# A Knowledge-Based Hybrid Approach on Particle Swarm Optimization Using Hidden Markov Models

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## Abstract

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## 1. Introduction

## 2. Related Work

## 3. Preliminaries

- Dynamic update: A random or deterministic updates the parameter value. This operation is performed without taking into account the search progress.
- Adaptive update: In this approach, parameter values evolve during the search progress. To change the parameter values, a function that mimics the behavior of the phenomenon is performed. For that, the memory of the search is mainly used. Hence, the parameters are associated with the representation and these are subject to updates in function of the problem’s solution.

## 4. Developed Solution

#### 4.1. Evolutionary Factor f

#### 4.2. Markov Models

#### 4.2.1. Markov Chains

#### 4.2.2. Hidden Markov Model

- Decoding.Given the parameters A, $\pi $, B, and the observed data O, estimate the optimal sequence of hidden states Q;
- Likelihood. Given an HMM $\lambda =(A,B)$ and a sequence of observations O, determine the probability that those observations belong to the HMM, $P\left(O\right|\lambda )$;
- Learning. Given a sequence of observations O and a set of states in the HMM, we learn its parameters A and B.

#### 4.3. HMM-PSO Integration

## 5. Experimental Setup

## 6. Results and Discussion

#### 6.1. Original PSO Comparison

#### 6.2. Exploration/Exploitation Balance

#### 6.3. Convergence Curves

#### 6.4. Results Discussion

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Example of PSO particle distribution: (

**a**) ${d}_{g}\approx {d}_{avg}$ exploration; (

**b**) ${d}_{g}\ll {d}_{avg}$ exploitation, convergence; and (

**c**) ${d}_{g}\gg {d}_{avg}$ jump out.

**Figure 3.**Markov chain. The arcs connecting the nodes/states indicate the transition probability between states.

**Figure 4.**Evolutionary states defined for the Adaptive PSO algorithm: S-$P{h}_{\{1,2,3,4\}}$ represent Exploration, Exploitation, Convergence, and Jump-out, respectively.

**Figure 7.**Exploration and exploitation percentage for original PSO. For small instances (scp41 and scpa1), the algorithm shows an exploitative behavior, for bigger instances (scpnre1 and scpnrh1), the algorithm shows an exploitative behavior. We can observe the transition between inner-phases at iterations 50, 300, and 600.

**Figure 8.**Exploration and exploitation percentage for HPSO. The algorithm shows a mostly exploitative behavior for small and big instances.

**Figure 9.**Convergence of PSO: The algorithm shows a premature convergence, with very few improvements after the first 50 iterations.

**Figure 10.**Convergence of HPSO. The algorithm shows improvements until approximately iteration 150, which represents 50% of the total iterations.

State (Inner-Phase) | Inertial Velocity w | Number of Particles np |
---|---|---|

Exploration | $\omega ={\omega}_{min}+({\omega}_{max}-{\omega}_{min})\xb7Rand(0,1)$ | $np-1$ |

Exploitation | $\omega =\frac{1}{1+\frac{3}{2}{exp}^{-2.6f}}$ | $np+1$ |

Convergence | $\omega ={\omega}_{min}$ | $np+1$ |

Jump out | $\omega ={\omega}_{max}$ | $np-1$ |

Orginal PSO Parameters | Proposed HPSO Parameters | ||
---|---|---|---|

Parameter | Value | Parameter | Value |

$\omega $ | $1-\frac{k}{L+1}$ | ${\omega}_{min}$ | 0.4 |

${\omega}_{max}$ | 0.9 | ||

$n{p}_{min}$ | 5 | $n{p}_{max}$ | 30 |

$n{p}_{max}$ | 50 | $n{p}_{max}$ | 30 |

$c1$ | $2.05\phantom{\rule{3.33333pt}{0ex}}rand(0,1)$ | $c1$ | $2.05\phantom{\rule{3.33333pt}{0ex}}rand(0,1)$ |

$c2$ | $2.05\phantom{\rule{3.33333pt}{0ex}}rand(0,1)$ | $c2$ | $2.05\phantom{\rule{3.33333pt}{0ex}}rand(0,1)$ |

iter. num. | 50 | iter. num. | 50 |

iter. num. | 250 | iter. num. | 250 |

Instance | Optimum | Best HPSO | Best PSO | Avg. HPSO | Avg. PSO | RPD HPSO | RPD PSO |
---|---|---|---|---|---|---|---|

scp41 | 429 | 429 | 430 | 429.81 | 432.419 | 0 | 0.233 |

scp51 | 253 | 253 | 255 | 253.68 | 260.71 | 0 | 0.791 |

scp61 | 138 | 138 | 140 | 138.19 | 140.871 | 0 | 1.449 |

scpa1 | 253 | 253 | 256 | 254.32 | 258.097 | 0.395 | 1.186 |

scpb1 | 69 | 69 | 71 | 69 | 91.129 | 0 | 2.899 |

scpc1 | 227 | 227 | 234 | 228.36 | 238.258 | 0 | 3.084 |

scpd1 | 60 | 60 | 79 | 60.13 | 123.323 | 0 | 31.667 |

scpnre1 | 29 | 29 | 85 | 29 | 106.871 | 0 | 193.103 |

scpnrf1 | 14 | 14 | 39 | 14 | 49.29 | 0 | 178.571 |

scpnrg1 | 176 | 176 | 348 | 178.17 | 480.839 | 0.568 | 97.727 |

scpnrh1 | 63 | 65 | 277 | 65.25 | 349.452 | 1.587 | 339.683 |

Instance | HPSO < PSO | PSO < HPSO |
---|---|---|

scp41 | 0.728 | 0.277 |

scp51 | 0.002 | 0.998 |

scp61 | 0.000 | 1.000 |

scpa1 | 0.000 | 1.000 |

scpb1 | 0.000 | 1.000 |

scpc1 | 0.000 | 1.000 |

scpd1 | 0.000 | 1.000 |

scpnre1 | 0.000 | 1.000 |

scpnrf1 | 0.000 | 1.000 |

scpnrg1 | 0.000 | 1.000 |

scpnrh1 | 0.000 | 1.000 |

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## Share and Cite

**MDPI and ACS Style**

Castillo, M.; Soto, R.; Crawford, B.; Castro, C.; Olivares, R.
A Knowledge-Based Hybrid Approach on Particle Swarm Optimization Using Hidden Markov Models. *Mathematics* **2021**, *9*, 1417.
https://doi.org/10.3390/math9121417

**AMA Style**

Castillo M, Soto R, Crawford B, Castro C, Olivares R.
A Knowledge-Based Hybrid Approach on Particle Swarm Optimization Using Hidden Markov Models. *Mathematics*. 2021; 9(12):1417.
https://doi.org/10.3390/math9121417

**Chicago/Turabian Style**

Castillo, Mauricio, Ricardo Soto, Broderick Crawford, Carlos Castro, and Rodrigo Olivares.
2021. "A Knowledge-Based Hybrid Approach on Particle Swarm Optimization Using Hidden Markov Models" *Mathematics* 9, no. 12: 1417.
https://doi.org/10.3390/math9121417