Novel Mathematical Model of Breast Cancer Diagnostics Using an Associative Pattern Classification
Abstract
:1. Introduction
2. Theoretical Description
2.1. Associative Memories
2.2. Associative Classification of Patterns
- A fundamental set of associations:
- The class to which each input pattern belongs is defined as:
- Compute the average vector as
- Translate all the patterns of the fundamental set with respect to the mean vector as
- Build matrix as
- Translate asPerform the following product
- Compute the components of class vector as
2.3. Numerical Example
- Computation of the average vector is
- Translation of the input patterns is
- Construction of matrix M is
- Translation of the vector is
- The product is gotten by
- Computation of the class vector is
- The index class of vector x is found according to the above discussion. Vector should be classified into class number one.
- Translation of the vector is
- The product is gotten by
- Computation of the class vector is
- The index of the class of vector x is found according to the above discussion. Vector should be classified into class number one.
- Learning and translation of the two input vectors provokes that they become the negative of each other. This is . Due to the fact that the output vectors for the two classes are orthogonal, matrix M will be composed of and its negative. This is . Note that between the two vectors there is a neutral position; this corresponds to vector .
- Classification of a non-distorted version of any of the input vectors’ translation provokes that it is first transformed to its translated original version. Multiplication of the association matrix M will always give a maximum value at the index class of the input vector.
- Classification of a distorted version of any of the input vectors’ translation provokes that it be first moved to one of the translated original versions. The moved vector could appear on one side or the other side of its corresponding translated original version. While the noise added to the input vector does not cause that its translated version does not surpasses the neutral position, the input vector will always be correctly classified. Of course, if translation of the input vector produces , then the class of the vector cannot be found because .
2.4. Wisconsin Breast Cancer Database
2.5. Minimum Distance Classifier
2.6. Naïve Bayes
2.7. K-Nearest Neighbor Classifier
2.8. Back-Propagation
2.9. Support Vector Machine (SVM)
2.10. C4.5
2.11. Comparison
3. Experimental Results
4. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
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# | Description | Type | Range of Values | Mean | Standard Deviation |
---|---|---|---|---|---|
1 | Clump thickness | Numeric | 1–10 | 4.418 | 2.816 |
2 | Uniformity of cell size | Numeric | 1–10 | 3.134 | 3.051 |
3 | Uniformity of cell shapes | Numeric | 1–10 | 3.207 | 2.972 |
4 | Marginal adhesion | Numeric | 1–10 | 2.807 | 2.855 |
5 | Single epithelial cell size | Numeric | 1–10 | 3.216 | 2.214 |
6 | Bare nuclei | Numeric | 1–10 | 3.545 | 3.644 |
7 | Bland chromatin | Numeric | 1–10 | 3.438 | 2.438 |
8 | Normal nucleoli | Numeric | 1–10 | 2.867 | 3.054 |
9 | Mitoses | Numeric | 1–10 | 1.589 | 1.715 |
10 | Class | Nominal | Benign, Malignant | ||
Class Distribution | Benign: 458 (65.5%) Malignant: 241 (34.5%) | ||||
Number of missing values | 16 | ||||
Number of instances | 699 |
Training-Test | Classifier | ||||||||
---|---|---|---|---|---|---|---|---|---|
ACP | MDC | NB | 1-NN | 2-NN | 3-NN | BP | SVM | C4.5 | |
1%–99% | 96.39 | 94.33 | 92.23 | 95.44 | 88.78 | 91.25 | 66.44 | 94.00 | 93.87 |
10%–90% | 97.16 | 96.06 | 95.60 | 94.96 | 93.52 | 95.69 | 95.52 | 96.20 | 91.93 |
30%–70% | 97.14 | 95.98 | 96.06 | 95.44 | 94.25 | 96.27 | 96.08 | 96.56 | 93.87 |
50%–50% | 97.31 | 96.05 | 96.15 | 95.62 | 94.84 | 96.55 | 96.37 | 96.75 | 94.32 |
70%–30% | 97.31 | 96.31 | 96.36 | 95.65 | 95.12 | 96.84 | 96.59 | 96.93 | 94.68 |
Validation | Classifier | ||||||||
---|---|---|---|---|---|---|---|---|---|
ACP | MDC | NB | 1-NN | 2-NN | 3-NN | BP | SVM | C4.5 | |
10-Folds | 97.13 | 95.91 | 96.07 | 95.28 | 94.81 | 96.60 | 96.40 | 96.62 | 95.01 |
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Santiago-Montero, R.; Sossa, H.; Gutiérrez-Hernández, D.A.; Zamudio, V.; Hernández-Bautista, I.; Valadez-Godínez, S. Novel Mathematical Model of Breast Cancer Diagnostics Using an Associative Pattern Classification. Diagnostics 2020, 10, 136. https://doi.org/10.3390/diagnostics10030136
Santiago-Montero R, Sossa H, Gutiérrez-Hernández DA, Zamudio V, Hernández-Bautista I, Valadez-Godínez S. Novel Mathematical Model of Breast Cancer Diagnostics Using an Associative Pattern Classification. Diagnostics. 2020; 10(3):136. https://doi.org/10.3390/diagnostics10030136
Chicago/Turabian StyleSantiago-Montero, Raúl, Humberto Sossa, David A. Gutiérrez-Hernández, Víctor Zamudio, Ignacio Hernández-Bautista, and Sergio Valadez-Godínez. 2020. "Novel Mathematical Model of Breast Cancer Diagnostics Using an Associative Pattern Classification" Diagnostics 10, no. 3: 136. https://doi.org/10.3390/diagnostics10030136