A grouping method based on grid density and relationship for crowd evacuation simulation

Physica A 473 (2017) 319–336

Contents lists available at ScienceDirect

Physica A journal homepage: www.elsevier.com/locate/physa

A grouping method based on grid density and relationship for crowd evacuation simulation Yan Li a,b , Hong Liu a,b,∗ , Guang-peng Liu a,b , Liang Li a,b , Philip Moore a,b,c , Bin Hu a,b,c a

School of Information Science and Engineering, Shandong Normal University, Jinan City, China

b

Shandong Provincial Key Laboratory for Novel Distributed Computer Software Technology, Jinan City, China

c

School of Information Science and Engineering, Lanzhou University, Lanzhou City, China

highlights • • • •

It presents an improved social force model which adds the force of group attraction. The force of group attraction is the synthesis of two forces. A grouping algorithm based on the grid density and relationship is proposed. We have tested the appropriate grid partition number and the relationship weight.

article

info

Article history: Received 21 September 2016 Received in revised form 30 November 2016 Available online 10 January 2017 Keywords: Crowd evacuation Social force model Computer simulation Grouping algorithm

abstract Psychological factors affect the movement of people in the competitive or panic mode of evacuation, in which the density of pedestrians is relatively large and the distance among them is small. In this paper, a crowd is divided into groups according to their social relations to simulate the actual movement of crowd evacuation more realistically and increase the attractiveness of the group based on social force model. The force of group attraction is the synthesis of two forces; one is the attraction of the individuals generated by their social relations to gather, and the other is that of the group leader to the individuals within the group to ensure that the individuals follow the leader. The synthetic force determines the trajectory of individuals. The evacuation process is demonstrated using the improved social force model. In the improved social force model, the individuals with close social relations gradually present a closer and coordinated action while following the leader. In this paper, a grouping algorithm is proposed based on grid density and relationship via computer simulation to illustrate the features of the improved social force model. The definition of the parameters involved in the algorithm is given, and the effect of relational value on the grouping is tested. Reasonable numbers of grids and weights are selected. The effectiveness of the algorithm is shown through simulation experiments. A simulation platform is also established using the proposed grouping algorithm and the improved social force model for crowd evacuation simulation. © 2017 Elsevier B.V. All rights reserved.

∗ Corresponding author: Hong Liu. Postal address: School of Information Science and Engineering, Shandong Normal University, No. 88, Wenhua East Road, Jinan City, China. E-mail address: [email protected] (H. Liu). http://dx.doi.org/10.1016/j.physa.2017.01.008 0378-4371/© 2017 Elsevier B.V. All rights reserved.

320

Y. Li et al. / Physica A 473 (2017) 319–336

1. Introduction The development of the economy and urbanization has resulted in a growing number of large-scale public events that attract large crowds. These events pose significant challenges in respect of crowd management; this has resulted in the growing research interest in simulating crowd behavior in relation to security challenges and evacuation scenarios. Public buildings may be subject to emergency situations, including fires and explosions, which instigate panic situations, in which, for example, stampedes and congestion can be experienced. These problems have historically resulted in large fatalities and personal injuries. Emergency evacuation of large crowds is a complex process. Crowd evacuation experiments are dedicated to several targets and are rich in data information. However, significant unavoidable challenges, including high cost and staff security, are experienced in attempts to conduct these experiments. Accordingly, computer simulation of crowd dynamics and the evacuation in large-scale public events has gained significant attraction given the implications for security and personal safety. Thus, computer simulations have become the main method for studying emergency evacuation [1–4]. Crowd motion is a complex physical process in the evacuation process. Its dynamic adjustments are directly or indirectly restricted by many factors. Interactions among the crowd and the psychological state of individuals are both important factors that may influence crowd motion. These factors contribute to several typical characteristics in crowd evacuation, including the following: 1. Clogging phenomenon: In emergency situations, individuals may react irrationally and move at a higher speed than normal. This feature results in pushing, congestion at egress points, and even crowd stampedes with resultant fatalities. 2. Mass behavior: In emergency situations, the panic emotion tends to generate and diffuse more easily than conventional evacuation; thus, ‘‘contagion’’ is higher. This phenomenon contributes to a series of non-adaptive crowd behaviors [5]. When the crowd is significantly large, the dissemination of information is limited by time and space. Individuals make decisions in an instant even with lack of information, which leads to ‘‘herding’’ [6,7], i.e., people tend to do what others do. In the same way, the emotion of people is easily affected by the panic of others. 3. Grouping behavior: In evacuating large-scale crowds, individuals tend to form a group with other people, and the group has a social relationship with themselves, such as family members and friends. This grouping sometimes helps in rapidly finding out the exit but does not necessarily accelerate the evacuation speed. Within a group, individuals interact, which demonstrate obvious nonlinear characteristics [8]. Behavioral characteristics [of individuals] in crowds may result in convergence, conflict, and balance and imbalance of various energies, as well as orientation and exclusion of groups [9]. The phenomena described above identify the difficulty in effectively modeling evacuation motion laws for crowds. Thus, establishing a reasonable simulation model for large-scale evacuation in unconventional emergencies is an urgent issue to be addressed. This model should integrate various uncertain factors, such as the psychological state(s), behavioral characteristics, motion laws, and interactions (as discussed in the subsequent sections of this paper). Significant research interest is given to the emerging application of psychosocial studies with computer simulation, which has been proposed as an effective approach for effective modeling of crowd dynamics in emergency situations [10–12]. Based on the social force model (SFM), an improved SFM (ISFM) is proposed to model crowds effectively, considering psychological factors [13,14]. In this paper, a crowd is divided into groups according to their social relations, and the attractiveness of groups is increased based on SFM. The force of group attraction is the synthesis of two forces; one is the attraction of the individuals generated by their social relations to gather, and the other is that of the group leader to the individuals within the group to ensure that the individuals follow the leader. The synthetic force determines the trajectory of the individuals. The evacuation process is demonstrated using ISFM. In ISFM, the individuals with close social relations gradually present a closer and coordinated action while following the leader. Based on ISFM, a novel grouping approach predicated on grid density and interpersonal relationships is proposed. In the proposed approach, the ‘‘plane space’’ is initially divided and modeled into a grid of cells. Subsequently, the relative density and relationship information of the non-empty grid cells are calculated. The grouping of data objects is then converted into clusters for grid cell. The proposed approach provides a number of benefits, such as the following:

• The number of grid cells is significantly smaller than the number of data objects; thus, the use of a grid data structure effectively reduces the computational complexity in grouping;

• Based on the selection process in core grid, outliers are effectively separated from the sparse grid and are then grouped with the classified center points of core grid. The approach used in classification is predicated on the K-medoids method, which is based only on the distance among data points and the relationships that exist among them. In the proposed approach, the grid grouping is reduced to the grouping of data objects. Thus, the proposed approach can effectively improve the accuracy of the grid-based grouping algorithm and is faster than the K-medoids algorithm. Several test cases are used to evaluate and demonstrate the power of the proposed approach. The remainder of this paper is organized as follows. Section 2 indicates the related works on SFM and clustering algorithms. Section 3 introduces the ISFM and the grouping of attractive force to SFM. In Section 4 a grouping algorithm based on grid density and relationship (GABGDR) is proposed. Section 5 discusses the conducted simulation experiments to show the efficiency of the proposed method. Finally, Section 6 concludes the paper and presents the future research focus.

Y. Li et al. / Physica A 473 (2017) 319–336

321

Fig. 1. Individuals are gathered as a group in the movement process.

2. Related works 2.1. SFM Force-based modeling, proposed by Helbing and Molnár [13,14], was originally derived from the studies of human social force. This model solves Newton’s equation to determine the position of each individual by considering repulsive interactions, friction forces, dissipation, and fluctuations. Later, this model was further applied and generalized to other simulation scenarios, including densely populated crowds [15], pedestrian evolution [16], and escape panic. This model can effectively capture phenomena, such as arching in the portals and the ‘‘faster-is-slower’’ effect. The SFM, as proposed by Molnar and Helbing in [13], is based on the formula of Newtonian mechanics to simulate crowd behavior. This method has attracted the attention of a few scholars who have attempted to improve the model. Helbing and Molnar proposed the SFM to simulate human behavior. The SFM incorporates the capability to reproduce phenomena, such as the ‘‘faster-is-slower’’ and the ‘‘arch of the exit’’. However, the SFM simplifies the behavioral characteristics of pedestrians. Despite its shortcomings in considering the relationships among individuals and the group characteristics, SFM can reproduce phenomena effectively. The video capture presented in Fig. 1 shows that the individuals gather as a group in the movement process. Force-based modeling and SFM have gained attraction and have been modified to investigate a range of evacuation problems [17–19]. The SFM, as modified by Moussaïd et al. [20], indicated the ‘‘small-group’’ phenomenon in a population. Johansson et al. [21] used the video capture analysis technology to modify SFM parameters, creating a model that more closely resembles crowd dynamics in a real evacuation scenario. However, crowd evacuation is a complex process and consists of a series of human behaviors, such as evacuation motion and behavioral reaction. Pure motion study estimates an ideal situation. Crowd motion is important in the evacuation process; many factors influence crowd dynamics in an evacuation. The time spent on other behaviors is arguably greater than the evacuation behavior. Therefore, physiological and psychological factors in large-scale crowd evacuation should be considered to conform to the random characteristics of crowd in evacuation. 2.2. Clustering method Clustering algorithms classify elements into categories or clusters based on their similarity. Several different clustering strategies have been proposed. In K-means [22] and K-medoids [23] methods, clusters are groups of data characterized by a small distance to the cluster center. An objective function, typically the sum of the distance to a set of putative cluster centers, is optimized [24–26] until the best cluster center candidates are found. However, these approaches cannot detect non-spherical clusters, because a data point is always assigned to the nearest center [27,28]. Clusters with an arbitrary shape are easily detected by approaches based on the local density of data points. In the densitybased spatial clustering of applications with noise (DBSCAN) [29], a density threshold is selected, the points in regions with densities lower than this threshold are discarded as noise, and different clusters are assigned to disconnected regions of high

322

Y. Li et al. / Physica A 473 (2017) 319–336

Fig. 2. Diagram of the original SFM.

density. This approach can form clusters of arbitrary size and shape, separate the environment noise, and use an efficient spatial index structure. The algorithm has relatively high efficiency, even for large data sets. The entire clustering process can be completed by scanning data sets. However, selecting different core neighborhood radii and core values in DBSCAN affects the results of the algorithm implementation, which is sensitive to parameter selection. This algorithm is unsuitable for cases in which density distribution is largely different in the cluster. The clustering by fast search and find of density peaks (CFSFDP) is a recently developed density-based clustering algorithm [30]. In CFSFDP, cluster centers are characterized by a higher density than those of their neighbors and by a relatively large distance from points with higher densities. Compared with DBSCAN, CFSFDP requires fewer parameters and is computationally inexpensive for its non-iteration. Alex et al. demonstrated the power of CFSFDP in many applications. However, CFSFDP does not perform well when more than one density peak exists for one cluster (‘‘no density peaks’’). The grid-based clustering method uses a multidimensional grid data structure to divide the space into a finite number of elements to form a grid structure that can be clustered. The main advantage of this algorithm is that the processing speed is fast, and the ‘‘noise’’ data can also have a good clustering characteristic. This algorithm can efficiently find the overall distribution of data sets and is commonly used in the pretreatment steps of data analysis. The grid-based clustering quality is greatly influenced by mesh size, i.e., small grid clustering leads to a high computational cost, a large mesh size, and a rough clustering quality. The grid-based clustering algorithm is a compromise between computational speed and precision. Common grid-based clustering algorithm includes the statistical information grid (STING) [31], wave cluster [32], and merging of adaptive finite interval algorithm [33]. Density-based clustering algorithms can effectively discover clusters of arbitrary shape but are inappropriate for cases in which large differences exist in the density distribution in a cluster. By contrast, grid-based clustering algorithms can significantly reduce data space, such that the microelements of a cluster are transformed from small-data objects to the ‘‘granularity’’ of large grid-data units. Thus, grid-based clustering can improve the efficiency of clustering and therefore may be used for the processing of data sets with high dimensions. In Section 4, we present the GABGDR for crowd evacuation simulation based on the SFM predicated on the existing density-based and grid-based algorithms. 3. ISFM with group attractive force We propose an ISFM, which introduces group characteristics, to effectively model the typical features of crowd dynamics in an evacuation scenario (e.g., mass behavior and grouping behavior). 3.1. SFM Helbing established SFM based on Newtonian mechanics and in accordance with the characteristics of collective behavior [13,14]. Social force refers to the force that one individual generates from the environment [including human and objects] instead of the physical force that directly applies to him. The different motivations of pedestrians and environmental influences imply four forces in the SFM, namely, (1) the driving force, (2) the interaction force among human beings, (3) the interaction force between individuals and obstacles, and (4) the disturbing force (see Fig. 2). The resultant force of the four forces affects pedestrians and contributes to acceleration. The internal driving force guides an individual to move toward the target However, before bodily contact, individuals [in a crowd] avoid interaction and collide with one another with the repulsive force. The interaction force between people and obstacles prevents individuals to collide with obstacles. This stage can be interpreted by the classic Newton’s second law and is expressed in Eq. (3.1): mi

dv ⃗i (t ) dt

= ⃗fi 0 +

 j(̸=i)

⃗fij +

 w

⃗fiw + ξ⃗ (t ) ,

(3.1)

Y. Li et al. / Physica A 473 (2017) 319–336

323

Fig. 3. Driving and disturbing forces.

Fig. 4. Physical forces between pedestrian i and other pedestrians j.

Fig. 5. Physical repulsion and friction among pedestrians.

mi is the mass of pedestrian i, and v ⃗i (t ) is the actual walking velocity. Eq. (3.1) shows that the motion of pedestrian i is affected by four types of force, which are the driving force of an individual ⃗ fi 0 , the interaction force between pedestrian i

and other pedestrians j(̸=i) ⃗ fij , the interaction force between pedestrian i and obstacles w ⃗ fiw , and disturbing force ξ⃗ (t ). The position of pedestrian i changes under the interactions of four forces. mi is the mass of pedestrian i, and v ⃗i represents the actual velocity of pedestrian i (see Fig. 3).





0 ⃗0 ⃗fi 0 = mi vi (t ) ei (t ) − v⃗i (t ) . τi

(3.2)

Eq. (3.2) describes the driving force ⃗ fi 0 of pedestrian i. In the moving process, pedestrian i adjusts his actual velocity v ⃗i (t ) constantly and moves toward the destination with a certain desired speed vi 0 (t ). τi is the characteristic time of pedestrian i. ⃗ ei 0 (t ) is the direction pointing from pedestrian i to the destination.

v⃗i (t ) =

d⃗ ri

(3.3)

dt

⃗fij = ⃗fij Psy + ⃗fij Touch .

(3.4)

Eq. (3.4) represents the interaction force imposed on pedestrian i by other pedestrians j. This interaction force includes two parts. One is the psychological forces in which pedestrian i tends to keep a velocity-dependent distance from other pedestrians j. The other is the physical forces between pedestrian i and other pedestrians j.

   ⃗fij Psy = Ai exp rij − dij /Bi n⃗ij ,

(3.5)

where Ai and Bi are constants. Ai represents the strength of interaction force, and Bi is the floating range force.  of repulsive  ri and rj are the radii for pedestrians i and j, respectively. rij = ri + rj is the sum of radii ri and rj . dij = ⃗ ri − ⃗ rj  denotes the

⃗ij = distance among the mass centers of pedestrians. n

⃗ri −⃗rj dij

⃗rj represent the position of pedestrians i and j, respectively.    ⃗fij Touch = g rij − dij kn⃗ij + κ 1vjit ⃗tij .

is the normalized vector pointing from pedestrians j to i. ⃗ ri and

(3.6)

Eq. (3.6) describes the physical force that pedestrian j imposes on pedestrian i. k and κ are constants, and 1vjit represents the tangential velocity difference between pedestrians i and j.

  1vjit = v⃗j − v⃗i · d⃗ij

(3.7)

⃗tij is the tangential direction from pedestrian i to pedestrian j. g (x) is zero when the pedestrians do not contact one another and is equal to x otherwise (see Figs. 4–6).

324

Y. Li et al. / Physica A 473 (2017) 319–336

Fig. 6. Interaction force between pedestrian and obstacle.

The interaction force between pedestrian and obstacle is expressed as follows:

⃗fiw = ⃗fiwPsy + ⃗fiwTouch ,

(3.8)

⃗fiwPsy = Ai exp [(ri − diw ) /Bi ] n⃗iw ,     ⃗fiwTouch = g (ri − diw ) kn⃗iw − κ v⃗i · ⃗tiw ⃗tiw .

(3.9) (3.10)

The parameters of Eqs. (3.8), (3.9), and (3.10) are similar to those of Eqs. (3.4), (3.5), (3.6), and (3.7) except that pedestrian j is replaced with obstacle W . 3.2. ISFM Crowd evacuation has evident social features, such as mass and grouping behaviors. In collective motions, the interactions among different individuals within the same group result in crowd behavior in the form of convergence, conflict, balance and imbalance of various energies, and orientation and exclusion from a group. These phenomena are apparent in crowd evacuation. Social psychology states that a swarm can be divided into small groups with different characteristics. Several common features among individuals exist within a small group. Observable facts indicate that the behavior of the same team of individuals is always consistent with crowd evacuation in a critical situation. Individuals negotiate with one another in the direction of movement and may leave the group together at some point. Therefore, each individual [in the same team] adjusts his speed to keep the consistency of the entire team. The proposed ISFM introduces group characteristics, in which the total force of the ISFM does not change (Eq. (3.1)). As mentioned earlier, the different motivations of pedestrians and environmental influences lead to four types of force in SFM, namely, the driving force, the interaction force among human beings, the interaction force between people and obstacles, and the disturbing force. Improvements are incorporated into five interaction forces among individuals, considering group dynamics and the influence of group characteristics, as shown in the following: mi

dv ⃗i (t ) dt

= ⃗fi 0 +

 j(̸=i)

⃗fij +

 w

⃗fiw +



⃗fij Group + ξ⃗ (t ) .

(3.11)

j(̸=i) Group

⃗fij In this paper, a group attractiveness force based on SFM is added to the proposed approach. This external force is obtained by adjusting the velocity of group members and points from the group members to the group center. The magnitude of this force is determined by the expectations that the group members desire to stay together. The expectation differs from group to group. For example, the attractiveness of a group that is composed of a mother and her children is much stronger than a group that is composed of colleagues (see Fig. 7). Group is presented in Eq. (3.12): The definition of ⃗ fij 

    ⃗fij group = wij · Ai · exp t ri − dij /Bi n⃗ij , (3.12) where wij = w × rel(i, j) (the test for reasonable w value is discussed in Section 5.2). t (x) is zero if the pedestrians do not ⃗ij is the normalized vector pointing from pedestrian i to pedestrian j. The contact one another and is equal to x otherwise. n ISFM with the increased group characteristics is the basis for GABGDR for crowd evacuation. Certain social relations among individuals in the group exist; thus, individual behavior is influenced by the movement of group members, which reflects the desire of the individual members of the group to not deviate from the group center. 4. GABGDR for crowd evacuation simulation GABGDR is proposed in this section in accordance with the ISFM with the grid density-clustering method. This method considers the influence of crowd evacuation relationship on population grouping. Moreover, GABGDR uses the meshing for evacuation in areas according to the grid density and relationship value to determine the core grid method accelerating the grouping process. This method fully considers the characteristics of human relations and has a fast packet rate. GABGDR is applicable to a plane room with doors and other regions, such as large flat stadiums, and does not consider the effect of obstacles. This method can be extended to square and other plane regions with multiple exits.

Y. Li et al. / Physica A 473 (2017) 319–336

325

Fig. 7. Modified SFM.

4.1. Formal description The formal description of the terms used for the proposed method is provided. Definition 4.1 (Plane Area). Plane area is defined as P = L × W . The plane is a rectangular flat area with length L and width W . Each of the points above can be represented by a coordinate (x, y), which consists of two real numbers. Definition 4.2 (Evacuee Data Set). Evacuee data set is defined as E = {ei , i = 1, 2, . . . , n}, where ei = (ei1 , ei2 , . . . , eim ) represents the ith evacuee in the data set, and pij indicates the ith evacuee with jth property. Definition 4.3 (Mapping of E → S). The mapping of evacuee data set E to plane area P E → S is defined as M = {ei (x, y)}, where ei is the ith evacuee in evacuee data set; 1 ≤ i ≤ n, (x, y) refers to the coordinates of two-dimensional plane, 1 ≤ x ≤ W and 1 ≤ y ≤ L; and ei (x, y) represents that the ith position of the evacuee on the flat area is (x, y). Definition 4.4 (Relational Value rel (el , em )). For the two evacuees el and em , the relational value indicates the close degree between the two evacuees in the range [0, 1] with a representation rel (el , em ). If the two evacuees do not have any relationship, then the relationship value is 0. If it is the mother-and-child or lover relations, then the relationship value tends to be 1. When the two evacuees have a close relationship, the relationship value between them is high. An emergency evacuation group usually comprises a guide or group leader. Each individual in the group establishes a relationship with the group leader, and their relationship value is 1.

 0.000    0.001–0.399 Relationship value range = 0.400–0.699   0.700–0.999  1.000

Strangers Colleagues Friends Families or lovers Group members and group leader.

(4.1)

Observations, statistics, and experimental results [20] imply that the type of individual groups varies because of the different social relations among individuals. For example, a family, friends, and working partners regard kinship, friendship, and work relations as a link, respectively. Different types of groups have different effects on pedestrian movement. As shown in Fig. 8, the individuals with different types of relationships have different attraction during evacuation. In this paper, the range of relationship value is determined according to the statistics and experimental results, especially the relationship value between each individual in the group and the group leader. Each group member is subjected to two types of attraction in the evacuation process; one is the social relation appeal, and the other is to follow the group leader out of danger as soon as possible to reach the safety zone of attraction. Each of the group members considers the two attractions, i.e., to follow the group leader while maintaining the social relations. This situation is expressed in Fig. 9. Group·Relation Individual pi is subjected to the attraction ⃗ fij of individual pj , and the degree of intimacy determines the size of Group·Relation

the attraction ⃗ fij . The length of the dotted line indicates the attraction degree, as shown in Fig. 9(a). Within the group, the leader has an attraction to the other individuals within the group to guide the movement of Group·Leader individuals. As shown in Fig. 9(b), individual pi is attracted by the force ⃗ fi from pi · Leader, which ensures that pi always tend to follow the movement of the leader in the group.

326

Y. Li et al. / Physica A 473 (2017) 319–336

Fig. 8. Schematic of the case in which individuals with different types of relationships have different attractions.

(a) Force ⃗ fij

Group·Relation

(b) Force ⃗ fij

.

(c) Synthetic force ⃗ fij

Group

Group·Leader

.

.

Fig. 9. Illustration of the case in which group members are subject to two types of attractive forces and form a synthetic force. Group·Relation Group·Leader from the leader of pi is subject to the attraction of the relationship within the group ⃗ fij and ⃗ fi Group Group the attraction. The synthesis of these two forces forms ⃗ fij (Fig. 9(c)). ⃗ fij and the other four types of social force (Eq. (3.11)) determine the state of the next moment of individual pi .

Definition 4.5 (Grid Partition). Grid partition means to divide the plane area P into h × k rectangular grids uniformly (as far as possible into the square), h × k = G. Partitioning method P is a rectangular planar√ area with length L and width W and is divided into G grids. The area of each grid is a = (L × W )/G, and the edge length is e = a. The rectangular area in horizontally divided into h = L/e and vertically divided into k = W /e. Definition 4.6 (Grid Cell). Without the loss of generality, the lower left corner of the area of plane coordinates is set as (0, 0).  The edge length   of each  rectangular grid ci,j is e and is expressed by the coordinates of the lower left and the upper right as xi−1 , yj−1 , xi , yj , xi = xi−1 + e, yj = yj−1 + e, 1 ≤ i ≤ h, 1 ≤ j ≤ k. Each rectangular grid is a grid cell.

Y. Li et al. / Physica A 473 (2017) 319–336

327

Definition 4.7 (Grid Density den(ci,j )). For a given grid cell ci,j , 1 ≤ i ≤ h, 1 ≤ j ≤ k, the number of individuals   ei (x, y) that  fall within the grid cell ci,j is counted. The number is determined by the x, y coordinates of the range in xi , yj , xi+1 , yj+1 , i.e., xi−1 < x ≤ xi , yi−1 < y ≤ yi . The total number of individuals in the grid cell is called grid density. Grid density den ci,j = count (pi (x, y)), xi−1 < x ≤ xi , yi−1 < y ≤ yi , count is the counting function.





Definition 4.8 (Grid Relationship Value rel(ci , j)). For a given grid cell ci,j , 1 ≤ i ≤ h, 1 ≤ j ≤ k, the sum of the relationship values of the individuals within the grid ci,j , divided by two, is called grid relationship value.





Grid relationship value rel ci,j =



rel (pl (x1 , y1 ) , pm (x2 , y2 ))



2,

(4.2)

(x1 ,y1 )∈ci,j (x2 ,y2 )∈ci,j

where xi−1 < x1 ≤ xi , xi−1 < x2 ≤ xi , yi−1 < y1 ≤ yi , yi−1 < y2 ≤ yi . The sum of the relationship values of the individuals within the grid ci,j is divided by two; because el (x1 , y1 ) is related with em (x2 , y2 ), and em (x2 , y2 ) has a symmetric relationship with el (x1 , y1 ). Therefore, the relationship value between el (x1 , y1 ) and em (x2 , y2 ) is counted twice when summed. Definition 4.9 (Adjacent Grid Cell). Adjacent grid cells are those with a direct connection with each other. The two kinds of connection relationship are edge sharing and common point connection. Definition 4.10 (Relationship–Density Value of Grid RelDen(ci,j )). The relationship–density value of the grid is the sum of the values of weighted relationship and weighted density. RelDen(ci,j ) = w1 × rel ci,j + w2 × den ci,j ,









(4.3)

where w2 = 1 − w1 . w1 and w2 are the weights of the relationship and density respectively; they are both generally set as 0.5. The values of w1 and w in Eq. (3.12) are the same. The test for reasonable ω value is indicated in Section 5.2. If fast grouping is needed, then the weight w2 is increased, and the weight w1 is decreased. Definition 4.11 (Relationship–Distance Value Between Two evacuees RelDist (el (x1 , y1 ) , em (x2 , y2 ))). The relationship– distance value between the two evacuees el (x1 , y1 ) and em (x2 , y2 ) is the sum of the values of the weighted relationship and weighted distance. RelDist (el (x1 , y1 ) , em (x2 , y2 )) = w1 × rel (el (x1 , y1 ) , em (x2 , y2 )) + w2 × dist (el (x1 , y1 ) , em (x2 , y2 )) ,

(4.4)

where w2 = 1 − w1 . dist (pl (x1 , y1 ) , pm (x2 , y2 )) =





(x2 − x1 )2 + (y2 − y1 )2 .





(4.5)



Definition 4.12 (Core Grid core ci,j ). The core grid core ci,j refers to the relational density value of a grid being higher than that of all the adjacent grids. Definition 4.13 (Wait Grouping Queue WaitQueu). The wait grouping queue is defined as WaitQueue. Each element w qi in WaitQueue is a non-empty grid cell, i.e., den(w qi ) ̸= 0. Definition 4.14 (Finish Grouping Queue FinishQueue). The finish grouping queue is defined as FinishQueue. Each element fqi in FinishQueue is a grid cell that has been grouped. 4.2. GABGDR for crowd evacuation simulation GABGDR initially divides the plane space into G grid cells. The data objects are then mapped into the corresponding grid cells, and the relative density and relationship information of the non-empty grid cells are calculated. Finally, the grouping of data objects is converted into clusters for the grid cell. The number of grid cells is significantly smaller than the number of data objects; thus, the use of the grid data structure effectively reduces the computational complexity of the grouping. The GABGDR is implemented according to the size of the grid density and the relationship attributes with its adjacent relation with the surrounding grid. The core grid is then selected. The outliers are effectively separated from the sparse grid and grouped with the core grid center points that have been classified. The K-medoids method is only based on the distance and relationship among data points. In this process, grid grouping is reduced to the grouping of data objects, which can effectively improve the accuracy of the gridbased grouping. GABGDR is implemented in three stages, namely, (1) initialization, (2) grid partition and calculation of density and relationship values, and (3) grouping based on grid density and relationship.

328

Y. Li et al. / Physica A 473 (2017) 319–336

4.3. Time complexity analysis of the GABGDR For the STING algorithm, the relationship–density value of each grid is calculated to create a grid table. Data are accessed once. The time complexity in the construction of a grid is O (n), where n is the number of individuals in the evacuee data set. In the sorting process, selecting and merging core grid, upon implementing the queue, will not repeat the search adjacent u×(u−1) grids. The worst-case time complexity is O( 2 ), where u is the number of elements in the queue WaitQueue, i.e., the number of non empty grids. In the processing of the remaining non-core grids, the time complexity mainly depends on the total number of individuals in the non-core grids n1 . Individuals should be assigned to the previously determined T core grids; thus, for the K-medoids method, the relationship–distance value with T core grid center should be calculated one at a time. The time complexity is O (T × n1 ). In sum, the time complexity of GABGDR is O (n + u × (u − 1)/2 + T × n1 ). n1 is significantly smaller than n because  T is a constant, and T × n1 may be incorporated into n. After simplification, the time complexity of GABGDR is O n + u2 . The computational complexity is greatly influenced by the number of non-empty grids. 5. Simulation experiments The following tests, as presented in the following sections, are conducted to validate the proposed method: (1) The test for GABGDR, the grid division number G; (2) The test for w in Eq. (3.12) (also w1 in Eqs. (4.3) and (4.4)); (3) The comparative experiment and analysis. The silhouette index was proposed for partitioning techniques by Rousseeuw in 1987 [34]. Each cluster is represented by a ‘‘silhouette’’, which is based on the comparison of its tightness and separation. The silhouette index is used to evaluate the clustering quality by interclass separation and intraclass compactness. In the sample space, the clustering quality is good if each cluster is separated from others, and each type of sample is close to others. The average silhouette width provides an evaluation of clustering validity. The following test cases 5.1, 5.2, and 5.3 are evaluated using the silhouette index and Matlab R2013a as the test platform.

Y. Li et al. / Physica A 473 (2017) 319–336

329

Table 1 Silhouette index similarity with the most suitable number of matches for each population size. Grid number 228

360

372

552

552

720

624

624

= 0.1 = 0.2 = 0.3 = 0.4 = 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 1.0

−0.0757 −0.03777 −0.05772 −0.02431 −0.08325 −0.09603 −0.08158 −0.08342 −0.02016 −0.05943

−0.06222 −0.09399 −0.05666 −0.07066 −0.08483 −0.03658 −0.05384 −0.02808 −0.01088

−0.06364 −0.02737 −0.02568 −0.03231 −0.00495 −0.02779

−0.06982 −0.06171 −0.04786 −0.05212 −0.05287 −0.03992

−0.04876 −0.04922 −0.04264 −0.01605 −0.03525 −0.00794

−0.09942 −0.05182 −0.03929 −0.01131

−0.05853 −0.05329

0.000532

0.037819

0.011318

0.01184 0.011835

0.002595 0.000826 0.024695 0.020929

−0.05111 −0.0419 −0.00609 −0.00653 −0.03179 −0.03398 −0.0018 −0.02082 0.034793 −0.00119

Population

N = 300

N = 400

N = 500

N = 600

N = 700

w w w w w w w w w w

−0.00535

0.010969

−0.03444

0.011715

0.028829

−0.02722 0.01269 0.035445

−0.0005 −0.03452 −0.02477

0.008329 0.058865 0.046104 0.013405 0.025869 0.043627

N = 800

N = 900

N = 1000

−0.00959

Fig. 10. Similarity with the most suitable grid number of matches for each population size.

5.1. Test case 1: the suitable grid division number G The proper grid division number G is tested in this experiment. The grid division number G is set according to the requirements of group division. The test set rules are that grid number G is 10 times of the number of teams T as a starting point, without surpassing population size N integer times for upper limit, and 10 equal division of its interval. From the data of the eight groups of experiments, the maximum values of the silhouette index similarity with different grid numbers, population sizes, and weights w are determined. The value of w is indicated in the right line, and the optimal matching population size n of the grid number is presented in the column of Table 1. Table 1 lists the most suitable grid number for each population size (300:228, 400:360, 500:372, 600:552, 700:552, 800:720, 900:624, 1000:624). Weight w , population size n (the suitable grid division number) and similarity are represented by the x-, y-, and z-axes, respectively. A 3D grid figure is shown in Fig. 10. 5.2. Test case 2: the test for w in Eq. (3.12) (and w1 in Eqs. (4.3) and (4.4)) The suitable weight w is tested in this experiment. With population size n = 1000, grid number from 120 to 960, weight w from 0.1 to 1.0, and interval of 0.1, Table 2 is obtained. Weight w , the number of grid division, and similarity are represented by the x-, y-, and z-axes, respectively. Fig. 11 shows the 3D grid figure. The line chart in Fig. 10 is generated with population size n = 1000, suitable grid number of 624, weight w from 0.1 to 1.0, and interval of 0.1. Fig. 8 shows that an increase in w value leads to reduced grid similarity, because the division of grids emphasizes the attention given to the relationship among individuals. When the value of w is 0.5, the similarity is higher than the average value. This value includes the relationship and distance simultaneously and is reasonable (see Fig. 12).

330

Y. Li et al. / Physica A 473 (2017) 319–336 Table 2 Similarity with the suitable weight values at different grids. Population n = 1000

Grid number 120

w w w w w w w w w w

= 0.1 = 0.2 = 0.3 = 0.4 = 0.5 = 0.6 = 0.7 = 0.8 = 0.9 = 1.0

204

288

372

456

540

624

708

792

876

960

−0.05818−0.05922−0.055560.00825 −0.034440.0083290.0553240.0368950.0122370.0258690.035959 −0.05759−0.05872−0.053290.005479−0.035980.0006220.0424660.02859 0.0019880.0170460.030538 −0.05764−0.05853−0.05401−0.00073−0.03734−0.007960.0012230.018954−0.02362−0.007910.014503 −0.05974−0.06342−0.0625 −0.0127 −0.05438−0.03582−0.04895−0.00798−0.04315−0.05791−0.02101 −0.07674−0.09882−0.09811−0.08502−0.07867−0.08517−0.08005−0.0602 −0.06103−0.08827−0.06513 −0.10028−0.1274 −0.12076−0.11092−0.10207−0.10636−0.10474−0.12755−0.07996−0.10851−0.08684 −0.12203−0.13245−0.14645−0.14011−0.13366−0.11649−0.12538−0.15803−0.10445−0.12672−0.0973 −0.13518−0.13991−0.1622 −0.15349−0.16865−0.13519−0.14862−0.18938−0.12797−0.14764−0.10613 −0.15712−0.15811−0.17967−0.17131−0.19692−0.15065−0.17019−0.2228 −0.15846−0.17832−0.12988 −0.19154−0.19418−0.20386−0.19907−0.2228 −0.17862−0.19582−0.25427−0.18837−0.20083−0.15836

0.1 0.0 -0.1 -0.2 -0.3 960

876

0.9 792

0.8 708

624

0.7 540

y--Grid Number

456

0.5 372

0.6

0.4 288

204

120

0.1

0.2

0.3

x--weight

Fig. 11. Similarity with population size N = 1000, grid number from 120 to 960, and weight w from 0.1 to 1.0.

Fig. 12. Silhouette index similarity for testing the weight w from 0.1 to 1.0.

5.3. Simulation experiment and comparative analysis Two comparative tests are performed to illustrate the feasibility of this study. First, the ISFM is compared with the original SFM. Second, the proposed grouping algorithm is compared with two classic classification algorithms, i.e., K-Mediods and STING. 5.3.1. Comparison between ISFM and the original SFM ISFM and the original SFM in scenes without obstacles are compared. The number of individuals from 100 to 500 and the average of 50 experiments in each group are considered to obtain the evacuation times of the two models in scenes with two exits (Table 3 and Fig. 13) and four exits (Table 4 and Fig. 14).

Y. Li et al. / Physica A 473 (2017) 319–336

331

Table 3 Evacuation times of the two models in the scene with two exits. The number of individuals

100 150 200 250 300 350 400 450 500

Time The original social force model

The improved social force model

32.60158 36.67354 40.77261 44.76593 50.84097 59.47532 69.62475 78.70953 88.62988

32.40182 35.80793 38.64656 42.44709 48.76953 55.37183 65.59564 73.38764 81.75719

Table 4 Evacuation times of the two models in the scene with four exits. The number of individuals

100 150 200 250 300 350 400 450 500

Time The original social force model

The improved social force model

21.0952704 22.6942640 25.7513047 28.6248530 32.1087139 36.4752157 40.4365680 44.8542300 51.0286431

21.5580923 22.4365800 25.2679967 27.3345876 29.7802225 33.1247580 36.2283403 40.1024586 44.3248958

Fig. 13. Evacuation times of the two models in the scene with two exits.

The formation of groups improves the evacuation efficiency because the groups at the exit do not have any competition. In addition, an individual is subjected to self-driving force while being attracted by other individuals in the group. This condition increases the cooperation force of the individual, thus increasing the movement speed of the individual and reducing the overall evacuation time. 5.3.2. Comparison between the proposed grouping algorithm and two classic classification algorithms The classic grid-based clustering algorithm STING and the partitional clustering algorithm K-Medoids are selected for the comparison with the proposed algorithm. The proposed algorithm combines the advantages of grid-based clustering and K-Medoids partitional clustering methods. The former part of the proposed algorithm adopts the idea of grid-based division to improve the grouping speed. The latter part of this algorithm considers the core grid as the center and implements the K-Medoids algorithm to group particles in sparse grids, ensuring partitioning accuracy. The combination of the two methods can effectively solve the problem of low classification accuracy of the grid-based clustering algorithm and the low speed of

332

Y. Li et al. / Physica A 473 (2017) 319–336

Fig. 14. Evacuation times of the two models in the scene with four exits.

Fig. 15. Running times of the three algorithms. Table 5 The running time of the three algorithms. The number of individuals

Time K-Mediods

STING

GABGDR

300 400 500 600 700 800 900 1000

0.1867 0.2669 0.4678 0.4870 0.5380 0.7727 0.8533 1.0017

0.0830 0.1040 0.1420 0.1700 0.2033 0.2066 0.2714 0.3114

0.0630 0.0820 0.1100 0.1710 0.1870 0.1880 0.1970 0.2190

the K-Medoids clustering method. Experiments were conducted to compare the running speed and the grouping accuracy of the three algorithms. The running times of the GABGDR, STING, and K-Medoids algorithm are compared according to the suitable weight value w and the appropriate grid number, with the same population size n. The same population size N from 300 to 1000 is used with an interval of 100. Each group is tested 10 times, with an average time for comparison. Table 5 and Fig. 15 show that the proposed algorithm does not present an obvious advantage in running speed compared with STING but indicates a significant advantage over K-Medoids. The grouping accuracies of GABGDR, STING, and K-Medoids algorithms are also compared by adopting the silhouette index. The same weight value w from 0.1 to 1.0 is used with an interval of 0.1. Each group is tested 10 times with an

Y. Li et al. / Physica A 473 (2017) 319–336

333

Table 6 The grouping accuracy of the three algorithms. The number of individuals

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Accuracy K-Mediods

STING

GABGDR

−0.0222 −0.0282 −0.0291 −0.034 −0.0396 −0.0544 −0.0588 −0.0688 −0.0872 −0.0979

0.0356 0.0201 0.0023 −0.0551 −0.0833 −0.1776 −0.1854 −0.1976 −0.2264 −0.2964

0.0553 0.0425 0.019 −0.0057 −0.0191 −0.0868 −0.0973 −0.1061 −0.1299 −0.1584

Fig. 16. Grouping accuracies of the three algorithms.

average grouping for comparison. Table 6 and Fig. 16 indicate that the proposed grouping algorithm is not as accurate as the K-Medoids algorithm. The K-Medoids algorithm is not affected by weight, and its grouping accuracy is relatively stable. However, the superiority of the grouping precision of our algorithm to that of STING becomes increasingly evident with an increase in weight value w . The comparison results confirm that GABGDR has obvious advantages in synthetically considering the grouping speed and precision for grouping a population with social relationship. 5.4. Simulation platform Microsoft.Net Framework 4.5, Microsoft XNA Game Studio 4.0, Microsoft Visual Studio 2012, OpenSceneGraph 3.2.1, and MonoGame 3.2 are adopted to develop a crowd evacuation animation platform, illustrating the performance efficiency of the proposed simulation model and algorithm for crowd evacuation. The computer crowd evacuation simulation from the plane area is set as 300 m × 250 m. ISFM parameters are set as follows. The population size is 300, Ai = 2 × 103 N, Bi = 0.08 m, τi = 0.5 s, vi0 = 1.5 m s−1 , k = 1.2 × 105 kg s−2 , κ = 2.4 × 105 kg m−1 s−1 , mi = 80 kg, and rij = 0.3 m. The desired velocity vi0 can reach more than 5 m s−1 . The normal walking speed of people is approximately 1 m s−1 and vi0 ≤ 1.5 m s−1 under nervous conditions. The initialization and grouping states are presented in Figs. 17 and 18, respectively. In Fig. 18, the same color indicates that the individuals are related. Individuals with relationship exist in the same group on the basis of the proposed method. In the process of moving toward the exits, the individuals with relationships remain in the same group; each individual in the same group tries to adjust his movement speed to keep the consistency with the entire group. This phenomenon is illustrated in Figs. 19–21, which show the process of each group moving to exits. The state of each group close to the exits is shown in Fig. 21. Figs. 17–21 indicate the following findings: (1) Physical connections exist in the group, such as the relatively close distance among the individuals in the same group and the queue to move forward.

334

Y. Li et al. / Physica A 473 (2017) 319–336

Fig. 17. State of the crowd evacuation simulation after initialization. A relationship exists among people with the same color. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 18. State after grouping. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 19. Status of each team moving to the exits based on ISFM. People with relationships (people with the same color) gradually move closer. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 20. State of each team moving toward the exits. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Y. Li et al. / Physica A 473 (2017) 319–336

335

Fig. 21. State in which each team closes to the exits. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(2) Social contacts occur in the group, such as families, friends, and colleagues. A closer relationship means a smaller distance among individuals during the evacuation process. (3) The individuals with the same characteristics are represented by the same color, such as the individuals in the same group. (4) The movement of the individuals in the same group is coordinated, and the movement tendency is always consistent. 6. Conclusions A new crowd evacuation simulation model and grouping algorithm is presented in this paper. The main contributions are summarized as follows: 1. Based on SFM, ISFM is proposed. The proposed ISFM considers the pre-existing relationships in a crowd (the population). Compared with the traditional SFM, the ISFM can effectively model crowd dynamics and reflect the individual response to the need to evacuate in an emergency. In ISFM, individuals react to an emergency based on their social relationships. Various relationships present constraints on crowd dynamics and evacuation. 2. GABGDR is proposed according to ISFM and the grid density clustering method; it draws the grid density using an approach predicated on the K-medoids classification method, to which a relationship component is added. This algorithm not only supports the ISFM for the computer simulation of crowd evacuation but also has a lower time complexity while maintaining accurate classification. 3. The appropriate grid partition number and the relationship weight are tested. The posited approach presented in this paper is evaluated using a simulation system. Experiments show that the proposed method is both practical and effective. Studies of crowd behavior with the related research into computer simulation provide an effective basis for architectural design and effective crowd management. Future research will continue to investigate crowd dynamics and study various psychological behaviors related to emergency situations and crowd evacuation dynamics to develop and perfect the computer simulation model in this paper. With reference to a current research and projected future work, the authors of this paper aim to develop a computer evacuation simulation system [35] that simulates more crowd evacuation phenomenon. Such a system will help improve architectural designs, because it relates to the timely and secured evacuation of large crowds. Acknowledgments This research is supported by the National Natural Science Foundation of China (61472232 and 61272094) and by the project of Taishan scholarship. References [1] J.K.K. Yuen, E.W.M. Lee, W.W.H. Lam, An intelligence-based route choice model for pedestrian flow in a transportation station, Appl. Soft Comput. 24 (2014) 31–39. [2] W.H. Li, J. Gong, P.H. Yu, et al., Simulation and analysis of congestion risk during escalator transfers using a modified social force model, Physica A 420 (2015) 28–40. [3] Q. Meng, X.B. Qu, Uncertainty propagation in quantitative risk assessment modeling for fire in road tunnels, IEEE Trans. Syst. Man Cybern. Part C Appl. Rev. 42 (6) (2012) 1454–1464. [4] A. Sagun, D. Bouchlaghem, C.J. Anumba, Computer simulations vs. building guidance to enhance evacuation performance of buildings during emergency events, Simul. Model. Pract. Theory 19 (3) (2011) 1007–1019. [5] K.H. Law, K. Dauber, X.S. Pan, Computational modeling of nonadaptive crowd behaviors for egress analysis. CIFE Seed Project Report, Stanford University, 2005. [6] R.M. Raafat, N. Chater, C. Frith, Herding in humans, Trends Cogn. Sci. 13 (2009) 420–428. [7] C. Saloma, G. Perez, Herding in real escape panic, Pedestrian Evacuation Dyn. 5 (2007) 471–479. [8] C.M. Henein, T. White, Macroscopic effects of microscopic forces between agents in crowd models, Physica A 373 (2007) 694–712.

336

Y. Li et al. / Physica A 473 (2017) 319–336

[9] K.A. Jehn, E.A. Mannix, The dynamic nature of conflict: A longitudinal study of intragroup conflict and group performance, Acad. Manag. J. 44 (2001) 238–251. [10] M.L. Xu, Y.P. Wu, P. Lv, et al., miSFM: On combination of mutual information and social force model towards simulating crowd evacuation, Neurocomputing 168 (30) (2015) 529–537. [11] N.W.F. Bode, E.A. Codling, Human exit route choice in virtual crowd evacuations, Anim. Behav. 86 (2013) 347–358. [12] X.X. Yang, H.R. Dong, Q.L. Wang, et al., Guided crowd dynamics via modified social force model, Physica A 411 (2014) 63–73. [13] D. Helbing, P. Molnar, Social force model for Pedestrian dynamics, Phys. Rev. E 51 (5) (1995) 4282–4286. [14] D. Helbing, I. Farkas, T. Vicsek, Simulating dynamical features of escape panic, Nature 407 (2000) 487–490. [15] N. Pelechano, J. Allbeck, N. Badler, Controlling Individual Agents in High-Density Crowd Simulation, Eurographics Association, SanDiego, California, 2007. [16] T.I. Lakoba, D. Kaup, N.M. Finkelstein, Modifications of the Helbing–Molnar– Farkas–Vicsek social force model for Pedestrian evolution, Simulation 81 (5) (2005) 339–352. [17] D.R. Parisi, M. Gilman, H. Moldovan, A modification of the social force model can reproduce experimental data of Pedestrian flows in normal conditions, Physica A 388 (2009) 3600–3608. [18] W.L. Zeng, H. Nakamura, P. Chen, A modified social force model for Pedestrian behavior simulation at signalized crosswalks, Pocedia- Soc. Behav. Sci. 138 (2014) 521–530. [19] X.X. Yang, H.R. Dong, et al., Guided crowd dynamics via modified social force model, Physica A 411 (2014) 63–73. [20] M. Moussaïd, N. Perozo, S. Garnier, D. Helbing, G. Theraulaz, The walking behavior of Pedestrian social groups and its impact on crowd dynamics, PLoS One 5 (4) (2010) e.0010047. [21] A. Johansson, D. Helbing, P. Shukla, Specification of the social force Pedestrian model by evolutionary adjustment to video tracking data, Adv. Complex Syst. 10 (2007) 271–288. [22] J. MacQueen, Some methods for classification and analysis of multivariate observations, in: L.M. Le Cam, J. Neyman (Eds.), Proceedings of the 15th Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, Univ. California Press, Berkeley, CA, 1967, pp. 281–297. [23] R. Scitovski, K. Sabo, Analysis of the k-means algorithm in the case of data points occurring on the border of two or more clusters, Knowl.-Based Syst. 57 (2014) 1–7. [24] B.J. Frey, D. Dueck, Clustering by passing messages between data points, Science 315 (2007) 972–976. [25] W.H. Li, Y. Yang, D. Yuan, Assuring cloud data reliability with minimum replication by proactive replica checking, IEEE Trans. Comput. 65 (5) (2016) 1494–1506. [26] L. Liu, S. Zhang, L. Diao, et al. Automatic verification of ‘isa’ relations based on features, in: Proceedings of Sixth International Conference on Fuzzy Systems and Knowledge Discovery, 2009, pp. 70–74. [27] Y. Yang, J. Jiang, Hybrid sampling-based clustering ensemble with global and local constitutions, IEEE Trans. Neural Netw. Learn. Syst. 27 (5) (2016) 952–965. [28] S. Khalid, Activity classification and anomaly detection using m-mediods based modelling of motion patterns, Pattern Recognit. 43 (2010) 3636–3647. [29] M. Ester, H.P. Kriegel, J. Sander, X.W. Xu, A density-based algorithm for discovering clusters in large spatial databases with noise, in: Proceedings of the Second International Conference on Knowledge Discovery and Data Mining, (KDD-96), AAAI Press, 1996, pp. 226–231. [30] A. Rodriguez, L. Alessandro, Clustering by fast search and find of density peaks, Science 344 (6191) (2014) 1492–1496. [31] W. Wang, J. Yang, R.R. Muntz, STING: A statistical information grid approach to spatial data mining, in: Proceedings of the 23rd International Conference on Very Large Data Bases, Morgan Kaufmann, Athens, 1997, pp. 186–195. [32] G. Sheikholeslami, S. Chatterjee, A.D. Zhang, Wave Cluster: A multi-resolution clustering approach for very large spatial databases, in: Proceedings of the 24th International Conference on Very Large Data Bases, Morgan Kaufmann, New York, 1998, pp. 428–439. [33] S. Giol, H. Nagesh, A. Choudhary, MAFIA: Efficient and scalable subspace clustering for very large data sets, in: Proceedings of 1999 International Conference of Data Engineering, ICDE’99, Sydney, 1999. [34] P.J. Rousseeuw, Silhouettes: A graphical aid to the interpretation and validation of cluster analysis, J. Comput. Appl. Math. 20 (1987) 53–65. [35] H. Liu, Y.L. Sun, Y.Y. Li, Modeling and path generation approaches for crowd simulation based on computational intelligence, Chin. J. Electron. 21 (4) (2012) 636–641.