias interval of the binomial distribution. This represents the sampled value within the alias interval.

6. A method for simulating Monte Carlo particle transport using improved probability density distribution sampling, characterized in that, Includes the following steps: Import particle transport data of the environment to be analyzed, wherein the particle transport data of the environment to be analyzed is particle transport data in the field of radiation protection or particle transport data in the field of irradiation processing; Determine the probability density distribution type of the particle transport data, and preprocess the particle transport data according to the probability density distribution type; A weighted aliasing sampling algorithm is used to optimize the Monte Carlo particle transport algorithm. The Monte Carlo particle transport algorithm involves sampling of probability density distributions in the source particle sampling, transport length sampling, reaction nuclide and reaction type sampling, post-reaction particle state sampling, and secondary particle state sampling processes. The weighted aliasing sampling algorithm is used to optimize the probability density distribution sampling in each process. The preprocessed particle transport data is calculated using the optimized Monte Carlo particle transport algorithm to obtain the radiation field calculation results of the environment to be analyzed. These radiation field calculation results are used for radiation dose control, irradiation process optimization, or emergency rescue decision-making. The data preprocessing module is specifically used to, when the probability density distribution type is determined to be a discrete distribution type, The discrete distribution type is represented as ,in, Represents the values ​​of discrete variables. This represents the probability of a discrete variable taking a value, and ; Multiply the probability of each value by the number of discrete distribution values. N Then the probability of the processed discrete distribution is ; The discrete distribution probabilities are assembled into an equally probable binomial distribution. Specifically, if the discrete distribution probability is less than 1.0, the difference in probability is subtracted from the discrete variable values ​​whose discrete distribution probabilities are greater than 1.0 to supplement them to an equally probable 1.0, until the discrete distribution probabilities corresponding to all discrete variable values ​​are filled to an equally probable 1.0, thus obtaining a new array of discrete distribution probabilities. and the corresponding filler item position array ,in, This indicates that discrete variable values ​​whose original probability is greater than or equal to 1.0 do not need to be filled. The data preprocessing module is further specifically used to, when determining that the probability density distribution type is a continuous distribution type... The continuous distribution type is represented as ,in, Represents the boundary values ​​of continuous variables. Represents the range of variables The probability of taking the value, and ; a. Multiply the probability of each value by the number of continuous distribution values. N That is, the probability of the continuous distribution after processing is ; b. Combine the continuous distribution probabilities into equal-probability binomial distributions. Specifically, if the continuous distribution probability is less than 1.0, subtract the difference in probability from the boundary values ​​of continuous variables with a continuous distribution probability greater than 1.0, and supplement them to equal probability 1.

0. Repeat steps a to b above until the continuous distribution probabilities corresponding to all continuous variable boundary values ​​are filled to equal probability 1.

0. Record the new variable critical point value, specifically: Assume the first i If the original probability of a column's value is greater than 1.0, move it to the column whose original probability is less than 1.

0. j Column, and by the first k Complete the column i If the probability of a column taking a value is 1.0, then the [number]th [column name]... i The critical value of the column variable is ， This yields a new continuous distribution probability array. And the binomial distribution of each column corresponds to the new variable critical point value. ,in, Representing the i The upper and lower limits of the binomial distribution are defined. If the current column contains only one term, then... All are -1, where, i Indicates the index of the column currently being processed. j The column index represents the probability of receiving additional data. k Auxiliary column index representing supplementary probability, Indicates the first i The left and right boundary values ​​of the column. Indicates the first k The left and right boundary values ​​of the column. Indicates the first i The probability values ​​after column preprocessing Indicates the first j The probability values ​​after column preprocessing Indicates the first k The probability values ​​after column preprocessing; The data preprocessing module is further specifically used to, when determining that the probability density distribution type is a function distribution type... Dividing the function distribution type into several uniformly distributed continuous spaces within the range of values ​​of the function's independent variable yields the distribution function. , distribution function Equivalent to ,in, Represents the range of variables The probability of taking the value of represents the distribution function P(x) within the range of the variable. Performing a definite integral within the variable gives the probability that the variable falls within the range of that variable. ,and .

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