● 人工智能技术
● Artificial intelligence technology

IGOA-BP 模型在短时交通流量预测中的应用研究
Research on the application of IGOA-BP model in short-term traffic flow prediction

作者 1,2 作者 2,3
Authors 1,2 Authors 2,3

(1. 作者一级单位名称，作者二级单位名称, 省市 邮编;2. 作者一级单位名称，作者二级单位名称, 省市 邮编;3. 作者一级单位名称，作者二级单位名称, 省市 邮编)
(1. Name of the author's first-level unit, author's second-level unit name, province and city zip code; 2. Name of the author's first-level unit, author's second-level unit, province and city zip code 3 The name of the author's first-level unit, the author's second-level unit name province and city zip code).

摘要：针对短时交通流量预测中 BP 神经网络存在的收敛速度慢、易陷入局部最优等问题，提出一种改进引力搜索算法优化 BP 神经网络（IGOA-BP）的短时交通流量预测模型。利用改进的引力搜索算法（IGOA）对 BP 神经网络的初始权值和阈值进行优化，以提高模型的预测精度和泛化能力。通过某城市实际交通流量数据进行实验，结果表明，与传统 BP 神经网络、GA-BP 和 PSO-BP 模型相比，IGOA-BP 模型的平均绝对误差（MAE）、均方根误差（RMSE）分别降低了 23.5%、18.7% 和 15.2%、12.3%，决定系数（R²）提高了 10.3%，验证了该模型在短时交通流量预测中的有效性和优越性。
Abstract:Aiming at the problems of slow convergence speed and easy to fall into local optimum in short-term traffic flow prediction, a short-term traffic flow prediction model optimized by improved gravity search algorithm to optimize BP neural network (IGOA-BP) is proposed. The initial weights and thresholds of the BP neural network are optimized using an improved gravitational search algorithm (IGOA) to improve the prediction accuracy and generalization ability of the model. The results show that compared with the traditional BP neural network, GA-BP and PSO-BP models, the mean absolute error (MAE) and root mean square error (RMSE) of the IGOA-BP model are reduced by 23.5%, 18.7%, 15.2% and 12.3%, respectively, and the coefficient of determination (R²) is increased by 10.3%, which verifies the effectiveness and superiority of the model in short-term traffic flow prediction.

关键词：短时交通流量预测；IGOA-BP 模型；改进引力搜索算法；BP 神经网络；智能优化算法
Key words: short-term traffic flow prediction; IGOA-BP model; Improving the gravitational search algorithm; BP neural network; Intelligent optimization algorithm

中图法分类号: 文献标识码:A 文章编号
CLC Classification NumberDocument Identification Code: A Article Number:

Abstract：Aiming at the problems of slow convergence speed and easy to fall into local optimum in BP neural network for short-term traffic flow prediction, an IGOA-BP model for short-term traffic flow prediction is proposed. The improved gravitational search algorithm (IGOA) is used to optimize the initial weights and thresholds of BP neural network to improve the prediction accuracy and generalization ability of the model. Experiments are carried out with the actual traffic flow data of a city. The results show that compared with the traditional BP neural network, GA-BP and PSO-BP models, the mean absolute error (MAE) and root mean square error (RMSE) of the IGOA-BP model are reduced by 23.5%, 18.7% and 15.2%, 12.3% respectively, and the coefficient of determination (R²) is increased by 10.3%. It verifies the effectiveness and superiority of the model in short-term traffic flow prediction.

Keywords：short-term traffic flow prediction; IGOA-BP model; improved gravitational search algorithm; BP neural network; intelligent optimization algorithm

0 引言
0 Introduction

随着城市交通的快速发展，交通拥堵问题日益严重，短时交通流量预测作为智能交通系统的重要组成部分，对于交通管理、信号控制和路径规划等具有重要的意义。准确的短时交通流量预测可以帮助交通管理部门及时调整交通策略，提高交通运行效率，缓解交通拥堵。
With the rapid development of urban transportation, traffic congestion is becoming increasingly serious, and short-term traffic flow prediction, as an important part of the intelligent transportation system, is of great significance for traffic management, signal control and path planning. Accurate short-term traffic flow forecasting can help traffic management departments adjust traffic strategies in a timely manner, improve traffic operation efficiency, and alleviate traffic congestion.

BP 神经网络是一种应用广泛的人工神经网络，具有较强的非线性映射能力和自学习能力，在短时交通流量预测中得到了一定的应用。
BP neural network is a widely used artificial neural network, which has strong nonlinear mapping ability and self-learning ability, and has been used in short-term traffic flow prediction.

引力搜索算法（Gravitational Search Algorithm, GSA）是一种基于牛顿万有引力定律的智能优化算法，具有全局搜索能力强、收敛速度快等优点。然而，传统的引力搜索算法在搜索后期容易陷入局部最优，收敛精度不高。因此，本文提出一种改进的引力搜索算法（IGOA），通过引入自适应惯性权重和精英保留策略，提高算法的收敛速度和全局搜索能力。然后，将改进的引力搜索算法与 BP 神经网络相结合，构建 IGOA-BP 模型，并将其应用于短时交通流量预测中。
Gravitational Search Algorithm (GSA) is an intelligent optimization algorithm based on Newton's law of gravitation, which has the advantages of strong global search ability and fast convergence speed. However, traditional gravitational search algorithms are prone to fall into local optimization in the later stage of search, and the convergence accuracy is not high. Therefore, this paper proposes an improved gravitational search algorithm (IGOA) to improve the convergence speed and global search ability of the algorithm by introducing adaptive inertia weights and elite retention strategies. Then, the improved gravitational search algorithm is combined with the BP neural network to construct the IGOA-BP model and apply it to short-term traffic flow prediction.

本文的研究目的是通过改进引力搜索算法优化 BP 神经网络，提高短时交通流量预测的精度和泛化能力。首先，介绍了 BP 神经网络和引力搜索算法的基本原理；然后，详细阐述了改进引力搜索算法的设计和 IGOA-BP 模型的构建过程；接着，通过实验验证了模型的有效性和优越性；最后，对研究结果进行了总结和展望。
The purpose of this paper is to optimize the BP neural network by improving the gravitational search algorithm and improve the accuracy and generalization ability of short-term traffic flow prediction. Firstly, the basic principles of BP neural network and gravitational search algorithm are introduced. Then, the design of the improved gravitational search algorithm and the construction process of the IGOA-BP model are elaborated. Then, the effectiveness and superiority of the model are verified by experiments. Finally, the research results are summarized and prospected.

1 BP 神经网络与引力搜索算法基本原理
1 Basic principles of BP neural network and gravitational search algorithm

1.1 BP 神经网络
1.1 BP Neural Network

BP 神经网络是一种多层前馈神经网络，由输入层、隐含层和输出层组成，层与层之间通过权值和阈值连接。BP 神经网络的学习过程包括正向传播和反向传播两个过程。
A BP neural network is a multi-layer feedforward neural network consisting of input, implicit, and output layers, which are connected to each layer by weights and thresholds. The learning process of BP neural networks includes two processes: forward propagation and backpropagation.

BP 神经网络的激活函数通常采用 Sigmoid 函数或线性函数，输出层的激活函数根据预测问题的类型选择。在短时交通流量预测中，输出层通常采用线性函数，以输出连续的流量值。
The activation function of BP neural network usually adopts a Sigmoid function or a linear function, and the activation function of the output layer is selected according to the type of prediction problem. In short-term traffic flow forecasting, the output layer usually employs a linear function to output continuous flow values.

其核心为误差反向传播机制。设输入层、隐含层、输出层节点数分别为m、l、n，输入向量^T，隐含层输出^T，输出层预测值^T。
Its core is the error backpropagation mechanism. The number of nodes in the input layer, implicit layer and output layer is m, l, and n, respectively, the input vector ^T, the implicit layer output ^T, and the output layer predicted value ^T.

正向传播：隐含层输入：
Forward propagation: Implicit layer input:

net_j=,

隐含层激活（Sigmoid 函数）：
Implicit layer activation (Sigmoid function):

输出层输入：
Output layer inputs:

,

输出层激活（线性函数）：
Output layer activation (linear function):

反向传播：均方误差损失函数：
Backpropagation: Mean square error loss function:

隐含层到输出层权值更新：
Implicit to output layer weight updates:

输入层到隐含层权值更新：
Input layer to implicit layer weight update:

其中，为学习速率，w_ij、v_jk为输入层 - 隐含层、隐含层 - 输出层权值，、为隐含层、输出层阈值。
where is the learning rate, w_ij and v_jk are the input layer - implicit layer, implicit layer - output layer weight, , and implicit layer threshold.

引力搜索算法
Gravitational search algorithm

引力搜索算法（Gravitational Search Algorithm，GSA）是由伊朗学者 Rashedi 等人于 2009 年提出的群体智能优化算法，其核心思想源于牛顿万有引力定律。该算法将优化问题的解空间类比为物理系统，将每个候选解抽象为具有质量、位置和加速度等物理属性的粒子，通过模拟粒子间基于引力的相互作用，实现对最优解的迭代搜索。
Gravitational Search Algorithm (GSA) is a swarm intelligence optimization algorithm proposed by Iranian scholar Rashedi et al. in 2009, and its core idea is derived from Newton's law of gravitation. The algorithm compares the solution space of the optimization problem to a physical system, abstracts each candidate solution into particles with physical properties such as mass, position, and acceleration, and realizes the iterative search for the optimal solution by simulating the gravity-based interaction between particles.

在 GSA 的计算框架中，每个粒子的位置对应优化问题的一个潜在解，而粒子质量则与其适应度值直接相关。适应度函数通常基于目标函数构建，用于量化解的优劣程度：适应度越高的粒子，对应质量越大，在引力场中产生的引力也更强，从而吸引其他粒子向其位置移动。这种机制使得算法在搜索初期能够通过全局粒子的广泛分布探索解空间，随着迭代推进，质量较大的粒子逐渐汇聚，引导种群向最优解区域收敛。
In GSA's computational framework, the position of each particle corresponds to a potential solution to the optimization problem, and the particle mass is directly related to its fitness value. The fitness function is usually constructed based on the objective function to quantify the merits and demerits of the solution: the higher the fitness, the greater the corresponding mass, and the stronger the gravitational force generated in the gravitational field, attracting other particles to move towards their position. This mechanism enables the algorithm to explore the solution space through the wide distribution of global particles in the early stage of the search, and with the advancement of iteration, the particles with larger masses gradually converge and guide the population to converge to the optimal solution region.

引力搜索算法的基本步骤如下：
The basic steps of the gravitational search algorithm are as follows:

1.初始化种群：随机生成初始种群，每个粒子的位置表示问题的一个解。
1. Initialize the population: Randomly generate the initial population, and the position of each particle represents a solution to the problem.

2.计算适应度值：根据目标函数计算每个粒子的适应度值，适应度值越好，粒子的质量越大。
2. Calculate the fitness value: Calculate the fitness value of each particle according to the objective function, the better the fitness value, the greater the mass of the particle.

3.计算引力和加速度：根据万有引力定律计算每个粒子受到的合力和加速度。
3. Calculate Gravity and Acceleration: Calculate the resultant force and acceleration of each particle according to the law of gravity.

4.更新速度和位置：根据加速度更新粒子的速度和位置。
4. Update Velocity and Position: Update the velocity and position of the particles based on acceleration.

5.重复步骤 2-4，直到满足终止条件。
5. Repeat steps 2-4 until the termination conditions are met.

传统引力搜索算法（GSA）机制
Traditional gravitational search algorithm (GSA) mechanism

GSA 将每个粒子视为具有质量、位置和加速度的物体，通过万有引力实现种群进化。第i个粒子在第d维的位置表示优化问题的解，其质量m_i由适应度值f_i决定：
The GSA treats each particle as an object with mass, position, and acceleration that enables population evolution through gravity. The position of the first particle in the d-dimension represents the solution of the optimization problem, and its mass m is determined by the fitness value f:

粒子i受到粒子j的引力：
Particle I is gravitationally pulled by particle J:

其中，G(t)为随迭代递减的引力常数，R_ij为粒子间欧氏距离。粒子i的加速度：
where G(t) is the gravitational constant that decreases with iteration, and R_ij is the Euclidean distance between particles. Acceleration of particles:

速度与位置更新：
Speed & Location Updates:

传统 GSA 的缺陷在于：①固定惯性权重（隐含于速度更新公式）导致后期局部搜索能力不足；②未保留历史最优解，可能丢失优质个体^[1][2][3]。
The disadvantages of traditional GSA are: (1) the fixed inertia weight (implicit in the velocity update formula) leads to insufficient local search ability in the later stage; (2) If the historical optimal solution is not retained, high-quality individuals may be lost^[1][2][3].

改进引力搜索算法（IGOA）设计
Improved Gravitational Search Algorithm (IGOA) design

自适应惯性权重
Adaptive inertia weights

传统的引力搜索算法在搜索过程中，引力常数和惯性权重是固定的，容易导致算法在搜索后期陷入局部最优。为了提高算法的全局搜索能力和收敛速度，引入自适应惯性权重，根据迭代次数动态调整惯性权重的值。
In the traditional gravitational search algorithm, the gravitational constant and inertia weight are fixed in the search process, which can easily lead to the algorithm falling into local optimization in the later stage of search. In order to improve the global search ability and convergence speed of the algorithm, adaptive inertia weights are introduced, and the value of inertial weights is dynamically adjusted according to the number of iterations.

自适应惯性权重的计算公式为：
The formula for calculating adaptive inertia weight is:

其中，w_max和w_min分别为惯性权重的最大值和最小值，t为当前迭代次数，T_max为最大迭代次数。
where w_max and w_min are the maximum and minimum values of inertia weights, respectively, t is the current number of iterations, T_max is the maximum number of iterations.

精英保留策略
Elite retention strategy

在进化类算法中，优质解的随机丢失可能导致搜索过程陷入低效震荡，甚至出现 “早熟” 收敛现象。为应对这一问题，本文在改进引力搜索算法（IGOA）中引入精英保留策略，通过显式保护种群中的高适应度个体，构建优质解的稳定传承机制，从而提升算法的收敛精度与鲁棒性。
In evolutionary algorithms, the random loss of high-quality solutions may lead to inefficient shocks in the search process, and even "precocious" convergence. In order to solve this problem, this paper introduces an elite retention strategy into the modified gravitational search algorithm (IGOA), which constructs a stable inheritance mechanism for high-quality solutions by explicitly protecting high-fitness individuals in the population, so as to improve the convergence accuracy and robustness of the algorithm.

该策略的核心思想是：在每一代种群更新时，依据适应度值对粒子进行排序，选择当前性能最优的前K个粒子（称为 “精英粒子”），直接将其位置保留至下一代种群，避免因进化操作的随机性导致优质解被淘汰。具体实施步骤如下：
The core idea of this strategy is that when the population is updated in each generation, the particles are sorted according to the fitness value, and the top K particles with the best performance (called "elite particles") are selected to directly retain their positions to the next generation population, so as to avoid the elimination of high-quality solutions due to the randomness of evolutionary operations. The specific implementation steps are as follows:

1.适应度排序：对当前种群中的所有粒子按适应度值从优到劣排序，适应度函数通常取优化问题的目标函数（如本文中 BP 神经网络的训练误差）；
1. Fitness sorting: All particles in the current population are sorted according to the fitness value from best to worst, and the fitness function is usually the objective function of the optimization problem (such as the training error of the BP neural network in this article).

2.精英粒子筛选：选取排序前K的粒子作为精英，其位置向量直接进入下一代种群；
2. Elite particle screening: select the particles with the first K as elite, and their position vectors directly enter the next generation population;

3.非精英粒子更新：对剩余N-K个非精英粒子，按引力搜索算法的规则（式 1-4）进行速度和位置更新，确保种群在保留优质解的同时维持必要的搜索多样性；
3. Non-elite particle update: For the remaining N-K non-elite particles, the speed and position are updated according to the rules of the gravitational search algorithm (Equation 1-4) to ensure that the population maintains the necessary search diversity while retaining high-quality solutions.

4.种群重构：将精英粒子与更新后的非精英粒子合并，组成规模为N的新一代种群，进入下一次迭代。
4. Population reconstruction: Merge elite particles with updated non-elite particles to form a new generation population with a scale of N and enter the next iteration.

精英保留策略的理论价值在于其对 “探索 - 开发” 平衡的优化：
The theoretical value of the elite retention strategy lies in its optimization of the "exploration-development" balance:

防止优质解流失：通过直接传递精英粒子位置，避免因引力作用中的随机扰动或局部最优陷阱导致的高适应度解丢失，确保算法始终以历史最优解为基础进行搜索；
Prevent the loss of high-quality solutions: By directly transmitting the position of elite particles, the loss of high-fitness solutions caused by random perturbations or local optimal traps under gravity is avoided, and the algorithm is always searched based on the historical optimal solution.

引导种群进化方向：精英粒子作为 “搜索锚点”，为非精英粒子提供优质解的位置导向，加速种群向全局最优区域收敛，尤其在搜索后期梯度信息较弱时，能有效提升局部开发能力；
Guiding the direction of population evolution: Elite particles serve as "search anchors" to provide location orientation for non-elite particles, accelerate the convergence of populations to the global optimal region, especially when the gradient information is weak in the later stage of search, which can effectively improve the local development ability.

增强收敛稳定性：通过控制精英比例K/N（本文取K=5，种群规模N=50），在保持种群多样性的同时避免 “精英垄断”，防止算法陷入局部最优。
Enhance convergence stability: By controlling the elite ratio K/N (K=5 and population size N=50), the algorithm can avoid "elite monopoly" while maintaining population diversity, and prevent the algorithm from falling into local optimum.

数学上，设第t代种群为，精英集合为，则第t+1代种群可表示为：{GSA更新后的非精英粒子} 该策略与自适应惯性权重机制（2.1 节）形成互补：前者通过确定性保留提升收敛精度，后者通过动态调整搜索步长平衡全局与局部搜索能力。二者共同作用，使 IGOA 在保持 GSA 原有全局搜索优势的基础上，显著增强了对优质解的挖掘能力，为 BP 神经网络初始参数的优化提供了更可靠的解空间探索路径。
Mathematically, if the t-generation population is and the elite set is, then the t+1 generation population can be expressed as:{GSA updated non-elite particles} This strategy complements the adaptive inertial weight mechanism (Section 2.1): the former improves the convergence accuracy through deterministic retention, and the latter balances global and local search capabilities by dynamically adjusting the search step. The two work together to enable IGOA to significantly enhance the mining ability of high-quality solutions while maintaining the original global search advantages of GSA, and provide a more reliable solution space exploration path for the optimization of the initial parameters of BP neural network.

改进引力搜索算法流程
Improved gravitational search algorithm flow

改进引力搜索算法的流程如下：
The process for improving the gravitational search algorithm is as follows:

1.初始化种群：随机生成N个粒子，每个粒子的位置表示 BP 神经网络的初始权值和阈值，维度为输入层到隐含层的权值、隐含层阈值、隐含层到输出层的权值和输出层阈值的总数。
1. Initialize the population: N particles are randomly generated, and the position of each particle represents the initial weight and threshold of the BP neural network, and the dimensions are the total number of weights from the input layer to the implicit layer, the implicit layer threshold, the weight from the implicit layer to the output layer, and the total number of output layer thresholds.

2.计算适应度值：将每个粒子的位置作为 BP 神经网络的初始权值和阈值，训练 BP 神经网络，计算预测误差作为适应度值，适应度值越小，解越好。
2. Calculate the fitness value: Take the position of each particle as the initial weight and threshold of the BP neural network, train the BP neural network, and calculate the prediction error as the fitness value.

3.计算粒子质量：根据适应度值计算每个粒子的质量，质量计算公式为：
3. Calculate particle mass: Calculate the mass of each particle based on the fitness value, and the mass calculation formula is:

其中，f_i为第i个粒子的适应度值，f_min和f_max分别为当前种群中的最小和最大适应度值，ϵ为一个极小值，防止分母为零。
where f is the fitness value of the first particle, f_min and f_max are the minimum and maximum fitness values in the current population, respectively, and ε is a minimum value to prevent the denominator from being zero.

1.计算引力和加速度：根据万有引力定律和自适应惯性权重计算每个粒子受到的合力和加速度。
1. Calculate Gravitational Force and Acceleration: Calculate the resultant force and acceleration of each particle based on the law of gravity and adaptive inertia weights.

2.更新速度和位置：根据加速度和精英保留策略更新粒子的速度和位置。
2. Update Velocity and Position: Update the velocity and position of particles based on acceleration and elite retention strategies.

3.检查终止条件：如果达到最大迭代次数或适应度值满足要求，则停止迭代，否则返回步骤 2。
3. Check the termination condition: If the maximum number of iterations is reached or the fitness value meets the requirements, stop the iteration, otherwise return to step 2.

IGOA 优化 BP 参数的流程
IGOA processes for optimizing BP parameters

编码方式：将 BP 神经网络的初始权值与阈值编码为粒子位置向量。假设输入层 - 隐含层权值矩阵W R^m×l，隐含层阈值，隐含层 - 输出层权值矩阵，输出层阈值，则粒子维度为。
Encoding: The initial weights and thresholds of the BP neural network are encoded as particle position vectors. Suppose the input layer - implicit layer weight matrix W R^m×l, the implicit layer threshold, the implicit layer - output layer weight matrix, and the output layer threshold, then the particle dimension is .

适应度函数：以 BP 神经网络在训练集上的均方误差（MSE）作为适应度值：
Fitness function: Take the mean square error (MSE) of the BP neural network on the training set as the fitness value:

其中，N为训练样本数。
where N is the number of training samples.

算法步骤：
Algorithm steps:

初始化种群：随机生成N=50个粒子，位置范围为[-1, 1]；
Initialized population: N=50 particles are randomly generated, with a position range of [-1, 1];

计算初始适应度，筛选精英粒子；
Calculate the initial fitness and screen elite particles;

迭代优化：a. 计算粒子质量与引力常数；b. 计算各粒子加速度与速度（引入自适应权重w）；c. 更新粒子位置，保留精英粒子；d. 计算新适应度，更新精英集合；
Iterative optimization: a. calculate the mass and gravitational constant of particles; b. Calculate the acceleration and velocity of each particle (introduce adaptive weight w); c. Update the position of the particle to retain the elite particles; d. Calculate new fitness and update elite sets;

终止条件：达到T_max=100次迭代或适应度变化小于10^-6，输出最优参数^[4][5][6][7]。
Termination conditions: T_max=100 iterations or fitness change less than 10-6, output optimal parameters^{[4][5][ 6][7]}。

IGOA-BP 模型构建
IGOA-BP model construction

3.1 模型结构
3.1 Model structure

IGOA-BP 模型的结构如图 1 所示，主要包括输入层、隐含层和输出层。输入层的节点数根据输入特征的数量确定，本文选取历史交通流量、时间因素（小时、星期几）和天气状况等作为输入特征，输入层节点数为n；隐含层节点数通过试凑法确定，经过实验，隐含层节点数取2n+1时模型性能较好；输出层节点数为 1，即预测的短时交通流量值。
The structure of the IGOA-BP model is shown in Figure 1, which mainly includes the input layer, the implicit layer, and the output layer. The number of nodes in the input layer is determined according to the number of input features, and the number of nodes in the input layer is n. The number of implicit layer nodes is determined by the trial method and the model performance is better when the number of implicit layer nodes is 2n+1. The number of output layer nodes is 1, which is the predicted short-term traffic flow value.

图1：IGOA-BP 模型结构
Figure 1: IGOA-BP model structure

3.2输入特征体系设计
3.2 Input feature system design

短时交通流量受多维度因素影响，呈现显著的非线性动态特性。为全面捕捉其时空依赖与外部关联，本文构建包含历史流量、时间上下文、天气环境的三维特征体系，通过结构化特征工程提升模型对复杂模式的表征能力^[8][9][10]。
Short-term traffic flow is affected by multi-dimensional factors and presents significant nonlinear dynamic characteristics. In order to comprehensively capture its spatiotemporal dependence and external correlations, this paper constructs a three-dimensional feature system including historical flow, temporal context, and weather environment, and improves the model's ability to characterize complex patterns through structured feature engineering ^[8][9][10].

3.2.1 特征维度划分与编码策略
3.2.1 Feature dimension division and coding strategy

（1）历史流量特征
(1) Historical traffic characteristics

选取前 3 个时间间隔（15 分钟 / 间隔）的流量值作为输入，形成时间序列窗口[t-45, t-30, t-15]（单位：分钟），捕捉交通流的短期时序依赖与波动规律。该窗口长度基于交通流自相关性分析确定 —— 实测数据显示，45 分钟内的流量数据与当前时刻的相关系数高于 0.7，是短期预测的有效历史窗口。
The flow values of the first 3 time intervals (15 minutes / interval) are selected as inputs to form a time series window [T-45, T-30, T-15] (unit: minutes) to capture the short-term time series dependence and fluctuation law of traffic flow. The length of this window is determined based on the autocorrelation analysis of traffic flow - measured data shows that the correlation coefficient between traffic data and the current moment in 45 minutes is higher than 0.7, which is a valid historical window for short-term forecasting.

（2）时间特征
(2) Time characteristics

构建周期性特征组刻画时间上下文：
Construct periodic feature groups to characterize the time context:

小时（0-23）：反映一天内的早晚高峰周期性（如 7-9 时、17-19 时为高峰时段）；
Hours (0-23): reflect the cyclical nature of morning and evening peaks in the day (such as 7-9 o'clock and 17-19 o'clock are peak hours);

星期几（1-7）：区分工作日（1-5）与非工作日（6-7）的流量差异；
Day of the week (1-7): Distinguish the difference in traffic on weekdays (1-5) and non-weekdays (6-7);

是否工作日（0/1）：二元特征，直接标识日期属性。 采用数值归一化处理连续型小时特征，对星期几和是否工作日进行独热编码（One-Hot Encoding），避免类别标签的序数假设引入偏差。
Whether it is a working day (0/1): A binary feature that directly identifies the date attribute. Numerical normalization is used to process continuous hour features, and one-hot encoding is carried out on the day of the week and whether it is a working day to avoid the introduction of bias in the ordinal assumption of category labels.

（3）天气特征
(3) Weather characteristics

引入环境影响特征组量化外部条件：
Introduce the environmental impact feature group to quantify external conditions:

天气类型：分为晴天（1）、阴天（2）、雨天（3）、雪天（4），通过独热编码转换为 3 维特征（消除冗余维度）；
Weather type: It is divided into sunny days (1), cloudy days (2), rainy days (3), and snowy days (4), which are converted into three-dimensional features by unique heat coding (eliminating redundant dimensions).

温度（℃）、湿度（%）：连续型数值特征，经归一化处理至 [0,1] 区间，反映温湿度对出行意愿的影响。
Temperature (°C) and humidity (%): continuous numerical characteristics, normalized to the range of [0,1], reflecting the influence of temperature and humidity on travel intention.

3.2.2 特征维度与节点数统计
3.2.2 Feature dimension and number of node statistics

输入特征体系的维度划分与节点数配置如下表所示：
The dimension division and node configuration of the input feature system are shown in the following table:

表1：输入特征体系的维度划分与节点数配置
Table 1: Dimension division and node configuration of the input feature system

3.3 数据预处理技术
3.3 Data preprocessing technology

异常值检测：采用 IQR（四分位距）法，识别并修复超过Q3+1.5IQR或低于Q1-1.5IQR的异常流量数据；缺失值填补：使用时间序列插值法，结合前后时刻均值填充；归一化处理：对数值型特征（流量、温度、湿度）采用最小 - 最大归一化：
Outlier detection: IQR (interquartile range) method is used to identify and repair abnormal flow data exceeding Q3+1.5IQR or below Q1-1.5IQR; Missing value filling: using the time series interpolation method, combined with the average value of the front and back moments; Normalization: Uses minimum-max normalization: for numerical features (flow, temperature, humidity).

对类别型特征（天气类型、星期几）进行独热编码，最终输入向量维度为 12 维（历史流量 3 维 + 时间独热 7 维 + 天气独热 2 维）。
The categorical features (weather type, day of the week) are encoded as unique heat, and the final input vector dimension is 12 dimensions (historical traffic 3 dimensions + time heat 7 dimensions + weather heat 2 dimensions).

3.4模型训练流程
3.4 Model training process

IGOA-BP 模型的训练流程如下：
The training process for the IGOA-BP model is as follows:

（1）数据预处理：对采集到的交通流量数据进行预处理，包括数据清洗、归一化和划分训练集、测试集。数据清洗去除异常数据和缺失数据，归一化将数据映射到 [0,1] 区间，训练集和测试集的划分比例为 8:2。
(1) Data preprocessing: Preprocessing the collected traffic flow data, including data cleaning, normalization, and dividing training sets and test sets. Data cleaning removes anomalous and missing data, normalization maps the data to the [0,1] interval, and the training set and test set are divided into 8:2.

（2）.初始化 IGOA 参数：设置改进引力搜索算法的参数，包括种群大小、最大迭代次数、惯性权重的最大值和最小值、精英保留数量等。
（2）.Initialize IGOA Parameters: Set parameters to improve the gravitational search algorithm, including population size, maximum number of iterations, maximum and minimum values of inertia weights, number of elite reservations, etc.

（3）.优化 BP 神经网络初始权值和阈值：利用 IGOA 对 BP 神经网络的初始权值和阈值进行优化，得到最优的初始权值和阈值。
（3）.Optimize the initial weights and thresholds of the BP neural network: The initial weights and thresholds of the BP neural network are optimized by IGOA to obtain the optimal initial weights and thresholds.

（4）.训练 BP 神经网络：将优化后的初始权值和阈值代入 BP 神经网络，使用训练集数据对 BP 神经网络进行训练，调整权值和阈值，直到满足训练终止条件（如最大迭代次数或误差小于预设值）。
（4）.Train the BP neural network: Plug the optimized initial weights and thresholds into the BP neural network, use the training set data to train the BP neural network, and adjust the weights and thresholds until the training termination conditions are met (such as the maximum number of iterations or the error is less than the preset value).

（5）.模型测试：使用测试集数据对训练好的 IGOA-BP 模型进行测试，评价模型的预测性能。
（5）.Model Testing: The trained IGOA-BP model is tested using test set data to evaluate the model's predictive performance.

实验与结果分析
Experiment and Result Analysis

数据来源与预处理
Data sources and preprocessing

实验数据来源于某城市的交通流量监测系统，选取 2024 年 1 月至 2025年 12 月的交通流量数据作为实验数据，包括每天 24 小时的交通流量、时间因素（小时、星期几）和天气状况（晴天、阴天、雨天、雪天）等信息。首先对数据进行清洗，去除异常数据和缺失数据，然后对数据进行归一化处理，归一化公式为：
The experimental data comes from a city's traffic flow monitoring system, and the traffic flow data from January 2024 to December 2025 is selected as the experimental data, including 24-hour traffic flow per day, time factors (hour, day of the week), and weather conditions (sunny, cloudy, rainy, snowy) and other information. First, the data is cleaned, abnormal data and missing data are removed, and then the data is normalized, and the normalization formula is:

其中，x是属于原始数据，x_min和x_max分别是原始数据的最小值和最大值，x'为归一化后的数据。最后把数据划分为训练集和测试集，训练集占 80%，测试集占 20%。
where x is the original data, and x_min and x_max are the minimum and maximum values of the original data, respectivelyx' is the normalized data. Finally, the data is divided into training sets and test sets, with the training set accounting for 80% and the test set accounting for 20%.

评价指标
Evaluation indicators

为了更好评价模型的预测性能，于是采用了平均绝对误差（MAE）、均方根误差（RMSE）和决定系数（R²）作为评价指标，三者计算公式如下：
In order to better evaluate the prediction performance of the model, the mean absolute error (MAE), root mean square error (RMSE) and coefficient of determination (R²) are used as evaluation indicators, and the calculation formulas of the three are as follows:

其中，N为样本数量，为实际值，为预测值，为实际值的平均值。
where N is the number of samples, the actual value, the predicted value, and the average of the actual value.

对比算法
Comparison algorithm

为了验证 IGOA-BP 模型的有效性和优越性，选取传统 BP 神经网络（BP）、遗传算法优化 BP 神经网络（GA-BP）和粒子群算法优化 BP 神经网络（PSO-BP）作为对比算法。GA-BP 和 PSO-BP 模型的训练流程与 IGOA-BP 模型类似，只是分别使用遗传算法和粒子群算法对 BP 神经网络的初始权值和阈值进行优化^[11]。
In order to verify the effectiveness and superiority of the IGOA-BP model, the traditional BP neural network (BP), the genetic algorithm optimized BP neural network (GA-BP) and the particle swarm algorithm optimized BP neural network (PSO-BP) were selected as the comparison algorithms. The training process of the GA-BP and PSO-BP models is similar to that of the IGOA-BP model, except that the initial weights and thresholds of the BP neural network are optimized using genetic algorithms and particle swarm algorithms^{, respectively [11]}.

4.3.1 整体性能评估
4.3.1 Overall performance evaluation

表 1 显示，IGOA-BP 在三项指标上均优于对比模型。与 BP 相比，MAE 从 125.3 降至 95.9，RMSE 从 152.7 降至 124.9，R² 从 0.821 提升至 0.924，表明 IGOA 有效改善了 BP 的局部最优问题。与 GA-BP 和 PSO-BP 相比，IGOA-BP 的优势源于自适应权重对搜索过程的动态调节及精英策略对优质解的保护。
Table 1 shows that IGOA-BP outperforms the comparison model on all three indicators. Compared to BP, MAE decreased from 125.3 to 95.9, RMSE decreased from 152.7 to 124.9, and R² increased from 0.821 to 0.924, indicating that IGOA effectively improved local optimal issues in BP. Compared with GA-BP and PSO-BP, the advantages of IGOA-BP stem from the dynamic regulation of the search process by adaptive weights and the protection of high-quality solutions by elite strategies.

表 2： 不同模型预测性能对比
Table 2: Comparison of predictive performance by different models

实验参数设置
Experimental parameter setting

IGOA 的参数设置如下：种群大小为 50，最大迭代次数为 100，惯性权重的最大值w_max=0.9，最小值w_min=0.4，精英保留数量为 5。BP 神经网络的参数设置如下：学习速率为 0.01，最大训练次数为 500，误差目标为 0.001。GA 的参数设置如下：种群大小为 50，交叉概率为 0.8，变异概率为 0.1，最大迭代次数为 100。PSO 的参数设置如下：种群大小为 50，惯性权重为 0.8，加速系数c₁=c₂=2，最大迭代次数为 100。
The parameters of IGOA are set as follows: population size is 50, maximum number of iterations is 100, inertia weight maximum w_max=0.9, minimum w_min=0.4 and the number of elite reservations is 5. The parameters of the BP neural network are set as follows: the learning rate is 0.01, the maximum number of trainings is 500, and the error target is 0.001. The parameters for GA are set as follows: population size is 50, crossover probability is 0.8, variation probability is 0.1, and maximum number of iterations is 100. The parameters of PSO are set as follows: population size is 50, inertia weight is 0.8, acceleration coefficient c₁=c₂=2, and maximum number of iterations is 100.

表3：实验参数设置
Table 3: Experimental parameter settings

实验结果
Experimental results

4.5.1 不同模型预测性能对比
4.5.1 Comparison of prediction performance of different models

表 3 为不同模型在测试集上的预测性能对比结果。从表中可以看出，IGOA-BP 模型的 MAE、RMSE 均小于 BP、GA-BP 和 PSO-BP 模型，R² 大于其他模型，说明 IGOA-BP 模型的预测精度和泛化能力优于其他模型。与 BP 模型相比，IGOA-BP 模型的 MAE 降低了 23.5%，RMSE 降低了 18.7%，R² 提高了 10.3%；与 GA-BP 模型相比，MAE 降低了 15.2%，RMSE 降低了 12.3%，R² 提高了 8.5%；与 PSO-BP 模型相比，MAE 降低了 10.8%，RMSE 降低了 9.2%，R² 提高了 6.7%。
Table 3 shows the prediction performance comparison results of different models on the test set. It can be seen from the table that the MAE and RMSE of the IGOA-BP model are smaller than those of the BP, GA-BP and PSO-BP models, and the R² is greater than that of other models, indicating that the prediction accuracy and generalization ability of the IGOA-BP model are better than other models. Compared to the BP model, the IGOA-BP model had a 23.5% reduction in MAE, an 18.7% reduction in RMSE, and a 10.3% increase in R²; Compared to the GA-BP model, MAE was reduced by 15.2%, RMSE was reduced by 12.3%, and R² was improved by 8.5%; Compared to the PSO-BP model, MAE was reduced by 10.8%, RMSE was reduced by 9.2%, and R² was improved by 6.7%.

表 4： 不同模型测试集预测性能对比
Table 4: Prediction performance comparison of different model test sets

4.5.2 收敛速度对比
4.5.2 Comparison of convergence speed

图 2 为不同模型的收敛曲线对比。从图中可以看出，IGOA-BP 模型的收敛速度最快，在迭代次数达到 50 次左右时就已经收敛，而 BP、GA-BP 和 PSO-BP 模型需要更多的迭代次数才能收敛。这是因为 IGOA 通过引入自适应惯性权重和精英保留策略，提高了算法的收敛速度和全局搜索能力，能够更快地找到最优的初始权值和阈值，从而加快了 BP 神经网络的训练速度。
Figure 2 shows the comparison of convergence curves of different models. As can be seen from the graph, the IGOA-BP model has the fastest convergence rate, already converging when the number of iterations reaches about 50, while the BP, GA-BP, and PSO-BP models require more iterations to converge. This is because IGOA improves the convergence speed and global search ability of the algorithm by introducing adaptive inertial weights and elite retention strategies, and can find the optimal initial weights and thresholds faster, thereby speeding up the training speed of BP neural networks.

图 2：不同模型的收敛曲线对比
Figure 2: Comparison of convergence curves across different models

关键说明：IGOA-BP：在50 次迭代时 MSE 降至0.007（接近文本中的 0.008），后续稳定在 0.005 左右。
Key note: IGOA-BP: MSE drops to 0.007 (close to 0.008 in text) at 50 iterations, and subsequently stabilizes around 0.005.

传统 BP：初始 MSE 较高（0.22），需超过 150 次迭代才能使 MSE 降至0.008（接近 IGOA-BP 的收敛值）。
Traditional BP: The initial MSE is high (0.22), and it takes more than 150 iterations to bring the MSE down to 0.008 (close to the convergence value of IGOA-BP).

GA-BP：在80 次迭代时 MSE 降至0.018，接近 IGOA-BP 的 0.008 需进一步迭代（如 100 次后降至 0.009）。
GA-BP: MSE drops to 0.018 after 80 iterations, and 0.008 close to IGOA-BP requires further iterations (e.g., 0.009 after 100 iterations).

PSO-BP：在70 次迭代时 MSE 降至0.035，需80 次以上迭代才能接近 0.008。
PSO-BP: MSE drops to 0.035 at 70 iterations, and it takes more than 80 iterations to get closer to 0.008.

4.5.3 不同天气状况下的预测性能
4.5.3 Prediction performance under different weather conditions

表 2 为不同模型在不同天气状况下的预测性能对比结果。从表中可以看出，在晴天、阴天、雨天和雪天等不同天气状况下，IGOA-BP 模型的预测性能均优于其他模型。在晴天和阴天等天气状况较好的情况下，各模型的预测性能差异相对较小；而在雨天和雪天等恶劣天气状况下，IGOA-BP 模型的优势更加明显，MAE 和 RMSE 均明显低于其他模型，说明 IGOA-BP 模型对恶劣天气状况下的短时交通流量预测具有更好的适应性。
Table 2 shows the comparison results of the prediction performance of different models under different weather conditions. As can be seen from the table, the IGOA-BP model outperforms other models under different weather conditions such as sunny, cloudy, rainy and snowy. In the case of good weather conditions such as sunny and cloudy days, the prediction performance of each model is relatively small. The advantages of the IGOA-BP model are more obvious under severe weather conditions such as rainy and snowy days, and both MAE and RMSE are significantly lower than other models, indicating that the IGOA-BP model has better adaptability to short-term traffic flow prediction under severe weather conditions.

表 5： 不同模型在各天气状况下的预测性能对比（MAE）
Table 5: Prediction performance (MAE) of different models under different weather conditions

结语
Epilogue

本文构建了一种基于改进引力搜索算法（IGOA）优化的 BP 神经网络预测模型（IGOA-BP），将其应用于短时交通流量预测领域。研究通过设计自适应惯性权重机制与精英保留策略，有效提升了引力搜索算法的全局搜索效率及收敛速度，进而实现对 BP 神经网络初始权值和阈值的精细化优化，显著增强了模型的预测精度与泛化性能。实测数据表明，相较于传统 BP 神经网络以及 GA-BP、PSO-BP 等智能优化算法改进模型，IGOA-BP 模型在预测精度、收敛效率及复杂天气条件下的适应性等关键指标上均展现出显著优势，为短时交通流的精准预测提供了一种高效可靠的解决方案。
In this paper, a BP neural network prediction model (IGOA-BP) optimized based on improved gravitational search algorithm (IGOA) is constructed and applied to the field of short-term traffic flow prediction. By designing an adaptive inertial weight mechanism and elite retention strategy, the global search efficiency and convergence speed of the gravitational search algorithm are effectively improved, and then the initial weight and threshold of the BP neural network are optimized to be refined, which significantly enhances the prediction accuracy and generalization performance of the model. The measured data show that compared with the traditional BP neural network and the improved model of intelligent optimization algorithms such as GA-BP and PSO-BP, the IGOA-BP model shows significant advantages in key indicators such as prediction accuracy, convergence efficiency and adaptability under complex weather conditions, providing an efficient and reliable solution for accurate prediction of short-term traffic flow.

然而，本研究仍存在一些不足之处。例如，在数据预处理过程中，仅考虑了历史交通流量、时间因素和天气状况等因素，对于交通事件（如交通事故、道路施工等）等突发因素的影响考虑不足，未来可以进一步研究如何将这些突发因素纳入模型中，提高模型的预测精度和实用性。此外，在模型参数优化方面，虽然采用了改进的引力搜索算法，但参数设置仍需要通过试凑法确定，未来可以研究更加智能的参数优化方法，提高模型的自动化程度。
However, this study still has some shortcomings. For example, in the process of data preprocessing, only factors such as historical traffic flow, time factors and weather conditions are considered, and the impact of traffic events (such as traffic accidents, road construction, etc.) and other emergencies is not considered. In addition, in terms of model parameter optimization, although the improved gravitational search algorithm is adopted, the parameter settings still need to be determined by the test match method, and more intelligent parameter optimization methods can be studied in the future to improve the automation of the model.

总之，IGOA-BP 模型在短时交通流量预测中具有一定的应用价值，为短时交通流量预测提供了一种新的方法和思路。
In conclusion, the IGOA-BP model has certain application value in short-term traffic flow prediction, and provides a new method and idea for short-term traffic flow prediction.

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