# Expressions of Rock Joint Roughness Coefficient Using Neutrosophic Interval Statistical Numbers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{T}be the chance that a particular trial results in a success, P

_{I}be the chance that a particular trial results in an indeterminacy (i.e., neither a success nor a failure), and P

_{F}be the chance that a particular trial results in a failure.

## 2. Neutrosophic Interval Probability

**Definition**

**1.**

^{L}, x

^{U}] be an interesting range of all the sample individuals. A NIP can be defined as P = <[x

^{L}, x

^{U}], (P

_{T}, P

_{I}, P

_{F})>, where P

_{T}is a truth-probability belonging to the determinate range, P

_{I}is an indeterminacy-probability belonging to the indeterminate range, and P

_{F}is a falsity-probability belonging to the almost impossible/failure range. Then, the sum of the three probabilities satisfies P

_{T}+ P

_{I}+ P

_{F}= 1.

^{U}(upper bound) and the minimum value x

^{L}(lower bound) in all trial data and the average value x

_{m}and standard deviation σ. Based on these statistical results, we propose the following calculation methods of NIP.

^{L}, x

^{U}] according to the maximum and minimum values (i.e., the upper and lower bounds) of the trial data. Then, in the interesting range of all the sample individuals we can calculate the truth-probability P

_{T}= n

_{T}/n, the indeterminacy-probability P

_{I}= n

_{I}/n, and the falsity-probability P

_{F}= n

_{F}/n from a statistical viewpoint, where n

_{T}implies the frequency in the robust/credible interval [x

_{m}− σ, x

_{m}+ σ], n

_{I}implies the frequency in the indeterminate/uncertain intervals [x

_{m}− 3σ, x

_{m}− σ) and (x

_{m}+ σ, x

_{m}+ 3σ], and n

_{F}indicates the frequency in the remaining/incredible intervals [x

^{L}, x

_{m}− 3σ) and (x

_{m}+ 3σ, x

^{U}].

**Example**

**1.**

_{T}= n

_{T}/n = 70/100 = 0.7, P

_{I}= n

_{I}/n = 25/100 = 0.25, and P

_{F}= n

_{F}/n =5/100 = 0.05.

^{L}, x

^{U}], (P

_{T}, P

_{I}, P

_{F})> = <[0,5], (0.7, 0.25, 0.05)>.

## 3. Neutrosophic Interval Statistical Number

^{L}, x

^{U}], (P

_{T}, P

_{I}, P

_{F})> and the ideal NIP be P

^{*}= <[x

^{L}, x

^{U}], (1, 0, 0)>. Then, the cosine measure value between P and P

^{*}[4] is defined as the confidence degree

_{m}is the average value/determinate part of N

_{e}and I is indeterminacy. Here, I may take the robust/credit interval [−σ, σ] based on a standard deviation σ.

^{L}, x

^{U}], (1, 0, 0)>, then N

_{e}= x

_{m}, which is degenerated to the classical average value (crisp value) x

_{m}with the maximum confidence degree; if e = 0 for P = <[x

^{L}, x

^{U}], (0, 0, 1)>, then N

_{e}= x

_{m}+ I, which is degenerated to a NN without confidence degree. However, when 0 < e < 1, the confidence degree of e can affect the indeterminate part (1 − e)I of the NISN N

_{e}.

_{e}= [1.97, 2.03].

_{m}, which is a specialty of the proposed neutrosophic interval statistical method.

## 4. Joint Roughness Coefficient Values Expressed by Using Neutrosophic Interval Statistical Number in Geotechnical Mechanics

_{m}and the standard deviation σ of each length L, and then the results of their statistical analysis are shown in Table 2.

_{e}= [7.6929, 7.7421].

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Frequency distributions in the different joint roughness coefficient (JRC) intervals for each length L.

n | x_{m} | σ | [x^{L}, x^{U}] | n_{T} in [1.5, 2.5] | n_{I} in [0.5, 1.5) and (2.5, 3.5] | n_{F} in [0, 0.5) and (3.5, 5] |
---|---|---|---|---|---|---|

100 | 2 | 0.5 | [0,5] | 70 | 25 | 5 |

Sample Length L | 10 cm | 20 cm | 30 cm | 40 cm | 50 cm | 60 cm | 70 cm | 80 cm | 90 cm | 100 cm |
---|---|---|---|---|---|---|---|---|---|---|

n | 187 | 85 | 51 | 39 | 34 | 34 | 34 | 34 | 34 | 34 |

x_{m} | 10.6035 | 9.9647 | 9.5320 | 8.8760 | 8.6121 | 8.6463 | 8.3931 | 8.1107 | 7.9051 | 7.7175 |

σ | 2.2090 | 1.6606 | 1.5695 | 1.5994 | 1.4899 | 1.5942 | 1.3637 | 1.2203 | 1.0893 | 1.0050 |

**Table 3.**The neutrosophic interval probability (NIP) and neutrosophic interval statistical number (NISN) of each length L.

L | P | N_{e} | |
---|---|---|---|

10 cm | <(6.62,16.41),0.6898,0.3102,0> | 10.6035 + 0.0879I | [10.4093,10.7978] |

20 cm | <(6.59,14.26),0.6706,0.3294,0> | 9.9647 + 0.1024I | [9.7946,10.1348] |

30 cm | <(6.81,14.24),0.7255,0.2745,0> | 9.5320 + 0.0647I | [9.4305,9.6336] |

40 cm | <(6.59,14.06),0.7180,0.2051,0.0769> | 8.8760 + 0.0435I | [8.8064,8.9456] |

50 cm | <(6.15,13.36),0.6765,0.2941,0.0294> | 8.6121 + 0.0837I | [8.4874,8.7367] |

60 cm | <(6.48,13.49),0.7353,0.2353,0.0294> | 8.6463 + 0.0483I | [8.5694,8.7233] |

70 cm | <(6.24,13.08),0.7647,0.2059,0.0294> | 8.3931 + 0.0350I | [8.3453,8.4409] |

80 cm | <(6.22,12.41),0.7647,0.2059,0.0294> | 8.1107 + 0.0350I | [8.0679,8.1534] |

90 cm | <(6.54,12.11),0.7647,0.2059,0.0294> | 7.9051 + 0.0350I | [7.8669,7.9433] |

100 cm | <(6.42,11.50),0.7941,0.1765,0.0294> | 7.7175 + 0.0245I | [7.6929,7.7421] |

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**MDPI and ACS Style**

Chen, J.; Ye, J.; Du, S.; Yong, R.
Expressions of Rock Joint Roughness Coefficient Using Neutrosophic Interval Statistical Numbers. *Symmetry* **2017**, *9*, 123.
https://doi.org/10.3390/sym9070123

**AMA Style**

Chen J, Ye J, Du S, Yong R.
Expressions of Rock Joint Roughness Coefficient Using Neutrosophic Interval Statistical Numbers. *Symmetry*. 2017; 9(7):123.
https://doi.org/10.3390/sym9070123

**Chicago/Turabian Style**

Chen, Jiqian, Jun Ye, Shigui Du, and Rui Yong.
2017. "Expressions of Rock Joint Roughness Coefficient Using Neutrosophic Interval Statistical Numbers" *Symmetry* 9, no. 7: 123.
https://doi.org/10.3390/sym9070123