The Algorithm That Maximizes the Accuracy of kClassification on the Set of Representatives of the k Equivalence Classes
Abstract
:1. Introduction
2. Theoretical Section
2.1. Formulation of the Problem and Proposed Idea
2.2. The Proposed Graph Algorithms
2.2.1. Maximin Algorithm
Algorithm 1 The Maximin Algorithm. 

2.2.2. Maximal Algorithm
Algorithm 2 Maximal Algorithm. 

2.2.3. Complexity of the Proposed Maximin and Maximal Algorithms
3. Materials and Methods
3.1. Data Simulation
3.2. Data Classification
 Linear: $(x,{x}^{\prime})$;
 RBF: $exp(\gamma x{x}^{\prime}{}^{2})$, where $\gamma $ must be greater than 0.
3.3. Proposed Algorithms’ Application and kClassification Accuracy Comparison
 kclassification based on the result of the combination obtained by the Maximin Algorithm (Algorithm 1)—corresponds to a kclique with the highest lowest weight of the clique edge (the highest worst accuracy of the binary classification);
 kclassification based on the result of the combination obtained by the Maximal Algorithm (Algorithm 2)—the obtained representatives’ combinations based on the kclique with a maximum total weight;
 The best kclassification accuracy from all possible combinations of representatives for a given simulation—the result of a complete enumeration (the bruteforce algorithm).
3.4. Runtime of Maximin and Maximal Algorithms and BruteForce Algorithm
4. Results
4.1. Data Simulation and Graph Representation
4.2. kClassification Accuracy on Simulated Data Using the Maximin and Maximal Algorithms and the Best kClassification Accuracy Obtained by the BruteForce Algorithm
4.3. Runtime on Simulated Data Using the Maximin and Maximal Algorithms and Complete Enumeration of the BruteForce Algorithm
5. Discussion
6. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
BCI  brain–computer interface 
EEG  electroencephalogram 
SVM  support vector machine 
RVM  relevance vector machine 
kNN  knearestneighbors algorithm 
LDA  linear discriminant analysis 
NPcomplete  nondeterministic polynomialtime complete 
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Number of Classes  A Set of Distributions Corresponds to the kClique from the Maximin Algorithm  A Set of Distributions Corresponds to a Maximal kClique from the Maximal Algorithm  The Best kClassification Accuracy from the BruteForce Algorithm  Median kClassification Accuracy  Minimal kClassification Accuracy 

kclassification accuracy (SVM with linear kernel)  
0.93  0.93  0.93  0.72  0.5  
0.98  0.98  0.98  0.75  0.34  
4  0.93  0.98  0.98  0.81  0.53 
0.97  0.97  0.97  0.75  0.5  
0.87  0.87  0.97  0.75  0.46  
0.83  0.85  0.89  0.59  0.3  
0.92  0.92  0.92  0.58  0.25  
8  0.89  0.85  0.9  0.59  035 
0.68  0.68  0.75  0.54  0.32  
0.76  0.78  0.82  0.55  0.26  
0.64  0.63  0.64  0.50  0.19  
16  0.54  0.48  0.54  0.27  0.13 
0.48  0.47  0.50  0.29  0.09  
0.49  0.49  0.49  0.28  0.15  
0.64  0.65  0.66  0.30  0.22  
kclassification accuracy (SVM with RBF kernel)  
0.98  0.98  0.98  0.73  0.33  
0.87  0.89  0.92  0.75  0.48  
4  0.87  0.87  0.87  0.67  0.42 
0.95  0.95  0.95  0.78  0.48  
0.87  0.87  0.94  0.68  0.41  
0.81  0.78  0.88  0.59  0.33  
0.82  0.88  0.88  0.58  0.33  
8  0.9  0.9  0.9  0.6  03 
0.68  0.79  0.8  0.55  0.28  
0.69  0.7  0.75  0.53  0.25  
0.63  0.65  0.66  0.51  0.23  
16  0.48  0.48  0.48  0.29  0.14 
0.53  0.47  0.53  0.28  0.12  
0.48  0.47  0.50  0.29  0.09  
0.62  0.64  0.65  0.31  0.23 
Runtime  

Number of Classes  Maximin Algorithm  Maximal Algorithm  BruteForce Algorithm 
4  3 ms  10 ms  700 ms 
8  30 ms  300 ms  10 min 
16  1200 ms  10 h  160 h 
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Bernadotte, A. The Algorithm That Maximizes the Accuracy of kClassification on the Set of Representatives of the k Equivalence Classes. Mathematics 2022, 10, 2810. https://doi.org/10.3390/math10152810
Bernadotte A. The Algorithm That Maximizes the Accuracy of kClassification on the Set of Representatives of the k Equivalence Classes. Mathematics. 2022; 10(15):2810. https://doi.org/10.3390/math10152810
Chicago/Turabian StyleBernadotte, Alexandra. 2022. "The Algorithm That Maximizes the Accuracy of kClassification on the Set of Representatives of the k Equivalence Classes" Mathematics 10, no. 15: 2810. https://doi.org/10.3390/math10152810