# Research on the Prediction Model of the Used Car Price in View of the PSO-GRA-BP Neural Network

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}can reach 0.984. Although the longest training time is 94.153 s, the overall prediction effect is significantly better than other used car price prediction models, providing a new idea and method for used car evaluation.

## 1. Introduction

- (1)
- In view of the shortcomings of the traditional BP neural network, use grey relation analysis to screen feature variables, combine with particle swarm optimization algorithm to further optimize, and apply the improved model to the evaluation of used car prices.
- (2)
- The PSO-GRA-BPNN established in this paper is compared with the traditional BP neural network, multiple linear regression, random forest [12], and support vector machine regression models [30], and the advantages of the prediction accuracy of this model are verified by actual cases, which provides a new idea and method for used car evaluation.

## 2. Data and Model Assumptions

#### 2.1. Analysis of Elements Making the Price of Used Cars Different

#### 2.2. Data Selection and Processing

- (1)
- For data columns that can be calculated quantitatively, such as displacement and new car price, use Lagrange interpolation to calculate their missing values, and delete missing values that are difficult to quantify such as brand and region.
- (2)
- According to the ranking of “100 Most Valuable Auto Brands in the World in 2021” released by Brand Finance, different brands are coded and digitized.
- (3)
- There are three driving modes: front-wheel drive, rear-wheel drive, and four-wheel drive, which are denoted by the numbers 0, 1, and 2.
- (4)
- The gearbox is divided into two types: automatic and manual, and it is converted into a Boolean type, that is, the number of automatic is 0, and the number of manual is 1.
- (5)
- The body structure is divided into the single box, hatchback, and sedan, delete a few single boxes, convert the hatchback to number 0, and the sedan to number 1.
- (6)
- The usage time is converted to the number of days since the data were acquired.
- (7)
- Since the National III cars are gradually being phased out, a small number of National III cars are deleted, and National IV is converted to the number 4, National V is converted to the number 5, and National VI is converted to the number 6.
- (8)
- The value is assigned according to the number of prefecture level cities in different provinces. Anhui Province is assigned 1–17, Zhejiang Province is assigned 18–28, Fujian Province is assigned 29–38, Jiangxi Province is assigned 39–51, Jiangsu Province is assigned 52–65, Shandong Province is assigned 66–81, and Shanghai is assigned 82.

#### 2.3. Analysis of Pearson Correlation

#### 2.4. Assumptions

- (1)
- Since the collected data are not the data after the actual transaction on the platform, we assume that every used car on the platform can be successfully traded at the page price.
- (2)
- The price of used cars is not only affected by their internal configuration but also by the external environment. We assume that the price of used car pages provided by the platform will not be affected by the external environment.

## 3. Methodology

#### 3.1. BP Neural Network

_{1}, x

_{2}, …, x

_{12}) and the used car price Y = (y) as the output vector and provide them to the established neural network; then, each unit’s output value ${Z}_{j}$ in the hidden layer is:

#### 3.2. BPNN Prediction Model Based on GRA Variable Screening

- Step 1: Determine the analysis index system and collect analysis data. Let m data sequences form the following matrix:

- Step 2: Determine the reference and comparison sequences in the system.

- Step 3: Apply dimensionless processing to the dependent and independent variable sequences, using the mean value processing approach described below.

- Step 4: Calculate the absolute difference between each compared sequence and the corresponding element of the reference sequence in turn.

- Step 5: Calculate the correlation coefficient between the compared sequence and the corresponding factor in the reference sequence.

- Step 6: Calculate the degree of association and sort the degree of association to obtain the relative closeness of each comparison sequence and the reference sequence.

#### 3.3. Construction of GRA-BPNN Prediction Model Based on PSO Optimization Algorithm

_{i}is the speed of particles; $\omega $ represents the inertia factor; c

_{1}, c

_{2}represent the learning factor; pbest represents the individual optimal solution; gbest represents the overall optimal solution; r represents a random constant between 0–1; z

_{i}means the particle position.

_{max}and minimum value v

_{min}of the speed, the value range of the corresponding setting position is z

_{min}-z

_{max}.

- Step 1: Define the PSO-GRA-BPNN model’s inputs and outputs.

- Step 2: Particle swarm initialization.

- Step 3: Calculate a single particle’s fitness value.

- Step 4: Update particle velocity and position.

- Step 5: Optimal particle count.

- Step 6: PSO-GRA-BPNN model training.

## 4. Results

#### 4.1. Model Evaluation Metrics

^{2}, are conducted predict to the model accuracy. The evaluation functions are:

#### 4.2. Model Building Process

#### 4.2.1. BPNN Prediction Model

_{test}is chosen as the final 1000 items of data, testing the prediction effect of the constructed model.

#### 4.2.2. GRA-BPNN Prediction Model

_{12}, displacement x

_{8}, mileage x

_{6}, gearbox x

_{3}, fuel consumption x

_{9}, and registration time x

_{7}are all greater than 0.7, while the correlation degree of drive mode x

_{2}, region x

_{11}, engine power x

_{4}, emission standard x

_{10}, body structure x

_{5}, and brand x

_{1}are less than 0.7. GRA is able to identify key factors affecting used car prices and provide more efficient inputs to predictive models. Therefore, choose the new car price, displacement, mileage, gearbox, fuel consumption, and registration time as the network input of the BPNN, set the same parameters and functions as the above BPNN, and construct a 6-10-1 three-layer structure GRA-BPNN used car price prediction model.

#### 4.2.3. PSO-GRA-BPNN Prediction Model

_{1}and c

_{2}, maximum speed V

_{max}, inertia weight ω, population size N, and number of iterations. The settings of the parameters are as follows [51,52,53,54]:

- (1)
- The learning factors c
_{1}and c_{2}: c_{1}and c_{2}are mainly the adjustment weights of particles that affect their own and group experience. If the value of c_{1}is zero, the particles only have group experience, and local convergence may occur; if the value of c_{2}is zero, information sharing cannot be carried out in the group, and you only have your own experience. If both are zero, particles cannot effectively obtain empirical information, and the motion of particle swarm will show a chaotic state. Therefore, the values of c_{1}and c_{2}can have a great impact on the whole model. In this paper, the values of c_{1}and c_{2}are taken as 1.49445. - (2)
- The maximum speed V
_{max}: During the effective search process of particles, the velocity of particle motion is usually described by V_{max}, and the search step size of particles is adjusted. When the value of V_{max}is too large, the position of the particle may exceed the optimal position; when the value of V_{max}is too small, the particle may have a problem of slow convergence. In this paper, V_{max}is chosen to be one. - (3)
- The inertia weight ω: The inertia weight ω will have a significant impact on the convergence of the particle swarm optimization algorithm. If ω is relatively small, the local search ability of the particle swarm optimization algorithm is strong, but the global search ability is low; if ω is large, the opposite characteristics are exhibited. Therefore, this paper selects the inertia weight as 0.9.
- (4)
- Population size N: Too small of a population size will lead to insufficient accuracy and will be extremely unstable, and too large of a population size will lead to performance degradation. For higher accuracy and stability, this paper takes n as 200.
- (5)
- Number of iterations: As can be seen from the Figure 4 below, the change trend of the error between the expected value and the absolute value of the actual output is that with the number of iterations from 0 to 20, and the overall error of the model decreases significantly. When the number of iterations is from 20 to 60, the slope of the curve tends to be zero, and the error of the model no longer decreases. If you continue to increase the number of iterations of the model, it will consume a lot of time resources, and the performance of the model will not be well-improved. Therefore, set the number of iterations to 60.

#### 4.3. Models Comparison

#### 4.3.1. Comparison of Results of Different BPNN Prediction Models

- (1)
- MAPE can more correctly depict the percentage difference between anticipated and true values, and the lower its value, the better the fitting effect. From the above table, we can see that the MAPE of PSO-GRA-BPNN is the smallest, GRA-BPNN is the second, and BPNN is the largest. This indicates that PSO-GRA-BPNN has high prediction accuracy.
- (2)
- MAE is the sum of the absolute values of the difference between the predicted value and the actual value. It measures the closeness of the predicted result to the actual dataset. The smaller the value and the closer to zero, the better the fitting effect. The MAE of PSO-GRA-BPNN is the smallest, and its value is 0.475, which indicates that the deviation of the predicted value of PSO-GRA-BPNN from the actual value is the smallest, and the actual prediction error value is the smallest.
- (3)
- R is between zero and one. The closer it is to one, the better the model’s prediction, and the closer it is to zero, the worse the model’s prediction. R is the smallest for BPNN with a value of 0.991, followed by GRA-BPNN with a value of 0.995, and the largest for PSO-GRA-BPNN with a value of 0.998. This indicates that the predicted value of PSO-GRA-BPNN has the strongest correlation with the true value.
- (4)
- R
^{2}is the goodness of fit, and the closer the value of the coefficient of determination is to one, the better the regression is. From the table, we can see that PSO-GRA-BPNN has the largest R^{2}value, 0.984, which is the best goodness of fit among these three models. - (5)
- From the training speed, it can be seen that PSO-GRA-BPNN has the slowest training speed with a time of 94.153 s, and GRA-BPNN has the fastest training speed with a time of 6.506 s. Using the GRA method can effectively shorten the running time of the model, but adding the PSO method increases the running time, although the prediction accuracy of PSO-GRA-BPNN is higher than that of BPNN and GRA-BPNN, and the time is lost.

#### 4.3.2. Comparison with Other Used Car Price Prediction Models

^{2}are the highest, but the training speed is the longest. At the same time, it can be seen from Figure 6 that most of the prediction points of the random forest model and the multiple regression model are above the 0% error bar, and the overall prediction value is high; the support vector machine regression model prediction points are more evenly distributed on both sides of the 0% error bar, but the overall deviation is large.

^{2}are the highest. The overall prediction effect is significantly better than the multiple linear regression model, random forest models, and support vector machine regression model.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**(

**a**) Comparison of the prediction error of low−priced models. (

**b**) Comparison of the prediction error of high−priced models.

Variable | Definition | Variable | Definition |
---|---|---|---|

x_{1} | Brand | x_{7} | Usage time |

x_{2} | Drive mode | x_{8} | Displacement |

x_{3} | Gearbox | x_{9} | Fuel consumption |

x_{4} | Engine power | x_{10} | Emission standard |

x_{5} | Body structure | x_{11} | Region |

x_{6} | Mileage | x_{12} | New car price |

Neurons in the Hidden Layer | MAPE | MAE | Training Speed | Number of Network Iterations |
---|---|---|---|---|

$l$ = 4 | 30.914% | 0.671 | 0.419 s | 13 |

$l$ = 5 | 32.873% | 1.256 | 1.103 s | 48 |

$l$ = 6 | 28.245% | 0.753 | 0.714 s | 24 |

$l$ = 7 | 51.237% | 1.493 | 0.542 s | 16 |

$l$ = 8 | 38.634% | 0.957 | 1.906 s | 62 |

$l$ = 9 | 43.670% | 1.374 | 1.788 s | 57 |

$l$ = 10 | 9.891% | 0.585 | 1.868 s | 54 |

$l$ = 11 | 13.934% | 0.816 | 2.261 s | 55 |

$l$ = 12 | 27.836% | 1.073 | 0.751 s | 14 |

$l$ = 13 | 20.897% | 1.016 | 1.714 s | 40 |

$l$ = 14 | 45.383% | 0.924 | 1.324 s | 17 |

Variable | Correlation | Ranking | Variable | Correlation | Ranking |
---|---|---|---|---|---|

x_{12} | 0.913 | 1 | x_{2} | 0.595 | 7 |

x_{8} | 0.861 | 2 | x_{11} | 0.591 | 8 |

x_{6} | 0.819 | 3 | x_{4} | 0.589 | 9 |

x_{3} | 0.775 | 4 | x_{10} | 0.578 | 10 |

x_{9} | 0.733 | 5 | x_{5} | 0.497 | 11 |

x_{7} | 0.720 | 6 | x_{1} | 0.472 | 12 |

Model | MAPE | MAE | R | R^{2} | Training Speed |
---|---|---|---|---|---|

BPNN | 9.891% | 0.585 | 0.991 | 0.912 | 12.549 s |

GRA-BPNN | 7.118% | 0.529 | 0.995 | 0.969 | 6.506 s |

PSO-GRA-BPNN | 3.936% | 0.475 | 0.998 | 0.984 | 94.153 s |

Model | MAPE | MAE | R | R^{2} | Training Speed |
---|---|---|---|---|---|

PSO-GRA-BPNN | 3.936% | 0.475 | 0.998 | 0.984 | 94.153 s |

Random Forest | 19.933% | 0.842 | 0.664 | 0.553 | 20.810 s |

Multiple Linear Regression | 33.977% | 1.097 | 0.949 | 0.917 | 1.914 s |

Support Vector Machines | 153.958% | 0.028 | 0.943 | 0.863 | 0.057 s |

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**MDPI and ACS Style**

Liu, E.; Li, J.; Zheng, A.; Liu, H.; Jiang, T.
Research on the Prediction Model of the Used Car Price in View of the PSO-GRA-BP Neural Network. *Sustainability* **2022**, *14*, 8993.
https://doi.org/10.3390/su14158993

**AMA Style**

Liu E, Li J, Zheng A, Liu H, Jiang T.
Research on the Prediction Model of the Used Car Price in View of the PSO-GRA-BP Neural Network. *Sustainability*. 2022; 14(15):8993.
https://doi.org/10.3390/su14158993

**Chicago/Turabian Style**

Liu, Enci, Jie Li, Anni Zheng, Haoran Liu, and Tao Jiang.
2022. "Research on the Prediction Model of the Used Car Price in View of the PSO-GRA-BP Neural Network" *Sustainability* 14, no. 15: 8993.
https://doi.org/10.3390/su14158993