# A Robust Algorithm for Classification and Diagnosis of Brain Disease Using Local Linear Approximation and Generalized Autoregressive Conditional Heteroscedasticity Model

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## Abstract

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## 1. Introduction

## 2. Methods and Materials

#### 2.1. Image Processing

#### 2.2. Discrete Wavelet Transform (DWT)

#### 2.3. Generalized Autoregressive Conditional Heteroscedasticity

Algorithm 1. GARCH |

1: Input: ${y}_{t},P,Q,dist$2: Output: ${a}_{i},{\u03f5}_{t}$3: Step 1: Estimate AR(q): 4: ${y}_{t}={a}_{0}+{a}_{1}{y}_{t-1}+\cdots .+{a}_{q}{y}_{t-q}+{\u03f5}_{t}$ 5: ${\widehat{\u03f5}}_{t}^{2}={\widehat{a}}_{0}+{{\displaystyle \sum}}_{i=1}^{q}\widehat{{a}_{i}}{\widehat{\u03f5}}_{t-i}^{2}$ 6: Step 2: Compute and plot the autocorrelations of ${\u03f5}^{2}$ by: 7: $\rho =\frac{{{\displaystyle \sum}}_{t=i+1}^{T}\left({\widehat{\u03f5}}_{t}^{2}-{\widehat{\sigma}}_{t}^{2}\right)\left({\widehat{\u03f5}}_{t-1}^{2}-{\widehat{\sigma}}_{t-1}^{2}\right)}{{{\displaystyle \sum}}_{t=1}^{T}{\left({\widehat{\u03f5}}_{t}^{2}-{\widehat{\sigma}}_{t}^{2}\right)}^{2}}$ 8: Step 3: null hypothesis states that there are no ARCH or GARCH errors |

#### 2.4. Local Linear Approximation

#### 2.5. K-Nearest Neighbour Algorithm

#### 2.6. Proposed Method

Algorithm 2. Presented |

1: Input:${y}_{m\times m}=\left\{m\times m\right\}\in {R}^{2}$2: Switch:3: Case 1: WGK4: Step 1: Wavelet decomposition for all images 5: Step 2: Calculate GARCH parameters for sub-bands of high-frequency detail of (HH1, HL1, LH1, HL2, LH2) 6: Step 3: Normalization of features 7: Step 4: Feature reduction using PCA and PCA+LDA 8: Step 5: Classification of Features using KNN 9: Case 2: D-WGK10: Step 1: Apply homomorphic filtering for all images 11: Step 2: Wavelet decomposition for all images 12: Step 2: Calculate GARCH parameters for all sub-bands of high-frequency detail of (HH1, HL1, LH1, HL2, LH2, LL2) 13: Step 3: Normalization of features 14: Step 4: Feature reduction using PCA and PCA+LDA 15: Step 5: Classification of Features using KNN 16: Case 3: WLK17: Step 1: Wavelet decomposition for all images 18: Step 2: Calculate LLA parameters 19: Step 3: Normalization of features 20: Step 4: Feature reduction using PCA and PCA+LDA 21: Step 5: Classification of Features using KNN 22: Comparison and analysis |

## 3. Results and Discussion

#### 3.1. Datasets

#### 3.2. Two-dimensional Discrete Wavelet Transforms (2D-DWT)

#### 3.3. Feature Reduction

- WGK: Using GARCH without LL2 + PCA
- WGK: Using GARCH without LL2 + PCA + LDA
- D-WGK: Using Homomorphic filtering + GARCH with LL2 + PCA
- WLK: Using LLA + PCA
- WLK: Using LLA + PCA + LDA

#### 3.4. The Classification Results

## 4. The Complexity Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Ethical Approval

## References

- Herszterg, I.; Poggi, M.; Vidal, T. Two-Dimensional Phase Unwrapping via Balanced Spanning Forests. INFORMS J. Comput.
**2019**, 31, 527–543. [Google Scholar] [CrossRef] - Won, D.; Manzour, H.; Chaovalitwongse, W. Convex Optimization for Group Feature Selection in Networked Data. INFORMS J. Comput.
**2020**, 32, 182–198. [Google Scholar] [CrossRef] - Zhang, Y.; Dong, Z.; Wu, L.; Wang, S. A hybrid method for MRI brain image classification. Expert Syst. Appl.
**2011**, 38, 10049–10053. [Google Scholar] [CrossRef] - Abdullah, N.; Ngah, U.K.; Aziz, S.A. Image classification of brain MRI using support vector machine. In Proceedings of the 2011 IEEE International Conference on Imaging Systems and Techniques, Penang, Malaysia, 17–18 May 2011; pp. 242–247. [Google Scholar]
- Gillis, N.; Vavasis, S.A. On the Complexity of Robust PCA and ℓ1-Norm Low-Rank Matrix Approximation. Math. Oper. Res.
**2018**, 43, 1072–1084. [Google Scholar] [CrossRef] [Green Version] - Abdulkareem, M.; Bakhary, N.; Vafaei, M.; Noor, N.M.; Mohamed, R.N. Application of two-dimensional wavelet transform to detect damage in steel plate structures. Measurement
**2019**, 146, 912–923. [Google Scholar] [CrossRef] - Talo, M.; Baloglu, U.B.; Yıldırım, Ö.; Rajendra Acharya, U. Application of deep transfer learning for automated brain abnormality classification using MR images. Cogn. Syst. Res.
**2019**, 54, 176–188. [Google Scholar] [CrossRef] - Abdelaziz Ismael, S.A.; Mohammed, A.; Hefny, H. An enhanced deep learning approach for brain cancer MRI images classification using residual networks. Artif. Intell. Med.
**2020**, 102, 101779. [Google Scholar] [CrossRef] - Chaplot, S.; Patnaik, L.M.; Jagannathan, N.R. Classification of magnetic resonance brain images using wavelets as input to support vector machine and neural network. Biomed. Signal. Process. Control
**2006**, 1, 86–92. [Google Scholar] [CrossRef] - Hackmack, K.; Paul, F.; Weygandt, M.; Allefeld, C.; Haynes, J.D. Multi-scale classification of disease using structural MRI and wavelet transform. Neuroimage
**2012**, 62, 48–58. [Google Scholar] [CrossRef] - Maitra, M.; Chatterjee, A. Hybrid multiresolution Slantlet transform and fuzzy c-means clustering approach for normal-pathological brain MR image segregation. Med. Eng. Phys.
**2008**, 30, 615–623. [Google Scholar] [CrossRef] - Ramathilagam, S.; Pandiyarajan, R.; Sathya, A.; Devi, R.; Kannan, S.R. Modified fuzzy c-means algorithm for segmentation of T1–T2-weighted brain MRI. J. Comput. Appl. Math.
**2011**, 235, 1578–1586. [Google Scholar] [CrossRef] - Rivest-Hénault, D.; Cheriet, M. Unsupervised MRI segmentation of brain tissues using a local linear model and level set. Magn. Reson. Imaging
**2011**, 29, 243–259. [Google Scholar] [CrossRef] [PubMed] - Hussain, S.J.; Savithri, A.S.; Devi, P.V.S. Segmentation of brain MRI with statistical and 2D wavelet features by using neural networks. In Proceedings of the 3rd International Conference on Trendz in Information Sciences & Computing (TISC2011), Chennai, India, 8–9 December 2011; pp. 154–159. [Google Scholar]
- Bhattacharyya, D.; Kim, T.-H. Brain Tumor Detection Using MRI Image Analysis; Springer: Berlin/Heidelberg, Germany, 2011; pp. 307–314. [Google Scholar]
- Kim, H.T.; Kim, B.Y.; Park, E.H.; Kim, J.W.; Hwang, E.W.; Han, S.K.; Cho, S. Computerized recognition of Alzheimer disease-EEG using genetic algorithms and neural network. Future Gener. Comput. Syst.
**2005**, 21, 1124–1130. [Google Scholar] [CrossRef] - Abásolo, D.; Hornero, R.; Espino, P.; Poza, J.; Sánchez, C.I.; de la Rosa, R. Analysis of regularity in the EEG background activity of Alzheimer’s disease patients with Approximate Entropy. Clin. Neurophysiol.
**2005**, 116, 1826–1834. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gholipour, A.; Estroff, J.A.; Barnewolt, C.E.; Connolly, S.A.; Warfield, S.K. Fetal brain volumetry through MRI volumetric reconstruction and segmentation. Int. J. Comput. Assist. Radiol. Surg.
**2011**, 6, 329–339. [Google Scholar] [CrossRef] [Green Version] - Zacharaki, E.I.; Kanas, V.G.; Davatzikos, C. Investigating machine learning techniques for MRI-based classification of brain neoplasms. Int. J. Comput. Assist. Radiol. Surg.
**2011**, 6, 821–828. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Fritzsche, K.H.; Stieltjes, B.; Schlindwein, S.; van Bruggen, T.; Essig, M.; Meinzer, H.P. Automated MR morphometry to predict Alzheimer’s disease in mild cognitive impairment. Int. J. Comput. Assist. Radiol. Surg.
**2010**, 5, 623–632. [Google Scholar] [CrossRef] [PubMed] - Devanand, D.P.; Bansal, R.; Liu, J.; Hao, X.; Pradhaban, G.; Peterson, B.S. MRI hippocampal and entorhinal cortex mapping in predicting conversion to Alzheimer’s disease. Neuroimage
**2012**, 60, 1622–1629. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zöllner, F.G.; Emblem, K.E.; Schad, L.R. SVM-based glioma grading: Optimization by feature reduction analysis. Z. Med. Phys.
**2012**, 22, 205–214. [Google Scholar] [CrossRef] [PubMed] - Afshar, P.; Mohammadi, A.; Plataniotis, K.N. Brain Tumor Type Classification via Capsule Networks. In Proceedings of the 2018 25th IEEE International Conference on Image Processing (ICIP), Athens, Greece, 7–10 October 2018; pp. 3129–3133. [Google Scholar]
- Mohan, G.; Subashini, M.M. MRI based medical image analysis: Survey on brain tumor grade classification. Biomed. Signal Process. Control
**2018**, 39, 139–161. [Google Scholar] [CrossRef] - Huang, H.; Meng, F.; Zhou, S.; Jiang, F.; Manogaran, G. Brain Image Segmentation Based on FCM Clustering Algorithm and Rough Set. IEEE Access
**2019**, 7, 12386–12396. [Google Scholar] [CrossRef] - Breton, M.; Frutos, J.d. Option Pricing Under GARCH Processes Using PDE Methods. Oper. Res.
**2010**, 58, 1148–1157. [Google Scholar] [CrossRef] - Zhao, Y.-B.; Luo, Z.-Q. Constructing New Weighted ℓ1-Algorithms for the Sparsest Points of Polyhedral Sets. Math. Oper. Res.
**2016**, 42, 57–76. [Google Scholar] [CrossRef] - Milstein, A.; Topol, E.J. Computer vision’s potential to improve health care. Lancet
**2020**, 395, 1537. [Google Scholar] [CrossRef] - Liu, D.; Oczak, M.; Maschat, K.; Baumgartner, J.; Pletzer, B.; He, D.; Norton, T. A computer vision-based method for spatial-temporal action recognition of tail-biting behaviour in group-housed pigs. Biosyst. Eng.
**2020**, 195, 27–41. [Google Scholar] [CrossRef] - Burrus, C.S.; Gopinath, R.A. Introduction to Wavelets and Wavelet Transforms: A Primer; Perason Prentice Hall: Upper Saddle River, NJ, USA, 1998. [Google Scholar]
- Bollerslev, T. Generalized autoregressive conditional heteroskedasticity. J. Econom.
**1986**, 31, 307–327. [Google Scholar] [CrossRef] [Green Version] - Engle, R.F. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica
**1982**, 50, 987–1007. [Google Scholar] [CrossRef] - Boker, S.M. Differential Models and “Differential Structural Equation Modeling of Intraindividual Variability”; American Psychological Association: 2001. Available online: https://psycnet.apa.org/record/2001-01077-006 (accessed on 2 June 2020).
- Lutu, P.E.N.; Engelbrecht, A.P. Base Model Combination Algorithm for Resolving Tied Predictions for K-Nearest Neighbor OVA Ensemble Models. INFORMS J. Comput.
**2012**, 25, 517–526. [Google Scholar] [CrossRef] - Fukunaga, K.; Narendra, P.M. A Branch and Bound Algorithm for Computing k-Nearest Neighbors. IEEE Trans. Comput.
**1975**, C-24, 750–753. [Google Scholar] [CrossRef] - Kalbkhani, H.; Shayesteh, M.G.; Zali-Vargahan, B. Robust algorithm for brain magnetic resonance image (MRI) classification based on GARCH variances series. Biomed. Signal. Process. Control
**2013**, 8, 909–919. [Google Scholar] [CrossRef] - Marti-nez, J.M.P.; Berlanga, R.; Aramburu, M.J.; Pedersen, T.B. Integrating Data Warehouses with Web Data: A Survey. IEEE Trans. Knowl. Data Eng.
**2008**, 20, 940–955. [Google Scholar] [CrossRef] - Johnstone, I.M.; Lu, A.Y. On Consistency and Sparsity for Principal Components Analysis in High Dimensions. J. Am. Stat. Assoc.
**2009**, 104, 682–693. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shi, Q.; Zhang, H. Fault diagnosis of an autonomous vehicle with an improved SVM algorithm subject to unbalanced datasets. IEEE Trans. Ind. Electron.
**2020**. [Google Scholar] [CrossRef] - Qi, Z.; Shi, Q.; Zhang, H. Tuning of digital PID controllers using particle swarm optimization algorithm for a CAN-based DC motor subject to stochastic delays. IEEE Trans. Ind. Electron.
**2019**, 67, 5637–5646. [Google Scholar] [CrossRef]

**Figure 5.**Normalized cumulative summation of eigenvalues of training data for GARCH (1, 1) with and without LL2 (Eight-classes).

**Figure 6.**Normalized cumulative summation of eigenvalues of training data for GARCH (1, 1) with and without LL2 (two-classes).

**Figure 7.**Normalized cumulative summation of eigenvalues of training data for the GARCH and LLA methods for different PCA and PCA + LDA methods (eight-classes).

**Figure 8.**Normalized cumulative summation of eigenvalues of training data for the GARCH and LLA methods for different PCA and PCA + LDA methods (two-classes).

Method | Class | Images | Features | Accuracy |
---|---|---|---|---|

Ref. * | 6 | 56 | 6 | 91.5 |

PCA + LDA (WGK) ** | 8 | 80 | 10 | 89.4 |

PCA (WGK) ** | 8 | 80 | 22 | 90.1 |

Proposed PCA + LDA (D-WGK) | 8 | 240 | 10 | 90.2 |

Proposed PCA (D-WGK) | 8 | 240 | 20 | 89.3 |

Proposed PCA + LDA (WLK) | 8 | 240 | 3 | 92.5 |

Proposed PCA (WLK) | 8 | 240 | 7 | 91.3 |

Diseases | TPR | TNR | PPV | ACC | FPR |
---|---|---|---|---|---|

Alzheimer | 0.933 | 1 | 0.903 | 0.967 | 0 |

Alzheimer+ | 0.933 | 1 | 0.875 | 0.967 | 0 |

Glioma | 0.900 | 1 | 1 | 0.950 | 0 |

Huntington | 0.967 | 1 | 0.906 | 0.983 | 0 |

Meningioma | 0.967 | 1 | 1 | 0.983 | 0 |

Pick | 0.867 | 1 | 0.839 | 0.933 | 0 |

Sarcoma | 0.833 | 1 | 0.926 | 0.917 | 0 |

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**MDPI and ACS Style**

Hamzenejad, A.; Jafarzadeh Ghoushchi, S.; Baradaran, V.; Mardani, A.
A Robust Algorithm for Classification and Diagnosis of Brain Disease Using Local Linear Approximation and Generalized Autoregressive Conditional Heteroscedasticity Model. *Mathematics* **2020**, *8*, 1268.
https://doi.org/10.3390/math8081268

**AMA Style**

Hamzenejad A, Jafarzadeh Ghoushchi S, Baradaran V, Mardani A.
A Robust Algorithm for Classification and Diagnosis of Brain Disease Using Local Linear Approximation and Generalized Autoregressive Conditional Heteroscedasticity Model. *Mathematics*. 2020; 8(8):1268.
https://doi.org/10.3390/math8081268

**Chicago/Turabian Style**

Hamzenejad, Ali, Saeid Jafarzadeh Ghoushchi, Vahid Baradaran, and Abbas Mardani.
2020. "A Robust Algorithm for Classification and Diagnosis of Brain Disease Using Local Linear Approximation and Generalized Autoregressive Conditional Heteroscedasticity Model" *Mathematics* 8, no. 8: 1268.
https://doi.org/10.3390/math8081268