# A Bat-Optimized One-Class Support Vector Machine for Mineral Prospectivity Mapping

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Geological and Geochemical Data

^{2}with a sampling density of 1–2 samples per 0.25 km

^{2}. The concentrations of 13 elements in each stream sediment sample were analyzed and tested by the Inner Mongolia Mineral Experiment Institute, China. The concentration of Au was analyzed by atomic absorption spectrometry (AAS), the concentrations of Hg and As were analyzed by atomic fluorescence spectrophotometry (AFS), and the concentrations of Ag, Sb, Mo, W, Cu, Pb, Zn, Bi, Ni, and Co were analyzed by inductively coupled plasma mass spectrometry (ICP-MS).

^{2}, which satisfies the condition that there is no more than one known mineral deposit in one cell.

#### 2.2. Receiver Operating Characteristic (ROC) Curve, Area under the Cuve (AUC), and Youden Index

_{p}true positive and t

_{n}true negative points in the study area. According to Chen [21], the AUC value of the continuous indicator can be expressed as

_{AUC}that conforms to the standard normal distribution [21]:

_{AUC}is the standard deviation of AUC, which can be calculated by

_{AUC}can be used to test whether there is a significant difference between the AUC value and 0.5 at the significance level of α = 0.05 [13,14,21,22,23]. According to the unit normal loss function, at the significance level of α = 0.05, the critical value of Z

_{UAC}is 1.96. If the Z

_{AUC}value calculated by Equation (2) is greater than the critical value of 1.96, the probability of a significant difference between the AUC value and the value of 0.5 is not less than 0.95.

#### 2.3. OCSVM

**x**that denotes the degree of cell

**x**being an outlier [13], ${\alpha}_{i}\text{}\left(i=1,2,\dots ,n\right)$ is the Lagrange parameter, $K\left(\xb7,\xb7\right)$ is the Gaussian kernel.

#### 2.4. Bat-Optimized OCSVM

_{min}= 0 means that a bat has just found the target and temporarily stops making any sound, and r

_{min}= 0 and r

_{max}= 1 respectively represent no pulse and the maximum emission rate. f

_{min}= 0 ≤ f ≤ 1 = f

_{max}corresponds to λ

_{max}≥ λ ≥ λ

_{min}. λ

_{max}represents the detectable range, and adjusting it only needs to change f because λ × f is constant [16,17].

**z**

_{l}in the d-dimensional search space after its loudness A

_{l}, emission rate r

_{l}, and frequency f

_{l}are randomly initialized. Each bat l flies randomly with velocity

**v**

_{l}at location

**z**

_{l}, searching for prey with a fixed frequency f

_{l}, varying wavelength λ

_{l}and loudness A

_{l}, and automatically adjusts wavelength λ

_{l}according to the degree at which it is approaching the prey [16,17].

**ε**∈ [–1, 1] is a d-dimensional random vector, and $\langle {A}^{t}\rangle $ is the average loudness of the L bats at iteration t.

_{l}and the emission rate r

_{l}of each bat l are updated accordingly as follows:

## 3. Mapping Mineral Prospectivity

#### 3.1. Geological Background and Mineralization

#### 3.2. Evidence Map Layers

_{AUC}s of all the geochemical elements are calculated, and those elements with Z

_{AUC}s greater than the critical value of 1.96 are selected for mapping mineral prospectivity. The selected elements are finally optimally converted into binary evidences using the Youden index.

_{AUC}s discussed in Section 2.2. Equation (1) was used to calculate the AUC value of each element according to the preprocessed data in Section 2.1. Then, Equation (2) was used to estimate the Z

_{AUC}value according to the AUC value. If the estimated value of Z

_{AUC}is greater than the critical value of 1.96 at the significance level of 0.05, the AUC value is considered to be significantly different from the value of 0.5. This means that the concentrations of the element are significantly spatially correlated to the known mineral deposits. In other words, the higher the concentration of the element in unit cells, the more likely the unit cells contain known mineral deposits. Table 3 lists the AUCs and Z

_{AUC}s of 13 elements estimated in this study. As can be seen from Table 3, the Z

_{AUC}s of Au, Co, Cu, Mo, Ni, and W are greater than the critical value of 1.96. Thus, the concentrations of these elements are significantly spatially correlated to the known mineral deposits.

#### 3.3. Mineral Target Extraction

_{min}= 0, f

_{max}= 1, A

_{min}= 0, A

_{max}= 1, and α = γ = 0.9. Figure 9a shows that in the optimization process of the bat algorithm, as the number of iterations increases, the AUC value of the OCSVM model becomes larger and larger. It can be seen from Figure 9a that after 23 iterations, the bat algorithm converges to AUC = 0.8649. The corresponding optimal values of μ and σ are respectively μ = 0.4276 and σ = 1.7559. The optimal threshold determined by using the Youden index is OT2 = 9.2496, and the corresponding maximum Youden index is MYI2 = 0.5763. The mineral targets extracted by the optimal threshold OT2 = 9.2496 are shown in Figure 8c.

_{min}= 0, f

_{max}= 1, A

_{min}= 0, A

_{max}= 1, and α = γ = 0.9. Figure 9b shows that in the optimization process of the bat algorithm, the AUC value of the OCSVM model increases with the increase of the number of iterations. It can be seen from Figure 9b that after 10 iterations, the bat algorithm converges to AUC = 0.8644. The corresponding optimal solution is μ = 0.4764 and σ = 1.3602. The optimal threshold determined by using the Youden index is OT3 = 101.4408, and the corresponding maximum Youden index is MYI3 = 0.5846. The mineral targets extracted by the optimal threshold OT3 = 101.4408 are shown in Figure 8d.

## 4. Results

_{AUC}value, and check whether there is a significant spatial relationship between the anomaly scores and the known mineral deposit locations [13,14,21,22,23].

_{AUC}s for the common and optimized OCSVMs were calculated using Equation (2). Table 6 lists the performance evaluation statistics of the common and optimized OCSVMs in mineral prospectivity mapping.

_{AUC}s of the common and trial and error-optimized OCSVMs are 4.8032 and 5.6029, and the Z

_{AUC}s of the bat-optimized OCSVMs are 5.8639 and 5.8483, respectively. These four Z

_{AUC}s are far higher than the critical value of 1.96. Therefore, both the common and optimized OCSVMs are significantly effective in predicting the known mineral deposit locations in the study area. In other words, the mineral targets predicted by the common and optimized OCSVMs are significantly spatially associated with known mineral deposits in the study area.

## 5. Discussion

_{min}, f

_{max}, A

_{min}, A

_{max}, α, and γ need to be defined for the bat algorithm. Among these parameters, only L and T maybe significantly affect the performance of the bat-optimized OCSVM. The other six parameters are usually defined as the default values suggested by Yang and Gandomi [17].

_{min}= 0, f

_{max}= 1, A

_{min}= 0, A

_{max}= 1, and α = γ = 0.9 was used to repeatedly optimize OCSVM parameters three times, to verify whether the bat algorithm can always converge to the global optimum. The results show that in each data modeling process, the bat algorithm converges to the same maximum AUC = 0.8649. Therefore, the bat algorithm can generally converge to the global maximum in OCSVM parameter optimization in mineral prospectivity mapping.

## 6. Conclusions

_{AUC}values of both the common and optimized one-class support vector machine models calculated in the case study are much higher than the critical value 1.96 at the significant level of 0.05. Therefore, the mineral targets predicted by both the common and optimized one-class support vector machine models are significantly spatially associated with known mineral deposits in the study area.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The concentrations of Au, Bi, Co, Cu, Mo, and Ni collected from the 6999 valid sampling locations in the study area.

**Figure 2.**The grid data of Au, Bi, Co, Cu, Mo, and Ni produced by the interpolation method of Inverse Distance to a Power in Surfer 12.

**Figure 3.**Mineral deposits and binary evidence map layers: (

**a**) the unit cell layer containing known mineral deposits, (

**b**) the Jinan Formation, (

**c**) porphyritic biotite granodiorite, (

**d**) porphyritic granodiorite, (

**e**) fine-grained monzonite, (

**f**) medium-fine-grained diorite, (

**g**) fault with 0.5 km buffer, (

**h**) troctolite boundary with 0.8 km buffer, (

**i**) porphyritic biotite granodiorite boundary with 0.1 km buffer, (

**j**) porphyritic granodiorite boundary with 0.6 km buffer, (

**k**) fine-grained monzonite boundary with 0.1 km buffer, (

**l**) medium-fine-grained diorite boundary with 1.0 km buffer, (

**m**) gold concentration anomalies, (

**n**) bismuth concentration anomalies, (

**o**) cobalt concentration anomalies, (

**p**) copper concentration anomalies, (

**q**) molybdenum concentration anomalies, and (

**r**) nickel concentration anomalies.

**Figure 8.**Mineral targets extracted by (

**a**) the common OCSVM, (

**b**) the trial and error-optimized OCSVM, (

**c**) the bat-optimized OCSVM 1, and (

**d**) the bat-optimized OCSVM 2.

**Figure 9.**The AUC value of the OCSVM model varies with iterations: (

**a**) the bat algorithm initialized with L = 20, T = 30, f

_{min}= 0, f

_{max}= 1, A

_{min}= 0, A

_{max}= 1, and α = γ = 0.9; and (

**b**) the bat algorithm initialized with L = 30, T = 20, f

_{min}= 0, f

_{max}= 1, A

_{min}= 0, A

_{max}= 1, and α = γ = 0.9.

The Algorithm for the Bat-Optimized OCSVM Model |
---|

Input: |

Binary data {x_{1}, x_{2}, …, x_{n}}; |

Binary ground truth data {d_{1}, d_{2}, …, d_{n}}. |

Output: |

Anomaly scores {f(x_{1}), f(x_{2}), …, f(x_{n})}. |

Algorithm: |

Initialization (): |

Randomly initialize the location and velocity of each bat z_{l} and v_{l}, (l = 1, 2, …, L); |

Define pulse frequency f_{l} at z_{l}, (l = 1, 2, …, L); |

Initialize emission rate r_{l} and the loudness A_{l}, (l = 1, 2, …, L). |

Evaluation (): |

Initialize the OCSVM model using z_{l}, (l = 1, 2, …, L); |

Train the OCSVM model on the binary data {x_{1}, x_{2}, …, x_{n}}; |

Compute the anomaly score of unit cell i using Equation (5), (i = 1, 2, …, n); |

Compute the AUC of the OCSVM model initialized by z_{l} (l = 1, 2, …, L) using Equation (1). |

While (t < T): |

Adjust the frequency of each bat f_{l} using Equation (6) (l = 1, 2, …, L); |

Update the velocity and location of each bat z_{l} and v_{l} using Equations (7) to (8) (l = 1, 2, …, L);Call Evaluation (). |

If (random < r_{l}): |

Select a location among the best locations; |

Generate a local location around the selected best location; |

Generate a new location according to Equation (9); Call Evaluation (). |

$\mathrm{If}\text{}(\mathrm{random}{A}_{l}\text{}\mathrm{and}\text{}\mathrm{the}\text{}\mathrm{AUC}\text{}\mathrm{for}\text{}{\mathit{z}}_{l}\text{}\mathrm{the}\text{}\mathrm{AUC}\text{}\mathrm{for}\text{}{\mathit{z}}_{*}$): |

Accept the new locations; |

Increase r_{l} and reduce A_{l} according to Equation (10); |

Rank the bats and find the current best ${\mathit{z}}_{*}$. |

Output the results. |

Linear Evidence | MYI | OBW (km) |
---|---|---|

Regional structure | 0.09887 | 0.5 |

Troctolite boundary | 0.04405 | 0.8 |

Mottled monzonite boundary | −0.01642 | 0.1 |

Porphyritic monzonite boundary | −0.03696 | 0.8 |

Stage II porphyritic monzonite boundary | −0.1287 | 0.1 |

Porphyritic biotite granodiorite boundary | 0.08729 | 0.1 |

Porphyritic granodiorite boundary | 0.2019 | 0.6 |

Fine-grained monzonite boundary | 0.1264 | 0.1 |

Medium-fine-grained monzonite boundary | −0.09409 | 0.1 |

Medium-fine-grained diorite boundary | 0.1831 | 1.0 |

Element | AUC | Z_{AUC} | Element | AUC | Z_{AUC} | Element | AUC | Z_{AUC} |
---|---|---|---|---|---|---|---|---|

Ag | 0.5268 | 0.3416 | Cu | 0.7222 | 2.8661 | Sb | 0.5802 | 1.0037 |

As | 0.6195 | 1.4889 | Hg | 0.5949 | 1.1835 | W | 0.6561 | 1.9531 |

Au | 0.6893 | 2.3958 | Mo | 0.7159 | 2.7727 | Zn | 0.6537 | 1.9217 |

Bi | 0.6620 | 2.0295 | Ni | 0.7619 | 3.4986 | |||

Co | 0.7007 | 2.5540 | Pb | 0.4327 | –0.9248 |

Element | MYI | OT | Element | MYI | OT | Element | MYI | OT |
---|---|---|---|---|---|---|---|---|

Au | 0.3483 | 0.6421 | Bi | 0.3357 | 0.1996 | Co | 0.3989 | 7.3378 |

Cu | 0.3889 | 11.0669 | Mo | 0.36406 | 1.1283 | Ni | 0.4715 | 10.5810 |

**Table 5.**The minimum and maximum values of the anomaly score generated by the OCSVM initialized with different values of σ.

σ | 0.05 | 0.1 | 0.5 | 1.0 | 5.0 | 10.0 | 50.0 | 100.0 | 500.0 | |
---|---|---|---|---|---|---|---|---|---|---|

Score | ||||||||||

Minimum | −300 | −300 | −200 | −60 | −5 | −5 | −5 | −5 | −5 | |

Maximum | 2000 | 1600 | 1200 | 460 | 95 | 95 | 95 | 95 | 95 |

Statistics | AUC | Z_{AUC} | MYI | OT | PGA (%) | Benefit (%) | PMT (s) |
---|---|---|---|---|---|---|---|

OCSVM0 | 0.8268 | 4.8032 | 0.5092 | 89.8292 | 29.61 | 93 | 47.73 |

OCSVM1 | 0.8567 | 5.6029 | 0.6214 | 144.3031 | 18.66 | 86 | n/a |

OCSVM2 | 0.8649 | 5.8639 | 0.5763 | 9.2496 | 19.84 | 93 | 24,856.56 |

OCSVM3 | 0.8644 | 5.8483 | 0.5846 | 101.4408 | 14.22 | 86 | 39,314.25 |

**Note:**MYI denotes maximum Youden index; OT denotes optimal threshold; PGA denotes the percentage of geological anomalies; PMT denotes program modeling time; OCSVM0 denotes the OCSVM initialized with the default parameters; OCSVM1 denotes the OCSVM optimized by trial and error; and OCSVM2 and OCSVM3 denote the OCSVMs optimized respectively by the bat algorithm with L = 20, T = 30 and the bat algorithm with L = 30, T = 20.

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## Share and Cite

**MDPI and ACS Style**

Chen, Y.; Wu, W.; Zhao, Q.
A Bat-Optimized One-Class Support Vector Machine for Mineral Prospectivity Mapping. *Minerals* **2019**, *9*, 317.
https://doi.org/10.3390/min9050317

**AMA Style**

Chen Y, Wu W, Zhao Q.
A Bat-Optimized One-Class Support Vector Machine for Mineral Prospectivity Mapping. *Minerals*. 2019; 9(5):317.
https://doi.org/10.3390/min9050317

**Chicago/Turabian Style**

Chen, Yongliang, Wei Wu, and Qingying Zhao.
2019. "A Bat-Optimized One-Class Support Vector Machine for Mineral Prospectivity Mapping" *Minerals* 9, no. 5: 317.
https://doi.org/10.3390/min9050317