Combining Grammatical Evolution with Modal Interval Analysis: An Application to Solve Problems with Uncertainty
Abstract
:1. Introduction
Related Work
2. Predictive Modeling by Grammatical Evolution
Context-Free Grammar
3. Interval Methods
3.1. Modal Interval Analysis
3.2. Modal Interval Extensions
3.3. Primary Theorems
3.4. Algorithm
4. Interval-Based Grammatical Evolution
4.1. Including Uncertainty in Grammatical Evolution
- Manage intervals as inputs of the system;
- Generate intervals as parameters;
- Compute intervals as basic operations.
4.2. Uncertainty in Context-Free Grammars
4.3. Intervalized Fitness Functions
4.4. Predictive Modeling Solutions
5. Implementation, Results, and Discussions
5.1. Illustrative Example
5.2. Modeling River Velocity
Algorithm 1: Context-free grammar used in our case study, where [Constant] is a non-terminal that generates real constants following a classical implementation via digit concatenation. |
[Expression] → [Term] | [Term][OperatorSimple][Expression] [Term] → ([Interval] [OperatorB] [OperatorComplex] (Bed Slope∧[Constant])) | ([Interval] [OperatorB] [OperatorComplex] (Flow Rate∧[Constant])) [OperatorComplex] → sqrt | sin | log | pow | [Constant]∧ | cos | [OperatorComplex][OperatorComplex] | [] [OperatorSimple] → [OperatorA] | [OperatorB] [OperatorA] → + | - [OperatorB] → * | ÷ |
5.3. Results
5.4. Discussion
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Hyperparameter | Value |
---|---|
Population size | 500 |
Generations | 100 |
Crossover prob. | 0.9 |
Mutation prob. | 0.03 |
K (weighting factor) | 10 |
Tournament size | 2 |
Max. Wraps | 1 |
Chromosome length | 50 |
Elitism | 1 |
N (number of executions) | 20 |
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Contreras, I.; Calm, R.; Sainz, M.A.; Herrero, P.; Vehi, J. Combining Grammatical Evolution with Modal Interval Analysis: An Application to Solve Problems with Uncertainty. Mathematics 2021, 9, 631. https://doi.org/10.3390/math9060631
Contreras I, Calm R, Sainz MA, Herrero P, Vehi J. Combining Grammatical Evolution with Modal Interval Analysis: An Application to Solve Problems with Uncertainty. Mathematics. 2021; 9(6):631. https://doi.org/10.3390/math9060631
Chicago/Turabian StyleContreras, Ivan, Remei Calm, Miguel A. Sainz, Pau Herrero, and Josep Vehi. 2021. "Combining Grammatical Evolution with Modal Interval Analysis: An Application to Solve Problems with Uncertainty" Mathematics 9, no. 6: 631. https://doi.org/10.3390/math9060631
APA StyleContreras, I., Calm, R., Sainz, M. A., Herrero, P., & Vehi, J. (2021). Combining Grammatical Evolution with Modal Interval Analysis: An Application to Solve Problems with Uncertainty. Mathematics, 9(6), 631. https://doi.org/10.3390/math9060631