# Convolutional Support Vector Models: Prediction of Coronavirus Disease Using Chest X-rays

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## Abstract

**:**

## 1. Introduction

## 2. Convolutional Networks

#### 2.1. Multi-Layer Perceptron Neural Networks

#### 2.2. Convolution Neural Networks

## 3. Support Vector Machines

#### 3.1. Support Vector Classifier

#### 3.2. Kernel Transformation

#### 3.3. Convolution Procedures and SVM

## 4. Simulation Study

#### Experimental Setup

- Accuracy (ACC): is the rate between correct predictions and total of predictions. It is sensitive to classes unbalancing.
- Sensitivity (SEN): known as recall, it is the rate of true positive predictions and all positive predictions.
- Specificity (SPC): it is the rate of true negative predictions and all negative predictions.
- Matthew’s correlation coefficient (MCC): this metric measures the correlation between true and predicted values. It may vary from −1 to 1 and the closer to 1 the correlation value is, the better is the predictions.
- F1 Score (F1): this metric is the harmonic mean of precision and recall. Precision indicates how many positive predictions are positive.

## 5. Real Data Study

#### 5.1. Data and X-ray Image Acquisition

#### 5.2. Predictive Models

## 6. Final Considerations

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Generic model architecture with convolutional layers. Source: prepared by the authors and inspired by [42].

**Figure 2.**Basic representation of a perceptron. Source: [44].

**Figure 3.**MLP architecture. Source: [44].

**Figure 4.**Convolutional neural network (CNN) architecture. Source: prepared by the authors and inspired by [42].

**Figure 5.**Support vector machine (SVM) representation showing the hyperplane, margins, and associated support vectors. Source: prepared by the authors.

**Figure 6.**Representation of a linear SVM using (

**a**) rigid margins and (

**b**) soft margins. Source: Adapted from [49] and prepared by the authors.

**Figure 7.**The kernel transformation (trick) process. A non-linear data space (left) is mapped, through a non-linear kernel, into a new linear separable data space (right). Source: Adapted from [49] and prepared by the authors.

**Figure 8.**Convolution support vector machine (CSVM) architecture. Source: prepared by the authors and inspired by [42].

**Figure 9.**Representation of a flattening layer. Source: adapted from [52] and prepared by the authors.

**Figure 10.**Synthetic 64 × 64 pixels images: (

**a**,

**b**) with $\mu $ difference in channel G equal to one standard deviation for classes 1 and 2; (

**c**,

**d**) with $\mu $ difference in channel G equals to three standard deviations for Classes 1 and 2. Source: prepared by the authors.

**Figure 11.**Representation of models. Source: adapted from [54] and prepared by the authors.

**Figure 12.**Example of X-rays: (

**a**,

**b**) coronavirus (COVID-19); (

**c**,

**d**) other diseases; and (

**e**,

**f**) healthy. Source: prepared by the authors.

**Figure 13.**Metadata provided by the COVID-19 reference dataset. Age histogram (

**a**) and gender pie chart (

**b**). Source: prepared by the authors.

**Figure 14.**Number of times which a method obtained greater accuracy (ACC) than others. The count summarizes 100 holdout values. It is clear the superiority of CSVM${}_{Pol}$ when it is compared with the other models. The darker is the table cell, the bigger is its score. Source: prepared by the authors.

**Figure 15.**Number of times in which a method obtained greater F1 Score than the others. The count summarizes 100 holdout values. It is clear the superiority of CSVM${}_{F1}$ when it is compared with the other models. The darker is the table cell, the bigger is its score. Source: prepared by the authors.

Kernel Type | $\mathit{K}(\mathit{x},{\mathit{x}}^{\prime})$ | Parameters |
---|---|---|

Linear | $\gamma (x\xb7{x}^{\prime})+d$ | $\gamma ,d$ |

Polynomial | ${(\gamma (x\xb7{x}^{\prime})+d)}^{q}$ | $\gamma ,d,q$ |

Gaussian | $exp\left(\frac{-\left|\right|x-{x}^{\prime}{\left|\right|}^{2}}{2{\gamma}^{2}}\right)$ | $\gamma $ |

Class 1 | Class 2 | |||
---|---|---|---|---|

$\mathit{\mu}$ Difference | Channel | $\mathit{\mu}$ | $\mathit{\sigma}$ | |

R | 128 | 128 | 25 | |

1SD | G | 119 | 137 | 18 |

B | 128 | 128 | 30 | |

R | 128 | 128 | 25 | |

3SD | G | 101 | 155 | 18 |

B | 128 | 128 | 30 |

**Table 3.**Simulation results for convolutional methods with $\mu $ difference equals to 1SD. The time column refers to the model’s training time. The suitable model performance is highlighted in bold for each case.

Samples | Method | ACC | SEN | SPC | MCC | F1 | Time |
---|---|---|---|---|---|---|---|

CSVM${}_{Gau}$ | 0.46 | 0.40 | 0.51 | −0.10 | 0.45 | 0.09 | |

CSVM${}_{Lin}$ | 0.45 | 0.41 | 0.49 | −0.10 | 0.44 | 0.09 | |

100 | CSVM${}_{Pol}$ | 0.45 | 0.41 | 0.49 | −0.10 | 0.45 | 0.09 |

CNN${}_{1}$ | 0.49 | 0.23 | 0.75 | −0.04 | 0.50 | 0.12 | |

CNN${}_{2}$ | 0.49 | 0.11 | 0.88 | −0.08 | 0.47 | 0.24 | |

CSVM${}_{Gau}$ | 0.52 | 0.50 | 0.54 | 0.05 | 0.50 | 0.10 | |

CSVM${}_{Lin}$ | 0.54 | 0.54 | 0.54 | 0.08 | 0.53 | 0.10 | |

300 | CSVM${}_{Pol}$ | 0.54 | 0.55 | 0.53 | 0.08 | 0.54 | 0.10 |

CNN${}_{1}$ | 0.51 | 0.48 | 0.55 | 0.03 | 0.49 | 0.20 | |

CNN${}_{2}$ | 0.50 | 0.08 | 0.91 | −0.07 | 0.40 | 0.53 | |

CSVM${}_{Gau}$ | 0.88 | 0.88 | 0.88 | 0.77 | 0.88 | 0.13 | |

CSVM${}_{Lin}$ | 0.88 | 0.87 | 0.88 | 0.76 | 0.88 | 0.12 | |

500 | CSVM${}_{Pol}$ | 0.87 | 0.86 | 0.88 | 0.74 | 0.87 | 0.12 |

CNN${}_{1}$ | 0.87 | 0.85 | 0.88 | 0.74 | 0.86 | 0.27 | |

CNN${}_{2}$ | 0.50 | 0.01 | 0.99 | 0.29 | 0.70 | 0.87 | |

CSVM${}_{Gau}$ | 0.98 | 0.98 | 0.98 | 0.97 | 0.98 | 0.24 | |

CSVM${}_{Lin}$ | 0.98 | 0.98 | 0.99 | 0.97 | 0.98 | 0.20 | |

1000 | CSVM${}_{Pol}$ | 0.98 | 0.98 | 0.98 | 0.97 | 0.98 | 0.22 |

CNN${}_{1}$ | 0.99 | 0.99 | 0.99 | 0.98 | 0.99 | 0.50 | |

CNN${}_{2}$ | 0.77 | 0.55 | 0.98 | 0.93 | 0.97 | 2.46 |

**Table 4.**Simulation results for convolutional methods with $\mu $ difference equals to 3SD. The time column refers to the model’s training time. The suitable model performance is highlighted in bold for each case.

Samples | Method | ACC | SEN | SPC | MCC | F1 | Time |
---|---|---|---|---|---|---|---|

CSVM${}_{Gau}$ | 0.45 | 0.41 | 0.49 | −0.10 | 0.45 | 0.09 | |

CSVM${}_{Lin}$ | 0.44 | 0.43 | 0.45 | −0.13 | 0.45 | 0.10 | |

100 | CSVM${}_{Pol}$ | 0.44 | 0.42 | 0.46 | −0.12 | 0.45 | 0.10 |

CNN${}_{1}$ | 0.48 | 0.22 | 0.74 | −0.13 | 0.50 | 0.13 | |

CNN${}_{2}$ | 0.50 | 0.16 | 0.84 | −0.12 | 0.56 | 0.24 | |

CSVM${}_{Gau}$ | 0.51 | 0.54 | 0.47 | 0.02 | 0.52 | 0.10 | |

CSVM${}_{Lin}$ | 0.51 | 0.51 | 0.50 | 0.19 | 0.51 | 0.10 | |

300 | CSVM${}_{Pol}$ | 0.51 | 0.52 | 0.51 | 0.03 | 0.51 | 0.10 |

CNN${}_{1}$ | 0.51 | 0.48 | 0.53 | 0.02 | 0.48 | 0.21 | |

CNN${}_{2}$ | 0.49 | 0.17 | 0.82 | −0.02 | 0.38 | 0.54 | |

CSVM${}_{Gau}$ | 0.88 | 0.88 | 0.88 | 0.78 | 0.88 | 0.12 | |

CSVM${}_{Lin}$ | 0.89 | 0.89 | 0.89 | 0.79 | 0.89 | 0.11 | |

500 | CSVM${}_{Pol}$ | 0.90 | 0.89 | 0.90 | 0.80 | 0.90 | 0.11 |

CNN${}_{1}$ | 0.89 | 0.89 | 0.89 | 0.79 | 0.89 | 0.28 | |

CNN${}_{2}$ | 0.88 | 0.87 | 0.90 | 0.80 | 0.89 | 0.95 | |

CSVM${}_{Gau}$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.18 | |

CSVM${}_{Lin}$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.16 | |

1000 | CSVM${}_{Pol}$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.16 |

CNN${}_{1}$ | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.48 | |

CNN${}_{2}$ | 0.99 | 0.99 | 1.00 | 1.00 | 1.00 | 2.65 |

COVID-19 | Other Diseases | Healthy | Total | |
---|---|---|---|---|

Quantity | 217 | 108 | 112 | 437 |

**Table 6.**Performance results. MLP1 corresponds to the model with more parameters, while MLP2 corresponds to the model with fewer parameters. The same applies to the CNN models. The time column refers to the training time.The suitable metrics are highlighted in bold.

Method | ACC | F1 | MCC | Time |
---|---|---|---|---|

MLP${}_{1}$ | 95.54 | 95.46 | 91.57 | 0.0422 |

MLP${}_{2}$ | 96.59 | 96.56 | 93.48 | 0.0370 |

CNN${}_{1}$ | 96.67 | 96.63 | 93.48 | 0.7792 |

CNN${}_{2}$ | 96.73 | 96.67 | 93.74 | 0.7585 |

SVM${}_{Lin}$ | 80.79 | 80.21 | 61.98 | 0.0074 |

SVM${}_{Pol}$ | 77.90 | 77.24 | 56.30 | 0.0076 |

SVM${}_{RBF}$ | 83.45 | 83.86 | 67.39 | 0.0067 |

CSVM${}_{Lin}$ | 98.00 | 97.97 | 96.11 | 0.0146 |

CSVM${}_{Pol}$ | 96.57 | 98.13 | 96.36 | 0.0143 |

CSVM${}_{Gau}$ | 98.14 | 96.59 | 93.34 | 0.0151 |

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Maia, M.; Pimentel, J.S.; Pereira, I.S.; Gondim, J.; Barreto, M.E.; Ara, A. Convolutional Support Vector Models: Prediction of Coronavirus Disease Using Chest X-rays. *Information* **2020**, *11*, 548.
https://doi.org/10.3390/info11120548

**AMA Style**

Maia M, Pimentel JS, Pereira IS, Gondim J, Barreto ME, Ara A. Convolutional Support Vector Models: Prediction of Coronavirus Disease Using Chest X-rays. *Information*. 2020; 11(12):548.
https://doi.org/10.3390/info11120548

**Chicago/Turabian Style**

Maia, Mateus, Jonatha S. Pimentel, Ivalbert S. Pereira, João Gondim, Marcos E. Barreto, and Anderson Ara. 2020. "Convolutional Support Vector Models: Prediction of Coronavirus Disease Using Chest X-rays" *Information* 11, no. 12: 548.
https://doi.org/10.3390/info11120548