# Forecasting by Combining Chaotic PSO and Automated LSSVR

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Initialization of the parameters: This is due to the possibility that it is unaware of the location of the global minimum at which the prior optimization problem was resolved.
- Characteristic extract or the characteristic evolution component can typically accomplish the decrease in data dimensionality. Often, this has been accomplished using characteristic extract methods like principal component analysis (PCA). The PCA is ineffective for this study’s objectives since it also wants to produce highly precise predictive models, not just reduce the dimensionality of the data. Nevertheless, the PCA doesn’t take into account the link for variables of input and variables of reply throughout the data reduction process, making it challenging to create a model that is extremely precise. Furthermore, if the input variables’ dimensionality is really high, it might be challenging to interpret the main components that are produced by the PCA. On the other hand, for data sets with high dimensionality, the PSO has been shown to perform better than other methods [11]. A simplified LSSVR model with improved generalization can be created by selecting more information for any data set provided when employing the fewest characteristics possible throughout the characteristic evolution phase.
- Another PSO is employed in the parameter evolution component to optimize the LSSVR’s parameters. Generally, LSSVR generalization ability is governed by the type of kernel, parameters’ kernel, and parameter’s upper bound. Every form of the kernel has benefits and drawbacks, hence a mixed kernel makes sense [12,13,14]. Additionally, computational time and complexity in the training of the algorithm equals the total execution generations multiplied by the number of total solutions and multiplied by the time complexity of the update for each solution.

- The CP-LSSVR is used to initialize the parameters for the parameters initialization issue of LSSVR applications.
- A binary PSO is utilized for feature selection in the input data to improve the model’s interpretability for the issue of requiring LSSVR to preprocess the input characteristics if the dimensions of input space or input characteristics are quite vast.
- A third PSO is applied to optimize its parameters to boost the LSSVR’s capacity for normalization.

## 2. SVR and LSSVR

_{k}. An error ${E}_{k}^{l}$ or ${E}_{k}^{l}$ has been typically tried to reduce in the objective function if α

_{k}is outside the tube. This is depicted in Figure 2. By minimizing the normalization term $\raisebox{1ex}{$\left|\right|a|{|}^{2}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ as well as the training error $C{\displaystyle \sum _{k=1}^{N}({E}_{k}^{u}+{E}_{k}^{l})}$, SVR prevents the training data to be underfitted and overfitted.

_{k}and x

_{s}are vectors in the input space;

**g**signifies the polynomial’s level and T presents the item of intercept constant in Equation (12) and σ

^{2}indicates the Gaussian kernel’s width in Equation (13).

## 3. LSSVR Based on Chaotic Particle Swarm Optimization (CPSO) Algorithm

#### 3.1. Automatic LSSVR Learning Paradigm

- Parameters initialization.
- It is required for LSSVR to preprocess the input characteristics if the dimensions of input space or input characteristics are quite vast, in order to improve the interpretability of the LSSVR-based forecasting model.
- This work adopts a mixed kernel model to get beyond the effect of kernel types because LSSVR normalization ability is frequently governed via (a) kernel type. The next two things, however, heavily rely on the researchers’ artistic ability. (b) kernel parameters: convex combination coefficients (λ
_{1}, λ_{2}, λ_{3}), and kernel parameters (**g**, σ). C is the upper bound parameter.

_{1}, λ

_{2}, λ

_{3}, g, σ, C) are optimized in the third section using PSO. Section 3.2, Section 3.3 and Section 3.4 define in full the contents of each section.

#### 3.2. Chaotic Sequences-Based Parameters Initialization

- Step 0. Generated by Logistic map chaotic sequence by Equation (15), following:

- Step 1. For the m particles in the D-dimensional space, the first generates a random initial value m:

- Step 2. Chaotic sequence to the initial value of m-Equation (15). At that point, m will be the trajectory after Z iterations.
- Step 3. Substituting the chaotic trajectory of the article from m in the selected Z iteration value into the Equation (16). One can compute ${x}_{\upsilon ,k}$

#### 3.3. PSO-Based Input Features Evolution

- To eliminate some less-important characteristics for decreasing the input characteristics’ size and enhance forecasting capability.
- Additionally, it is to pinpoint several crucial characteristics that influence model performance, hence bringing down model complexity.

^{th}solution’s objective.

Algorithm 1: PSO—Based Input Features Evolution |

Goal: Reducing (1-hit ratio)Input: training data set. Output: The PSO-LSSVR’s characteristics set. BEGINEstablish the population While (number of generations, or the halting requirement is not fulfilled) For i = 1 (particles’ number) When one’s fitness level X _{k} exceeds another’s p-best,next update p-best _{k} = X_{k}For υ belongs to neighborhood of X _{k}If fitness X _{υ} is higher than fitness of g-best,Next update g-best = X _{υ}then υ Every dimension g ${V}_{k,g}(h+1)=a{V}_{k,g}(h)+{\tau}_{1}{c}_{1}({P}_{k,g}-{x}_{k,g}(h))+{\tau}_{2}{c}_{2}({P}_{j,g}-{x}_{k,g}(h))$ $S({V}_{k,g}(h+1))=\frac{1}{1+{\pi}^{-{V}_{k,g}(h+1)}}$ when rand() < $S({V}_{k,g}(h+1))$ then ${X}_{k,g}(h+1)=1$ else ${X}_{k,g}(h+1)=0$ Next g Next k Next generation till the ending criteria END |

_{m}(m = 1, 2,…, N) denotes this characteristic has been chosen as a claimed characteristic for the next renewal as {1} otherwise, denotes this characteristic has not been chosen as a claimed characteristic for the next renewal as {0}.

#### 3.4. PSO-Based Parameters Optimization

## 4. Experiment Findings

#### 4.1. Benchmark Datasets and Compared Approaches

#### 4.2. Characteristic Selection Using PSO

- Step 1: Use the initial value that the chaotic map process created in step 1 (for instance, the dataset Auto-Mpg Figure 5).
- Step 2: Assess the fitness function.
- Step 3: Before MCN.set $w=0.9-0.5\xb7j/MCN,j=\mathrm{iteration}$, choose the number of input characteristics using Binary PSO.
- Step 4: Selected training data $k\%$ (k = 50, 60,…, 100) are used to train the LSSVR with PSO.
- Step 5: The trained LSSVR was tested for a set size.

#### 4.3. PSO-Based Parameter Optimization for CP-LSSVR

#### 4.4. Comparisons and Discussion

- The proposed CP-LSSVR performs best among the comparable methodologies for Servo in SSR/SST = 1.7869,
- SVR performs best among the comparable methodologies for Boston Housing Data in both SSE/SST = 0.1274 and SSR/SST = 0.9032, Servo in SSE/SST = 0.1315, Concrete Compressive Strength in SSR/SST = 0.9425, Auto Price in SSE/SST = 0.1278.
- LSSVR performs best among the comparable methodologies for Auto-Mpg in both SSE/SST = 0.1064 and SSR/SST = 0.9897, machine CPU in both SSE/SST = 0.1017 and SSR/SST = 0.9877, Concrete Compressive Strength in SSE/SST = 0.1226, Auto Price in SSR/SST = 0.9952.

- The CP-LSSVR has an SVR feature that can get beyond some of the BP-neural network’s drawbacks, like overfitting and local minima.
- The chaotic PSO parameter optimization method can help improve the normalization effect. Fourth, the character development in the CP-LSSVR can quickly identify important factors that influence model performance, improving the LSSVR’s interpretability.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Example of the nonlinear transformation used to get from the input space to the characteristic space (d = 2; m = 3), where colors represent different characteristic and symbol $\varphi (\xb7)$ signifies a sequence of nonlinear transformations.

No. | Data Sets | Observations | Training (100%) | Testing | Attributions |
---|---|---|---|---|---|

1 | Boston Housing Data | 506 | 304 | 202 | 13 |

2 | Auto-Mpg | 398 | 239 | 159 | 8 |

3 | machine CPU | 209 | 125 | 84 | 6 |

4 | Servo | 167 | 100 | 67 | 4 |

5 | Concrete Compressive Strength | 1030 | 618 | 412 | 8 |

6 | Auto Price | 159 | 95 | 64 | 14 |

Attribution | Boston Housing Data | Auto-Mpg | Machine CPU | Servo | Concrete Compressive Strength | Auto Price |
---|---|---|---|---|---|---|

1 | CRIM | cylinders | MYCT | motor | Cement | normalized-losses |

2 | ZN | displacement | MMIN | screw | Blast Furnace Slag | wheel-base |

3 | INDUS | horsepower | MMAX | pgain | Fly Ash | length |

4 | CHAS | weight | CACH | vgain | Water | width |

5 | NOX | acceleration | CHMIN | Superplasticizer | height | |

6 | RM | model year | CHMAX | Coarse Aggregate | curb-weight | |

7 | AGE | origin | Fine Aggregate | engine-size | ||

8 | DIS | car name | Age | bore | ||

9 | RAD | stroke | ||||

10 | TAX | compression-ratio | ||||

11 | PTRATIO | horsepower | ||||

12 | B | peak-rpm | ||||

13 | LSTAT | city-mpg | ||||

14 | highway-mpg | |||||

15 | price |

No. | Data Sets | Number of Particles | Iteration | c_{1} | c_{2} | w_{0} | u |
---|---|---|---|---|---|---|---|

1 | BostonHousing Data | 50 | 200 | 2 | 2 | 0.9 | 4 |

2 | Auto-Mpg | 50 | 200 | 2 | 2 | 0.9 | 4 |

3 | machine CPU | 50 | 200 | 2 | 2 | 0.9 | 4 |

4 | Servo | 50 | 200 | 2 | 2 | 0.9 | 4 |

5 | Concrete Compressive Strength | 50 | 200 | 2 | 2 | 0.9 | 4 |

6 | Auto Price | 50 | 200 | 2 | 2 | 0.9 | 4 |

Training (%) | Selected Feature ID | #Features |
---|---|---|

50 | 1, 2, 4, 5, 8, 9, 11, 12 | 8 |

60 | 1, 2, 4, 6, 8, 13 | 6 |

70 | 1, 2, 3, 9, 11 | 5 |

80 | 1, 2, 4, 8, 9 | 5 |

90 | 1, 2, 4, 6, 9, 11 | 6 |

100 | 1, 2, 4, 8, 9, 11 | 6 |

Average | 6.0000 |

Training (%) | Selected Feature ID | #Features |
---|---|---|

50 | 1, 2, 3, 4, 5, 8 | 6 |

60 | 1, 3, 4, 6, 8 | 5 |

70 | 1, 4, 7, 8 | 4 |

80 | 2, 3, 4, 6 | 4 |

90 | 1, 2, 4, 5, 7 | 5 |

100 | 1, 4, 6, 8 | 4 |

Average | 4.6667 |

Training (%) | Selected Feature ID | #Features |
---|---|---|

50 | 1, 2, 3, 5, 6 | 5 |

60 | 1, 2, 4, 6 | 4 |

70 | 2, 3, 4, 6 | 4 |

80 | 1, 2, 4 | 3 |

90 | 1, 3, 4, 6 | 4 |

100 | 1, 2, 3, 4 | 4 |

Average | 4.0000 |

Training (%) | Selected Feature ID | #Features |
---|---|---|

50 | 1, 2, 3, 4 | 4 |

60 | 1, 2, 3 | 3 |

70 | 1, 2, 3, 4 | 4 |

80 | 1, 2, 3, 4 | 4 |

90 | 1, 2, 3, 4 | 4 |

100 | 1, 2, 3, 4 | 4 |

Average | 3.8333 |

Training (%) | Selected Feature ID | #Features |
---|---|---|

50 | 2, 3, 4, 5, 6, 8 | 6 |

60 | 1, 2, 3, 6, 7, 8 | 6 |

70 | 1, 2, 3, 7 | 4 |

80 | 1, 2, 4, 7 | 4 |

90 | 1, 2, 4, 6, 7 | 5 |

100 | 1, 2, 4, 7 | 4 |

Average | 4.8333 |

Training (%) | Selected Feature ID | #Features |
---|---|---|

50 | 1, 2, 3, 4, 5, 7, 9, 10, 12, 13, 14 | 11 |

60 | 2, 3, 4, 6, 7, 9, 10, 12, 13 | 9 |

70 | 1, 3, 4, 510, 12, 13 | 7 |

80 | 1, 2, 4, 5, 7, 10, 12, 13 | 8 |

90 | 1, 2, 4, 5, 7, 10, 12, 13 | 8 |

100 | 1, 2, 4, 5, 10, 12, 13 | 7 |

Average | 8.3333 |

Training (100%) | Weights of Mixed Kernel | Kernel Parameters | Upper Bound | |||
---|---|---|---|---|---|---|

lada_1 | lada_2 | lada_3 | d | sigma | C | |

Boston Housing Data | 0.2140 | 0.3711 | 0.4148 | 2.8264 | 0.4331 | 3.0288 |

Auto-Mpg | 0.2970 | 0.3417 | 0.3613 | 2.8863 | 0.3420 | 2.2992 |

machine CPU | 0.1661 | 0.2654 | 0.5684 | 2.5853 | 0.4018 | 3.4588 |

Servo | 0.3223 | 0.2743 | 0.4034 | 2.6001 | 0.5901 | 2.4512 |

Concrete Compressive Strength | 0.2827 | 0.3197 | 0.3976 | 2.7553 | 0.6408 | 3.2270 |

Auto Price | 0.3466 | 0.3828 | 0.2705 | 2.3582 | 0.3772 | 2.4982 |

Data Sets | Regressor | SSE/SST | SSR/SST |
---|---|---|---|

Boston Housing Data | SVR | 0.1274 | 0.9032 |

LSSVR | 0.1293 | 0.8964 | |

PSO-LSSVR | 0.9091 | 0.1720 | |

CP-LSSVR | 0.9900 | 0.1484 | |

CP-LSSVR | 1.0000 | 0.1276 | |

CP-LSSVR | 1.0030 | 0.1302 | |

CP-LSSVR | 1.0437 | 0.1563 | |

Auto-Mpg | SVR | 0.1134 | 0.9873 |

LSSVR | 0.1064 | 0.9897 | |

PSO-LSSVR | 0.9941 | 0.4608 | |

CP-LSSVR | 0.9560 | 0.5095 | |

CP-LSSVR | 1.0409 | 0.4609 | |

CP-LSSVR | 1.0071 | 0.4688 | |

CP-LSSVR | 1.0010 | 0.4884 | |

machine CPU | SVR | 0.1048 | 0.9813 |

LSSVR | 0.1017 | 0.9877 | |

PSO-LSSVR | 0.9585 | 0.0064 | |

CP-LSSVR | 0.9652 | 0.0103 | |

CP-LSSVR | 0.9552 | 0.0104 | |

CP-LSSVR | 0.9700 | 0.0055 | |

CP-LSSVR | 0.9547 | 0.0058 | |

Servo | SVR | 0.1315 | 0.9774 |

LSSVR | 0.1331 | 0.9756 | |

PSO-LSSVR | 0.9713 | 1.7185 | |

CP-LSSVR | 1.0034 | 1.7251 | |

CP-LSSVR | 1.0044 | 1.7869 | |

CP-LSSVR | 1.0043 | 1.6734 | |

CP-LSSVR | 1.0234 | 1.7577 | |

Concrete Compressive Strength | SVR | 0.1237 | 0.9425 |

LSSVR | 0.1226 | 0.9338 | |

PSO-LSSVR | 0.9395 | 0.2030 | |

CP-LSSVR | 0.9604 | 0.1944 | |

CP-LSSVR | 0.9802 | 0.1836 | |

CP-LSSVR | 0.9700 | 0.1942 | |

CP-LSSVR | 0.9692 | 0.1936 | |

Auto Price | SVR | 0.1278 | 0.9821 |

LSSVR | 0.1288 | 0.9952 | |

PSO-LSSVR | 0.9913 | 0.1858 | |

CP-LSSVR | 0.9843 | 0.1803 | |

CP-LSSVR | 0.9950 | 0.1982 | |

CP-LSSVR | 1.0515 | 0.1639 | |

CP-LSSVR | 1.0562 | 0.1862 |

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**MDPI and ACS Style**

Yeh, W.-C.; Zhu, W.
Forecasting by Combining Chaotic PSO and Automated LSSVR. *Technologies* **2023**, *11*, 50.
https://doi.org/10.3390/technologies11020050

**AMA Style**

Yeh W-C, Zhu W.
Forecasting by Combining Chaotic PSO and Automated LSSVR. *Technologies*. 2023; 11(2):50.
https://doi.org/10.3390/technologies11020050

**Chicago/Turabian Style**

Yeh, Wei-Chang, and Wenbo Zhu.
2023. "Forecasting by Combining Chaotic PSO and Automated LSSVR" *Technologies* 11, no. 2: 50.
https://doi.org/10.3390/technologies11020050