# Anomaly Detection in Automotive Industry Using Clustering Methods—A Case Study

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## Abstract

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## Featured Application

**Parts grouping “Clustering” by master data similarity on automotive industry. Anomaly detection on material classification. Machine learning clustering algorithms comparison—Study Case. Clustering quality index techniques comparison.**

## Abstract

## 1. Introduction

## 2. Clustering Algorithms

#### 2.1. K-Means

**c**

_{k}is the centroid of cluster k,

**x**

_{i}is the i-th object of the k-th cluster, K is the number of clusters, and n

_{q}is the number of elements on each cluster and

_{k}is the number of objects in the respective cluster.

#### 2.2. K-Medoids

**X**is stored in K clusters. This algorithm minimizes the differences between each object in a cluster and its representative object (centroid), similar to the K-Means.

_{k}samples from the data group, drawn randomly. This is done to minimize the effects of K-Means random initialization [35].

#### 2.3. Fuzzy C-Means (FCM)

_{ij}∈ [0, 1] is the degree of membership, ${{\displaystyle \sum}}_{j=1}^{{n}_{k}}{u}_{ij}$, ∀i, and $0<{{\displaystyle \sum}}_{i=1}^{N}{u}_{ij}<N$, being N the number of objects. Note that each pattern is named i for each cluster j and n

_{k}is the number of objects in the respective cluster.

**J**

_{m}described in Equation (4):

#### 2.4. Hierarchical Clustering

#### 2.5. Density-Based Spatial Clustering of Applications with Noise (DBSCAN)

#### 2.6. Self Organizing Maps (SOM)

^{2}neurons are arranged in a grid or a plane, the network is called two-dimensional since it maps high-dimensional input vectors to a two-dimensional surface. For a given network, the input vector

**x**= (x

_{1}, x

_{2}, …, x

_{n}) has a fixed dimension n. These n components are connected to each neuron in the matrix. A synaptic weight

**w**

_{ij}is defined for connecting the i-th component of the input vector to the j-th neuron. Therefore, a n-dimensional vector

**w**

_{j}of synaptic weights is associated with each neuron j [52].

**w**

_{ij}are the weights at the initial iteration t = 0, which are small randomly generated values, t is the number of the current iteration, and

**x**

_{j}is a randomly selected vector from the training data set.

_{0}is the initial learning rate and T the number of iterations, both user-defined data.

_{0}being the initial topological neighborhood.

#### 2.7. Particle Swarm Optimization (PSO)

**pBest**) reached until the current iteration, and the best position represents social learning achieved considering, for example, the entire population (

**gBest**). Together, cognitive and social learning are used to calculate the velocity of particles and their next position [13].

_{k}centroids concatenated in a vector. The PSO can be applied directly, using the position and velocity, according to Equations (11) and (12), respectively:

**x**

_{i}is the position of the particle,

**v**

_{i}is the speed,

**pBest**

_{i}is the best position ever found by each particle,

**gBest**is the best position found by the group, q

_{1}and q

_{2}are two previously defined constants, and r

_{1}and r

_{2}are two numbers randomly generated in the interval [0, 1].

_{max}; + v

_{max}] [54].

#### 2.8. Genetic Algorithm (GA)

#### 2.9. Differential Evolution (DE)

**x**, where i denotes each agent and G represents the generation of the population, is used. As in GA, the operators are crossover, mutation, and selection [62].

_{i,G}**x**

_{r}

_{1},

**x**

_{r}

_{2}, and

**x**

_{r}

_{3}. In the next step, the mutation operator is applied, generating a new donor vector

**v**

_{i,G}through Equation (13):

_{1}, r

_{2}, r

_{3}$\in $ {1, …, NP}.

**u**

_{i,G}is created by the binomial combination of the target vector

**x**

_{i,G}and the donor vector

**v**

_{i,G}elements. Each element of the trial vector comes from

**x**

_{i,G}with crossover probability C

_{r}$\in $ [0, 1] selected along with the population, as in Equation (14) [62]:

**x**

_{i,G}is compared with the trial vector

**u**

_{i,G}

_{,}and the corresponding vector with the best fitness value is taken into next-generation, agreed approach, as in Equation (15):

## 3. Clustering Metrics

#### 3.1. Sum of Squared Errors (SSE)

**X**= {

**x**

_{1},

**x**

_{2}, …,

**x**

_{n}} as a dataset with n samples. Suppose that the samples in

**X**present rigid labels belonging to clusters k without overlap, being

**K**= {

**c**

_{1},

**c**

_{2}, …,

**c**

_{k}} the centroids. The clustering algorithm seeks to find the ideal partition

**P**= {P

_{1}, P

_{2}, …, P

_{m}}, positioning the k centroids iteratively [61]. This metric can be defined by Equation (1) [10], being the same used as the objective function to the K-Means.:

#### 3.2. Sum of Squares within Clusters (SSW)

#### 3.3. Sum of Squares between Clusters (SSB)

**c**

_{k}is the centroid of cluster k and

**c**

_{1}are the other clusters [10].

#### 3.4. Calinski-Harabasz Index (CH)

#### 3.5. WB Index

#### 3.6. Silhouette Index (SI)

_{i}is given by Equation (20):

_{k}is the number of objects in a specific cluster, b

_{i}is given by Equation (21):

## 4. Case Study, Results, and Discussions

#### 4.1. Industry Data

#### 4.2. Pre Processing Stages

_{j}is the input data, $\overline{{x}_{j}}$ and ${\sigma}_{j}$ are the samples mean and the standard deviation of the j-th attribute, respectively. The transformed dimension presents a zero mean and a variance of 1.

- (a)
- Four dimensions sepal length, sepal width, petal length, and petal width without PCA application;
- (b)
- PCA application using only the first principal components;
- (c)
- PCA application using only the first and second principal components.

#### 4.3. Computational Results

#### 4.4. PCA Assessment for the Best Algorithms

#### 4.5. Company Feedback on Results

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The final solution of the hierarchical clustering method showing a 2D dataset and corresponding dendrogram.

**Figure 6.**SSE index evaluation over a different number of groups considering nine clustering methods.

Algorithm | (a) | (b) | (c) |
---|---|---|---|

GA | 89.78% | 92.00% | 88.67% |

FCM | 89.78% | 89.33% | 91.33% |

K-Medoids | 89.78% | 89.33% | 91.33% |

K-Means | 89.78% | 89.33% | 91.33% |

Hierarchical | 89.78% | 89.33% | 90.00% |

PSO | 89.11% | 90.00% | 88.67% |

DE | 89.11% | 90.00% | 88.67% |

SOM | 85.33% | 85.33% | 85.33% |

DBSCAN | 67.78% | 68.00% | 67.33% |

**Table 2.**Overall ranking using Borda count method considering different indexes and algorithm results.

Algorithm | SSE | SSW | SSB | WB | CH | SI | Total |
---|---|---|---|---|---|---|---|

Hierarchical | 8 | 8 | 7 | 8 | 8 | 9 | 48 |

K-Medoids | 9 | 9 | 8 | 9 | 9 | 3 | 47 |

K-Means | 7 | 7 | 6 | 7 | 7 | 8 | 42 |

GA | 6 | 6 | 5 | 6 | 6 | 7 | 36 |

PSO | 5 | 5 | 4 | 5 | 5 | 6 | 30 |

DE | 4 | 4 | 3 | 4 | 4 | 4 | 23 |

DBSCAN | 1 | 1 | 9 | 1 | 1 | 5 | 18 |

SOM | 3 | 3 | 2 | 3 | 3 | 2 | 16 |

FCM | 2 | 2 | 1 | 2 | 2 | 1 | 10 |

**Table 3.**Ranking using Borda count method considering results from Hierarchical, K-Means and K-Medoids with and without PCA application.

Algorithm | SSE | SSW | SSB | WB | CH | SI | Total | |
---|---|---|---|---|---|---|---|---|

With PCA | K-Medoids | 3 | 3 | 3 | 3 | 3 | 1 | 16 |

Hierarchical | 2 | 2 | 2 | 2 | 2 | 3 | 13 | |

K-Means | 1 | 1 | 1 | 1 | 1 | 2 | 7 | |

Without PCA | K-Medoids | 3 | 3 | 3 | 3 | 3 | 2 | 17 |

Hierarchical | 2 | 2 | 1 | 1 | 1 | 3 | 10 | |

K-Means | 1 | 1 | 2 | 2 | 2 | 1 | 9 |

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**MDPI and ACS Style**

Guerreiro, M.T.; Guerreiro, E.M.A.; Barchi, T.M.; Biluca, J.; Alves, T.A.; de Souza Tadano, Y.; Trojan, F.; Siqueira, H.V.
Anomaly Detection in Automotive Industry Using Clustering Methods—A Case Study. *Appl. Sci.* **2021**, *11*, 9868.
https://doi.org/10.3390/app11219868

**AMA Style**

Guerreiro MT, Guerreiro EMA, Barchi TM, Biluca J, Alves TA, de Souza Tadano Y, Trojan F, Siqueira HV.
Anomaly Detection in Automotive Industry Using Clustering Methods—A Case Study. *Applied Sciences*. 2021; 11(21):9868.
https://doi.org/10.3390/app11219868

**Chicago/Turabian Style**

Guerreiro, Marcio Trindade, Eliana Maria Andriani Guerreiro, Tathiana Mikamura Barchi, Juliana Biluca, Thiago Antonini Alves, Yara de Souza Tadano, Flávio Trojan, and Hugo Valadares Siqueira.
2021. "Anomaly Detection in Automotive Industry Using Clustering Methods—A Case Study" *Applied Sciences* 11, no. 21: 9868.
https://doi.org/10.3390/app11219868