Radial basis function neural network learning method for breast cancer discrete feature data classification

Abstract

The present application provides a kind of radial basis function neural network learning method for breast cancer discrete feature data classification, belong to artificial intelligence mode data fitting, identification and classification field.The method of the present application simultaneously determines multiple center points in each iteration process, which greatly speeds up the model training speed, reaches 50-320 times of the traditional OLS method;Because the effectiveness of the center point and width value determined by the method, the size of the model is significantly reduced, only a few center points are used to improve the accuracy of the model;In the case of the same model parameters, the model accuracy obtained by the method will be improved by 3 orders of magnitude;At the same time, since the method is automatically calculated by the base function width value, it can be used in different types of data and applications, without different parameter settings for different applications, simplifying the use difficulty of the method.

Description

Technical Field

[0001] This invention belongs to the field of artificial intelligence (artificial neural network) pattern data fitting, recognition and classification, and specifically relates to a novel learning method of radial basis function (RBF) neural network (NN) for fitting and classifying discrete feature data of breast cancer. Background Technology

[0002] Artificial intelligence-based breast cancer classification can provide a wealth of relevant and referential information, offering significant auxiliary applications and clinical value for diagnosis and subsequent treatment. Discrete feature breast cancer data simplifies data structure and acquisition, reduces data storage space, and accelerates data processing, but it places certain demands on the classification algorithm model. This invention proposes a radial basis function neural network learning algorithm to classify discrete feature data of breast cancer.

[0003] Radial basis function (RBF) neural networks are a class of universal approximators that, under certain conditions, can approximate any continuous function or mapping with a certain accuracy. RBF neural networks are simple in structure, powerful in performance, and have significant application value, finding wide application in various fields, including system identification and modeling, nonlinear system control, and pattern recognition and classification. Models based on RBF neural networks exhibit good generalization ability and serve as the foundation for further implementation of many practical applications and functions.

[0004] The RBF network model has a simple structure, consisting of three layers of neurons: an input layer, hidden layers, and an output layer. The input layer has the same number of nodes as the input data dimension, providing the input data for the entire network. Each neuron in the hidden layer is connected to the input layer, and its activation function is the radial basis function (RBF). The connection coefficient between the hidden layer and the input layer is the center point of the RBF. The output layer provides the network output, which can be multi-dimensional, where each dimension of the output is a linear combination of the outputs of the hidden layer nodes. The RBF network model requires determining appropriate network parameter values ​​based on the training data (including input patterns and target output values) for a specific problem. These parameters include the number, location (center point), and width of the RBF in the hidden layer nodes, as well as the linear connection coefficients between the hidden and output layers. The suitability of these parameter values ​​ultimately determines the performance of the RBF network model in handling a specific problem, such as the accuracy of fitting or classification.

[0005] RBF network parameter training methods have unique characteristics and approaches. Currently, there are various RBF network parameter training methods. Schwenker et al. provided a good and systematic summary of RBF network training methods according to the number of stages required for radial basis function network training, proposing the concepts of one-stage, two-stage, and three-stage learning methods for training RBF networks. One-stage learning trains some or all of the RBF network parameters in a single learning process, such as the linear connection coefficients of the output layer or the parameters of the hidden layers. Specific learning algorithms include Support Vector Learning (SVLM) or Orthogonal Least Squares (OLS) to determine the output layer connection coefficients, or direct computation of pseudo-inverses, etc. When using these methods to train the output layer parameter values ​​within one stage, the parameters of the hidden layers of the network model are directly determined through certain specific methods. One-stage training methods can determine all model parameters through gradient descent or orthogonal decomposition algorithms. Two-stage learning trains the hidden and output layer parameters of the RBF network separately in two separate learning processes. The hidden layer parameters, including the position and width of the radial basis functions, can be obtained through unsupervised clustering methods (such as k-means) or supervised methods (such as Learned Vector Quantization (LVQ) or decision trees), while the basis function width needs to be determined through further statistical calculations after obtaining the center points. Three-stage training methods fine-tune the model parameters obtained in the two-stage training using an algorithm similar to backpropagation. In fact, for any RBF network model, the obtained parameter values ​​can be used as initial values ​​for further optimization and fine-tuning through iterative methods similar to gradient descent.

[0006] In the aforementioned methods for training RBF networks, a common key factor is determining appropriate basis function center points and width values. Extensive research in the field has also shown that this plays a crucial role in improving the accuracy and performance of RBF network models. Current methods for determining center points include uniform sampling or random selection of data points in the input dataset, and determining center points through unsupervised or supervised methods. These methods suffer from poor model accuracy. There are various methods for determining basis function width values, including pre-setting a uniform fixed value, automatically calculating a uniform value based on the dataset, automatically determining a width value for each basis function, automatically determining a width value for each dimension of the basis function, and Mahalanobis distance with absolute shape. However, obtaining the optimal basis function width for better model accuracy remains an important problem to be solved. Regularization methods and orthogonal decomposition methods are also commonly used to improve RBF network performance. Orthogonal decomposition has the advantages of high accuracy and small model size compared to other methods, but it also requires a large amount of memory. In response to the objective needs of improving model accuracy and reducing the computational and memory requirements of the model, how to further optimize the center point position and basis function width is a problem that needs to be addressed and urgently solved in training and using RBF neural network models. Summary of the Invention

[0007] To address the problems existing in the practical application of RBF neural networks, the purpose of this invention is to construct a novel learning method for RBF neural network models, which can quickly obtain high-precision RBF neural network models with smaller sizes, thereby improving the performance of RBF network models in applications.

[0008] The technical solution of the present invention:

[0009] A radial basis function neural network learning method for classifying discrete feature data of breast cancer is described below:

[0010] (I) Constructing a radial basis function neural network

[0011] The RBF network consists of an input layer, hidden layers, and an output layer. The input layer provides the input data; the hidden layers contain multiple radial basis function nodes, mapping the input data space to the feature space (the output of the hidden basis functions); the output layer is a linear model, achieving linear fitting or classification of the feature space data. A Gaussian function φ(x) is used as the radial basis function, and the training dataset D contains N input-output pairs {x...}. i ,y i |i=1,2,…,N}, where x i For m0-dimensional input data, y iFor m2-dimensional output data, the output of the j-th (out of m1, indicating that the hidden layer feature space is m1-dimensional) radial basis (Gaussian) function of the hidden layer is expressed as:

[0012]

[0013] Among them, c j Let ||·|| be the center point of the j-th Gaussian function, where ||·|| represents the second norm (Euclidean distance), i.e., the distance between the i-th input data and the j-th center point, and σ. j Let be the width of the j-th Gaussian function. The output layer can achieve multi-dimensional output; the k-th dimension output can be represented as:

[0014]

[0015] Where, ω jk (1≤j≤m1,1≤k≤m2) are the linear combination coefficients of the output layer linear model. Each dimension of the output layer is connected to the outputs of the m1 hidden layer radial basis functions, for a total of m1m2 connection coefficients.

[0016] definition

[0017]

[0018] The dimension is m2xm1

[0019]

[0020] Y = [y1, y2, ..., y N ] T The dimension is N x m2 (2)

[0021] Then there is

[0022] φΩ T =Y (3)

[0023] Where φ is a matrix formed by the output values ​​of the radial basis functions of the hidden layer, with dimensions N x m1, and its elements are φ. ij , 1≤i≤N, 1≤j≤m1. The task of the RBF network model learning method is to determine all basis functions φ based on the input dataset used for training through a learning method. ij The parameter values ​​and the output layer connectivity coefficient Ω.

[0024] The described radial basis function (RBF) neural network learning method estimates the center points and Gaussian function widths based on the extreme points of the model residuals and their local domain ranges. It iteratively calculates and incrementally determines all center points and widths, ultimately presenting a training scheme for the RBF network. This scheme includes two steps in each iteration: first, updating the RBF network parameters (center points and Gaussian function widths) through a nested second-layer (inner layer) iteration process; second, performing performance evaluation calculations on the updated network. This two-step iterative training process is repeated until the iteration stops meeting a predetermined condition. This condition can be that the model performance reaches a set requirement, or the number of hidden layer nodes (number of basis functions) reaches a preset value.

[0025] (II) The construction of the radial basis function neural network learning method includes the following steps:

[0026] 1. Network performance evaluation: In the nth iteration (n≤1), the center point is first determined through inner nested iterations (details below). The corresponding Gaussian function width is These two parameters determine the radial basis Gaussian function of the hidden layer. Input data X = [x1, x2, ..., x...] N ] T After radial basis function mapping, the output data of the hidden layer is φ. n ,pass (where Y = [y1, y2, ..., y) N ] T Methods such as the expected output value vector and least squares (LS) can be used to determine the estimated values ​​of the coefficients of the linear model in the output layer. This completes the construction of the entire model parameters in the nth iteration (at this point, the model has parameter C). n σ n , After determining all model parameters in the nth iteration, the residual corresponding to the k-th dimension output when the model is input with the i-th data is: (in, The input data x corresponding to the network model i,k,n The total error of all dimensions of the output (the output value) is represented in vector form as: The total error of all data outputs is expressed as: in Model performance is evaluated using the normalized root mean square error (NRMSE), which is the square root of the model error energy as a percentage of the total energy of the input dataset. where ∑ represents the sum over the vector elements by traversing indices i and k. When training the model, the model performance information can be directly obtained through a single NRMSE value.

[0027] 2. Network update criterion: If the normalized root mean square error after the nth iteration is greater than the preset target error threshold NRMSE > ∈ and the number of center points of the model's radial basis function is less than the preset threshold m 1,n < m1, the model needs to perform the (n + 1)th iteration, further add center points and determine the corresponding Gaussian function width values. Then repeat the network performance evaluation calculation process in the above nth iteration to determine whether a further (n + 2)th iteration is required, and continue like this until the iteration condition is no longer satisfied at the end. In the (n + 1)th iteration process, first obtain the residual e n , and determine the center points to be added and the corresponding Gaussian function width values through the method of the second-layer inner nested iteration.

[0028] 3. Inner nested iteration to update network parameters (calculate the position and width of the basis function): Define ‖E n ‖ 2 =[||e 1,n || 2 ,||e 2,n || 2 ,…,||e N,n || 2 T , the preset threshold (such as ), in actual operation, the elastic setting of the threshold can be achieved by changing the proportional coefficient (i.e., 0.1 in the previous formula) to adapt to the situation where the residual gradually becomes smaller and it is difficult to find effective center points. Then the inner iteration first divides the input data into two parts according to the residual norm vector ‖E n ‖ 2 and the preset threshold :

[0029] 1) The non-zero residual part for solving the local domain center point (residual extreme point), corresponding to the input data point X ={x nz 丨i = 1,2,…,N} (including the center point and neighborhood points); nz,i

[0030] 2) The part for providing zero error value points, corresponding to the input data point X ={x z 丨i = 1,2,…,N} (including the boundary points of the neighborhood range and the points with low fitting error). z,i

[0031] X nz ​Let x be the input data point corresponding to the maximum residual value. max By calculating x max With X z The distance d between mz It can be determined that the distance from x max The most recent residual norm ||e n || 2 Zero point x z x max With x z The corresponding distance is d mz,min x max and d mz,min This refers to the newly added center point and its corresponding Gaussian function width. To improve computational speed, model performance, and the efficiency of adding center points, a second (inner) nested layer can be selected to iteratively add center points (the maximum number of nested iterations can be limited, i.e., the maximum number of center points added in each outer iteration; generally, the maximum number of center points preset for the entire method can be used). The specific process is as follows: calculate x... max and X nz The distance d between mm , will X nz The middle has d mm <d mz,min After removing the input data points, a new dataset point X is obtained. nz and its corresponding residual norm ||e n || 2 The new X nz Maximum point x max As a further added center point; by repeating the above process, the new corresponding Gaussian function width value d can be determined. mz,min This process can be iterated and repeated until the entire original X has been traversed. nz ={x nz,i The method identifies local residual maxima in the interval |i = 1, 2, ..., N}, or iterates only a specific number of times (adding only a specific number of center points) according to given constraints. This allows for the addition of multiple center points and the determination of the corresponding Gaussian function width during the (n+1)th iteration. The iterative calculation continues until the result satisfies the preset condition, i.e., until NRMSE ≤ ∈ or m 1,n ≥m1, and finally obtain the model parameter center point C and the width σ of the Gaussian radial basis function.

[0032] To obtain a more effective model, reduce model performance error, decrease the number of radial basis functions, and improve model generalization ability, this modeling method further adjusts σ, ultimately adopting λσ as the width value of the Gaussian radial basis functions. The value of the width adjustment factor λ is closely related to the number of centroids. When the number of centroids is sufficient, a larger λ value will achieve better modeling results; insufficient centroids or a small λ value will lead to reduced model performance and relatively large errors. To strike a trade-off between model performance and the number of centroids, an appropriate λ value can be selected based on the expected model performance error (normalized root mean square error, NRMSE). Generally, λ = 5 to 6 will achieve very good model performance. If a smaller model size is desired (i.e., fewer centroids and radial basis functions), a suitable λ value between 1 and 5 can even be used.

[0033] The beneficial effects of this invention are that it constructs a novel learning method for radial basis function neural networks, which can achieve tasks such as fitting and classifying specific input data. Compared with existing methods (OLS), this method has several significant advantages when training the radial basis function neural network model, including faster training speed, smaller model size, higher model accuracy, simplicity (no need to set internal key parameters), and automatic calculation of internal key parameters, thus making it applicable to different types of datasets. Specifically, because this invention determines multiple centroids simultaneously in each iteration, it greatly accelerates the model training speed, reaching 50-320 times that of traditional OLS methods; due to the effectiveness of the centroids and width values ​​determined by this method, the model size is significantly reduced, using only a small number (half) of centroids to achieve improved model accuracy; with the same model parameters, the accuracy obtained by this method is improved by three orders of magnitude; furthermore, because this method automatically calculates the basis function width values, it can be used for different types of data and applications, eliminating the need for different parameter settings for different applications and simplifying the method's usage. Attached Figure Description

[0034] Figure 1 This is a flowchart of the radial basis function neural network learning method of the present invention;

[0035] Figure 2 This is a schematic diagram illustrating how the parameter values ​​of the radial basis function neural network learning method of the present invention change according to the input data during the model training process;

[0036] Figure 3This section presents the output boundary fitting curve of a radial basis function neural network (RBF) for classifying one-dimensional discrete feature data of breast cancer. The horizontal axis represents the discrete feature values, and the vertical axis represents the category to which the feature belongs. Dark gray dots (y=0) represent tumors of category 0 (benign), and light gray dots (y=1) represent tumors of category 1 (malignant). The size of the dots represents the number of discrete feature data samples. The solid curve represents the output boundary fitting curve of the RBF neural network obtained after training based on the input training dataset, i.e., the dark gray and light gray discrete data points. Based on this curve and its distance from 0 or 1, the category corresponding to any discrete input feature data can be determined. The black square represents the center point of the trained RBF neural network model.

[0037] Figure 4 Example of classification results for 9-dimensional discrete feature data of breast cancer; Detailed Implementation

[0038] To facilitate understanding by those skilled in the art, the present invention will be further described below with reference to embodiments and accompanying drawings. The content mentioned in the embodiments is not intended to limit the present invention.

[0039] This invention relates to a radial basis function neural network model learning method, the specific implementation process of which is as follows: Figure 1 As shown, the implementation process, its underlying principles, and the corresponding data are as follows: Figure 2 As shown. In a specific implementation case involving discrete feature data, the implementation method includes the following steps:

[0040] 1. A training dataset for the network model is formed from the acquired discrete pattern feature data (both input patterns and output targets are discrete). Because it is discrete, there is some overlap in the discrete feature data across different categories; that is, the same input feature data corresponds to different categories with different probabilities. Here, we take a breast cancer dataset provided by the University of Wisconsin (http: / / archive.ics.uci.edu / ml / datasets / Breast+Cancer+Wisconsin+%28Original%29) as an example. This dataset calculates nine features of breast tumor cell morphology and provides a benign or malignant label (0 or 1). The nine features include mass thickness, cell size uniformity, cell shape uniformity, edge adhesion, size of a single epithelial cell, naked nucleus, pale chromatin, normal nucleoli, and mitosis. Each sample feature takes an integer value between 1 and 10. These discrete data constitute the feature distribution of benign or malignant breast cancer. The input discrete feature data of breast tumors constitute the network input dataset X, and the benign or malignant label of the tumor constitutes the network output Y. The distribution of the one-dimensional data is shown below. Figure 3 As shown.

[0041] 2. Standardize the data in the dataset before classification, and standardize both the feature data and the label data to between 0 and 1.

[0042] 3. After determining the discrete feature training data X and Y, perform the first outer iteration calculation process of the radial basis function neural network learning method. In the n = 1 loop, first calculate Y0 corresponding to the input data X. The input-output correspondence can be obtained according to linear least squares or mean estimation or simply set Y0 = 0; initialize the residual square and the standardized root mean square

[0043] 4. In the nth (n > 1) first outer iteration, based on ‖e n-1 ‖ 2 , obtain the updated network center point C n and the Gaussian radial basis function width value λσ n (where λ is the width adjustment factor, which is preset to a suitable value according to the expected model scale and performance, and is set to 3 in the method); the process of updating the network center point C n and the Gaussian function width value λσ n is as follows:

[0044] 4.1 Divide the input dataset X into n ‖ 2 and according to the residual norm ‖e and At the same time, update ‖e n ‖ 2 ←‖e n ‖ 2 丨x∈X nz ;

[0045] 4.2 Determine the input data point x n ‖ 2 with the maximum residual value according to ‖e nz and X max = arg x max(‖e n ‖ 2 );

[0046] 4.3 Calculate the distance d max between x z and X mz ={d mz,i =|x max -x z,i |丨i = 1,2,…,N}, and its minimum distance is d mz,min = min(d mz );

[0047] 4.4 If x max is not the center point of the existing model, then x max and λd mz,min are a newly added center point c n,p and the corresponding Gaussian radial basis function width λσ n,p , p = 1, 2, … P n ;

[0048] 4.5 Calculate the distance d max between x nz and X mm ={d mm,i =|x max -x nz,i | | i = 1, 2, …, N}, remove the processed residual extreme points and their neighborhoods in X nz , update X nz ←arg x (d mm >d mz,min ) and its corresponding residual ‖e n ‖ 2 ←‖e n ‖ 2 | x ∈ X nz ;

[0049] 4.6 If the number of added center points (P n ) is less than the preset maximum number added each time (generally, the maximum number of model center points preset for the whole method can be used) and X nz is not empty, then jump to step 4.2 to continue adding center points; otherwise, this inner loop process ends.

[0050] 5. Based on the center point C n , width λσ n , input data X, calculate the output value φ n of the hidden layer basis function, and then use and the least squares method to determine the output layer coefficient estimate value

[0051] 6. Calculate the residual square of the radial basis function neural network model and where

[0052] 7. If the normalized root mean square error NRMSE ≥ ∈ (set according to the problem requirements, set to 0 in this example) and m 1,nIf m1, jump to step 4 to continue the outer iteration process; otherwise, a model that meets the error or hidden layer node number requirements has been obtained, and go to step 8;

[0053] 8. Obtain the center point C of the model parameters and the width λσ of the Gaussian radial basis function, and the method completes the calculation of the model parameters.

[0054] 9. According to the obtained model parameters C, λσ and the estimated value of the output layer coefficients The output value of any input data can be calculated. The performance of the model center point and the classification boundary curve on one of the one-dimensional coordinate axes is as Figure 3 shown.

[0055] 10. Determine the category corresponding to the input pattern data by calculating the output value of the RBF neural network model and finding the category value with the smallest distance from it. The classification result is as Figure 4 shown.

[0056] The specific application ways of the present invention are numerous. The above description is only the preferred embodiment of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements can be made, and these improvements should also be regarded as the protection scope of the present invention.

Claims

1. A radial basis function neural network learning method for classifying discrete feature data of breast cancer, characterized in that, The specific implementation process is as follows: (a) Constructing a radial basis function neural network The RBF network consists of an input layer, hidden layers, and an output layer; the input layer provides the input data; the hidden layer contains multiple radial basis function nodes, which realize the mapping from the input data space to the feature space; The output layer is a linear model, achieving linear fitting or classification of the feature space data; a Gaussian function φ(x) is used as the radial basis function, and the training dataset D contains N input-output data pairs. ,in Input data for dimension m0, For m²-dimensional output data, the output of the j-th radial basis function of the hidden layer is represented as: (1) in, Let j be the center point of the Gaussian function. Here, σ represents the second norm, which is the distance between the i-th input data point and the j-th center point. j The width of the j-th Gaussian function; the output layer implements multi-dimensional output, and the k-th dimension output is represented as: in, The linear combination coefficients of the output layer linear model. Each dimension of the output layer is connected to the outputs of the m1 hidden layer radial basis functions, with a total of m1m2 connection coefficients; definition (2) Then there is (3) in The matrix formed by the output values ​​of the radial basis functions of the hidden layer has dimensions N x m1, where the elements are... , The task of the RBF network model learning method is to determine all basis functions based on the input dataset used for training through a learning method. Parameter values ​​and output layer connectivity coefficients The value; The radial basis function (RBF) neural network learning method estimates the center point and Gaussian function width based on the extreme points of the model residuals and their local domain ranges. It iteratively calculates and incrementally determines all center points and width values, ultimately providing a training scheme for the RBF network. This scheme includes two steps in each iteration: first, updating the RBF network parameters (center point and Gaussian function width) through a nested second-layer iteration process; second, performing performance evaluation calculations on the updated network. This two-step iterative training process is repeated until the iteration stops meeting a predetermined condition. Specifically, the iteration stops when the model performance reaches a set requirement or the number of hidden layer nodes reaches a preset value. (II) The construction of the radial basis function neural network learning method includes the following steps:

1. Network performance evaluation: In the nth iteration, First, the center point is determined through inner nested iterations. The corresponding Gaussian function width is These two parameters determine the radial basis Gaussian function of the hidden layer; input data The output data of the hidden layer is obtained after radial basis function mapping. ,pass The least squares method is used to determine the estimated coefficients of the linear model of the output layer. This completes the construction of the entire model parameters in the nth iteration, at which point the model has parameters. , , After determining all model parameters in the nth iteration, the residual corresponding to the k-th dimension output when the model is input with the i-th data is: ,in, Input data for the network model The output value, and the total error of all dimensions, are represented in vector form as: The total error of all data outputs is expressed as ,in Model performance is evaluated using the normalized root mean square error, which is the square root of the percentage of model error energy to the total energy of the input dataset. , where ∑ represents summing over the indices i and k of the vector elements; when training the model, model performance information can be obtained directly through a single NRMSE value; 2. Network update criterion: If the normalized root mean square error after the nth iteration is greater than the preset target error threshold and the number of center points of the model's radial basis function is less than the preset threshold m 1,n <m1, the model needs to perform the (n + 1)th iteration, further add center points and determine the corresponding Gaussian function width values; then repeat the network performance evaluation calculation process in the above nth iteration to determine whether a further (n + 2)th iteration is required, and continue like this until the iteration condition is no longer satisfied at the end; in the (n + 1)th iteration process, first obtain the residual based on the result of the nth iteration , and determine the center points to be added and the corresponding Gaussian function width values through the method of the second - layer inner nested iteration; 3. Inner nested iterative update of network parameters: Definition Preset threshold In practice, the threshold is set flexibly by changing the scaling factor to adapt to situations where the residual gradually decreases and it becomes difficult to find an effective center point; the inner iteration first relies on the residual norm vector. and preset threshold The input data is divided into two parts: 1) Used to solve for the center point of a local domain The non-zero residual part corresponds to the input data point ; 2) Used to provide zero error value points Part, corresponding to input data points ; The input data point corresponding to the maximum residual value is denoted as Through calculation and Distance between Determine the distance Recent residual norm Zero point , and The corresponding distance is ; and This involves adding a new center point and its corresponding Gaussian function width; selecting a second level of nesting and iteratively adding center points in a loop; specifically, calculating... and Distance between ,Will It has After removing the input data points, a new dataset is obtained. and its corresponding residual norm new Maximum point As a further added center point; the new corresponding Gaussian function width value is determined by repeating the above process. This process continues iteratively until the entire original sequence has been traversed. The method identifies local residual maxima points, or, based on provided constraints, iterates only a specific number of times, i.e., adds only a specific number of center points, thereby completing the addition of multiple center points and determining the corresponding Gaussian function width value during the (n+1)th iteration; the entire iterative calculation continues until the calculation result meets the preset conditions, i.e., until... or Finally, the model parameter center point 𝐂 and the width of the Gaussian radial basis function are obtained. ; Before using the radial basis function neural network, λ was further adjusted, and finally λ was adopted as the width value of the Gaussian radial basis function; where an appropriate value of λ was selected according to the expected model performance error. The dataset calculated nine features of breast tumor cell morphology and provided a benign or malignant label for the tumor, including 0 or 1. The nine features include mass thickness, cell size uniformity, cell shape uniformity, edge adhesion, size of a single epithelial cell, naked nucleus, pale chromatin, normal nucleolus, and mitosis. Each feature value is an integer in the range of 1-10. These discrete data constitute the feature distribution of benign or malignant breast cancer.

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