#
Homogenous Granulation and Its Epsilon Variant^{ †}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Granular Rough Inclusions

#### 2.1. $\epsilon $–Modification of the Standard Rough Inclusion

#### 2.2. Covering of Universe of Training Objects

#### 2.3. Granular Reflections

## 3. Homogenous Granulation

#### 3.1. Simple Example of Homogenous Granulation

- ${g}_{0.385}\left({u}_{1}\right)=({u}_{1},{u}_{6},{u}_{10},{u}_{11},{u}_{12},{u}_{18},{u}_{20}),$
- ${g}_{0.462}\left({u}_{2}\right)=({u}_{2},{u}_{3},{u}_{4},{u}_{5},{u}_{9},{u}_{23}),$
- ${g}_{0.539}\left({u}_{3}\right)=({u}_{2},{u}_{3},{u}_{5}),$
- ${g}_{0.615}\left({u}_{4}\right)=\left({u}_{4}\right),$
- ${g}_{0.539}\left({u}_{5}\right)=({u}_{3},{u}_{5},{u}_{21},{u}_{23}),$
- ${g}_{0.462}\left({u}_{6}\right)=({u}_{4},{u}_{6},{u}_{16},{u}_{20},{u}_{21}),$
- ${g}_{0.539}\left({u}_{7}\right)=({u}_{7},{u}_{15},{u}_{17}),$
- ${g}_{0.462}\left({u}_{8}\right)=({u}_{7},{u}_{8},{u}_{13}),$
- ${g}_{0.462}\left({u}_{9}\right)=({u}_{2},{u}_{4},{u}_{9}),$
- ${g}_{0.615}\left({u}_{10}\right)=\left({u}_{10}\right),$
- ${g}_{0.385}\left({u}_{11}\right)=({u}_{1},{u}_{6},{u}_{11},{u}_{12},{u}_{20}),$
- ${g}_{0.385}\left({u}_{12}\right)=({u}_{1},{u}_{11},{u}_{12},{u}_{18},{u}_{20}),$
- ${g}_{0.615}\left({u}_{13}\right)=\left({u}_{13}\right),$
- ${g}_{0.385}\left({u}_{14}\right)=({u}_{14},{u}_{15},{u}_{24}),$
- ${g}_{0.615}\left({u}_{15}\right)=\left({u}_{15}\right),$
- ${g}_{0.539}\left({u}_{16}\right)=\left({u}_{16}\right),$
- ${g}_{0.539}\left({u}_{17}\right)=({u}_{7},{u}_{15},{u}_{17}),$
- ${g}_{0.389}\left({u}_{18}\right)=({u}_{1},{u}_{2},{u}_{6},{u}_{10},{u}_{12},{u}_{18},{u}_{20},{u}_{21},{u}_{23}),$
- ${g}_{0.615}\left({u}_{19}\right)=\left({u}_{19}\right),$
- ${g}_{0.462}\left({u}_{20}\right)=({u}_{1},{u}_{6},{u}_{11},{u}_{12},{u}_{18},{u}_{20}),$
- ${g}_{0.462}\left({u}_{21}\right)=({u}_{3},{u}_{5},{u}_{6},{u}_{21},{u}_{23}),$
- ${g}_{0.615}\left({u}_{22}\right)=\left({u}_{22}\right),$
- ${g}_{0.462}\left({u}_{23}\right)=({u}_{2},{u}_{3},{u}_{5},{u}_{21},{u}_{23}),$
- ${g}_{0.462}\left({u}_{24}\right)=({u}_{7},{u}_{15},{u}_{24}),$

- ${g}_{0.462}\left({u}_{2}\right)=({u}_{2},{u}_{3},{u}_{4},{u}_{5},{u}_{9},{u}_{23}),$
- ${g}_{0.539}\left({u}_{3}\right)=({u}_{2},{u}_{3},{u}_{5}),$
- ${g}_{0.462}\left({u}_{6}\right)=({u}_{4},{u}_{6},{u}_{16},{u}_{20},{u}_{21}),$
- ${g}_{0.462}\left({u}_{8}\right)=({u}_{7},{u}_{8},{u}_{13}),$
- ${g}_{0.385}\left({u}_{12}\right)=({u}_{1},{u}_{11},{u}_{12},{u}_{18},{u}_{20}),$
- ${g}_{0.385}\left({u}_{14}\right)=({u}_{14},{u}_{15},{u}_{24}),$
- ${g}_{0.539}\left({u}_{17}\right)=({u}_{7},{u}_{15},{u}_{17}),$
- ${g}_{0.385}\left({u}_{18}\right)=({u}_{1},{u}_{2},{u}_{6},{u}_{10},{u}_{12},{u}_{18},{u}_{20},{u}_{21},{u}_{23}),$
- ${g}_{0.615}\left({u}_{19}\right)=\left({u}_{19}\right),$
- ${g}_{0.462}\left({u}_{21}\right)=({u}_{3},{u}_{5},{u}_{6},{u}_{21},{u}_{23}),$
- ${g}_{0.615}\left({u}_{22}\right)=\left({u}_{22}\right),$

## 4. Epsilon Variant of Homogenous Granulation

- ${g}_{0.571429}\left({u}_{1}\right)=\left({u}_{1}\right),$
- ${g}_{0.5}\left({u}_{2}\right)=({u}_{2},{u}_{4},{u}_{15},{u}_{21}),$
- ${g}_{0.571429}\left({u}_{3}\right)=({u}_{3},{u}_{9},{u}_{19},{u}_{20}),$
- ${g}_{0.5}\left({u}_{4}\right)=({u}_{1},{u}_{2},{u}_{4},{u}_{6},{u}_{21}),$
- ${g}_{0.5}\left({u}_{5}\right)=({u}_{5},{u}_{10},{u}_{19},{u}_{24}),$
- ${g}_{0.5}\left({u}_{6}\right)=({u}_{1},{u}_{4},{u}_{6}),$
- ${g}_{0.5}\left({u}_{7}\right)=\left({u}_{7}\right),$
- ${g}_{0.5}\left({u}_{8}\right)=({u}_{8},{u}_{9},{u}_{11},{u}_{17}),$
- ${g}_{0.642857}\left({u}_{9}\right)=({u}_{9},{u}_{10},{u}_{11},{u}_{17},{u}_{19},{u}_{20}),$
- ${g}_{0.642857}\left({u}_{10}\right)=({u}_{9},{u}_{10},{u}_{19}),$
- ${g}_{0.642857}\left({u}_{11}\right)=({u}_{9},{u}_{11},{u}_{17},{u}_{19},{u}_{20}),$
- ${g}_{0.642857}\left({u}_{12}\right)=\left({u}_{12}\right),$
- ${g}_{0.571429}\left({u}_{13}\right)=\left({u}_{13}\right),$
- ${g}_{0.428571}\left({u}_{14}\right)=({u}_{2},{u}_{14},{u}_{16},{u}_{21}),$
- ${g}_{0.5}\left({u}_{15}\right)=({u}_{2},{u}_{12},{u}_{15},{u}_{21}),$
- ${g}_{0.5}\left({u}_{16}\right)=({u}_{1},{u}_{14},{u}_{16}),$
- ${g}_{0.642857}\left({u}_{17}\right)=({u}_{9},{u}_{11},{u}_{17},{u}_{20}),$
- ${g}_{0.642857}\left({u}_{18}\right)=\left({u}_{18}\right),$
- ${g}_{0.571429}\left({u}_{19}\right)=({u}_{3},{u}_{9},{u}_{10},{u}_{11},{u}_{17},{u}_{19},{u}_{20},{u}_{24}),$
- ${g}_{0.642857}\left({u}_{20}\right)=({u}_{9},{u}_{11},{u}_{17},{u}_{19},{u}_{20}),$
- ${g}_{0.5}\left({u}_{21}\right)=({u}_{2},{u}_{4},{u}_{14},{u}_{15},{u}_{21}),$
- ${g}_{0.642857}\left({u}_{22}\right)=\left({u}_{22}\right),$
- ${g}_{0.642857}\left({u}_{23}\right)=\left({u}_{23}\right),$
- ${g}_{0.642857}\left({u}_{24}\right)=\left({u}_{24}\right),$

- Covering granules: ${g}_{0.5}\left({u}_{2}\right)=({u}_{2},{u}_{4},{u}_{15},{u}_{21}),$
- ${g}_{0.571429}\left({u}_{3}\right)=({u}_{3},{u}_{9},{u}_{19},{u}_{20}),$
- ${g}_{0.5}\left({u}_{5}\right)=({u}_{5},{u}_{10},{u}_{19},{u}_{24}),$
- ${g}_{0.5}\left({u}_{6}\right)=({u}_{1},{u}_{4},{u}_{6}),$
- ${g}_{0.5}\left({u}_{7}\right)=\left({u}_{7}\right),$
- ${g}_{0.5}\left({u}_{8}\right)=({u}_{8},{u}_{9},{u}_{11},{u}_{17}),$
- ${g}_{0.642857}\left({u}_{12}\right)=\left({u}_{12}\right),$
- ${g}_{0.571429}\left({u}_{13}\right)=\left({u}_{13}\right),$
- ${g}_{0.5}\left({u}_{16}\right)=({u}_{1},{u}_{14},{u}_{16}),$
- ${g}_{0.642857}\left({u}_{18}\right)=\left({u}_{18}\right),$
- ${g}_{0.642857}\left({u}_{20}\right)=({u}_{9},{u}_{11},{u}_{17},{u}_{19},{u}_{20}),$
- ${g}_{0.5}\left({u}_{21}\right)=({u}_{2},{u}_{4},{u}_{14},{u}_{15},{u}_{21}),$
- ${g}_{0.642857}\left({u}_{22}\right)=\left({u}_{22}\right),$
- ${g}_{0.642857}\left({u}_{23}\right)=\left({u}_{23}\right),$

## 5. Description of Classifier Used for Evaluation of the Granulation

- Step 1.
- The training granular decision system $({G}_{{r}_{gran}}^{trn},A,d)$ and the test decision system $({U}_{tst},A,d)$ are given, where A is a set of conditional attributes, d is the decision attribute, and ${r}_{gran}$ a granulation radius.
- Step 2.
- Classification of test objects, by means of granules of training objects, is performed as follows.

#### Parameter Estimation in $kNN$ Classifier

## 6. The Results of Experiments

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Simple demonstration of granulation for objects represented by the pairs of attributes. In the picture we have objects of two classes, circles and triangles. Granulating the decision system in homogenous way we can obtain ${g}_{0.5}\left(ob1\right)=\{ob1,ob5\}$, ${g}_{1}\left(ob2\right)=\{ob2\}$, ${g}_{0.5}\left(ob3\right)=\{ob3\}$, ${g}_{1}\left(ob4\right)=\{ob4\}$, ${g}_{0.5}\left(ob1\right)=\{ob5,ob1\}$. The set of possible radii is $\{\frac{0}{2},\frac{1}{2},\frac{2}{2}\}$.

**Figure 2.**Exemplary toy demonstration for objects represented as pairs of attributes. We have two decision concepts: circles and rectangles. Epsilon homogenous granules can be ${g}_{0.5}^{\epsilon}\left(ob1\right)=\{ob1,ob5\}$, ${g}_{1}^{\epsilon}\left(ob2\right)=\{ob2\}$, ${g}_{0.5}^{\epsilon}\left(ob3\right)=\{ob3\}$, ${g}_{1}^{\epsilon}\left(ob4\right)=\{ob4\}$, ${g}_{0.5}^{\epsilon}\left(ob1\right)=\{ob5,ob1\}$. The set of possible radii is $\{\frac{0}{2},\frac{1}{2},\frac{2}{2}\}$. The descriptors can be shifted in the range determined by $\epsilon $ and still were treated as indiscernible.

${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ | ${\mathit{b}}_{3}$ | ${\mathit{b}}_{4}$ | ${\mathit{b}}_{5}$ | ${\mathit{b}}_{6}$ | ${\mathit{b}}_{7}$ | ${\mathit{b}}_{8}$ | ${\mathit{b}}_{9}$ | ${\mathit{b}}_{10}$ | ${\mathit{b}}_{11}$ | ${\mathit{b}}_{12}$ | ${\mathit{b}}_{13}$ | d | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${u}_{1}$ | $74.0$ | $0.0$ | $2.0$ | $120.0$ | $269.0$ | $0.0$ | $2.0$ | $121.0$ | $1.0$ | $0.2$ | $1.0$ | $1.0$ | $3.0$ | 1 |

${u}_{2}$ | $65.0$ | $1.0$ | $4.0$ | $120.0$ | $177.0$ | $0.0$ | $0.0$ | $140.0$ | $0.0$ | $0.4$ | $1.0$ | $0.0$ | $7.0$ | 1 |

${u}_{3}$ | $59.0$ | $1.0$ | $4.0$ | $135.0$ | $234.0$ | $0.0$ | $0.0$ | $161.0$ | $0.0$ | $0.5$ | $2.0$ | $0.0$ | $7.0$ | 1 |

${u}_{4}$ | $53.0$ | $1.0$ | $4.0$ | $142.0$ | $226.0$ | $0.0$ | $2.0$ | $111.0$ | $1.0$ | $0.0$ | $1.0$ | $0.0$ | $7.0$ | 1 |

${u}_{5}$ | $43.0$ | $1.0$ | $4.0$ | $115.0$ | $303.0$ | $0.0$ | $0.0$ | $181.0$ | $0.0$ | $1.2$ | $2.0$ | $0.0$ | $3.0$ | 1 |

${u}_{6}$ | $46.0$ | $0.0$ | $4.0$ | $138.0$ | $243.0$ | $0.0$ | $2.0$ | $152.0$ | $1.0$ | $0.0$ | $2.0$ | $0.0$ | $3.0$ | 1 |

${u}_{7}$ | $60.0$ | $1.0$ | $4.0$ | $140.0$ | $293.0$ | $0.0$ | $2.0$ | $170.0$ | $0.0$ | $1.2$ | $2.0$ | $2.0$ | $7.0$ | 2 |

${u}_{8}$ | $63.0$ | $0.0$ | $4.0$ | $150.0$ | $407.0$ | $0.0$ | $2.0$ | $154.0$ | $0.0$ | $4.0$ | $2.0$ | $3.0$ | $7.0$ | 2 |

${u}_{9}$ | $40.0$ | $1.0$ | $1.0$ | $140.0$ | $199.0$ | $0.0$ | $0.0$ | $178.0$ | $1.0$ | $1.4$ | $1.0$ | $0.0$ | $7.0$ | 1 |

${u}_{10}$ | $48.0$ | $1.0$ | $2.0$ | $130.0$ | $245.0$ | $0.0$ | $2.0$ | $180.0$ | $0.0$ | $0.2$ | $2.0$ | $0.0$ | $3.0$ | 1 |

${u}_{11}$ | $54.0$ | $0.0$ | $2.0$ | $132.0$ | $288.0$ | $1.0$ | $2.0$ | $159.0$ | $1.0$ | $0.0$ | $1.0$ | $1.0$ | $3.0$ | 1 |

${u}_{12}$ | $71.0$ | $0.0$ | $3.0$ | $110.0$ | $265.0$ | $1.0$ | $2.0$ | $130.0$ | $0.0$ | $0.0$ | $1.0$ | $1.0$ | $3.0$ | 1 |

${u}_{13}$ | $70.0$ | $1.0$ | $4.0$ | $130.0$ | $322.0$ | $0.0$ | $2.0$ | $109.0$ | $0.0$ | $2.4$ | $2.0$ | $3.0$ | $3.0$ | 2 |

${u}_{14}$ | $56.0$ | $1.0$ | $3.0$ | $130.0$ | $256.0$ | $1.0$ | $2.0$ | $142.0$ | $1.0$ | $0.6$ | $2.0$ | $1.0$ | $6.0$ | 2 |

${u}_{15}$ | $59.0$ | $1.0$ | $4.0$ | $110.0$ | $239.0$ | $0.0$ | $2.0$ | $142.0$ | $1.0$ | $1.2$ | $2.0$ | $1.0$ | $7.0$ | 2 |

${u}_{16}$ | $64.0$ | $1.0$ | $1.0$ | $110.0$ | $211.0$ | $0.0$ | $2.0$ | $144.0$ | $1.0$ | $1.8$ | $2.0$ | $0.0$ | $3.0$ | 1 |

${u}_{17}$ | $67.0$ | $1.0$ | $4.0$ | $120.0$ | $229.0$ | $0.0$ | $2.0$ | $129.0$ | $1.0$ | $2.6$ | $2.0$ | $2.0$ | $7.0$ | 2 |

${u}_{18}$ | $51.0$ | $0.0$ | $3.0$ | $120.0$ | $295.0$ | $0.0$ | $2.0$ | $157.0$ | $0.0$ | $0.6$ | $1.0$ | $0.0$ | $3.0$ | 1 |

${u}_{19}$ | $64.0$ | $1.0$ | $4.0$ | $128.0$ | $263.0$ | $0.0$ | $0.0$ | $105.0$ | $1.0$ | $0.2$ | $2.0$ | $1.0$ | $7.0$ | 1 |

${u}_{20}$ | $57.0$ | $0.0$ | $4.0$ | $128.0$ | $303.0$ | $0.0$ | $2.0$ | $159.0$ | $0.0$ | $0.0$ | $1.0$ | $1.0$ | $3.0$ | 1 |

${u}_{21}$ | $71.0$ | $0.0$ | $4.0$ | $112.0$ | $149.0$ | $0.0$ | $0.0$ | $125.0$ | $0.0$ | $1.6$ | $2.0$ | $0.0$ | $3.0$ | 1 |

${u}_{22}$ | $53.0$ | $1.0$ | $4.0$ | $140.0$ | $203.0$ | $1.0$ | $2.0$ | $155.0$ | $1.0$ | $3.1$ | $3.0$ | $0.0$ | $7.0$ | 2 |

${u}_{23}$ | $47.0$ | $1.0$ | $4.0$ | $112.0$ | $204.0$ | $0.0$ | $0.0$ | $143.0$ | $0.0$ | $0.1$ | $1.0$ | $0.0$ | $3.0$ | 1 |

${u}_{24}$ | $58.0$ | $1.0$ | $3.0$ | $112.0$ | $230.0$ | $0.0$ | $2.0$ | $165.0$ | $0.0$ | $2.5$ | $2.0$ | $1.0$ | $7.0$ | 2 |

${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ | ${\mathit{b}}_{3}$ | ${\mathit{b}}_{4}$ | ${\mathit{b}}_{5}$ | ${\mathit{b}}_{6}$ | ${\mathit{b}}_{7}$ | ${\mathit{b}}_{8}$ | ${\mathit{b}}_{9}$ | ${\mathit{b}}_{10}$ | ${\mathit{b}}_{11}$ | ${\mathit{b}}_{12}$ | ${\mathit{b}}_{13}$ | d | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${g}_{0.462}\left({u}_{2}\right)$ | $65.0$ | $1.0$ | $4.0$ | $120.0$ | $177.0$ | $0.0$ | $0.0$ | $140.0$ | $0.0$ | $0.4$ | $1.0$ | $0.0$ | $7.0$ | 1 |

${g}_{0.539}\left({u}_{3}\right)$ | $65.0$ | $1.0$ | $4.0$ | $120.0$ | $177.0$ | $0.0$ | $0.0$ | $140.0$ | $0.0$ | $0.4$ | $2.0$ | $0.0$ | $7.0$ | 1 |

${g}_{0.462}\left({u}_{6}\right)$ | $53.0$ | $0.0$ | $4.0$ | $142.0$ | $226.0$ | $0.0$ | $2.0$ | $111.0$ | $1.0$ | $0.0$ | $2.0$ | $0.0$ | $3.0$ | 1 |

${g}_{0.462}\left({u}_{8}\right)$ | $60.0$ | $1.0$ | $4.0$ | $140.0$ | $293.0$ | $0.0$ | $2.0$ | $170.0$ | $0.0$ | $1.2$ | $2.0$ | $3.0$ | $7.0$ | 2 |

${g}_{0.385}\left({u}_{12}\right)$ | $74.0$ | $0.0$ | $2.0$ | $120.0$ | $269.0$ | $0.0$ | $2.0$ | $159.0$ | $0.0$ | $0.0$ | $1.0$ | $1.0$ | $3.0$ | 1 |

${g}_{0.385}\left({u}_{14}\right)$ | $56.0$ | $1.0$ | $3.0$ | $130.0$ | $256.0$ | $0.0$ | $2.0$ | $142.0$ | $1.0$ | $0.6$ | $2.0$ | $1.0$ | $7.0$ | 2 |

${g}_{0.539}\left({u}_{17}\right)$ | $60.0$ | $1.0$ | $4.0$ | $140.0$ | $293.0$ | $0.0$ | $2.0$ | $170.0$ | $1.0$ | $1.2$ | $2.0$ | $2.0$ | $7.0$ | 2 |

${g}_{0.385}\left({u}_{18}\right)$ | $71.0$ | $0.0$ | $4.0$ | $120.0$ | $269.0$ | $0.0$ | $2.0$ | $121.0$ | $0.0$ | $0.0$ | $1.0$ | $0.0$ | $3.0$ | 1 |

${g}_{0.615}\left({u}_{19}\right)$ | $64.0$ | $1.0$ | $4.0$ | $128.0$ | $263.0$ | $0.0$ | $0.0$ | $105.0$ | $1.0$ | $0.2$ | $2.0$ | $1.0$ | $7.0$ | 1 |

${g}_{0.462}\left({u}_{21}\right)$ | $59.0$ | $1.0$ | $4.0$ | $112.0$ | $234.0$ | $0.0$ | $0.0$ | $161.0$ | $0.0$ | $0.5$ | $2.0$ | $0.0$ | $3.0$ | 1 |

${g}_{0.615}\left({u}_{22}\right)$ | $53.0$ | $1.0$ | $4.0$ | $140.0$ | $203.0$ | $1.0$ | $2.0$ | $155.0$ | $1.0$ | $3.1$ | $3.0$ | $0.0$ | $7.0$ | 2 |

**Table 3.**Training data system $({U}_{trn},A,d)$, (a sample from australian credit dataset), for $\epsilon =0.05$.

${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ | ${\mathit{b}}_{3}$ | ${\mathit{b}}_{4}$ | ${\mathit{b}}_{5}$ | ${\mathit{b}}_{6}$ | ${\mathit{b}}_{7}$ | ${\mathit{b}}_{8}$ | ${\mathit{b}}_{9}$ | ${\mathit{b}}_{10}$ | ${\mathit{b}}_{11}$ | ${\mathit{b}}_{12}$ | ${\mathit{b}}_{13}$ | d | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${u}_{1}$ | 1 | $20.17$ | $8.17$ | 2 | 6 | 4 | $1.96$ | 1 | 1 | 14 | 0 | 2 | 60 | 159 | 1 |

${u}_{2}$ | 1 | $34.92$ | 5 | 2 | 14 | 8 | $7.5$ | 1 | 1 | 6 | 1 | 2 | 0 | 1001 | 1 |

${u}_{3}$ | 1 | $58.58$ | $2.71$ | 2 | 8 | 4 | $2.415$ | 0 | 0 | 0 | 1 | 2 | 320 | 1 | 0 |

${u}_{4}$ | 1 | $29.58$ | $4.5$ | 2 | 9 | 4 | $7.5$ | 1 | 1 | 2 | 1 | 2 | 330 | 1 | 1 |

${u}_{5}$ | 0 | $19.17$ | $0.58$ | 1 | 6 | 4 | $0.585$ | 1 | 0 | 0 | 1 | 2 | 160 | 1 | 0 |

${u}_{6}$ | 1 | $23.08$ | $2.5$ | 2 | 8 | 4 | $1.085$ | 1 | 1 | 11 | 1 | 2 | 60 | 2185 | 1 |

${u}_{7}$ | 0 | $21.67$ | $11.5$ | 1 | 5 | 3 | 0 | 1 | 1 | 11 | 1 | 2 | 0 | 1 | 1 |

${u}_{8}$ | 1 | $27.83$ | 1 | 1 | 2 | 8 | 3 | 0 | 0 | 0 | 0 | 2 | 176 | 538 | 0 |

${u}_{9}$ | 1 | $41.17$ | $1.33$ | 2 | 2 | 4 | $0.165$ | 0 | 0 | 0 | 0 | 2 | 168 | 1 | 0 |

${u}_{10}$ | 1 | $41.58$ | $1.75$ | 2 | 4 | 4 | $0.21$ | 1 | 0 | 0 | 0 | 2 | 160 | 1 | 0 |

${u}_{11}$ | 1 | $22.5$ | $0.12$ | 1 | 4 | 4 | $0.125$ | 0 | 0 | 0 | 0 | 2 | 200 | 71 | 0 |

${u}_{12}$ | 1 | $33.17$ | $3.04$ | 1 | 8 | 8 | $2.04$ | 1 | 1 | 1 | 1 | 2 | 180 | 18028 | 1 |

${u}_{13}$ | $1.234$ | $22.08$ | $11.46$ | 2 | 4 | 4 | $1.585$ | 0 | 0 | 0 | 1 | 2 | 100 | 1213 | 0 |

${u}_{14}$ | 0 | $58.67$ | $4.46$ | 2 | 11 | 8 | $3.04$ | 1 | 1 | 6 | 0 | 2 | 43 | 561 | 1 |

${u}_{15}$ | 1 | $33.5$ | $1.75$ | 2 | 14 | 8 | $4.5$ | 1 | 1 | 4 | 1 | 2 | 253 | 858 | 1 |

${u}_{16}$ | 0 | $18.92$ | 9 | 2 | 6 | 4 | $0.75$ | 1 | 1 | 2 | 0 | 2 | 88 | 592 | 1 |

${u}_{17}$ | 1 | 20 | $1.25$ | 1 | 4 | 4 | $0.125$ | 0 | 0 | 0 | 0 | 2 | 140 | 5 | 0 |

${u}_{18}$ | 1 | $19.5$ | $9.58$ | 2 | 6 | 4 | $0.79$ | 0 | 0 | 0 | 0 | 2 | 80 | 351 | 0 |

${u}_{19}$ | 0 | $22.67$ | $3.8$ | 2 | 8 | 4 | $0.165$ | 0 | 0 | 0 | 0 | 2 | 160 | 1 | 0 |

${u}_{20}$ | 1 | $17.42$ | $6.5$ | 2 | 3 | 4 | $0.125$ | 0 | 0 | 0 | 0 | 2 | 60 | 101 | 0 |

${u}_{21}$ | 1 | $41.42$ | 5 | 2 | 11 | 8 | 5 | 1 | 1 | 6 | 1 | 2 | 470 | 1 | 1 |

${u}_{22}$ | 1 | $20.67$ | $1.25$ | 1 | 8 | 8 | $1.375$ | 1 | 1 | 3 | 1 | 2 | 140 | 211 | 0 |

${u}_{23}$ | 1 | $48.08$ | $6.04$ | 2 | 4 | 4 | $0.04$ | 0 | 0 | 0 | 0 | 2 | 0 | 2691 | 1 |

${u}_{24}$ | 0 | $28.17$ | $0.58$ | 2 | 6 | 4 | $0.04$ | 0 | 0 | 0 | 0 | 2 | 260 | 1005 | 0 |

${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ | ${\mathit{b}}_{3}$ | ${\mathit{b}}_{4}$ | ${\mathit{b}}_{5}$ | ${\mathit{b}}_{6}$ | ${\mathit{b}}_{7}$ | ${\mathit{b}}_{8}$ | ${\mathit{b}}_{9}$ | ${\mathit{b}}_{10}$ | ${\mathit{b}}_{11}$ | ${\mathit{b}}_{12}$ | ${\mathit{b}}_{13}$ | d | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${g}_{0.5}\left({u}_{2}\right)$ | 1 | $34.92$ | 5 | 2 | 14 | 8 | $7.5$ | 1 | 1 | 6 | 1 | 2 | 0 | 1001 | 1 |

${g}_{0.571429}\left({u}_{3}\right)$ | 1 | $58.58$ | $2.71$ | 2 | 8 | 4 | $0.165$ | 0 | 0 | 0 | 0 | 2 | 320 | 1 | 0 |

${g}_{0.5}\left({u}_{5}\right)$ | 0 | $19.17$ | $0.58$ | 2 | 6 | 4 | $0.21$ | 1 | 0 | 0 | 0 | 2 | 160 | 1 | 0 |

${g}_{0.5}\left({u}_{6}\right)$ | 1 | $20.17$ | $8.17$ | 2 | 6 | 4 | $1.96$ | 1 | 1 | 14 | 1 | 2 | 60 | 159 | 1 |

${g}_{0.5}\left({u}_{7}\right)$ | 0 | $21.67$ | $11.5$ | 1 | 5 | 3 | 0 | 1 | 1 | 11 | 1 | 2 | 0 | 1 | 1 |

${g}_{0.5}\left({u}_{8}\right)$ | 1 | $27.83$ | $1.33$ | 1 | 2 | 4 | $0.165$ | 0 | 0 | 0 | 0 | 2 | 176 | 1 | 0 |

${g}_{0.642857}\left({u}_{12}\right)$ | 1 | $33.17$ | $3.04$ | 1 | 8 | 8 | $2.04$ | 1 | 1 | 1 | 1 | 2 | 180 | 18,028 | 1 |

${g}_{0.571429}\left({u}_{13}\right)$ | $1.234$ | $22.08$ | $11.46$ | 2 | 4 | 4 | $1.585$ | 0 | 0 | 0 | 1 | 2 | 100 | 1213 | 0 |

${g}_{0.5}\left({u}_{16}\right)$ | 0 | $20.17$ | $8.17$ | 2 | 6 | 4 | $1.96$ | 1 | 1 | 14 | 0 | 2 | 60 | 561 | 1 |

${g}_{0.642857}\left({u}_{18}\right)$ | 1 | $19.5$ | $9.58$ | 2 | 6 | 4 | $0.79$ | 0 | 0 | 0 | 0 | 2 | 80 | 351 | 0 |

${g}_{0.642857}\left({u}_{20}\right)$ | 1 | $22.5$ | $1.33$ | 2 | 4 | 4 | $0.165$ | 0 | 0 | 0 | 0 | 2 | 168 | 1 | 0 |

${g}_{0.5}\left({u}_{21}\right)$ | 1 | $34.92$ | 5 | 2 | 14 | 8 | $7.5$ | 1 | 1 | 6 | 1 | 2 | 0 | 1001 | 1 |

${g}_{0.642857}\left({u}_{22}\right)$ | 1 | $20.67$ | $1.25$ | 1 | 8 | 8 | $1.375$ | 1 | 1 | 3 | 1 | 2 | 140 | 211 | 0 |

${g}_{0.642857}\left({u}_{23}\right)$ | 1 | $48.08$ | $6.04$ | 2 | 4 | 4 | $0.04$ | 0 | 0 | 0 | 0 | 2 | 0 | 2691 | 1 |

**Table 5.**Estimated parameters for $kNN$ based on $5\times CV5$ cross–validation, data from UCI Repository [20].

$\mathit{Name}$ | $\mathit{Optimal}\phantom{\rule{4pt}{0ex}}\mathit{k}$ |
---|---|

$Australian$-$credit$ | 5 |

$Car\phantom{\rule{4pt}{0ex}}Evaluation$ | 8 |

$Diabetes$ | 3 |

$German$-$credit$ | 18 |

$Heartdisease$ | 19 |

$Hepatitis$ | 3 |

$Nursery$ | 4 |

$SPECTF\phantom{\rule{4pt}{0ex}}Heart$ | 14 |

**Table 6.**Basic information about datasets-[20].

$\mathit{Name}$ | $\mathit{Attr}\phantom{\rule{4pt}{0ex}}\mathit{Type}$ | $\mathit{Attr}\phantom{\rule{4pt}{0ex}}\mathit{no}.$ | $\mathit{Obj}\phantom{\rule{4pt}{0ex}}\mathit{no}.$ | $\mathit{Class}\phantom{\rule{4pt}{0ex}}\mathit{no}.$ |
---|---|---|---|---|

Australian-credit | $categorical,\phantom{\rule{4pt}{0ex}}integer,\phantom{\rule{4pt}{0ex}}real$ | 15 | 690 | 2 |

$Car\phantom{\rule{4pt}{0ex}}Evaluation$ | $categorical$ | 7 | 1728 | 4 |

$Diabetes$ | $categorical,\phantom{\rule{4pt}{0ex}}integer$ | 9 | 768 | 2 |

German-credit | $categorical,\phantom{\rule{4pt}{0ex}}integer$ | 21 | 1000 | 2 |

$Heartdisease$ | $categorical,\phantom{\rule{4pt}{0ex}}real$ | 14 | 270 | 2 |

$Hepatitis$ | $categorical,\phantom{\rule{4pt}{0ex}}integer,\phantom{\rule{4pt}{0ex}}real$ | 20 | 155 | 2 |

$Nursery$ | $categorical$ | 9 | 12,960 | 5 |

$SPECTF\phantom{\rule{4pt}{0ex}}Heart$ | $integer$ | 45 | 267 | 2 |

**Table 7.**The result for dynamic granulation; $5\times CV5$ method with $kNN$ classifier; $acc\_5CV5=average\phantom{\rule{4pt}{0ex}}accuracy$, $GS\_size=granular\phantom{\rule{4pt}{0ex}}decision\phantom{\rule{4pt}{0ex}}system\phantom{\rule{4pt}{0ex}}size$, $TRN\_size=training\phantom{\rule{4pt}{0ex}}set\phantom{\rule{4pt}{0ex}}size$, $TRN\_reduction=reduction\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}object\phantom{\rule{4pt}{0ex}}number\phantom{\rule{4pt}{0ex}}in\phantom{\rule{4pt}{0ex}}training\phantom{\rule{4pt}{0ex}}size$, $radii\_range=spectrum\phantom{\rule{4pt}{0ex}}of\phantom{\rule{4pt}{0ex}}radii$.

$\mathit{Name}$ | $\mathit{acc}$ | $\mathit{GS}\_\mathit{size}$ | $\mathit{TRN}\_\mathit{size}$ | $\mathit{TRN}\_\mathit{reduction}$ | $\mathit{radii}\_\mathit{range}$ |
---|---|---|---|---|---|

Australian-credit | $0.835$ | $286.52$ | 552 | $48.1\%$ | ${r}_{u}\ge 0.5$ |

$Car\phantom{\rule{4pt}{0ex}}Evaluation$ | $0.797$ | $728.5$ | 1382 | $47.3\%$ | ${r}_{u}\ge 0.667$ |

$Diabetes$ | $0.653$ | $488.9$ | 614 | $20.4\%$ | ${r}_{u}\ge 0.25$ |

German-credit | $0.725$ | $513.3$ | 800 | $35.8\%$ | ${r}_{u}\ge 0.6$ |

$Heartdisease$ | $0.833$ | $120.5$ | 216 | $44.2\%$ | ${r}_{u}\ge 0.461$ |

$Hepatitis$ | $0.88$ | $46.16$ | 124 | $62.8\%$ | ${r}_{u}\ge 0.579$ |

$Nursery$ | $0.607$ | $9009.1$ | 10368 | $13.1\%$ | ${r}_{u}\ge 0.875$ |

$SPECTF\phantom{\rule{4pt}{0ex}}Heart$ | $0.763$ | $138.75$ | 214 | $35.2\%$ | ${r}_{u}\ge 0.068$ |

**Table 8.**Summary of results for $kNN$ Classifier, granular and non granular case, $acc$ = accuracy of classification, $red$ = percentage reduction in object number, r = granulation radius, $method$ = variant of Naive Bayes classifier, $nil.acc$ = non granular case.

$\mathit{Name}$ | $\mathit{acc},\mathit{red},\mathit{r}$ | $\mathit{nil}.\mathit{acc}$ |
---|---|---|

$Australian$-$credit$ | $0.851,71.86,0.571$ | $0.855$ |

$Car\phantom{\rule{4pt}{0ex}}Evaluation$ | $0.865,73.23,0.833$ | $0.944$ |

$Diabetes$ | $0.616,74.74,0.25$ | $0.631$ |

$German$-$credit$ | $0.724,59.85,0.65$ | $0.73$ |

$Heartdisease$ | $0.83,67.69,0.538$ | $0.837$ |

$Hepatitis$ | $0.884,60,0.632$ | $0.89$ |

$Nursery$ | $0.696,77.09,0.875$ | $0.578$ |

$SPECTF\phantom{\rule{4pt}{0ex}}Heart$ | $0.802,60.3,0.114$ | $0.779$ |

**Table 9.**The result for homogenous granulation ($HG$) and for epsilon homogenous granulation ($\epsilon -HGS$); $5\times CV5$; $HG\_acc=$ average accuracy for $HG$, $\epsilon -HG\_acc$ average accuracy for $\epsilon -HGS$, $HGS\_size=$$HG$ decision system size, $\epsilon -HGS\_size=$$\epsilon -HGS$ decision system size, $TRN\_size=training\phantom{\rule{4pt}{0ex}}set\phantom{\rule{4pt}{0ex}}size$, $H{G}_{T}RN\_red=$ reduction in object number in training set for $HG$, $\epsilon -HGS\_size=$ reduction in object number in training set for $\epsilon -HGS$, $HG\_r\_range=$ spectrum of radii for $HG$, $\epsilon -HG\_r\_range=\phantom{\rule{3.33333pt}{0ex}}$ spectrum of radii for $\epsilon -HGS$, $dat{a}_{1}$ = Australian-credit, $dat{a}_{2}$ = German-credit, $dat{a}_{3}$ = Heartdisease, $dat{a}_{4}$ = Hepatitis.

${\mathit{Data}}_{1}$ | ${\mathit{Data}}_{2}$ | ${\mathit{Data}}_{3}$ | ${\mathit{Data}}_{4}$ | |
---|---|---|---|---|

$HG\_acc$ | $0.835$ | $0.725$ | $0.833$ | $0.88$ |

$\epsilon -HG\_acc$ | $0.842$ | $0.725$ | $0.831$ | $0.87$ |

$HGS\_size$ | $286.52$ | $513.3$ | $120.5$ | $46.16$ |

$\epsilon -HGS\_size$ | $274.52$ | 503 | $109.4$ | $46.2$ |

$TRN\_size$ | 552 | 800 | 216 | 124 |

$H{G}_{T}RN\_red$ | $48.1\%$ | $35.8\%$ | $44.2\%$ | $62.8\%$ |

$\epsilon H{G}_{T}RN\_red$ | $50.3\%$ | $37.1\%$ | $49.4\%$ | $62.7\%$ |

$HG\_r\_range$ | ${r}_{u}\ge 0.5$ | ${r}_{u}\ge 0.6$ | ${r}_{u}\ge 0.461$ | ${r}_{u}\ge 0.579$ |

$\epsilon -HG\_r\_range$ | ${r}_{u}\ge 0.571$ | ${r}_{u}\ge 0.65$ | ${r}_{u}\ge 0.615$ | ${r}_{u}\ge 0.579$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Ropiak, K.; Artiemjew, P.
Homogenous Granulation and Its Epsilon Variant. *Computers* **2019**, *8*, 36.
https://doi.org/10.3390/computers8020036

**AMA Style**

Ropiak K, Artiemjew P.
Homogenous Granulation and Its Epsilon Variant. *Computers*. 2019; 8(2):36.
https://doi.org/10.3390/computers8020036

**Chicago/Turabian Style**

Ropiak, Krzysztof, and Piotr Artiemjew.
2019. "Homogenous Granulation and Its Epsilon Variant" *Computers* 8, no. 2: 36.
https://doi.org/10.3390/computers8020036