Trajectory Forecasting for Human Mobility Considering Movement Patterns and the Heterogeneous Effects of Geographical Environments via Potential Fields

Abstract

Trajectory forecasting for human mobility plays a critical role in the effective management and sustainable development of urban transportation, which aligns with the advocacy of Sustainable Development Goals (SDGs). Although several approaches have been developed in other trajectory forecasting applications, such as autonomous driving and intelligent robotics, there remain limitations in forecasting trajectories of human mobility. This is because they do not adequately consider the prior knowledge of human movement patterns and the heterogeneous effects of geographical environments. Therefore, in this study, we propose an environment-driven trajectory forecasting method that can adapt to distinct movement patterns. First, the indicator systems, which systematically summarize the heterogeneous effects of different environmental factors on human mobility, are, respectively, constructed for the convergence, divergence, and leadership patterns. Then, based on the corresponding indicator system, the potential field is generated, representing the calibrated probability of the human mobility direction under the environmental effects. A gradient descent algorithm is finally employed on the potential field to forecast the next-step mobility location. Extensive experiment results demonstrated the satisfactory performance of our proposed method under different movement patterns. Compared to other baselines, our proposed method also shows advantages in both long-term and real-time forecasting.

1. Introduction

Under the advocacy of the United Nations Sustainable Development Goals (SDGs), current urban development has emphasized a sustainable orientation to better improve the quality of people’s lives [1]. The optimization of urban transportation, which is closely linked to people’s daily travel activities, serves as a critical task in urban development. As cities expand and populations increase, the most vital challenge for urban traffic management is how to accommodate the escalation and diversification of people’s travel demand. Traffic management needs to alleviate the increasingly prominent human–land contradiction, such as traffic congestion, infrastructure limitations, and resource inefficiency [2]. Therefore, to meet sustainable needs, the optimization of urban transportation should be people-centered, striving to maximize the convenience and efficiency of human mobility [3]. This calls for further optimizing the layout of urban transportation infrastructure and improving the dynamic allocation of public transportation resources [4]. In this context, accurate understanding and forecasting the dynamics of human mobility becomes essential. This serves as a foundation for achieving effective and targeted traffic optimizations.

With the rapid development of mobile sensing devices, the mobility of city dwellers can be captured and recorded in real time, generating a wealth of trajectory data [5]. These trajectory data contain valuable insights into people’s travel purposes, such as work, study, leisure, and other daily activities [6]. Analyzing human mobility from trajectory data is essential for urban managers to gain a deeper understanding of urban transportation dynamics [7]. Among various data mining techniques, spatio-temporal forecasting for human mobility trajectory holds a significant position since it can particularly provide future information for downstream decision-making tasks. Thereby, it can enable more rational planning and more efficient management of urban transportation. Examples include the optimization of traffic signal timing, the construction of bike-sharing infrastructure, the design of pedestrian-friendly spaces, and the planning of metro station layouts. All of these contribute to promoting the sustainable development of the urban transportation system [8,9,10].

However, achieving accurate trajectory forecasting for human mobility still poses challenges, arising from two major aspects. First, depending on the different travel purposes, the trajectories of human mobility can conform to different prior movement patterns [11]. In our study, we focus on the following three important patterns: the convergence pattern, the divergence pattern, and the leadership pattern, as illustrated in Figure 1. The convergence pattern reflects the travel trend of people moving and concentrating towards a specific destination over a period of time [12]. The destinations in the convergence pattern are typically important public places such as workplaces, schools, and commercial centers. For instance, when students are going to school, their trajectories often align with the convergence pattern, as they head toward the same location. In contrast to the convergence pattern, the divergence pattern reflects the movement of people from a centralized start location to dispersed destination locations [12]. For instance, after school, students might disperse in various directions to go home. The leadership pattern reflects that the people’s mobility behaviors are influenced by a leader, with individuals following the leader’s movement in a specific formation [13]. For instance, if a teacher leads a group of students on an excursion, their trajectories exemplify the leadership pattern, where the teacher influences the movement of the students. To improve forecasting performance, it is crucial to incorporate prior knowledge of different movement patterns into the forecasting modeling process [14]. Second, human mobility is strongly influenced by the geographical environments [15]. How to consider the heterogeneous effects of geographical environments is also of significance. As demonstrated in Figure 2, various environmental factors, such as roads, buildings, and water bodies, can significantly influence human mobility. Some environmental factors provide opportunities for human mobility (e.g., the well-developed road networks), while others act as obstacles (e.g., the water bodies). The heterogenous effects we considered are reflected not only in different types of environmental factors but also in different movement patterns. Particularly, the same environmental factor may have varying degrees of effect on human mobility under different movement patterns. For instance, restricted areas deter the convergence of people, while some people may still pass through in the divergence pattern, as long as they choose routes that minimize exposure. Steep areas significantly deter convergence due to physical exertion, but in a leadership pattern, if the leader chooses the steep path, followers may be motivated to follow, reducing the impact on their trajectories. Therefore, the heterogeneous effects of different environmental factors under distinct movement pattern scenarios should be well considered.

Currently, numerous trajectory forecasting methods have been developed and applied in various fields, such as autonomous driving [16], traffic management [17], and intelligent robotics [18]. Existing trajectory forecasting methods can be generally categorized into three types: traditional methods, machine learning methods, and deep learning methods. Traditional methods model the movement process of moving objects through physical or mathematical formulas, thereby achieving future trajectory forecasting. Such methods are characterized by their reliance on prior knowledge while typically offering the advantages of simplicity and efficiency. Representatives include the Kalman filtering method [19,20], Hidden Markov Models (HMMs) [21], particle filtering algorithms [22], etc. Machine learning methods achieve trajectory forecasting through efficient data-driven learning. Such methods implicitly learn the movement trends of the moving objects from large amounts of historical trajectory data, acquiring the capacity to handle more complex nonlinear relationships. Representatives include the Random Forest method [23], the Gradient Boosting Decision Tree method [24], and the K-Nearest Neighbors method [25]. The deep learning method is a vital branch of the machine learning method. Such methods can perform deep and nonlinear feature extraction on historical trajectory states and thereby forecast future trajectory states through mapping functions. With their strong capability in automatic feature learning, deep learning methods can achieve high-precision forecasting performance. Meanwhile, data intensiveness is also a major characteristic of deep learning methods. Representatives include Long Short-Term Memory (LSTM) [26], Gated Recurrent Unit (GRU) [27,28], and Seq2Seq [29]. However, existing methods face several limitations when applied to the forecasting scenario of human mobility that arises from daily urban activities. Most of the existing approaches fail to simultaneously consider the prior knowledge of human movement patterns and the heterogeneous effects of environmental factors. Thereby, they lack the ability to accurately and robustly forecast human mobility trajectories in the complex urban environment [30].

In our study, we formulate human mobility actions in daily urban activities as reactions to specific environmental factors. Consequently, we developed an environment-driven trajectory forecasting method that adapts to complex forecasting needs. Specifically, we begin by classifying different movement patterns; notable examples include convergence, divergence, and leadership. Since the influence of the geographical environments on human mobility varies depending on the prior patterns, we established several indicator systems for each movement pattern based on prior knowledge to quantify these heterogeneous effects. During the forecasting process, these indicator systems are utilized to generate potential fields, representing the calibrated probability of the next-step mobility direction. A gradient descent algorithm is then employed on these potential fields to determine the next-step mobility location. By treating the forecasted location as the current state of the trajectory, the method can support robust long-term forecasting. Furthermore, since there are no parameters in this environment-driven method that need to be trained, efficient real-time forecasting is also feasible.

The contributions of this study can be summarized as follows:

(1)

Human mobility in daily urban activities is significantly influenced by geographical environments. Hence, we developed an environment-driven trajectory forecasting method. Specifically, a potential field model and a gradient descent algorithm to simulate the heterogeneous effects of geographical environments during the forecasting process.

(2)

In order to incorporate the prior knowledge about different movement patterns, the indicator system is tailored to each type of movement pattern, which systematically summarizes the types of environmental factors that impact human mobility and their specific effects.

(3)

The effectiveness of the proposed method is validated on the synthetic trajectory dataset containing multiple movement patterns. Experimental results demonstrate that compared to data-driven approaches, the proposed method can achieve a stable forecasting performance under different environmental conditions and movement patterns.

The remainder of this paper is organized as follows. Section 2 provides a review of related work in the field of human mobility trajectory forecasting. Section 3 outlines the study area and the datasets used in this study. Section 4 provides a detailed description of the proposed human mobility trajectory forecasting method. Section 5 presents and discusses the experimental results. Finally, Section 6 concludes the paper and suggests future research directions.

2. Related Works

Trajectory forecasting, particularly in the context of human mobility arising from daily urban activities, has been widely researched across various domains. Existing forecasting methods can be generally divided into three major categories: traditional methods, machine learning methods, and deep learning methods. Each of these methods has its strengths and weaknesses, particularly in terms of robust long-term and real-time dynamic forecasting.

2.1. Traditional Methods

Traditional methods for human trajectory forecasting primarily rely on statistical models and mathematical techniques that utilize historical data. These models often focus on creating deterministic or probabilistic frameworks to forecast human mobility.

One of the earliest and most prominent methods for human mobility trajectory forecasting is the Kalman Filter method, which is widely used in linear dynamic systems [31,32]. For instance, Motai et al. [20] proposed a novel trajectory forecasting method that combines Kalman filtering and optical flow, enabling reliable real-time trajectory forecasting in dynamic environments. Additionally, Markov Chains and HMMs have also been employed to forecast trajectories based on previous states [20,33]. Qiao et al. [34] developed a hybrid Markov-based model for predicting non-Gaussian mobility data, achieving over 56% accuracy by adapting to individual mobility patterns and spatio-temporal characteristics. Furthermore, Lv et al. [35] utilized HMMs to enhance spatio-temporal mobility forecasting, demonstrating that incorporating individuals’ living habits improves mobility forecasting accuracy across diverse groups. Another notable advancement was introduced by Fox et al. [22], who applied particle filters as an improvement over Kalman filters, particularly beneficial for real-time applications like robot localization. This method proved effective in handling nonlinear, non-Gaussian noise, making it suitable for dynamic daily activities, such as real-time obstacle monitoring.

While traditional methods excel in real-time forecasting with linear and structured environments, they often falter when dealing with complex and highly variable environments. Their lack of adaptability hinders their ability to account for intricate movement patterns and long-term trends in human mobility.

2.2. Machine Learning Methods

With the rise in computational power, machine learning techniques have gained prominence in trajectory forecasting for human mobility. These methods are capable of handling more complex data but typically require large datasets for accurate forecasting.

Papathanasopoulou et al. [23] employed Random Forest algorithms to classify human movement based on factors such as gender, walking pace, and distraction, achieving high accuracy. However, their study was limited by the use of experimental data from a controlled environment, which may not fully capture the variability of real-world pedestrian movements. Similarly, He et al. [24] applied Gradient Boosting Decision Trees to predict human mobility patterns, demonstrating the model’s capability to capture complex, nonlinear relationships between environmental factors and human movement, though they faced challenges with dataset sparsity and missing values, which may impact prediction accuracy. Additionally, Yu et al. [25] employed K-Nearest Neighbors (KNN) for short-term traffic condition predictions, achieving effective results by incorporating spatial and temporal parameters. While the model demonstrated good accuracy, it struggled with long-term predictions. Cho et al. [36] investigated the influence of social ties on human mobility forecasting, revealing significant impacts of social connections on movement patterns, thus enhancing forecasting accuracy. Moreover, Song et al. [11] examined the predictability of human mobility using large-scale location-based data, concluding that individual mobility is highly predictable, which has significant implications for forecasting models.

While machine learning methods provide greater flexibility and are able to capture nonlinear relationships, they often require extensive training data and can be computationally intensive, which limits their real-time applicability.

2.3. Deep Learning Methods

The advent of deep learning has revolutionized trajectory forecasting. Deep neural networks are capable of modeling complex movement patterns in human daily activities. However, these models come with challenges, particularly when real-time predictions are required.

Alahi et al. [26] proposed a Long Short-Term Memory (LSTM)-based model to predict human trajectories in crowded environments, where the LSTM excels in learning long-term dependencies, making it ideal for sequential data, such as trajectory prediction. Graph Convolutional Networks (GCNs) have also been effectively utilized in pedestrian trajectory prediction by modeling complex interactions and long-term tendencies, with innovations like Sparse GCN (SGCN) [27] and attention-based GCN (AVGCN) [28] further enhancing accuracy in dynamic environments. Karatzoglou et al. [29] introduced Seq2Seq models for long-term human mobility forecasting, effectively translating input sequences into output sequences for more accurate forecasting in continuous time-series data. Vemula et al. [37] implemented attention-based models for trajectory prediction, allowing the model to focus on the most relevant parts of the data. This approach enhanced both the accuracy and interpretability of forecasting. Further advancements include the work by Jia et al. [38], where they proposed an Attention-LSTM model for aircraft trajectory forecasting, demonstrating higher forecasting accuracy by focusing on influential factors. In another study, Lin et al. [39] explored the use of LSTMs with spatial-temporal attention mechanisms for trajectory forecasting, highlighting the model’s ability to explain the influence of historical trajectories and surrounding environments. Additionally, research by Lv et al. [40] and Peng et al. [41] has shown promising results in the application of LSTM and attention mechanisms for trajectory forecasting, respectively.

Deep learning methods offer unparalleled accuracy in capturing complex movement patterns and making long-term forecasts [42,43]. However, the computational demands of these models, combined with the large datasets required for training, often hinder real-time applicability [44].

In summary, traditional methods provide effective real-time trajectory forecasting but have limitations in handling complex movement patterns with robustness over long-term periods. Machine learning methods present greater flexibility but usually at the cost of computational efficiency. Deep learning techniques deliver high accuracy, particularly for long-term forecasting, but also struggle with maintaining computational efficiency. Therefore, our study aims to fill these gaps by integrating potential field theory with movement patterns, improving forecasting performance in terms of both efficiency and robustness. The potential field method, which simulates the heterogeneous effects of geographical environments, ensures computational feasibility for dynamic real-time applications, while incorporating pattern-based knowledge enhances long-term forecasting accuracy, minimizing the compounding errors typical of data-driven approaches.

3. Study Area and Data Description

3.1. Study Area

The study area is the Zhongzheng District of Taipei, an urban region known for its dense population. The area’s urban landscape features a mix of natural, economic, and cultural areas, making it ideal for studying trajectories of human mobility arising from daily urban activities. As shown in Figure 3, the area’s diverse features, including roads, buildings, and water bodies, create an environment for examining how geographical environments affect human mobility trajectories.

3.2. Synthetic Trajectory Dataset

We used a synthetic trajectory dataset to evaluate forecasting performance via a hybrid simulation framework. This framework is driven by prior knowledge and practical data to generate a reliable and adaptive dataset. Specifically, the prior knowledge is utilized to define basic simulation rules for each type of movement pattern. And the practical dataset, GeoLife [45,46,47], is utilized to provide reference information, improving the simulation reliability within our target study area. To ensure the applicability of the GeoLife dataset to our target study area, we carefully considered the similarities between Beijing (the primary location of GeoLife trajectories) and Taipei. Both cities are highly urbanized, and economically developed, and exhibit similar industrial structures and urban mobility patterns. These shared characteristics support the transferability of human mobility knowledge derived from the GeoLife dataset to our target study area, thereby enhancing the reliability of the synthetic trajectory dataset in representing realistic movement scenarios [48,49,50].

The prior knowledge of different movement patterns is summarized as follows: First, the convergence pattern is typically destination-oriented. It requires the fastest arrival to the destination, which is generally an important public place. Second, for the divergence pattern, on the contrary, its starting location is generally a public place. And without an urgent need for arrival, it would first pass through certain waypoints for other activities before ultimately reaching the destination. Third, the leadership pattern is typically leader mobility-oriented. It primarily focuses on the movement speed of the leader and the queue of the followers. Therefore, we clarify the key points for different movement patterns’ simulation, which is briefly presented in

Table 1. Key points for different movement patterns’ simulation based on prior knowledge.

Movement Pattern	Prior Knowledge	Key Points for Simulation
Convergence pattern	destination-oriented; public places as destination locations; urgency	the selection of the starting location and the destination location; the simulation of the fastest arrival to the destination
Divergence pattern	public places as starting locations; meaningful waypoints that represent participation in other activities	the selection of the starting location, meaningful waypoints, and the destination location
Leadership pattern	leader mobility-orientated; meaningful movement attributes	the simulation of the leader’s and follower’s movement attributes

. The specific simulation rules for each pattern are introduced in the following sections.

(1)

Trajectory simulation for the convergence pattern

To simulate the convergence pattern, we first extract reference information from the GeoLife dataset, including the alternative types of starting location and destination location (Algorithm 1). Then, based on the reference information and POI dataset, the starting location and destination location are randomly selected for a trajectory sample (Algorithm 2, lines 3–9). Third, the A* method is used to generate the shortest path from the starting location to the destination location (Algorithm 2, line 11). Finally, with the sparsification and randomization on the path (Algorithm 2, lines 13–18), the trajectory sample can be generated.

Algorithm 1: Reference information extraction from the GeoLife dataset for the convergence pattern

Input: GeoLife trajectory dataset ${\mathcal{D}}^{GL}={{traj}_{\left(i\right)}^{GL}}_{i=1}^{N_{GL}}$; POI dataset in Beijing ${POIs}^{BJ}$ minimum frequency threshold $f_{min}$

Output: alternative set for the selection of starting and destination location types ${\mathcal{A}}^{con}={{<t^{start}\left(i\right),t^{des}\left(j\right)>}_{\left(k\right)}}_{k=1}^N$ 1.  Initialization: pair samples set ${\mathcal{A}}_{temp}\leftarrow$ 2.    For ${traj}_{\left(i\right)}^{GL}=\{q_{\left(i\right)}^1=\left(x_1,y_1\right),q_{\left(i\right)}^2,\dots ,q_{\left(i\right)}^n=\left(x_n,y_n\right)$ in ${\mathcal{D}}^{GL}$ do 3.    //Identify the types of the starting location and the destination location 4.    $t_{\left(i\right)}^{start}={PO{I}_{T}ype}_{\left(k\right)}$, where $k=\underset{k}{\mathrm{argma}x}\left({\displaystyle \frac{N_k}{N_{nei\left(q_{\left(i\right)}^1\right)}}}\right)$ //the semantics type of location is determined by the POI type with the highest proportion its neighborhood. 5.    $t_{\left(i\right)}^{des}={PO{I}_{T}ype}_{\left(k\right)}$, where $k=\underset{k}{\mathrm{argma}x}\left({\displaystyle \frac{N_k}{N_{nei\left(q_{\left(i\right)}^n\right)}}}\right)$ 6.    //select potential trajectory in convergence pattern 7.    If $t_{\left(i\right)}^{des}$ in {Type_education, Type_workplace, Type_commercial, Type_ tourist} do 8.     Add <$t_{\left(i\right)}^{start},t_{\left(i\right)}^{des}$> to ${\mathcal{A}}_{temp}$ 9.    End if 10. End for 11. //mining meaningful starting and destination location pair types from ${\mathcal{A}}_{temp}$ 12. Initialization: alternative selection set: ${\mathcal{A}}^{con}\leftarrow$ 13. For each pair type <$t^{start}\left(i\right),t^{des}\left(j\right)$> occurred in ${\mathcal{A}}_{temp}$: 14.  //$t^{start}\left(i\right)$ represents the $i$-th type starting location and $t^{start}\left(j\right)$ represents the $j$-th type destination location 15.  If $Count\left(<t^{start}\left(i\right),t^{des}\left(j\right)>\right)>f_{min}$ do 16.     Add $<t^{start}\left(i\right),t^{des}\left(j\right)>$ to ${\mathcal{A}}^{con}$ 17.  End if 18.  End for 19.  Return alternative set for the selection of starting and destination location types ${\mathcal{A}}^{con}$

Algorithm 2: Trajectory sample simulation for the convergence pattern

Input: alternative set for the selection of starting and destination location types ${\mathcal{A}}^{con}={{<t^{start}\left(i\right),t^{des}\left(j\right)>}_{\left(k\right)}}_{k=1}^N$; POI dataset in Taipei ${POIs}^{TP}$; minimum distance threshold $d_{min};$ road network $\mathcal{G}$; sampling frequency $f$; gauss noise parameters $\left(\mu ,\sigma \right)$

Output: simulated trajectory sample ${traj}^{con}=q^1=\left(x_1,y_1\right),q^2,\dots ,q^n=\left(x_n,y_n\right)$ 1. Initialization: ${traj}^{con}\leftarrow$ 2. //Selection of the starting location and destination location 3. Randomly select a pair $<t^{start}\left(i\right),t^{des}\left(j\right)>$ from ${\mathcal{A}}^{con}$ 4. Do 5.   Randomly sample a ${POI}_{k1}\left(i\right)$ from ${POIs}^{TP}$//$i$ represent the type index same as $t^{start}\left(i\right)$ 6.   $q^{start}\leftarrow q\left({POI}_{k1}\left(i\right)\right)$ 7.   Randomly sample a ${POI}_{k2}\left(j\right)$ from ${POIs}^{TP}$ //$j$ represent the type index same as $t^{start}\left(i\right)$ 8.   $q^{\mathrm{des}}\leftarrow q\left({POI}_{k2}\left(j\right)\right)$ 9.  Until $distance\left(q^{\mathrm{start}},q^{\mathrm{des}}\right)>d_{\min }$ //ensure the trajectory length 10. //Path generation 11. $path=A_a^{\ast }lgorithm\left(q^{\mathit{start}},q^{\mathit{des}},\mathcal{G}\right)$ 12. //Sparsification and randomization 13. ${traj}_{temp}^{con}=Sparsificatio{n}_{s}ampl{e}_{a}lgorithm\left(path,f\right)$ 14. For ${q^{\prime }}^i=\left(x_i^{\prime },y_i^{\prime }\right)$ in ${traj}_{temp}^{con}$ do 15.  $x_i=x_i^{\prime }+\mathrm{random}.\mathrm{gauss}\left(\mu ,\sigma \right)$ 16.  $y_i=y_i^{\prime }+\mathrm{random}.\mathrm{gauss}\left(\mu ,\sigma \right)$ 17.  Add $q^i=\left(x_i,y_i\right)$ to ${traj}^{con}$. 18.  End for 19.  Return ${traj}^{con}$

(2)

Trajectory simulation for the divergence pattern

To simulate the divergence pattern, we first extract reference information from the GeoLife dataset, including the alternative types of starting location, waypoints, and destination location (Algorithm 3). It is worth noting that waypoint detection considers both movement attributes and semantic information (Algorithm 3, lines 9–12). Then, based on the reference information and POI dataset, starting location waypoints and destination locations are randomly sampled. Third, we generate a path using the A* method according to the order of these sampling points. And the trajectory sample is finally generated with a sparsification and randomization operation (these steps are similar to those in Algorithm 2).

Algorithm 3: Reference information for the divergence pattern

Input: GeoLife trajectory dataset ${\mathcal{D}}^{GL}={{traj}_{\left(i\right)}^{GL}}_{i=1}^{N_{GL}}$; POI dataset in Beijing ${POIs}^{BJ}$; time threshold and distance threshold for stay-point detection algorithm $timeThreh,distThreh$; minimum frequency threshold $f_{min}$

Output: alternative set for the selection of starting location, waypoints and destination location types ${\mathcal{A}}^{div}={{<t^{start}\left(i\right),t_1^{wp}\left(p\right),t_2^{wp}\left(l\right),..,t^{des}\left(j\right)>}_{\left(k\right)}}_{k=1}^N$ 1. Initialization: sequence samples set ${\mathcal{A}}_{temp}\leftarrow$ 2. alternative set ${\mathcal{A}}^{div}\leftarrow$ 3.   For ${traj}_{\left(i\right)}^{GL}=\{q_{\left(i\right)}^1=\left(x_1,y_1\right),q_{\left(i\right)}^2,\dots ,q_{\left(i\right)}^n=\left(x_n,y_n\right)$ in ${\mathcal{D}}^{GL}$ do 4.   //Identify the types of the starting location and the destination location 5.   $t_{\left(i\right)}^{start}={PO{I}_{T}ype}_{\left(k\right)}$, where $k=\underset{k}{\mathrm{argma}x}\left({\displaystyle \frac{N_k}{N_{nei\left(q_{\left(i\right)}^1\right)}}}\right)$ //the semantics type of location is determined by the POI type with the highest proportion its neighborhood. 6.   $t_{\left(i\right)}^{des}={PO{I}_{T}ype}_{\left(k\right)}$, where $k=\underset{k}{\mathrm{argma}x}\left({\displaystyle \frac{N_k}{N_{nei\left(q_{\left(i\right)}^n\right)}}}\right)$ 7.   //select potential trajectory in divergence pattern 8.   If $t_{\left(i\right)}^{start}$ in {Type_education, Type_workplace, Type_commercial, Type_ tourist} do 9.     //preliminary waypoints detection based on movement attributes of the trajectory 10.     $\left\{q_{\left(i,1\right)}^{wp},q_{\left(i,2\right)}^{wp},\dots ,q_{\left(i,m\right)}^{wp}\right\}=sta{y}_{p}oin{t}_{d}etectio{n}_{a}lgorithm\left({traj}_{\left(i\right)}^{GL},timeThreh,distThreh\right)$ 11.     //meaningful waypoints filter based on semantics information of surrounding POIs, only certain types of waypoints are retained. 12.     $\left\{t_{\left(i,1\right)}^{wp},t_{\left(i,2\right)}^{wp},\dots ,t_{\left(i,m^{\prime }\right)}^{wp}\right\}=filte{r}_{a}lgorithm\left(\left\{q_{\left(i,1\right)}^{wp},q_{\left(i,2\right)}^{wp},\dots ,q_{\left(i,m\right)}^{wp}\right\},{POIs}^{BJ}\right)$ 13.     Add <$t_{\left(i\right)}^{start},t_{\left(i,1\right)}^{wp},t_{\left(i,2\right)}^{wp},\dots ,t_{\left(i,m^{\prime }\right)}^{wp},t_{\left(i\right)}^{des}$> to ${\mathcal{A}}_{temp}$ 14.  End if 15. End for 16. //mining meaningful starting location, waypoints and destination location sequence types from ${\mathcal{A}}_{temp}$ 17. Initialization: alternative selection set: ${\mathcal{A}}^{div}\leftarrow$ 18. For each sequence type $<t^{start}\left(i\right),t_1^{wp}\left(p\right),t_2^{wp}\left(l\right),..,t^{des}\left(j\right)>$ occurred in ${\mathcal{A}}_{temp}$: 19.  //$i,p,l,j$ represent the type indexes 20.  If $Count\left(<t^{start}\left(i\right),t_1^{wp}\left(p\right),t_2^{wp}\left(l\right),..,t^{des}\left(j\right)>\right)>f_{min}$ do 21.     Add $<t^{start}\left(i\right),t_1^{wp}\left(p\right),t_2^{wp}\left(l\right),..,t^{des}\left(j\right)>$ to ${\mathcal{A}}^{div}$ 22.  End if 23.  End for 24.  Return ${\mathcal{A}}^{div}$

(3)

Trajectory simulation for the leadership pattern

To simulate the leadership pattern, we first extract reference information from the GeoLife dataset, including the alternative selections of movement speeds (Algorithm 4). Then, the leader trajectory is first generated (Algorithm 5, lines 4–18). The generation steps are similar to those previously described, with one major difference in the sparsification operation. Specifically, when sampling trajectory points along the path, the reference movement speeds should be taken into account (Algorithm 5, lines 13–14). Subsequently, the final follower trajectory is generated considering the following queue parameters (Algorithm 5, lines 20–22).

Algorithm 4: Reference information for the leadership pattern

Input: GeoLife trajectory dataset ${\mathcal{D}}^{GL}={{traj}_{\left(i\right)}^{GL}}_{i=1}^{N_{GL}}$

Output: alternative set for the selection of movement speeds in trajectory ${\mathcal{A}}^{lea}={{<v_{\left(1\right)},v_{\left(2\right)},..v_{\left(n\right)}>}_{\left(k\right)}}_{k=1}^N$ 1. Initialization: alternative set ${\mathcal{A}}^{lea}\leftarrow$ 2. Sample a number of walking-type trajectories ${{\mathcal{D}}^{\prime }}^{GL}$ from ${\mathcal{D}}^{GL}$ 3.   For ${traj}_{\left(i\right)}^{GL}=\{q_{\left(i\right)}^1=\left(x_1,y_1\right),q_{\left(i\right)}^2,\dots ,q_{\left(i\right)}^n=\left(x_n,y_n\right)$ in ${{\mathcal{D}}^{\prime }}^{GL}$ do 4.   //Calculate movement speeds between trajectory locations 5.   $k=1$ 6.   While $k<n$ do 7.     $v_{\left(k\right)}=dist\left(q_{\left(i\right)}^k,q_{\left(i\right)}^{k+1}\right)/tim{e}_{d}elta\left(q_{\left(i\right)}^k,q_{\left(i\right)}^{k+1}\right)$ 8.     $k=k+1$ 9.   End while 10. Add $<v_{\left(1\right)},v_{\left(2\right)},..v_{\left(n\right)}>$ to ${\mathcal{A}}^{lea}$ 11. End for 12. Return  ${\mathcal{A}}^{lea}$

Algorithm 5: Trajectory sample simulation for the leadership pattern

Input: alternative set for the selection of movement speeds in trajectory ${\mathcal{A}}^{lea}={{<v_{\left(1\right)},v_{\left(2\right)},..v_{\left(n\right)}>}_{\left(k\right)}}_{k=1}^N$; POI dataset in Taipei ${POIs}^{TP}$; minimum distance threshold $d_{min};$ road network $\mathcal{G}$; gauss noise parameters $\left(\mu ,\sigma \right)$; lag distance and offset angle for the followers’ queue $d_{lag},{\theta }_{off}$

Output: simulated trajectory sample ${traj}^{lea}=q^1=\left(x_1,y_1\right),q^2,\dots ,q^n=\left(x_n,y_n\right)$ 1.  Initialization: ${traj}^{lea}\leftarrow$ 2.  //Generation of leader trajectory 3.  //Selection of the starting and destination locations 4.  Do 5.   Randomly sample a ${POI}_{k1}$ from ${POIs}^{TP}$ 6.   $q^{start}\leftarrow q\left({POI}_{k1}\right)$ 7.   Randomly sample a ${POI}_{k2}$ from ${POIs}^{TP}$ 8.   $q^{\mathrm{des}}\leftarrow q\left({POI}_{k2}\right)$ 9.  Until $distance\left(q^{\mathrm{start}},q^{\mathrm{des}}\right)>d_{\min }$ 10. //Path generation 11. $path=A_a^{\ast }lgorithm\left(q^{\mathit{start}},q^{\mathit{des}},\mathcal{G}\right)$ 12. //Sparsification 13. Randomly select a sequence $<v_{\left(1\right)},v_{\left(2\right)},..v_{\left(n\right)}>$ from ${\mathcal{A}}^{lea}$ 14. ${traj}_{temp}^{lea}=Sparsificatio{n}_{s}ampl{e}_{a}lgorithm\left(path,<v_{\left(1\right)},v_{\left(2\right)},..v_{\left(n\right)}>\right)$ 15. //Randomization 16. For ${q^{″}}^i=\left(x_i^{″},y_i^{″}\right)$ in ${traj}_{temp}^{lea}$ do 17.  $x_i^{\prime }=x_i^{\prime \prime }+\mathrm{random}.\mathrm{gauss}\left(\mu ,\sigma \right)$ 18.  $y_i^{\prime }=y_i^{″}+\mathrm{random}.\mathrm{gauss}\left(\mu ,\sigma \right)$ 19.  //Generation of follower trajectory 20.  $x_i=x_i^{\prime }+d_{lag}\ast sin\left({\theta }_{off}\right)$ 21.  $y_i=y_i^{\prime }+d_{lag}\ast cos\left({\theta }_{off}\right)$ 22.  Add $q^i=\left(x_i,y_i\right)$ to ${traj}^{lea}$. 23.  Return ${traj}^{lea}$

Finally, according to the defined simulation rules, we generated 100 trajectory samples for each type of movement pattern. Therefore, a synthetic trajectory dataset with diverse movement patterns and an abundant number of samples can be constructed for the subsequent experiments.

3.3. Environment Datasets

In our study, the environmental factors include nine categories, namely plain areas, roads, steep areas, forests, grass lawns, water bodies, buildings, restricted areas, and accident black spots. The environmental factors are acquired from multi-source geographical data. A detailed description of each category of environmental factors is presented in

Table 2. Description of environmental datasets.

Dataset	Year	Source	Description
Plain areas	2023	Geospatial Data Cloud (https://www.gscloud.cn/home, accessed on 1 December 2023)	Extracted from the 30 m resolution DEM data. The specific extraction operations include slope calculation, reclassification, and raster-to-polygon conversion.
Roads	2024	OSM (https://www.openstreetmap.org/, accessed on 1 March 2024)	Extracted from the road data provided by OSM. The road types we considered include primary, secondary, tertiary, residential, footway, etc.
Steep areas	2023	Geospatial Data Cloud	Extracted from the 30 m resolution DEM data. The specific extraction operations include slope calculation, reclassification, and raster-to-polygon conversion.
Forests	2024	OSM	Extracted from the land use data provided by OSM.
Grass lawns	2024	OSM	Extracted from the land use data provided by OSM.
Water bodies	2024	OSM	Extracted from the water data provided by OSM.
Buildings	2024	OSM	The buildings’ data were downloaded from OSM.
Restricted areas	2024	OSM	Extracted from the land use data provided by OSM. Restricted areas specifically refer to areas that are inaccessible due to sensitive factors such as political or military considerations.
Accident black spots	2024	TomTom (https://www.tomtom.com/traffic-index/, accessed on 1 March 2024)	Quantified based on traffic index data, with data reference provided by TomTom, a well-known map company. Areas with a high propensity for traffic incidents are regarded as accident black spots.

. And the spatial distribution of environmental data is visualized in Figure 4.

4. Methodology

4.1. Overall Framework

In our study, we propose an environment-driven trajectory forecasting method for human mobility arising from daily urban activities. In all, we formulate human mobility actions in daily urban activities as reactions to specific environmental factors. Firstly, as illustrated in Figure 5, considering the influence of the geographical environments on human mobility varies depending on the prior patterns, we develop different indicator systems for the convergence, divergence, and leadership patterns. These indicator systems comprehensively summarize the heterogeneous effects of environmental factors. Under the guidance of the indicator system of the corresponding movement pattern, the potential field is first generated to quantify the comprehensive effect of all environmental factors. Specifically, the potential field modeling includes two aspects. On the one hand, attractive and repulsive potential functions are constructed, respectively, to simulate the attraction or repulsion effect of environmental factors on human mobility. On the other hand, the weights of different environmental factors are determined in a hybrid manner. Thereby, a comprehensive potential field can be generated through the weighted sum manner, representing the calibrated probability of the next-step mobility direction. Finally, a gradient descent algorithm is then employed on the potential field to determine the next-step mobility location.

4.2. Indicator System Construction

The indicator system is developed to systematically identify what types of environmental factors are influencing human mobility and what specific effects they have. Specifically, the indicator system is structured into three levels.

The first level of the indicator system reflects the attractive or repulsive effect of environmental factors. Attractive factors, such as accessible roads and destinations, have a positive effect on human mobility. People are drawn to these factors for convenience or safety during the travel process. In contrast, repulsive factors, such as restricted areas, water bodies and obstacles, negatively influence the decisions of human mobility, pushing them toward alternative routes. Attraction and repulsion are the most fundamental manifestations of the heterogeneous effects of environmental factors.

The second level of the indicator system further refines the specific aspects of how attractive and repulsive factors affect human mobility. Attractive factors can be further refined into travel opportunities and travel purposes. Their detailed meanings are presented as follows:

• Travel Opportunities: Refers to the availability of paths or plain areas that facilitate easy movement through the urban spaces. Roads and open spaces offer flexibility and convenience in route selection, influencing decision-making and efficiency.
• Travel Purpose: Encompasses the motivations behind the movement, influencing trajectories in various ways. It can lead to convergence when individuals aim for a common destination, divergence when they disperse from a central point or a starting location, and leadership patterns when others follow a leader’s path. It helps shape movement patterns in daily activities and social contexts, reflecting both individual goals and group behavior.

Repulsive factors can be further refined into travel cost, public ethics, obstacles, and safety concerns. Their detailed meanings are presented as follows:

• Travel Cost: Represents the effort required to traverse specific terrains, such as steep inclines or forested areas. High travel costs are associated with challenging terrains that impede movement, making these areas less likely to be chosen for travel.
• Public Ethics: Social norms, such as avoiding walking on grass lawns, influence pedestrian trajectories. Ethical considerations and public rules can create restrictions on movement, even if these paths offer shortcuts.
• Obstacles: Physical barriers like water bodies or buildings prompt people to adjust their routes, avoiding direct paths to prevent collisions with these obstacles. These obstacles require detours but do not necessarily alter the overall trend of travel, simply influencing the specific paths taken.
• Safety Concerns: Relates to the perceived risk associated with certain areas, such as restricted areas or accident black spots. People are more likely to avoid unsafe routes, opting for paths that offer better security and lower risks [51].

The first and second levels of the indicator system provide a comprehensive understanding of the heterogeneous effect of environmental factors on human mobility. Then, the third level of the indicator system presents the specific environmental factors corresponding to the aforementioned classification.

The indicator systems for convergence, divergence, and leadership patterns differ in their focus on movement patterns and interactions with the environment. For the convergence pattern, factors attract people to a common point (destination), while restricted areas or obstacles act as repulsive forces. In divergence, the indicators emphasize dispersal opportunities and multiple path choices. For leadership patterns, the system focuses on how people follow a leader, with the leader’s influence being more important than other environmental factors. Each system reflects how people interact with their surrounding environments in different movement patterns. The indicator system constructed for the convergence pattern is shown in Figure 6. For divergence and leadership, the primary differences lie in two aspects: the travel purpose and the weight of environmental factors. Specifically, in convergence, the travel purpose is a shared destination; in divergence, it is the start location; and in leadership, it revolves around the leader’s trajectory. Additionally, the weights assigned to environmental factors vary across these patterns to reflect their unique influences on each pattern. The detailed calculation manner for these weights is provided in Section 4.3.3.

4.3. Potential Field Modeling

4.3.1. Overview for Modeling Approach

Potential field modeling is employed to comprehensively and quantitatively simulate the effect of environmental factors on human mobility [52,53]. The potential field method conceptualizes the environment as a field of forces, where environmental factors typically exert two types of forces, i.e., attractive or repulsive forces, on human mobility. The combined effects of these two types of forces finally determine the trajectories on which people are likely to travel [54].

Based on the abovementioned indicator system constructed in a top-down manner, we can now calculate the potential field, which comprehensively forms the effect of all environmental factors on human mobility from a bottom-up perspective. Specifically, the potential $V_{\left(x,y\right)}$ at a location $q_{\left(x,y\right)}=\left(x,y\right)$ is computed as a weighted sum of the potentials from each type of environmental factor,

$$V_{\left(x,y\right)}={\sum }_{i=1}^{n}w\left(i\right)·U_{\left(x,y\right)}\left(i\right),$$

(1)

where $V_{\left(x,y\right)}$ is the total potential at the location $q_{\left(x,y\right)}$. $n$ is the total number of environmental factors. $w\left(i\right)$ is the weight associated with the $i$-th factor, reflecting its relative importance. $U_{\left(x,y\right)}\left(i\right)$ is the potential contribution from the $i$-th environmental factor. It can be divided into two categories, i.e., attractive potential $U_{att}$ and repulsive potential $U_{rep}$. They use a different computation manner. The next two subsections would describe in detail how to construct the two types of potential functions and how to determine the weights of the environmental factors.

4.3.2. Attractive and Repulsive Potential Functions

Considering the attractive force can pull the people towards the attractive factors, the attractive potential $U_{att}$ can be typically calculated by the following potential function:

$$U_{\left(x,y\right)}\left(att\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{k}_{att}{{ d}_0}^2& ,if‖q_{\left(x,y\right)}-q_{\left(x,y\right)}^{att}‖\le { d}_0\hfill \\ {\displaystyle \frac{1}{2}}{k}_{att}{‖q_{\left(x,y\right)}-q_{\left(x,y\right)}^{att}‖}^2& ,if{ d}_0<‖q_{\left(x,y\right)}-q_{\left(x,y\right)}^{att}‖\le r\hfill \\ 0& ,if‖q_{\left(x,y\right)}-q_{\left(x,y\right)}^{att}‖>r\hfill \end{array}\right.,$$

(2)

where $U_{\left(x,y\right)}\left(att\right)$ is the attractive potential at location $U_{\left(x,y\right)}\left(att\right)$. $k_{att}$ is the attraction gain coefficient. $q_{\left(x,y\right)}^{att}$ represents the location of the attractive factors closest to the location $q_{\left(x,y\right)}$. $‖q_{\left(x,y\right)}-q_{\left(x,y\right)}^{att}‖$ represents the distance between location $q_{\left(x,y\right)}$ and its closest attractive factors.${ d}_0$ is the trigger threshold. $r$ is the influence range of the attractive factors. Once the people get into the effect range of attractive factors, the attractive potential would significantly decrease as the distance between the people and the attractive factors gets smaller. It can ensure that the people are increasingly attracted as they get closer to the attractive factors.

On the contrary, the repulsive force is designed to push the people away from the repulsive factors. The repulsive potential $U_{rep}$ can be typically calculated by the following potential function:

$$U_{\left(x,y\right)}\left(rep\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{2}}{k}_{rep}{\left({\displaystyle \frac{1}{{ d}_0}}-{\displaystyle \frac{1}{r}}\right)}^{2 } ,if‖q_{\left(x,y\right)}-q_{\left(x,y\right)}^{rep}‖\le { d}_0\\ {\displaystyle \frac{1}{2}}{k}_{rep}{\left({\displaystyle \frac{1}{‖q_{\left(x,y\right)}-q_{\left(x,y\right)}^{rep}‖}}-{\displaystyle \frac{1}{r}}\right)}^{2 } ,if{ d}_0<‖q_{\left(x,y\right)}-q_{\left(x,y\right)}^{rep}‖\le r\\ 0 ,if‖q_{\left(x,y\right)}-q_{\left(x,y\right)}^{rep}‖>r\end{array}\right.,$$

(3)

where $U_{\left(x,y\right)}\left(rep\right)$ is the repulsive potential at location $q_{\left(x,y\right)}$. $k_{rep}$ is the repulsion gain coefficient. $q_{\left(x,y\right)}^{rep}$ represents the location of the repulsive factors closest to the location $q_{\left(x,y\right)}$. $‖q_{\left(x,y\right)}-q_{\left(x,y\right)}^{rep}‖$ represents the distance between location $q_{\left(x,y\right)}$ and its closest repulsive factors.${ d}_0$ is the trigger threshold, and $r$ is the influence range of the repulsive factors. This repulsive potential would significantly increase as the people get closer to a repulsive factor, thus exerting a repulsive force that pushes the people away from potential collisions.

The meanings and setting values of the hyperparameters in Formulas (2) and (3) are further explained in this paragraph. Taking the convergence pattern as an example, the distances calculated in Equations (2) and (3) are based on the shortest distance from the nearest line or polygon geometry. Notably, there is a small threshold value $d_0$ where distances within this range are assigned a value of $d_0$ to prevent the potential from becoming infinitely large, especially when the point is inside the polygon. Conversely, $r$ represents the influence range; beyond this range, the potential is set to zero. The gain coefficients ($k_{att}$/$k_{rep}$) and influence radius ($r$) for each factor are set as constants for all environmental factors to simplify the model’s complexity and enhance its generalization capabilities. Constant hyperparameters are commonly used in spatial interaction and potential field models, such as the gravitational coefficient in human mobility studies, to represent stable relationships between human and environmental factors [55,56]. Similarly, the fixed influence radius helps clearly define the spatial scope of environmental factors’ effects, similar to the service radius concept in urban facilities planning [57]. Except for the travel purpose, it has a global influence and is not restrained by a finite radius.

4.3.3. Manner for Determining the Weights of Environmental Factors

In addition, the other important elements in Equation (1), namely the weights for different types of environmental factors, are required to be determined. Each environmental factor’s potential field is represented as a 2D potential field array. The weights are used to aggregate these arrays into a composite potential field for subsequent trajectory forecasting. In our study, the weights for different types of environmental factors are determined by a combined weighting manner. Specifically, this manner integrates both the subjective expert scoring method and the objective entropy weight method. For each type of movement pattern, the details of the weighting manner are described as follows.

(1)

Subjective expert scoring method

The expert scoring method incorporates domain knowledge to determine weights [58]. A structured questionnaire was distributed to 20 domain experts, who evaluated the factors across five levels of importance: Highly Important (Score 5), Moderately Important (Score 4), Neutral (Score 3), Slightly Important (Score 2), and Not Important (Score 1). The detailed results of the expert scoring survey are shown in Appendix A. It is worth noting that the expert scoring survey results are, respectively, obtained for the convergence, divergence, and leadership patterns. For the $i$-th environmental factors, its total score evaluated by an expert is calculated by

$$\mathrm{s}\left(i\right)={\sum }_{p=1}^{N_p}{c}_p\left(i\right)·s_p,$$

(4)

where $\mathrm{s}\left(i\right)$ represents the total score of the $i$-th environmental factors. $N_p$ represents the number of importance levels. $c_p\left(i\right)$ represents the number of $p$-th importance levels received from experts for the $i$-th environmental factor. $s_p$ represents the basic score of the $p$-th importance level. Then, the weight for the $k$-th environmental factor is calculated by the proportion of its score in the total score, using the following equation:

$$w_s\left(i\right)={\displaystyle \frac{\mathrm{s}\left(i\right)}{\sum _{j=1}^n\mathrm{s}\left(j\right)}},$$

(5)

where $w_s\left(i\right)$ represents the weight for the $i$-th environmental factor determined by the subjective expert scoring method.

(2)

Objective entropy weight method

The entropy weight method is a data-driven method that objectively reflects the importance of environmental factors based on their variability across data samples [59]. The detailed calculation steps are as follows. Firstly, it needs to perform the data normalization operation. The raw data of each environmental factor is normalized to ensure comparability,

$$U_{\left(x,y\right)}^{norm}\left(i\right)={\displaystyle \frac{U_{\left(x,y\right)}\left(i\right)-U^{min}\left(i\right)}{U^{max}\left(i\right)-U^{min}\left(i\right)}},$$

(6)

where $U_{\left(x,y\right)}^{norm}\left(i\right)$ represents the normalized potential value, and $U^{min}\left(i\right)$ and $U^{max}\left(i\right)$ are the minimum and maximum potential values for the $i$-th environmental factor, respectively. Then, it needs to calculate the information entropy. The entropy $E\left(i\right)$ for the $i$-th environmental factor is computed as follows:

$$E\left(i\right)=-{\displaystyle \frac{1}{\ln \left(N_x·N_y\right)}}{\sum }_{y=1}^{N_y}{\sum }_{x=1}^{N_x}{p}_{\left(x,y\right)}\left(i\right)\ln \left(p_{\left(x,y\right)}\left(i\right)\right),$$

(7)

where $p_{\left(x,y\right)}\left(i\right)={\displaystyle \frac{U_{\left(x,y\right)}^{norm}\left(i\right)}{\sum _{y=1}^{N_y}{\sum }_{x=1}^{N_x}{U}_{\left(x,y\right)}^{norm}\left(i\right)}}$ represents the proportion of the sample in location $q_{\left(x,y\right)}$ for the $i$-th environmental factor. $N_x$ and $N_y$ represent the numbers of locations in width and height, respectively.

Finally, it needs to determine the objective weight based on information entropy. For a certain environmental factor, the smaller its information entropy value, the greater the dispersion degree it has, and the more important information it reflects. Therefore, it requires a larger weight. Then, the objective weight for the $i$-th environmental factor $w_o\left(i\right)$ is calculated as follows:

$$w_o\left(i\right)={\displaystyle \frac{1-E\left(i\right)}{{\sum }_{j=1}^n\left(1-E\left(j\right)\right)}}.$$

(8)

(3)

Weight combination

The final weight for each factor is a linear combination of objective and subjective weights and is as follows:

$$w\left(i\right)=\alpha \cdot w_s\left(i\right)+\left(1-\alpha \right)\cdot w_o\left(i\right),$$

(9)

where $\alpha$ represents the proportion of reliance on entropy weighting, set at 0.5 in this study based on empirical validation. The weights are applied in the potential field model to scale the influence of each environmental factor’s potential field when summing their contributions. This ensures that both data-driven insights and expert judgment are reflected in the trajectory forecasting. This combined approach balances objectivity and domain relevance, enhancing model robustness and interpretability. The calculated results of the combined weights are also presented in Appendix A.

4.4. Gradient Descent Algorithm for Trajectory Forecasting

Based on the modeling result of the potential field, the gradient descent algorithm is utilized to implement the trajectory forecasting process. In the potential field, as illustrated in the pseudo-code of Algorithm 6, people travel from a starting location. During the travel process, people tend to move towards locations with low potential while simultaneously avoiding locations with high potential until they ultimately reach the destination location. To achieve the optimal trajectory forecasting, the gradient descent algorithm can be applied. First, in the initialization step, the location $q_{\left(x_0,y_0\right)}$ is set to $q^{start}$, the iteration index $i$ is initialized to 0, and then $q_{\left(x_0,y_0\right)}$ is added to the trajectory sequence $traj$ (Algorithm 6, lines 1–2). In the while loop, the next location $q_{\left(x_{i+1},y_{i+1}\right)}$ is calculated at each step by subtracting $\alpha \nabla V_{\left(x_i,y_i\right)}$ from $q_{\left(x_i,y_i\right)}$ and is added to the trajectory sequence $traj$ (Algorithm 6, lines 4–5). Here, the direction of the gradient $\nabla V_{\left(x_i,y_i\right)}$ indicates the direction in which the potential changes fastest. The learning rate $\alpha$ is determined based on the walking speed of pedestrians in the simulation. It controls the step size of each iteration, ensuring that the movement is neither too fast nor too slow [60]. The loop runs until the Euclidean distance between $q_{\left(x_i,y_i\right)}$ and $q^{des}$ is less than a step size or $i$ exceeds maximum iterations (Algorithm 6, line 3). An example of the trajectory forecasting using the gradient descent algorithm is also visualized in Figure 7.

Algorithm 6: Gradient descent algorithm for trajectory forecasting

Input: starting location $q^{start}$, destination location $q^{des}$, potential filed $V$ which simulates environment effects, learning rate $\alpha$, step size $s$, maximum iterations $n$

Output: forecasted trajectory $traj=q_{\left(x_0,y_0\right)}=q^{start},q_{\left(x_1,y_1\right)},q_{\left(x_2,y_2\right)},\dots ,q_{\left(x_n,y_n\right)}=q^{des}$ Initialization: $q_{\left(x_0,y_0\right)}\leftarrow q^{start}$, $i\leftarrow 0$ Add $q_{\left(x_0,y_0\right)}$ to trajectory sequence $traj$ While Dist($q_{\left(x_i,y_i\right)},q^{des}$) < $s$ and $i<n$ do $q_{\left(x_{i+1},y_{i+1}\right)}=q_{\left(x_i,y_i\right)}-\alpha \nabla V_{\left(x_i,y_i\right)}$ Add $q_{\left(x_{i+1},y_{i+1}\right)}$ to trajectory sequence $traj$ $i=i+1$ End while Return forecasted trajectory $traj$

5. Experiments and Discussions

5.1. Evaluation Metrics

The forecasting performance of the proposed method is evaluated using the following three metrics:

(1)

Average Displacement Error (ADE) [61]: ADE measures the mean squared distance between the forecasted location and the actual location. It is calculated as follows:

$$ADE={\displaystyle \frac{1}{n·t_{pred}}}{\sum }_{i=1}^n{\sum }_{t=1}^{t_{pred}}‖Y_i\left(t\right)-\widehat{Y_i}\left(t\right)‖,$$

(10)

where $Y_i\left(t\right)$ is the actual location at time $t$ for the $i$-th target, $\widehat{Y_i}\left(t\right)$ is the forecasted location, $n$ is the number of targets, and $t_{pred}$ is the number of forecasting time steps.

(2)

Maximum Absolute Error (MaxAE) [62]: MaxAE is the maximum observed distance between the forecasted and actual trajectory. It is calculated as:

$$MaxAE={max}_{i,t}‖Y_i\left(t\right)-\widehat{Y_i}\left(t\right)‖,$$

(11)

where $Y_i\left(t\right)$ is the actual location at time $t$ for the $i$-th target, $\widehat{Y_i}\left(t\right)$ is the forecasted location.

(3)

Miss Rate (MR) [63]: MR is the proportion of trajectories where the forecasted location exceeds a threshold distance from the actual position, as illustrated in Figure 8. The formula is as follows:

$$MR={\displaystyle \frac{N^{miss}}{N}},$$

(12)

where $N$ is the total number of trajectories, and $N^{miss}$ is the number of missed forecasted locations (those exceeding the threshold).

These metrics help in assessing the overall accuracy (ADE), worst-case performance (MaxAE), and model failures (MR).

5.2. Comparisons with Baseline Methods

To assess the effectiveness of the potential field approach, a comparison was made with Kalman Filter and LSTM baselines using metrics including ADE, MaxAE, and MR, as presented in Figure 9. Multi-step prediction involves forecasting several future time steps based on previous observations and predictions, allowing for a comprehensive evaluation of each model’s performance over extended horizons. The results indicate that the potential field model generally outperforms the other two methods in long-term predictions.

As also shown in the examples in Figure 10, one of the key advantages of the potential field approach is its ability to maintain low ADE and MaxAE across multiple forecasting steps. This demonstrates its robustness and reliability in accurately forecasting human mobility trajectories, even when the forecasting time step extends or when situations such as turns are encountered. Although the potential field model may not outperform LSTM in short-term forecasting (one, three, or five steps), it excels in longer-term forecasting (ten steps), showcasing its stability and long-term advantages. Furthermore, the potential field model exhibits a lower MR, indicating that it effectively minimizes the occurrence of significant forecasting errors.

In contrast, the Kalman filter, despite being widely used in various prediction applications, exhibits notably higher values for ADE, MaxAE, and MR across all three metrics. These errors increase significantly as the prediction step increases. The primary reason for this performance drop might be explained from two aspects. On the one hand, Kalman filters are designed for systems with linear dynamics and Gaussian noise, making them less effective in real-world urban environments where the environment is complex and highly nonlinear. On the other hand, the Kalman filter does not account for the underlying movement patterns, such as convergence, divergence, or trendsetter behaviors. This absence of consideration for human movement patterns and complex environmental factors means that the Kalman filter is unable to capture the full range of influences that drive mobility, which is essential for accurate trajectory forecasting in urban environments. Furthermore, the potential field model incorporates both complex environmental factors and prior movement patterns, which contributes to its superior performance in long-term forecasting. Although LSTM models are capable of learning from historical data and performing well in short-term predictions, they also struggle to maintain accuracy over longer prediction steps, primarily due to error accumulation.

Overall, the potential field method stands out for its ability to provide stable and accurate trajectory forecasting, making it particularly suitable for daily urban activities where understanding human mobility is critical for traffic management and safety. This trend highlights the model’s potential as a preferred choice for urban trajectory forecasting.

5.3. Forecasting Accuracy Results Under Different Movement Patterns

Based on the experimental results from

Table 3. The comparison of experimental results under different movement patterns: convergence, divergence, and leadership.

	ADE	MaxAE	MR
convergence	9.96	141.42	11.25%
divergence	6.22	141.42	9.30%
leadership	8.24	141.42	10.25%

, which compares the forecasting performance of the model based on potential fields under different movement patterns, slight variations are observed. For the ADE, the convergence pattern shows an average trajectory error of 9.96, which is slightly higher than the divergence (6.22) and leadership patterns (8.24). However, the MR and MaxAE show no significant differences across the patterns, indicating that the prediction algorithm maintains strong stability and robustness across different movement patterns.

Upon further analysis, the MR for all patterns remains within the range of 9.30% to 11.25%, showing that the forecasting algorithm adapts well to different movement patterns and maintains a low error rate in most cases, which is crucial for real-world applications. In terms of MaxAE, all movement patterns recorded the same error of 141.42. While the divergence pattern is characterized by its inherent uncertainty and exploratory nature, which might suggest variability, the consistent MaxAE across all three modes indicates that the prediction model effectively handles the complexities of each movement pattern. This uniformity reinforces the model’s robustness and reliability in trajectory forecasting, demonstrating that it can maintain performance despite the distinct behaviors associated with convergence, divergence, and leadership patterns.

In conclusion, the prediction model based on potential fields demonstrates strong prediction accuracy and stability across various movement patterns, making it an effective method for trajectory forecasting.

5.4. Capability Analysis of Real-Time Forecasting

The potential field approach has demonstrated remarkable efficacy in real-time trajectory forecasting applications. In contrast to deep learning models that often grapple with latency due to their intensive computational requirements, the potential field method offers expedited predictions, achieving a response time of less than 1 s per update. This swift response capability is crucial for real-time systems, as it allows for dynamic adjustment of forecasting based on the latest environmental data. In this study, with historical data as a reference, the method is capable of providing long-term trajectory forecasting for more than 30 min based on positional sampling intervals of 3 min. Moreover, the average accuracy of these forecasting processes is maintained at no less than 85%. This level of accuracy underscores the method’s reliability and effectiveness in long-term trajectory forecasting, which is particularly valuable for applications requiring precise and sustained predictive insights. This level of performance is highly beneficial for applications that require precise and timely trajectory guidance over extended periods. As depicted in Figure 11, the potential field method’s efficiency is evident, making it an ideal choice for systems that demand both dynamic real-time and robust long-term forecasting capabilities.

5.5. Evaluation of Data Sensitivity

This section presents an analysis of the model’s performance under varying conditions of data availability in a fifteen-step prediction. With results summarized in

Table 4. The comparison of experimental results under different missing conditions of environmental data.

Condition	ADE	MaxAE	MR
Full data	25.97	1358.31	11.71%
Missing Travel Opportunities data	28.34	1364.73	11.77%
Missing Travel Cost data	28.34	1364.73	11.78%
Missing Public Ethics data	28.10	1364.73	11.77%
Missing Obstacles data	29.08	1400.00	12.20%
Missing Safety Concerns data	29.21	1408.01	12.05%

, by comparing the model’s accuracy with complete environmental data against the conditions lacking specific environmental data categories, we assess the impact of each indicator on trajectory forecasting.

The results show that the model maintains robust forecasting accuracy even when some environmental data are unavailable. The absence of “Travel Opportunities” and “Travel Cost” data results in slight increases in ADE to 28.34 and MaxAE to 1364.73, with minimal changes in MR. Similarly, the lack of “Public Ethics” data shows a comparably smaller impact, with ADE increasing to 28.10, indicating that these factors, while influential, do not critically impair the model’s performance. However, the absence of “Obstacles” data significantly affects forecasting accuracy, with ADE rising to 29.08, MaxAE to 1400.00, and MR increasing to 12.20%. These results underline the critical role of obstacle-related data, such as water bodies and buildings, in shaping pedestrian trajectories. Furthermore, missing “Safety Concerns” data also leads to a notable degradation in performance, with ADE increasing to 29.21 and MR to 12.05%, reflecting the importance of accurately accounting for safety-related factors in trajectory forecasting.

Furthermore, we compare the results of the experiments with combined missing data, as shown in Figure 12. Both rows and columns represent the missing data categories. Hence, the results are symmetric matrices and the values on the diagonal correspond to the cases where only a single category of data is missing. The results of combined missing data further validate the conclusion drawn from the cases of single-category missing data. When data on “Travel Opportunities”, “Travel Cost”, and “Public Ethics” are missing in pairwise combinations, the three metrics (ADE, MaxAE, and MR) are generally smaller. This indicates that the absence of these data combinations has a relatively smaller impact on the results. Conversely, when data on “Obstacles” and “Safety Concerns” are missing in combination, the three metrics are generally larger. This significant increase in the metrics suggests that the lack of information regarding obstacles and safety has a substantial negative impact on the model’s performance, highlighting the crucial role these two factors play in accurately predicting the trajectories.

In summary, while the model demonstrates resilience in the face of environmental data absence, the type of missing data significantly influences its forecasting performance. The experiments underscore the necessity of incorporating diverse environmental factors for improved accuracy and reliability in trajectory forecasting. Future research will investigate methods for mitigating data sensitivity, such as data imputation techniques or integrating complementary data sources to further enhance model robustness under variable data conditions.

6. Summary and Outlook

6.1. Conclusions

This paper presents a novel environment-driven trajectory forecasting method that leverages potential field theory to model human mobility in urban environments. The methodology integrates the prior knowledge of movement patterns and the heterogeneous effects of geographical environments to provide robust and adaptive trajectory predictions for daily urban activities. The main contributions of the study are summarized as follows:

(1)

The study introduces a unique approach to incorporate prior movement patterns, including convergence, divergence, and leadership, into the trajectory forecasting process. This integration allows the model to account for distinct mobility behaviors, enhancing its applicability across various scenarios.

(2)

By systematically constructing indicator systems for environmental factors, the method quantifies the effects of heterogeneous geographical environments on human mobility. The combination of attractive and repulsive potential fields provides a robust framework for modeling complex environmental effects.

(3)

The proposed method employs a hybrid weighting approach, combining expert scoring and entropy weight methods to assign weights to environmental factors. This ensures both data-driven objectivity and domain-specific relevance, resulting in a calibrated potential field that reflects real-world dynamics.

(4)

The proposed methodology is extensively evaluated on synthetic trajectory datasets representing various movement patterns. It demonstrates superior performance in long-term forecasting compared to Kalman filter and LSTM models, showcasing its robustness and reliability.

6.2. Implications for Practice

The practical future implications of this study are substantial, particularly in aspects such as effective traffic management, dynamic resource allocation, and sustainable urban development.

In terms of effective traffic management, the model can be used to forecast human mobility paths in high-density areas, informing traffic signal timings and road designs. This capability allows city planners to anticipate pedestrian navigation at critical intersections, thereby reducing congestion and enhancing safety. However, as the complexity of the environment increases, computational demands may grow significantly due to the combinatorial nature of pedestrian interactions. To address this challenge, strategies such as modular zoning, which involves dividing large areas into smaller computational units, or integrating edge computing technologies, can help maintain real-time responsiveness in practical applications.

For resource allocation, the model facilitates the strategic placement of resources such as bike-sharing stations or pedestrian signage. By forecasting how people are likely to move through different spaces, city services can be dynamically distributed to optimize facility usage and minimize waste. However, when applied at a city scale, such as across multiple commercial or residential districts, computational bottlenecks may arise due to complex spatial topologies, including elevated walkways or multi-layer road systems. A prioritized approach that focuses on transient high-traffic zones, such as subway stations during rush hours, rather than attempting city-wide synchronization, can enhance computational efficiency while preserving the model’s utility.

In the context of sustainable urban development, the model aids in optimizing urban designs to promote walking and cycling and reduce vehicular congestion. By strategically positioning green corridors, bike lanes, and walking trails based on predicted pedestrian paths, more eco-friendly transportation options can be encouraged in cities. However, simulating extended time horizons, such as annual predictions, introduces dynamic variables like weather and events, which may strain computational resources. To mitigate this, incremental updates that refine model parameters iteratively using historical data, rather than performing full recomputations, offer a practical solution to reduce computational overhead in long-term scenarios.

Overall, the study underscores the critical role of human mobility trajectory prediction in achieving sustainable urban environments, supporting the shift towards more livable and eco-friendly cities. It underscores the importance of balancing precision and computational efficiency, particularly in large-scale or long-term applications. Faced with more complex environments, combinations with other lightweight algorithms can also be considered.

6.3. Research Deficiency

While the model demonstrated strong performance, it does have some limitations. First, the model relies heavily on the accuracy and completeness of environmental data, such as roads, buildings, and other geographical features. In practical applications, these data may be outdated, incomplete, or unavailable in real time, which could compromise the reliability of the model. Second, the current indicator system assumes static environmental conditions, which do not account for dynamic changes like temporary infrastructure modifications, weather conditions, or road closures. These changes can have a significant impact on human mobility patterns and are not yet fully addressed in the current framework. Finally, the evaluation of the model primarily relies on the synthetic trajectory dataset. Despite the representative advantages of synthetic datasets, they may not fully reflect the complexity and variability of real-world human mobility. The validation under large-scale real-world trajectory remains to be further realized.

6.4. Future Research

To address these limitations, future research will focus on the following directions:

(1)

The integration of dynamic environmental data, such as real-time traffic conditions, weather updates, or temporary road closures, will be explored. Incorporating these dynamic inputs into the potential field modeling process would allow the model to better adapt to rapidly changing urban environments, significantly improving its robustness and applicability in complex environments.

(2)

Recognizing the need for real-world validation, future efforts will include the collection of real-world trajectory data. This will involve conducting volunteer-based studies, where participants provide detailed travel purposes and mobility trajectories through surveys and GPS tracking. Additionally, collaborations with transportation agencies and urban planning organizations will be pursued to access comprehensive datasets, ensuring that the model is tested and validated in diverse and practical scenarios.

(3)

Future studies will also aim to integrate social and behavioral factors into the model. Factors such as group dynamics, individual preferences, and social interactions could provide a richer understanding of human mobility and lead to more accurate trajectory forecasting.

(4)

Exploring hybrid models that combine the advantages of deep learning for feature extraction and the potential field method for real-time decision-making is another promising direction. Such approaches could enhance both short-term accuracy and long-term stability in trajectory forecasting, particularly in highly dynamic urban environments.