A Markovian measure for evaluating accessibility to urban  opportunities

<jats:p>Accessibility is a fundamental notion in urban planning and its related fields. While accessibility is dynamic and varies during different time moments, most of the accessibility metrics are static and do not take this variation into account. In doing so, to address the questions of (1) how accessible urban opportunities are in different time moments and (2) how accessibility value of a person to a certain place changes regarding his/her spatiotemporal restrictions in time instants, this article—by using semi-Markovian and Brownian Bridge stochastic processes—offers a probabilistic time-dependent accessibility model that blends the magnitude of opportunities magnitude with the probability of individuals visiting.&#x0D; &#x0D; To show the model’s applicability, it was applied on a hypothetical case, along with two common accessibility metrics, and the outputs were compared. Then the proposed model was implemented in a study area for measuring temporal accessibility in two real policies made for daily markets in Isfahan, Iran. The first policy that presented the model application for analytical purposes was “market exclusion and area expansion,” and the second policy that depicted the model implementation for normative usage was “new market location.”&#x0D; &#x0D; Results of the model execution on the hypothetical cases indicated there was a significant difference between the outputs of the common metrics and the ones of the proposed model. In addition, in the study area, the first policy generated higher total accessibility value in comparison with the second policy when market 2 was excluded and the area for market 8 was doubled.</jats:p>

section_content: Understanding how people move and how they can access to the opportunities is necessary for comprehending the function of human settlements and providing livable places (Tenkanen, 2017). In this way, study of moving objects has become a noteworthy area of research in transportation and GISience (Laube, Imfeld, & Weibel, 2005; Li, Zhang, Wang, & Zeng, 2011). Time geography is a pivotal concept, both theoretically and practically, in these scientific scopes and is established around the notion of space-time prisms (STPs) that provides analytical tools for binding temporal aspects and location-based urban constructs such as accessibility. Classical STPs, demarcating reachable locations by a moving object given its origin, destination, and travel time budget, assume that all places within the boundary of a STP are equally accessible. This presumption neglects the interior properties of STPs as well as the fact that the probability of being in different locations inside a STP is not equal (Song & Miller, 2014; Song, Miller, Zhou, & Proffitt, 2016). 

To differentiate locations within the prisms, recent studies have attempted to develop probabilistic models (Downs & Horner, 2012; Horner & Downs, 2014; Neutens, Schwanen, Witlox, & De Maeyer, 2012; Song & Miller, 2014; Song, et al., 2016; Winter & Yin, 2010, 2011). A group of studies has revolved around the statistical time geographic constructs. Downs (2010) integrated kernel density estimation (see: (Xie & Yan, 2008) ) within STPs and proposed a time-geographic density estimation (TGDE) to identify probable locations of objects having the time budget between two benchmark points. Downs and Horner (2012) proffered a network-based TGDE which was able to produce potential path trees by using data gathered for vehicle's movement. Another group of research applied movement principles to estimate the visiting probability of mobile objects within STPs. Using Random Walk (RW) theory, Winter and Yin (2011) elaborated a mathematical framework to distinguish locations of an object, on the basis of the movement probability. Following the works of Winter and Yin, (2010, 2011), Song and Miller (2014) established models compatible with discrete and continuous times for simulating visiting probability in planar STPs. Song et al. (2016) moved the previous research forward adapting Brownian Bridge (BB) and semi-Markovian (SM) stochastic processes for modeling the probability of non-vehicular and vehicular movements within network-based space-time prisms (NTPs), respectively. 

As accessibility is closely related to time geography, in that temporal features affect where and when individuals participate in activities and how transportation systems work (Tenkanen, 2017), recently, several measures have been developed for calculation of accessibility, applying space-time models (Church & Marston, 2003; Delafontaine, Neutens, Schwanen, & Van de Weghe, 2011; El-Geneidy & Levinson, 2006; Ettema & Timmermans, 2007; Kim & Kwan, 2003; Kwan, Murray, O'Kelly, & Tiefelsdorf, 2003; Kwan, 1998; Miller, 1999; Miller & Wu, 2000; Neutens, Schwanen, Witlox, & De Maeyer, 2010; O'Sullivan, Morrison, & Shearer, 2000). Despite the variety of modeling approaches, these models can be generally classified in two categories: individual-and opportunity-based measures (Chen et al., 2016; Miller, 2007). While in opportunity-based measures accessibility value relates to activity places, in individual-based models, accessibility is calculated by considering spatial and personal restrictions. Measures in the first category are simple, easy to use, and aggregate but measures in the second category are disaggregate, representing time and space characteristics in details. However, individual-based measures commonly enlist deterministic STPs, they do not support the dynamics underlying accessibility changes, and cause accessibility value to be overestimated (Song et al. 2016). 

In addition to the main classes of the accessibility models, another group of metrics, called blended measures, can be identified. These models attempt to put both person-and place-based approaches together for representing characteristics of these perspectives in a unified structure. For example, Downs and Horner (2012) combined a statistical function that estimates location of an individual on a road network with attractiveness of opportunities reachable in a given time window. However, despite the necessity of more investigations in this scope of research (Horner & Downs, 2014), a dynamic probabilistic accessibility model incorporating movement principles along with characteristics of urban opportunities has not yet been elaborated and applied in practice. Therefore, this paper, that is an extension to the work of Song et al. (2016), aims to offer an accessibility model linking attractiveness of opportunities to transportation system by using BB and SM processes for tackling the following questions: (1) on the opportunity side, how accessible are urban opportunities in different moments of time? (or how accessibility of an opportunity has been changed during a time section?), and (2) on the individual side, how much is the accessibility of a person to a certain place regarding his/her spatiotemporal restriction and attendance probability in time instants? To apply the proposed measure and show its analytical and practical implications, two policies made by municipality of Isfahan in response to the issue of improving service delivery to the citizens by daily markets are considered. The first, "market exclusion and area expansion", is an application for analytical purpose and the second, "new market location", is an example of normative usage. 

The rest of this paper is organized as follows. Section 2 outlines the background and concepts leading this research. Section 3 presents the model development process and analysis. Section 4 shows model implementation in the study area and results. Ultimately, the conclusions and further works are given in Section 5.
section_name: Introduction

section_content: Background and concepts
section_name: 2

section_content: The notion of time geography, created by Hägerstraand (1970) and then operationalized by Lenntorp (1977), was established to describe activity pattern of individuals by capturing the spatiotemporal facets and the constraints imposed on a person by transportation and activity systems (Cerdá, 2009). These constraints are: (1) capability constraints which encompass physical and psychological requirements to engage in an activity; (2) coupling constraints which enfold spatiotemporal requirements that makes it possible to participate in different activities with other people; and (3) authority constraints which include laws that limit an individual's access to opportunities (Malekzadeh, 2015; Miller, 2007). Time geography is constituted around the central concept of STP in which if origin and destination, available time for movement, and maximum moving speeds from an origin to a destination are given, STP will delimit all reachable locations and will determine the remaining available time to spend at each opportunity (Kwan & Weber, 2008; Miller, 1991 Miller,, 1999;; Miller & Bridwell, 2009; Song & Miller, 2014). By iterating the calculation of STP for a number of individuals, their choice set and movement pattern can be tracked (Horner & Downs, 2014). There are two classes of STPs: (1) punctiform prism which works on the basis of the opportunities Euclidian distance between origin and destination (see: (Miller, 1999; Song, 2015) ), and (2) NTP that is congruent with the topology of the road network (Kim & Kwan, 2003). STPs have been applied for studying various topics related to transportation by modeling both the physical social interaction (Farber, Neutens, Miller, & Li, 2013 ) and the virtual one (Yu & Shaw, 2008) to formation of travelers' choice set (Chen & Kwan, 2012) to elaboration of mobility system (Winter & Raubal, 2006) and computation of accessibility as space-time concepts are appropriate for describing accessibility in the individual level (Malekzadeh, 2015; Song, 2015).
section_name: Time geography

section_content: Accessibility is a key concept in urban planning, transportation geography, and the other related fields. This notion is a multi-dimensional construct including individual, temporal, network, and land-use components (Geurs & Ritsema van Eck, 2001; Geurs & Van Wee, 2004). There is a wide variety of interpretations of accessibility from the opportunity attractiveness (Hansen, 1959) to the connectivity level of different locations (Ingram, 1971), to the ease of a place to be reached by different individuals in a certain geographical area (Cascetta, Cartenì, & Montanino, 2013). However, after the paradigm shift from mobility to accessibility and by emerging sustainable transportation theories, this term is often defined as the ease of participating in different activities (Banister, 2008; Ferreira, Beukers, & Te Brömmelstroet, 2012; Handy, 2002; Song, 2015). There are two main branches of accessibility models in the literature: opportunity-and individual-based measures. 

Opportunity-based measures compute accessibility at a given location (Li et al., 2011). Distancebased (Geurs & Van Wee, 2004), cumulative-based (Breheny, 1978; Envall, 2007 ), gravity-based (Hanson & Schwab, 1987; Knox, 1978; Makri & Folkesson, 1999; Shen, 1998), and place-rank models (El-Geneidy & Levinson, 2006) are of the main categories of the opportunity-based measures. Although these models are simple in application and can be used to evaluate the impact of policy-related changes on the transportation or land-use system, these measures do not take individual preferences and spatiotemporal constraints into account, and are aggregate in nature (Chen et al., 2016; El-Geneidy & Levinson, 2006; Miller, 2007). 

Individual-based measures, in contrast to opportunity-based metrics, make one able to consider movement constraints and other factors related to travel-activity system (Horner & Downs, 2014; Neutens et al., 2010). While individual-based measures connect all components of accessibility to each other (Geurs & Van Wee, 2004), these measures need high-resolution data, and aggregation of their results together with relating person-level outputs to locations still remains a demanding scope for further research (Delafontaine, Neutens, & Van de Weghe, 2012; Horner & Downs, 2014; Neutens, Schwanen, & Witlox, 2011).
section_name: Accessibility models

section_content: Several studies have been dedicated to combining time geography with accessibility measurement. According to Lenntorp (1977) and Burns (1979 ), Miller (1991) and Miller and Wu (2000) developed a computational process to operationalize STPs for calculating accessibility value in real study areas. Kwan (1998) conducted a study to compare person-based metrics with place-based ones. She found that the results of these measures are widely different from each other. In line with these works, Neutens et al. (2010) did a comprehensive study including a large set of measures and analytically compared the outputs. They indicated that using different metrics makes different insights into the accessibility and service delivery analysis. It must be noted that in the aforementioned and other related papers, the accessibility of a person to opportunities distributed between two anchors pointing mandatory activities is computed (Delafontaine et al., 2012; Kim & Kwan, 2003; Miller & Wu, 2000) and the models can be applied to assess accessibility in the case of simulating potential trips, whether travel data is available or not. 

In addition to the above mentioned studies, sporadic progresses were made on blending peopleand place-based models. Based on the perspective of statistical time geography, Horner and Downs (2014) enlisted network-based TGDE developed by Downs and Horner (2012) and applied it to identify links with more traversing probability and combined the attending probability with the opportunities attractiveness located along the network as a measure of accessibility. Similar to STP-based models, their metric can be applied in retrospective and prospective ways. In the same vein, some works modeled visiting probability of each segment of a street network by using movement principles. Following Song and Miller (2014) and Winter and Yin (2010, 2011), Song et al. (2016) established a robust model to determine the probability of each link by BB and SM processes. Although this model shows the links more likely to be passed by a mobile object, it does not differentiate more probable opportunities to be visited and it is not applied to develop blended accessibility measures. The next section shows a blended probabilistic accessibility measure in coincidence with the movement principles.
section_name: Related works

section_content: Recent work by Song et al. (2016) proposed models, on the basis of BB and SM stochastic processes, respectively for non-vehicular and vehicular moving objects in the street network. These models are applied to quantify visiting probability of edges and the most probable ones in the context of a street network. These probabilities are computed by the below equations (equation 1-equation 6):

(1)

(2)

(3) (4)

(5) According to Song et al. (2016), for non-vehicular movements, due to the limited time budget, maximum moving speed, and pressure of arriving at the destination before the planned time, probability of movements does not obey the standard distribution and is expected to be biased toward the destination. Thus, the probability of moving along edge ij is considered as the joint probability of that edge at t D -t and t-t O (equation 2 and equation 3). Accordingly, the normalized edge probabilities are calculated by equation 1. For vehicular movement, because of one-way and turn restrictions, and speed limits, the mobile object is not free to move backward and forward. Hence, in this case, instead of BB, Markov process is adopted for considering movement probability and holding time in an integrated framework. In this regard, the diffusion process from the origin, which shows the dependency of the passing probability to the available arrival time at that location, is modeled by equation 5 and similarly, the diffusion process from that location to the destination is represented by equation 6. Therefore, the probability of movement from vertex i to vertex j and vertex j to vertex i along the edge ij at t is measured by equation 4. 

For demonstrating accessibility of an object to edges of a road network relative to activities, equations 1 and 4 are adopted. According to Horner and Downs (2014) An example of accessibility calculation with this metric is elaborated for an individual with a hypothetical trip. This person starts his/her trip from an origin (red spot in Figure 1 ) to a destination (blue spot in Figure 1 ) within 20 minutes in a street network including 3064 links. Figure 1 illustrates the distribution of stationary activities (larger ones have higher attractiveness) and the layout of the road network. Figure 2(a) shows the accessibility level of opportunities in 5 min, 10 min, and 15 min time steps. According to equation 8, AA l t for this traveler can be calculated as the sum of perceived accessibility, that is equal to 0. 000159, 0. 000206, and 0. 000232 for 5 min, 10 min, and 15 min time lags, respectively. 

To put the model in the context of accessibility modeling, it is useful to refer to the works of Lenntorp (1977), Kwan (1998), and studies by El-Geneidy and Levinson (2006) and Neutens et al., (2010). The proposed measure is in line with their pieces of research where two approaches may be identified. The first is enumerating the number of opportunities in the feasible opportunity space (FOS) under a person's budget constraint and the second is summing the attraction of activities within FOS. In both of these approaches, the only characteristic that matters for considering an opportunity in accessibility calculation is its existence in FOS. None of the identified activities are weighed in these measures, despite the fact that all activities in STP or NTP are not equally accessible (e. g., with the same attractiveness, opportunities which are closer to the boundary of NTP/STP are less accessible than that of more distant ones (Song et al., 2016; Winter & Yin, 2010) ). The main differences of the proffered model are as follows (1) the binary nature of the previous metrics leading to overestimation effect was mitigated by adding a correction factor, and (2) individual-and opportunity-related aspects were linked in a unified dynamic framework. Figure 2
section_name: Model development and analysis

section_content: To show the applicability of the developed metric in real environments, in this part of paper, study area and constraints affecting the available time and the characteristics of land-use and transportation systems are represented.
section_name: Model specification and application

section_content: Food desert has negative impact on residents and increases the risk of getting some diseases such as: diabetes, cancer, and hyper-tension (Bazzano et al., 2002; He et al., 2004; Hung et al., 2004; Leal & Chaix, 2011). In this regard, it is very important to provide spatial access to healthy food for citizens (Widener, Farber, Neutens, & Horner, 2015). To provide accessibility to healthy food for residents of Isfahan, the municipality started to establish and manage Kowsar retail markets that supply daily and weekly needs of different households with various socio-economic levels from low-to middle-and highincome families (Chegounian, 2016). Recently, municipality pursues two policies targeting enhancement of temporal accessibility to these markets in a WH (work to home) trip sequence taking place in a time window starting at 5:00 p. m. and ending at 8:00 p. m. The reason for opting to WH trip chain and this time span is that the most frequent shopping from these markets are carried out in WH trips from 5:00 p. m. to 8:00 p. m. The aim of the first policy, "market exclusion and area expansion", is to determine and exclude a market with the smallest temporal accessibility value and to analyze the effect of expanding the area of each remained market on the level of accessibility. The purpose of the second policy, "new market construction", is to suggest place or places with the highest potential accessibility for building a new market with predefined level of attractiveness. It is apparent that for assessing the above mentioned policies, a space-time measure, which is able to calculate accessibility value in different moments of time, is required. Hence, in this paper, the spatial accessibility of people living in the city of Isfahan to these markets is pondered by the proposed model to show its applicability in real situations. 

Configuration of the streets and passing time through each link were acquired from Transportation Research Center of Isfahan University of Technology (TRCIUT). Area of the markets that was consid- ered as the representative of opportunities attractiveness was obtained from the city municipality (Table 2 and Figure 3 ). As Kowsar markets are highly vehicular-dependent and trips by non-vehicular modes are very rare (Chegounian, 2016), non-vehicular model was excluded from the computation process and λ, the parameter of vehicular model, was specified to be 0. 016 (see Appendix A for the parameter estimation process). Visiting probability and accessibility to the markets were calculated by a program developed in MATLAB and ArcGIS 10. 3 was applied for output visualization. The produced program was distributed on 60 processors and processing was about 7 hours and 24 minutes that is a significant amount of time-saving, compared to a desktop PC (personal computer).
section_name: Study area and data preparation

section_content: Results of the model implementation are presented as follows:
section_name: Results

section_content: Given the origin and destination of 26646 WH trips, accessibility of individuals to the markets was calculated by equation 8, and accessibility of opportunities was computed through equation 9. Table 3 shows PA k t along the rank of markets on the basis of the accessibility values. Figure 4 portrays accessibility distribution at t=45, t=90, and t=135 after departure time. This figure displays that markets 4 and 2 had the maximum and minimum accessibility value among the other opportunities, respectively. In addition, it could be seen that, compared to the traditional models, activities with higher attractiveness did not necessarily have higher accessibility level. For instance, market 6 had the ninth rank in terms of attractiveness while it had the tenth and eleventh ranks in terms of accessibility level. For more analysis of this observation, another simulation was executed. Figure 5 illustrates the results of the model execution for five activities distributed in the context of street network along with their attraction and temporal accessibility. In this figure, opportunity T2 has larger attraction than T1, and T1 is adjacent to a link with higher level of visiting probability. At the moments of t=45 and t=135 and in total case, accessibility of T1 with smaller attraction was larger than T2. To test the correlation between attraction and accessibility, rho coefficient was calculated by Spearman's test. The outputs represented that the correlation between attraction and level of accessibility was not significant at t=45, at t=135, and in the case of total accessibility (Table 4 ), but at t = 90, correlation between attraction and accessibility was confirmed. This result is analytically justifiable. Links of a street network have visiting probability depending on the spatiotemporal constraints. By changing costs of links, the shortest paths vary from the origin to the starting point of a link and from the end point of a link to the destination. As a place with larger attraction may be located next to a link with smaller value of visiting probability and accessibility of the place is measured by the combination of visiting probability and attractiveness, its level of accessibility may be smaller than a place situated by a link with larger visiting value and smaller attraction. Furthermore, in models like NUMOA, opportunities in NTP with larger attraction certainly have higher accessibility than opportunities with smaller attraction. However, regarding the effect of visiting probability and the number of individuals in the calculation process, accessibility of a more attractive location may be adjusted and be equal or even smaller than opportunities with smaller attractiveness. After identification of the market with the smallest accessibility value, i. e., market 2, municipality of Isfahan needs to see how the accessibility changes if this market is closed and in exchange, the area of one of the remained markets be doubled. In this regard, the model ran for each of the defined area expansion scenarios, i. e., area duplication of one market while the area of the others remains the same. Table 5 reports changes of accessibility versus variations in the opportunities attractiveness. 

The comparison of the results of each scenario with the base values that are accessibility of opportunities after excluding market 2 without change in attractiveness of the other markets along with time-steps-based comparison of cases, revealed that if the area of market 7 is doubled, accessibility in two time instants will be maximized. However, the maximum total accessibility is when market 8 is doubled. Minimum accessibility values, in terms of both total and the number of instants with maximum accessibility, occurred when market 4 was expanded. Furthermore, it can be seen that if the area of markets 4, 7, and 8 changes, attractiveness of the other opportunities will be modified and consequently the accessibility values will be varied. However, in the other scenarios only the accessibility of the expanded market changed and the accessibility of the other markets remains the same as the base values (See appendix B for the outputs of accessibility calculation in details). . 4566 393. 9331 213. 3298 845. 7195 256. 7852 422. 5974 225. 1921 904. 5747 246. 7788 412. 2062 222. 7337 881. 7187 119. 2283 196. 9665. 6726 414. 6748 221. 6747 889. 0221 252. 2714 416. 9805 224. 6321 893. 884 250. 5152 412. 3862 223. 3007 886. 2021 267. 1989 441. 36 243. 9298 952. 4887 To identify the most balanced case, standard deviation of the accessibility values at each time step was calculated. Table 6 denotes the values calculated for this statistical indicator. The results depict that the most balanced state is achieved when area of market 4 is doubled. However, this scenario gives the smallest total accessibility value. It must be noted that there are some items left for further research. For example, as well as visiting probability, attractiveness of opportunities might be changed during the time and might be dynamic. In addition, attractiveness can be bounded by capacity variable and competition effect.
section_name: First policy: Market exclusion and area expansion

section_content: Second policy: New market location

Isfahan has a monocentric spatial structure and a large portion of shopping activities for supplying daily and weekly needs is done by WH trips in the afternoon times. These trips are mainly originated in the central areas and destined in the peripheral districts. In this way, the municipality likes to have a set of locations including areas in L={1000 m 2 ‚ 2000 m 2 ‚ 3000 m 2 } with the highest level of potential accessibility to respond the shopping demand of citizens and to deliver better services to people with this kind of trip. Having this set of locations, the local government is able to choose a solution from the others, considering construction factors such as: available monetary budget, land ownership, land price, etc. 

In doing so, places with areas equal to the elements in L with the largest visiting probability values need to be selected to maximize equation 7. As the area of markets was given, firstly, visiting probability of each segment in the street network was calculated at three time instants. Figure 6 (a) illustrates the visiting probability of links in the study area and Figure 6 (b) shows the links that are common among the simulated states with the highest probability values. Secondly, locations with the area of 1000, 2000, and 3000 square meters adjacent to the segments with the highest probability were determined. Figure 7 displays the founded places and their rank of accessibility in each group of L independent from the other identified locations for new construction and other markets in the study area. 

To compare the first policy with the second policy, the accessibility of markets shown in Figure 7 was computed considering the other opportunities in the study area when market 2 was removed from the set of available markets without any intervention in other conditions of the current state. shows that construction of market L_3000_3 gives the largest accessibility value, and the best value of the first policy that is area expansion of the market with ID 8 has better value than the best of the second policy. 

It must be noted that these policies were analyzed in terms of accessibility, and for selecting a case in each policy, the results can be combined with the cost-benefit studies including other socio-economic factors. Integrating these complementary studies into the model structure can be investigated in future works.
section_name: 4.2.2

section_content: Considering the literature on accessibility modeling and time geography, this paper has fostered an accessibility measure using the frameworks of Horner and Downs (2014) and Song et al. (2016). This measure can be applied to analyze the accessibility in both retrospective and prospective ways and mitigate the overestimation effect existed in the previous metrics by taking the movement principles and stochastic processes into account. In addition, in the proposed model, it was attempted to connect people-and place-based measures together because despite the necessity of working in this area (Horner & Downs, 2014), scant studies throughout the literature have been dedicated to develop blended models (Miller & Wu, 2000; Neutens, Delafontaine, Scott, & De Maeyer, 2012). The model can be aggregated over individuals or places to derive individual-or opportunity-based accessibility value. It means that the model can play a role in both person-and place-based arenas. 

As mentioned before, the proposed model has some similarity with previous models such as NUMO and NUMOA, but the context of the adopted approach (appending a probabilistic component to the model's structure) in pondering and mapping accessibility makes it different from its precedents. However, it must not be said that this model is not necessarily better or worse than other methods, but it is another perspective alongside the other approaches at least. 

Several applications can be considered for the proposed model. For example, it can be regarded for analyzing the effects of changes in the road network (e. g., adding a new road segment or increase in streets' level of service) on individual's route choice and consequently accessibility or the effect of constructing a new opportunity on the accessibility level. Along with the policy analysis applications, the model can be adopted for policy-making purposes. For instance, by using optimization techniques, an analyst can suggest the optimum travel time, departure time, or arrival time for maximum level of accessibility or find the best location of activities for improving accessibility value. These sorts of reports can guide managers and planners to schedule activities and make activity distribution plans. 

The following extensions can be regarded for future works. Although in this model accessibility of individuals traveling by vehicular mode or non-vehicular mode can be analyzed, considering the effect of mode transfer on the accessibility level and form of NTPs seems valuable as people who choose public transport for their movement mostly switch between walking, rail, and bus modes. The proposed model is a cumulative-based one, but this approach can be applied to construct measures in Burns-Miller framework (see: (Neutens et al., 2010) Difference between the socio-economic characteristics of individuals or social groups (in terms of gender, age, etc. ) with their level of accessibility which reveals the way they use the space and their differences in accessibility to opportunities. In addition, the measure can be used for analyzing the potential of social interaction between different people as the model considers visiting probability and it is able to differentiate between places in terms of attending possibility. Further extension is to incorporate the effect of changes in attractiveness of opportunities during the time and cost-benefit analysis in the model structure.
section_name: Discussion and conclusion

section_name: Acknowledgements

section_content: The authors would like to thank the general editor, Prof. David M. Levinson, and the anonymous reviewers for their insightful and constructive comments.
section_name: Acknowledgements