Scaling laws of spatial visitation frequency

Computers, Environment and Urban Systems 64 (2017) 332–343

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Computers, Environment and Urban Systems journal homepage: www.elsevier.com/locate/ceus

Scaling laws of spatial visitation frequency: Applications for trip frequency prediction Zhong Zheng a, Suhong Zhou b,c,⁎ a b c

Urban Planning Group, Eindhoven University of Technology, Eindhoven, The Netherlands School of Geography and Planning, Sun Yat-sen University, Guangzhou, PR China Guangdong Key Laboratory for Urbanization and Geo-simulation, Guangzhou, PR China

a r t i c l e

i n f o

Article history: Received 12 July 2016 Received in revised form 10 April 2017 Accepted 11 April 2017 Available online xxxx Keywords: Human mobility Scaling laws Urban dynamic Transport prediction Built environment

a b s t r a c t The relationship of the built environment to human travel is one of the mainstream themes in urban studies. It provides a foundation for transport prediction. The existing literature is limited in accuracy when predicting spatial temporal travels from built environment. Understanding the scaling laws of spatial visitation frequency sheds new light on the issue. The scaling laws connect travel and the built environment by ordered-rankings, which make it possible to predict the number of arrivals from environmental variables. This research analyses the scaling laws of dynamic spatial visitation frequency using taxis' global positioning system (GPS) records, and proposes a model to predict spatial temporal arrivals from points of interest (POIs). The results show that: (i) the scaling law of spatial visitation frequency is exponential; (ii) the exponential scaling law is explained by the linear preferential attachment effect and a logarithmic travel growth process; (iii) the exponential scaling law is not sensitive to time; (iv) the proposed model predicts spatial temporal arrivals with high accuracy (R2 N 0.6). © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Uncovering the relationship between the built environment and human travel is of research interest in urban studies. Understanding the influence of the built environment has the potential to provide guidance for transport prediction. Travel is generally defined as the displacement of individuals and measured by frequency, length, modes and purposes (Handy, Boarnet, Ewing, & Killingsworth, 2002). The built environment is measured by environmental variables such as density, diversity, design, destination accessibility, distance to transit (Ewing & Cervero, 2010), and land-use (Crane, 2012). The analysis of the built environment and travel relationship can be in the aggregate level of traffic zones or in the disaggregate level of the individual/household (Handy, 1996). Examining the influence of the built environment on travel is a long research tradition in both aggregate analysis (Aljoufie, Zuidgeest, Brussel, van Vliet, & van Maarseveen, 2013; Ewing, Hamidi, Gallivan, Nelson, & Grace, 2014; Holtzclaw, Clear, Dittmar, Goldstein, & Haas, 2002) and disaggregate analysis (Cervero & Kockelman, 1997; Chao & Qing, 2011; Crane & Crepeau, 1998; Dieleman, Dijst, & Burghouwt, 2002; Ewing & Cervero, 2001; Fan & Khattak, 2009; Meurs & Haaijer, 2001; Zhang, 2005). Despite the appeal of this fruitful research, there are still challenges in predicting travel frequency with high accuracy ⁎ Corresponding author at: School of Geography and Planning, Sun Yat-sen University, 135 Xingang Xi Road, Guangzhou, PR China. E-mail addresses: [email protected] (Z. Zheng), [email protected] (S. Zhou).

http://dx.doi.org/10.1016/j.compenvurbsys.2017.04.004 0198-9715/© 2017 Elsevier Ltd. All rights reserved.

because of the limit of predictability in human mobility (Lu, Wetter, Bharti, Tatem, & Bengtsson, 2013; Song, Qu, Blumm, & Barabási, 2010). Generally, spatial travel frequency is assumed to be a function of environmental variables, and a rigorous mathematical relationship is required. However, it is not fully clear which particular mathematical form (linear, power, exponential, polynomial, etc.) is the most appropriate for the prediction function. Predicting travel frequency from environmental variables is also highly dependent on the spatial context. As geographical heterogeneity varies in different spatial contexts, it is challenging to summarise a universal regularity of the built environment and travel frequency relationship. Scaling laws of human mobility in complex systems may shed light on this issue. The scaling laws uncover statistical patterns of human mobility by finding probabilistic distributions of mobility variables. The main research interests in the scaling laws of mobility are spatial density (Makse, Havlin, & Stanley, 1995; Rozenfeld, Rybski, Andrade, Batty, Stanley et al., 2008; Yuan, Raubal, & Liu, 2012) and movement displacement (Calabrese, Di Lorenzo, & Ratti, 2010; Jiang, Yin, & Zhao, 2009; Krings, Calabrese, Ratti, & Blondel, 2009; Liang, Zheng, Lv, Zhu, & Xu, 2012; Liu, Kang, Gao, Xiao, & Tian, 2012; Yan, Han, Wang, & Zhou, 2013; Zheng, Rasouli, & Timmermans, 2016). Visitation frequency, or spatial travel density (see Liu, Gong, Gong, & Liu, 2015), reveals the probability of an individual's arrival at a specific location. The higher visitation frequency a location has, the more likely it is to attract visitors. The frequency-ranking law defines a relationship between visitation frequency and an ordered-ranking of a location. As the mathematical

Z. Zheng, S. Zhou / Computers, Environment and Urban Systems 64 (2017) 332–343

relationship in scaling laws is rigorous, visitation frequency can be estimated from the given ordered-ranking of a location. At the same time the rankings can be empirically estimated from built environment variables. Two relationships are accordingly built: visitation frequency and locational ranking, locational ranking and the built environment. Therefore, a transport prediction model can be hopefully built using the scaling laws. There is great value for doing so. Theoretically, this approach is advantageous for traditional models as it introduces locational rankings. First, a rigorous mathematical relationship can be defined. Travel frequency is seen as a function of ordered-ranking, rather than environmental variables. It is easier to find and explain the mathematical regularity of the frequency-ranking law. Second, the relationship of locational ranking and the built environment is more relaxed. It is not necessary to apply a mathematical relationship to rankings and environmental variables. All locations are ranked via criteria of environmental variables. If the environmental criteria of location A are greater than location B, the ranking of A is then superior to B. Rankings provide relative comparisons among locations. For A to be superior to B only requires greater environmental criteria for A, regardless of how much greater A is. Such a relaxed relationship adapts to geographical heterogeneity: locational rankings, which are estimated from environmental variables, may differ in different spatial contexts. The geographical heterogeneity effect is reduced as long as locational rankings are properly estimated. Practically, predicting visitation frequency from spatial facilities, improves adjusting planning and transportation policies for urban planners. Cities in developing countries are experiencing fast changes in land uses and spatial facilities, thus raising great challenges in transportation planning and management. New town construction and city regeneration generate great traffic volumes. The prediction model can provide support in forecasting traffic volumes from land-use changes and newly constructed facilities. The purpose of this research is to find the dynamic scaling laws in spatial visitation frequency, and to propose a transport prediction model. Human travel in this research is represented by arrival behaviour, and arrivals are identified by drop-off points from taxis' global positioning system (GPS) records. Generally, transport can be predicted from trip generation (Moreira-Matias, Gama, Ferreira, MendesMoreira, & Damas, 2013), trip attraction (Giannotti et al., 2011), or both (Black, 2003). The ability to attract travellers directly reflects the influence of the built environment on travel. The number of spatial arrivals, or spatial visitation frequency, is a straightforward measurement of the attractiveness of the built environment. This research uses points of interest (POIs) to represent the built environment. Spatial density of POIs reflects spatial attractiveness, and types of POIs reflect spatial function. Moreover, accessibility of POIs is sensitive to time of day, such that different POIs types have different opening time windows and temporal attractiveness (Gong, Liu, Wu, & Liu, 2015). Therefore, using POIs as a representation of the built environment, means the prediction model is sensitive to dynamic daily travels. The contribution of this work is accordingly twofold. First, the scaling law of spatial visitation frequency and its underlying mechanism is uncovered. The key to understanding the scaling laws lies in the preferential attachment mechanism and travel growth process. Second, a transport demand prediction model is built. Spatial arrivals depend on locational rankings, and locational rankings depend on POIs density. The innovation is in the simplicity of the model: it predicts dynamic spatial arrivals from POIs density. 2. Related works 2.1. Built environment and human travel The relationship of the built environment and human travel has a long research tradition in transportation and urban studies. A review of relevant works can be found in Handy (1996) for early studies and Ewing and Cervero (2010) for recent studies. This section briefly introduces new research trends since 2010.

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With the development of information and communication technology (ICT), collecting large volume data becomes an alternative approach to studying human travel. Various data sources (floating car data, mobile phones, transit check-ins, social media, etc.) provide new insights into human travel. There are two aspects to innovations from the usage of big data. First, it is possible to discover spatial distribution of daily travels. Traditional travel diary surveys use random sampling to represent the whole population. However a small sample cannot fully represent the spatial travel patterns of the whole population. Alternatively big data are good representatives of spatial travel patterns. Second, points of interest (POIs) have the potential to be new measurements of the built environment. Traditionally, the built environment is characterised by variables beginning with ‘D’, the original ‘three Ds’ of density, diversity and design (Cervero & Kockelman, 1997), and later ‘six Ds’ of additional destination accessibility, distance to transit and demand management (Ewing & Cervero, 2010). The function of the built environment is important. Residential and work function can be measured by population density and job density, but commercial, recreational, public service, etc. functions are less discussed. POIs data of spatial facilities are good supplements for the functions of the built environment. Veloso, Phithakkitnukoon, and Bento (2011) formulated a taxi trip prediction model based on POIs. Given the drop-off area type (identified by POIs), temporal variables and weather condition, the probability of pick-up area type can be predicted with an accuracy of 54%. Huang, Li, and Yue (2010) identified POIs temporal attractiveness from GPS traced data. It contributes to a shortcoming of environmental variables - not sensitive to time. Thus it is possible to predict temporal trips from POIs. Several works examined the qualitative relationship of travel and the built environment. For example, Liu, Kang, et al. (2012) compared the spatial density of taxi trips and population density. They found taxi trip distribution is more concentrated in the city centre rather than according to population distribution. Using mobile phone massive data, Kang, Ma, Tong, and Liu (2012) examined how urban morphology influences human travels. Large or less compact cities have larger numbers of trips, and an irregular shape of a city constrains local movements but brings more long trips. Peng, Jin, Wong, Shi, and Lio (2011) identified taxi trips purposes as commuting, business and other, and examined the interaction with locational function. Pan, Qi, Wu, Zhang, and Li (2013) found taxi travel dynamics exhibited clear patterns corresponding to land-use classes. Liu, Wang, Xiao, and Gao (2012) examined how landuse types affected daily trip generation using taxi GPS traces. Rich literature examined the influence of the built environment on travel, while the usage of big data makes it possible to discover the reverse relationship - identifying environmental information from travel. Four types of land-use (residential, commercial, recreational and industrial) were successfully identified from the spatial and temporal distribution of taxi trips (Liu, Wang, et al. 2012). In other research (Pan et al., 2013) land-use classes were identified from taxi pick-up points (PUPs) and drop-off points (DOPs) with an accuracy of 95%. Yuan, Zheng, and Xie (2012) used 2-year POIs datasets and 3-month taxi GPS trajectory datasets and formulated a topic-based inference model. The result showed a region is represented by a distribution of human mobility patterns. Using London subway ‘Oyster’ card data, Roth, Kang, Batty, and Barthélemy (2011) found a poly-centric structure for large flows, and a mixed hierarchy structure for small flows. Taxi trips well reflect urban space function and temporal variation. From the above brief review, it can be seen that the quantitative relationship of trip frequency (from big data) and the built environment is less discussed. Our research contributes to current studies by formulating a quantitative model linking trip frequency and POIs density. 2.2. Scaling laws in urban studies Spatial visitation frequency can be measured by two approaches: frequency-ranking laws or frequency probability distributions. The

334


frequency-ranking law, or Zipf's law, defines the relationship between visitation frequency and an ordered-ranking of a location. The frequency probability distribution depicts the probability of visitation frequency at a location. The scaling laws of frequency-ranking vary across empirical studies, while frequency probability distribution is generally believed to follow a power law. The frequency-ranking laws can be a power law, an exponential law, or an exponential/power law with a cut-off. For example, the spatial visitation frequency based on mobile phone users' traces followed a power law (Song, Koren, Wang, & Barabasi, 2010). It could be explained by the preferential return effect and power law growth of distinct visited locations. The model from Hu, Zhang, Huan, and Di (2011) concluded that the home-return constraint is of importance and has significant influence on optimising mobility scaling laws. However, a study of GPS records of private cars found that spatial visitation frequency by ranking followed an exponential law (Giannotti et al., 2011). Rank-ordered incoming and outgoing flows of subway stations also displayed an exponential decay pattern. Based on the exponential decay pattern, multiple centres were defined depending on the time of day (Roth et al., 2011). Hasan, Lafayette, and Ukkusuri (2013) used locationbased social media check-in data to analyse human mobility patterns, and concluded that visitation frequency followed an exponential power law. But for top-ranked locations, visitation frequency decayed according to a power law, while it had a cut-off when ranking increased. Frequency probability distribution generally follows a power law, although in some cases it has a cut-off. Using social media check-in data from 370 Chinese cities, it was found that the frequency distribution of check-ins in cities followed a power law, and the distribution of visited cities also followed a power law (Liu, Sui, Kang, & Gao, 2014). The scaling law can be explained using the size-ranking distribution of Chinese cities. From empirical research on taxi trips, the probability distribution of the visiting frequency for different locations displayed a power law distribution, and the relative deviation of traffic followed a normalised binomial distribution (Peng et al., 2011). The probability of passenger flows between two subway stations in London was distributed by a power law with exponent around 1.3 (Roth et al., 2011). This work uncovered that the frequency-ranking law and frequency probability distribution could be different, with an exponential law for the former and a power law for the latter. Lü, Zhang, and Zhou (2013) gave an explanation. Assuming that Zipf's law and the frequency probability distribution both follow a power law, then when the exponent of frequency

distribution is close to 1, the exponent of Zipf's law diverges so that it becomes an exponential law or in another form. Both of the two measurements can be applied to uncover human mobility patterns, but this research concentrates on the frequencyranking law. For the purpose of transport prediction, the frequencyranking law can predict aggregate travels explicitly and directly based on a location's ordered-ranking. The scaling law of frequency-ranking varies from different empirical evidence, therefore it arises the following questions: Under what conditions does the visitation frequency power/exponentially decay? Can visitation frequency be explained by a universal mechanism? Is the scaling law of visitation frequency sensitive to time? The answers of these questions still remain unclear, and they are the foundation of the transport prediction model. 3. Data Recently, the development of information and communication technology (ICT) and location-based services (LBS) have provided opportunities for analysing human mobility regularities and modelling individual travel behaviour using ‘big data’. Taxi GPS records are an example of big spatial-temporal data used to represent human mobility (Liu, Wang, Xiao, & Gao, 2012; Zhou, Fang, Thill, Li, & Li, 2015). The study area explored in this paper is the inner city of Guangzhou, China (Fig. 1). Guangzhou is the most populous city in South China. The inner city is a highly developed and urbanised part of the whole city, with 7.73 million inhabitants and a 1210.2 km2 area. Taxi GPS records were collected on Monday, 11th May 2009. The data set recorded trajectories of 13,000 anonymous taxis in Guangzhou. Most of the taxi records (95.8%) were recorded in the inner city. A GPS log was installed in each taxicab. The GPS information on position coordinates, time, velocity and carriage status was collected every 20 s (Table 1). In one day, 24.4 million GPS records were examined. Pickup points (PUPs) and drop-off points (DOPs) were then identified by using the shift in carriage status (for example, from vacant to occupied). As spatial visitation measures the attractiveness of destinations, only drop-off records were used to analyse the data. The inner city space was divided into 1326 spatial grids (1 km2). Spatial visitation frequency, or spatial arrivals, is measured by number of DOPs in each grid. Researchers detected 429,772 drop-off records, which were divided later into twelve 2-hour time periods to explore dynamic mobility. Fig. 2 maps the data.

Fig. 1. Study area.


335

Table 1 Example of a taxi GPS record. Field

Date

Time

Longitude

Latitude

Speed

Status

Example

11/05/2009

8/30/21

113.2318

23.1721

19

1

4. The scaling law of spatial visitation frequency 4.1. The scaling law model Taxi spatial arrival is a highly stochastic process. Let L1 , L2 , … LS denote visited grid locations. Spatial visitation frequency is defined as the number of trips f(r) arrived at the rth most visited location Lr. The scaling law of f(r) to r is explained by reference to two components: the preferential attachment mechanism and the distinct visited locations growth process. Preferential attachment (Barabási & Albert, 1999), or preferential return (Song, Koren, et al., 2010), or the richget-richer effect (Lü et al., 2013), refers to the probability that a location being chosen as the destination is proportional to the location's previous visitation frequency (Song, Koren, et al., 2010):

Based on Eq. (2) the relationship of f(r) to r is (Song, Koren, et al., 2010):

pðLr Þ f ðr Þ

f ðr Þ 1=S−1 ðr Þ

ð1Þ

where p(Lr) is the probability that a taxi arrives at the location Lr. The distinct visited locations growth process encompasses how the number of distinct visited locations S(n) grows with total individual travels n. Under the preferential attachment assumption, the discovery time (the time of the first visit) k(Lr) of location Lr, the frequency-ranking r, and distinct visited locations S(nr) have the following relationship: Sðnr Þ ¼ kðLr Þ ¼ r

ð2Þ

The explanation of Eq. (2) is as follows. Suppose during a period of observation, there are n total taxi arrivals and S distinct visited locations (L1 , L2 , … LS). The earlier a location is visited, the more likely it is to be visited again. The rank of visitation frequency r equals its first visitation time k(Lr). At the time location Lr is first visited by the trip nr, it is the kth discovered location, so distinct visited locations S(nr) are now equal to k(Lr).

Fig. 2. Destinations of taxis in the inner city.

Fig. 3. Spatial visitation frequency by rankings.

ð3Þ

Using the above equations, the scaling law of f(r) to r can be examined in relation to the linear preferential attachment effect and the growth of distinct visited locations S(n). That is, linear preferential attachment determines Eq. (3), and the scaling laws of Eq. (3) determine the scaling laws of visitation frequency f(r) to ranking r. To examine the scaling law of visitation frequency, a frequencyranking model f(r) to r is applied. It ranks travel frequency in each spatial grid and then fits the frequency and ranking by a distribution. The distribution of f(r) to r is the frequency-ranking model. To explain the scaling law of visitation frequency, the preferential attachment effect and travel growth process are examined. According to the preferential attachment effect, the probability that a traveller arrives at a grid Li is proportional to the grid's previous visitation frequency f(r). It is tested by considering the relationship between the average arrival probability g(f) and frequency f, as proposed by Lü et al. (2013).The travel growth process is defined by the growth of distinct visited locations S(n) with total individual trips n. The growth process and stable status of S(n) jointly explain the scaling law of visitation frequency. The model above was then used for twelve time periods. The fundamental differences between the 12 models explained the dynamic change process: whether the models were from the same distribution or not, whether parameter differences were from scaling parameters or shaping parameters or both. If different distributions or shaping parameters were observed, a dynamic and thus changing scaling law

Fig. 4. Spatial visitation frequency at different spatial scales.

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However, the linear preferential attachment effect does not change the visitation frequency distribution. Suppose at time t we observe n trips, with visitation frequencies at location Li and Lj being fn(i) and fn(j) respectively. After a period of time t′, n′ trips are observed. Then we have: 0 n −n f n ðiÞ 0 f n0 ðiÞ f ðiÞ n 0 ¼ ¼ n n −n f n0 ð jÞ f n ð jÞ f n ð jÞ f n ð jÞ þ n0 f n ðiÞ þ

Fig. 5. Linear preferential attachment effect.

would be shown, whereas if only different scaling parameters were observed, the scaling law was considered to be stable by time. 4.2. The scaling law is exponential Travel frequency decay f(r) is shown in the semi-log plot in Fig. 3. The linear relationship indicates an exponential decay effect. The result is surprising as it follows an almost perfect exponential law f(r) ~ α−r. The exponential law is light-tailed, which implies the number of locations to be visited is limited. For taxi travels, the number of candidate destinations is no doubt limited. As taxi travel is constrained by cost, time budgets and city boundaries, almost all taxi trips are within Guangzhou city's boundary and most of them (95.8%) arrive at the inner city. The modifiable areal unit problem (MAUP) is an issue to be considered. Measurement on different spatial scales (units area) may result in statistical bias of spatial point frequency. Thus, spatial visitation frequency was examined using different spatial scales: on spatial grids of 25 km2, 4 km2, 1 km2 and 0.25 km2. As shown in Fig. 4, the linear trend in the semi-log plot confirms an exponential decay of visitation frequency exists for all illustrated spatial scales. Therefore the exponential law of visitation frequency is convincing at different spatial scales. Whether the preferential attachment effect existed in our observations needed to be confirmed. Using the concept of taxi travel's preferential attachment meant the probability of a location chosen as a destination was linearly proportional to previous travel frequency in the location. As explained in the method section, the average arrival probability g(f) should be linear to travel frequency f. A linear relationship can be examined from the exponent in a power law relationship: gð f Þ f

λ

ð4Þ

where λ = 1 confirms a linear relationship, g(f) ~ f. In the log-log scale of Eq. (4), the curve slope is the exponent λ. Fig. 5 shows that for all trips λ = 0.9856, which is very close to 1. It indicates g(f) ~ f, that a linear preferential attachment effect is confirmed.

ð5Þ

The visitation frequency distribution at time t′ is the same as at time t without considering arrivals at unvisited locations. It is the new arrival at an unvisited location that changes the visitation frequency distribution. The probability of visiting a new location is found in the distinct visited location growth process. Fig. 6 shows the process. Different stages are observed in the growth of distinct visited locations S(n). In the early stages the linear growth in the log-log scale predicts a power law growth, especially a short period of linear growth as the slope equals 1 at the beginning. In the late stage a linear growth trend exists in the semi-log plot, reflecting a logarithmic growth. Lü et al. (2013) explained the mechanism of growth of S(n) in the following equation: ε Vε SðnÞ ¼ V 1− Vε þ n

ð6Þ

where V is the number of grids; Vε is the joint visitation probability of all candidate grids; and ε is initial attractiveness. The growth of S(n) has three stages: (i) in the early stage S(n) ≈ n; (ii) as n grows and ε is very small SðnÞ ≈ V ε ln ð1 þ Vtε Þ; (iii) as n grows very large S(n) ≈ V. The linear part in stage (i) and logarithmic part in stage (ii) were consistent with research observations (Fig. 6). Observations of spatial visitation frequency happen in the last moment of travel growth (stage ii), therefore there is no doubt that it is within the logarithmic growth stage. Substituting the logarithmic growth of S(n) into Eq. (3) we find: f ðr Þ α −r

ð7Þ

Eq. (7) explains the exponential decay in spatial visitation frequency. Taxi travel frequency decays as an exponential law. The exponential decay rate (the slope in Fig. 3) where α′ = 0.0062 indicates the decay effect. The exponential decay rate comes from Eq. (8), where the ‘lg’ is the logarithm with the base of 10: α 0 ¼ − lgðα Þ

ð8Þ

Travel growth has a linear preferential attachment effect, which is confirmed from the exponent λ = 1 in Eq. (4). The probability of a visit to a specific location is linearly proportional to previous travel frequency. Until now there has been general knowledge about all taxi trips,

Fig. 6. Distinct visited location growth (left: log-log scale; right: semi-log scale).


337

Fig. 7. Spatial visitation frequency by time periods.

but this research satisfies curiosity about the temporal differences: travel frequency decay rates and distinct visited location growth rates. 4.3. Temporal differentiation of the scaling law In this part, temporal differences of visitation frequency are examined across 12 two-hour time periods. The exponential decay rate α′ in the frequency-ranking model explains the decay difference. The larger α′ is, the greater the rate of travel decay. Fig. 7 and Table 2 show the difference. Surprisingly, again the temporal differences are so small that observations across different time periods have no significant decay differences. Fig. 7 plots f(r) in a semi-log plot, showing the lines are almost parallel with slopes between 0.0059 and 0.0066. The decay rates are very close, with their only differences being from scale parameters (intercepts in Fig. 7). In other words, taxi trips in all time periods have the same exponential decay rate. Travel growth further confirms the similarity across different time periods. Table 2 shows λ values in 12 time periods ranging from 0.9648 to 1.0179, very close to 1. Thus the linear preferential attachment effect exists in all time periods. There is no significant difference in the spatial visitation scaling law among the different time periods. Distinct visited locations also grow according to the same pattern in different time periods (Fig. 8): for each time period at the beginning S(n) grows linearly, and after n = 1000 trips, S(n) grows logarithmically. The scope of function value S(n) is within 364 and 503. The same logarithmic growth rate explains the same frequency decay rate in all time periods. As individual trips grow, after 1000 trips the visitation frequency decays exponentially regardless of the time periods. It has been confirmed that both the travel frequency decay rate and travel growth patterns do not change through time. Therefore taxi travel patterns are quite clear: temporal travel demand determines the overall size of total trips; linear preferential attachment and logarithmic Table 2 Decay parameters and linear effect by time periods. Time period

α′

λ

01:00–03:00 03:00–05:00 05:00–07:00 07:00–09:00 09:00–11:00 11:00–13:00 13:00–15:00 15:00–17:00 17:00–19:00 19:00–21:00 21:00–23:00 23:00–01:00

0.0059 0.0063 0.0062 0.0062 0.0061 0.0060 0.0062 0.0063 0.0063 0.0066 0.0062 0.0062

1.0003 0.9829 1.0179 0.9948 1.0082 0.9960 1.0071 1.0114 0.9830 0.9648 1.0116 0.9773

location growth determine the exponential scaling law. Space is like a stable attractor, pulling in travellers according to the same scaling law. Both the frequency decay rate and travel growth rate are temporally stable. 5. Prediction on spatial temporal arrivals The analysis has shown that the scaling law in spatial visitation frequency is stable. The scaling law is always exponential regardless of what spatial scale or areal unit is used, and which time period it is in. The stability in space and time can be explained by the general mechanism: linear preferential attachment and logarithmic location growth. Understanding these scaling properties provides a solid foundation for the transport prediction model. 5.1. The prediction model The purpose of the proposed model is to predict the number of arrivals at a spatial location based on its built environment. The built environment can be represented by various environmental variables. Choosing proper variables is important for locational rankings estimation. Points of interest (POIs) have the opportunity to be good environmental variables (Di Lorenzo, Reades, Calabrese, & Ratti, 2012). POIs correspond to density and diversity in environmental variables (Ewing & Cervero, 2010), and POIs can generally reflect temporal accessibility due to their open time. Meanwhile spatial arrivals of travellers closely relate to spatial facilities, which can also be represented by POIs. The ordered-ranking of a location is a bridge linking spatial arrivals and the built environment. The prediction model consists of two steps: calculating spatial arrivals from locational rankings, and estimating locational rankings from POIs. The first step defines a mathematical relationship between visitation frequency and ordered-ranking, and the second step determines the ordered-ranking of a location from its POIs density. In this section both steps are presented and merged into a spatial arrival prediction model. Through understanding the scaling law and its underlying mechanism, an exponential prediction model is presented thus: f ðr Þ ¼ α SðnÞ−r

ð9Þ

where f(r) is the number of predicted spatial arrivals. Eq. (9) satisfies one visitation in the highest ranking location LS(n). As the proposed model should predict the exact arrivals at every location, assigning rankings to all locations is necessary. The second step of the model is to infer locational rankings from POIs density. The POIs data of Guangzhou were provided by a mapping and navigation company ‘Daodaotong’, which provides the original map

338


Fig. 8. Distinct location growth by time periods (left: log-log scale; right: semi-log scale).

data for China's biggest search engine (www.baidu.com). A ‘point of interest’ has a spatial location and facility type attributes. Facility type attributes are classified into 65 categories according to social and economic sectors, such as factories, offices, restaurants, shops, hospitals and so on. Points of interest are further aggregated into seven types (Table 3). A higher spatial density of POIs attracts more arrivals. Each type has the potential to attract travellers, and attractiveness is assumed to be temporally different. To better present the model the following assumptions are made: first, locational rankings are negatively correlated to POIs density. High POIs density results in a small ranking number (top-ranking); second, POIs types have weighted impact on locational rankings; third, the weighted impact of POIs types differs in different time periods. Using POIs density by functional types as environmental variables, locational rankings are a function of these variables. Weighted parameters of variables reflect environmental impact on rankings. Let r be the ranking of location Lr, and Xr = {xrk, 1 ≤ k ≤ K} be a set of POI density variables in K types at the rth-ranked location. The ranking r and POI density set have the following relationship: K

r ¼ hðV r Þ; V r ¼ ∑ wk xrk

ð10Þ

k¼1

where Vr is the overall environmental criteria of location Lr , measured by a weighted average of POI density in K types, h() is a monotonic function of the criteria Vr to assess the ranking r from Vr , {wk, 1 ≤ k ≤ K} is a set of weighted parameter of POIs density in type k, xrk is the density of POIs in type k at location r. All locations are ranked in non-increasing order of the environmental criteria, that Vr ≥ V(r + 1) for r = 1 , 2 , 3 , … , S(n) − 1. After all locations are ranked by the environmental criteria V r , the locational rankings r are assessed to these locations. Substituting Eq. (10) into Eq. (9) it becomes: f ðr Þ ¼ α

SðnÞ−h

K

∑k¼1 wk xrk

parameters α is the shaping parameter regarding the visitation frequency decay rate, S(n) is the scale parameter regarding the number of visited locations, and {wk} are a set of weighted parameters of POIs types. Parameters were estimated for 12 time periods. 5.2. Parameters estimation 5.2.1. Estimation of α and S(n) The summation of all visited locations' arrivals equal to total trips (given in observations): SðnÞ

∑ α SðnÞ−r ¼ n

ð12Þ

r¼1

and the equation 1−α SðnÞ ¼n 1−α

ð13Þ

solves the parameter α according to the geometric sequences summation formula. The distinct visited location S(n) is estimated from: SðnÞ ¼ a lgðnÞ þ b

ð14Þ

where parameters a and b are estimated from observations by least square. The result of estimated parameters is shown in Table 4. The number of travels n is the input data; locations to be visited S(n) is estimated from Eq. (14); the exponential base α is solved from Eq. (13). The estimated exponential base α is within 1.0128 and 1.0132 by time periods, consistent with the conclusion that visitation frequency decays at a similar rate through time.

ð11Þ

Eq. (11) is the model to predict spatial arrivals based on POIs density. In Eq. (11), several parameters, α , S(n) , {wk} are estimated. In these Table 3 POIs types and number. POIs type

POIs number

Work Home Transportation Dining Shopping Recreation Public service

97611 10236 23251 49282 7138 18503 24438

Table 4 Estimation results of α and S(n). Time periods n

Observed S(n) Predicted S(n) Observed α Predicted α

01:00–03:00 03:00–05:00 05:00–07:00 07:00–09:00 09:00–11:00 11:00–13:00 13:00–15:00 15:00–17:00 17:00–19:00 19:00–21:00 21:00–23:00 23:00–01:00

447 364 395 459 503 501 499 486 484 461 501 399

18,128 8861 11,277 31,932 50,843 44,216 53,309 40,074 49,712 45,309 52,347 13,764

424 375 391 464 496 486 499 495 494 488 498 405

1.0137 1.0146 1.0144 1.0144 1.0141 1.0139 1.0144 1.0146 1.0146 1.0153 1.0144 1.0144

1.0129 1.0127 1.0128 1.0131 1.0132 1.0131 1.0132 1.0131 1.0132 1.0131 1.0132 1.0128


339

Table 5 Estimation results of {wk}. Time periods

k=1 Work

k=2 Home

k=3 Transportation

k=4 Dining

k=5 Shopping

k=6 Recreation

k=7 Public service

01:00–03:00 03:00–05:00 05:00–07:00 07:00–09:00 09:00–11:00 11:00–13:00 13:00–15:00 15:00–17:00 17:00–19:00 19:00–21:00 21:00–23:00 23:00–01:00

0.0026 0.0247 0.1257 0.0315 0.0694 0.0703 0.0237 0.2429 0.2199 0.0173 0.0030 0.0046

0.1206 0.1291 0.1377 0.0302 0.1643 0.0987 0.0670 0.0881 0.0262 0.0082 0.0244 0.0045

0.3381 0.2625 0.3379 0.2183 0.3089 0.1934 0.3187 0.3049 0.3174 0.3253 0.2957 0.2964

0.2528 0.2635 0.1656 0.2627 0.0688 0.2387 0.4167 0.0166 0.0607 0.0307 0.2298 0.3213

0.0060 0.0063 0.1882 0.1628 0.0909 0.2477 0.1405 0.3070 0.1238 0.2354 0.1097 0.1674

0.1583 0.0131 0.0180 0.2849 0.2796 0.0730 0.0159 0.0133 0.2495 0.3802 0.3156 0.1858

0.1217 0.3009 0.0270 0.0096 0.0180 0.0781 0.0174 0.0271 0.0025 0.0030 0.0218 0.0198

5.2.2. Estimation on {wk} Parameters {wk} are estimated from observations, given spatial rankings and POIs density by types. Traditionally, regression analysis is a model to estimate the relationship between independent and dependent variables. When the dependent variable is the orderedranking of each location, neither linear regression nor ordinal regression are applicable. Linear programing provides an approach to measure weights of POIs density variables. It has been applied to estimate weighted parameters from ordinal ranked dependent variables and multidimensional independent variables. Proposed by Zappe, Webster, and Horowitz (1993), the estimation of {wk} is to maximise Z, subject to K ∑ wk xrk −xðrþ1Þk ≥Z; r ¼ 1; 2; 3; …; Sn −1

ð15Þ

k¼1

and K

∑ wk ¼ 1

ð16Þ

k¼1

The rankings of all locations have the constraint - when a location's environmental criteria (weighted average POIs density) are greater, it is superior to ordered-ranking. Eq. (15) satisfies the constraint. Maximising Z value reaches an optimal solution. The estimation was conducted in MATLAB R2013b. Table 5 shows results of {wk} that show different POI types have different functional and temporal impact. Generally transportation POIs have the most influence on locational rankings (0.1934–0.3379). Transportation facilities, including railway stations, metro stations, etc., have great attraction for taxi travellers. Work and home POIs, comparatively, have less impact on spatial rankings. It may be caused by the mismatch between spatial distribution of taxi arrivals and work/home POIs. From 1:00 to 7:00 impact of home POIs is slightly more, which could be

explained by people's returning home behaviour. However the high impact of work POIs in the evening peak (15:00–19:00) seems to violate common sense. Dining, shopping and recreation POIs have a different temporal impact on spatial rankings. The impact of dining POIs is at breakfast time (5:00–9:00), lunch time (11:00–15:00) and dinner time (21:00–5:00). A slight time lag exists at dinner time as the impact of dinner POIs begins later than people's usual dinner time. The explanation might be that other activities are more important than to have dinner at that time. In the Chinese tradition people like to have a night snack, thus night restaurants are still open late. For shopping, parameters are low at closing time (1:00–5:00) and relatively high during opening time (5:00–1:00). Shopping activities may occur in groceries, supermarkets or shopping malls, so a long period (20 h) of impact is observed. For recreation, a low parameter value is seen at closing time (3:00–7:00) and a relatively high value at opening time (17:00–1:00), which is easy to understand. Another high value in the morning (7:00–11:00) may result from activities of non-working people. The public service POIs generally have little influence, expect late at night (1:00–5:00). POIs density may not be the perfect measure of environmental variables, since not all weighted parameters can be explained in a convincing way. As an explorative analysis, the aim of using POIs as environmental variables is to examine how well the proposed prediction model can behave, rather than showing POIs as the perfect measure of environmental variables. It is entirely possible for other environmental variables to perform better in locational rankings estimation. 5.3. Prediction results Based on Eq. (11) and estimated parameters, spatial temporal arrivals are predicted. The prediction is validated by R2. The results of prediction are shown in Table 6 and Fig. 9. Table 6 compares the mean and standard deviation value of predictions and observations, and it

Table 6 Prediction results. Time periods

01:00–03:00 03:00–05:00 05:00–07:00 07:00–09:00 09:00–11:00 11:00–13:00 13:00–15:00 15:00–17:00 17:00–19:00 19:00–21:00 21:00–23:00 23:00–01:00

Observations

R2

Predictions

Mean

Std.

Mean

Std.

40.55 24.34 28.54 69.57 101.08 88.26 107.04 103.03 102.71 98.28 104.49 34.50

58.84 31.82 53.00 144.99 222.87 185.36 232.71 201.11 177.71 164.78 175.06 46.21

39.53 24.32 28.39 69.58 100.48 84.92 105.56 98.45 101.68 94.68 103.56 33.59

54.40 28.23 35.33 98.56 152.77 128.15 163.97 145.30 150.44 134.55 157.01 42.13

0.7427 0.6629 0.4486 0.6254 0.5979 0.6144 0.6075 0.6514 0.7421 0.7351 0.7901 0.7177


Fig. 9. Prediction results of spatial temporal arrivals.

340


341

Table 7 Comparison of scaling laws. Scaling laws

Source

Dataset

Candidate locations

Exponential Exponential Exponential with cut-off Power Power with cut-off

This study Giannotti et al. (2011) Roth et al. (2011) Song, Koren, et al. (2010) Hasan et al. (2013)

Taxi GPS Private car GPS Subway Mobile phone Twitter

Limited Limited Super limited Unlimited Between limited and unlimited

presents R2 for all time periods. Fig. 9 shows the comparison of predicted and observed arrivals at each location at all time periods. The horizontal axis represents locations, and is ordered by predicted location rankings. The vertical axis represents spatial arrivals of prediction and observation at each location. The proposed model has a good overall performance, such that R2 is generally above 0.6. The prediction model behaves differently according to the time. In the night time (17:00– 3:00) the model generates spatial arrivals with higher accuracy (R2 N 0.7177), however in the day time (5:00–17:00) the accuracy is relatively low, with R2 between 0.4486 and 0.6514. In Fig. 9 some observation points are highly above the predicted curve in the day time. These highly transport intensive locations, which are not captured by the prediction model, cause the decline in prediction accuracy. As mentioned above, there is no temporal difference in the visitation frequency decay rate and travel growth process, but temporal difference in actual visitation still exists. One of the differences is the visitation frequency of top-ranking locations. From observations (Fig. 3 and Fig. 7), top-ranked locations attract more travel than the exponential law. The thickness in the curve head (top-ranked locations) might be explained by a super preferential attachment effect. Although we confirm the linear preferential attachment effect from all observed locations, it remains possible that preferential attachment is more than linear for a few top-ranked locations. We call it ‘super preferential’ attachment where arrivals at a location are more than linear proportional to its previous arrivals (λ N 1). Travel is more concentrated in a few locations in the day time, while travel is distributed relatively sparsely in the evening. In the morning, travel is distributed mainly in the business area of the city, while travel purposes vary more in the evening as people have more locations to go to. Therefore, our proposed model performs better in the evening. 6. Discussion The research presented in this paper has a different finding to that of most activity mobility work. Temporal differences in human mobility are rather small. It is interesting to uncover a simple regularity in a complex urban system, and to apply it to transport prediction. However the exponential scaling law of this study is limited to the study area, Guangzhou. It needs further testing to determine whether the exponential law of travel frequency is a universal law. Using data from different cities is a possible solution. We admit collecting data from other cities to confirm the exponential law is beyond our current capacity. Alternatively, a glimpse of previous research provides deeper understanding of the scaling law. The exponential law is consistent with the result from Giannotti et al. (2011), and partly consistent with the exponential law with a cutoff from Roth et al. (2011), but different from the empirical research of Song, Koren, et al. (2010) and Hasan et al. (2013). According to Song, Koren, et al. (2010), visitation frequency follows a power law: f(r) ~ r−β. The power law results from the growth of distinct visited locations S(n), which satisfies S(n) ~ n1/β. Comparatively, the growth of distinct visited locations in this research is S(n) ~ n/ lg(α). The visitation frequency of this research accordingly follows an exponential law: f(r) ~ α−r. The difference underlying these scaling laws, whether it is a power law or an exponential law, is in distinct visited locations growth. The exponential law is observed in this study and also by Giannotti et al. (2011). Both research projects used GPS records of floating car data. Each vehicle had a home location, so is constrained within a city's

range. Even if taxi destinations are determined by passengers, taxi drivers do not like to drive far beyond the city boundary due to the high cost of returning. Limited candidate locations make the visitation frequency law exponential. Observations from subway station flows follow an exponential law with a cut-off (Roth et al., 2011). Because candidate locations are more limited for subway stations, there is no probability of visiting a new station. Therefore a sharp cut-off effect happens in the tail rankings. A power law is observed by examining mobile phone data (Song, Koren, et al., 2010). The dataset has no limits on users' locations, thus people could travel freely. The unlimited candidate locations result in a heavy tail for frequency distribution, thus approximating a power law. An exponential power law exists for Twitter social media data (Hasan et al., 2013). Although potential candidate check-in locations are unlimited, the data was collected from three US cities. Candidate locations were between unlimited and limited, so the result displays a power law trend with a cut-off. A general comparison is shown in Table 7. Although this research found different empirical scaling laws in different cities, the underlying explanation is consistent. The scaling law is determined by distinct visited location growth. Since the linear preference attachment effect and logarithm distinct visited location growth are confirmed, the exponential law of travel frequency is a confident conclusion. We discovered the travel and built environment relationship from a new perspective: the relationship of trip frequency and POIs density. Traditional researches based on small sample data investigated length, modes and purposes of human travel. The trip frequency, as another important measurement of human travel, is not fully captured by small sample data. GPS big data contribute to measurement of trip frequency. Meanwhile, POIs have the potential to be new environmental variables. Environmental variables can be described by words beginning with ‘D’the ‘six Ds’ variables (Ewing & Cervero, 2010). POIs data belong to the ‘density’ and ‘diversity’ of the ‘six Ds’. POIs extend the residential and work density to the density of all types of human activity. POIs data are also sensitive to temporal activities (Gong et al., 2015; Huang et al., 2010). 7. Conclusion To understand the human mobility dynamic, this study examined the scaling laws of spatial visitation frequency, and proposed a transport prediction model. The study was conducted in the inner city of Guangzhou using one day's taxi GPS records. The city space was divided into spatial grids; taxi arrivals in each grid were counted as visitation frequency. A frequency-ranking model, depicting how spatial visitation frequency decays with rankings, was formulated to find the underlying scaling laws. The model was then used with 12 time periods to uncover dynamic scaling laws. Based on that a transport prediction model was proposed, which predicted spatial temporal arrivals from POIs density. The research revealed that the scaling law of visitation frequency is exponential, that is, that spatial arrivals decay exponentially with locational rankings. It is explained by the linear preferential attachment effect and logarithmically distinct visited locations growth. Travel frequency decay rates for the 12 time periods were found to be almost the same. The linear preferential attachment effect existed in the 12 time periods, and the distinct visited location growth in the 12 time periods followed the same laws. These two effects explain why the scaling

342


laws are not sensitive to time. Based on these findings a transport prediction model was proposed. Spatial visitation is estimated from its locational ranking, and the spatial ranking is estimated from spatial density of POIs. Assuming ordered-ranking is a function of environmental variables, which are represented by POIs density by functional types, and a linear programing estimates weighted parameters of these variables. As weighted parameters are temporally dynamic, the model is thus able to capture dynamic change. Finally, the proposed model predicts spatial temporal arrivals according to a spatial location's POIs density. Predicted trips matched observations well in almost all time periods. Our research contributed to transport prediction models. Linking space to travel is always a difficult task as spatial heterogeneity is unpredictable. The scaling laws of spatial visitation frequency provides a bridge between travel and the built environment. An ideal travel prediction model should make predictions as accurately as possible without much pre-knowledge. Our intention was to assign trips to urban space, while only knowing the total amount of taxi trips and environmental variables. Although spatial temporal arrivals are predicted from the proposed model, applying the prediction to practice is still difficult. One research limitation is in the distinct visited location growth. To predict the number of distinct visited locations is difficult without empirical evidence. Simply applying the estimated parameters to other cities may be statistically problematic. Uncovering a universal regularity in the distinct visited locations growth process of human movements could be an approach to solve the problem. A future direction is to compare parameters in distinct visited location functions using data from different cities. Another research limitation is the measurement of environmental variables. In this research, the density of POIs was used as environmental variables to estimate locational rankings. However, weighted parameters of POIs variables are not fully convincing. Great opportunities exist for finding proper environmental variables from assigning locations' visitation rankings. The ‘six Ds’ variables (density, diversity, design, destination accessibility, distance to transit and demand management) are all possible environmental variables included in the POIs data source. From this, a universal trip frequency prediction model could be hopefully built in the future. Acknowledgement This research was supported by Natural Science Foundation of China (41522104, 41271166), the Fundamental Research Funds for the Central Universities of China (15lgjc24) and China Scholarship Council (201306380083). References Aljoufie, M., Zuidgeest, M., Brussel, M., van Vliet, J., & van Maarseveen, M. (2013). A cellular automata-based land use and transport interaction model applied to Jeddah, Saudi Arabia. Landscape and Urban Planning, 112(1), 89–99http://doi.org/10.1016/j. landurbplan.2013.01.003. Barabási, A. -L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286, 509–512http://doi.org/10.1126/science.286.5439.509. Black, W. R. (2003). Transportation: A geographical analysis. New York: Guilford Press. Calabrese, F., Di Lorenzo, G., & Ratti, C. (2010). Human mobility prediction based on individual and collective geographical preferences. IEEE conference on intelligent transportation systems, proceedings, ITSC (pp. 312–317). 77 Massachusetts avenue, Cambridge, MA, United States: Senseable City Laboratory, Massachusetts Institute of Technology. Cervero, R., & Kockelman, K. (1997). Travel demand and the 3Ds: Density, diversity, and design. Transportation Research Part D: Transport and Environment, 2(3), 199–219http://doi.org/10.1016/S1361-9209(97)00009-6. Chao, L., & Qing, S. (2011). An empirical analysis of the influence of urban form on household travel and energy consumption. Computers, Environment and Urban Systems, 35(5), 347–357http://doi.org/10.1016/j.compenvurbsys.2011.05.006. Crane, R. (2012). The influence of urban form on travel: An interpretive review. Journal Of Planning Literature, 15(1), 2–23http://doi.org/10.1177/08854120022092890. Crane, R., & Crepeau, R. (1998). Does neighborhood design influence travel?: A behavioral analysis of travel diary and GIS data. Transportation Research Part D: Transport and Environment, 3(4), 225–238http://doi.org/10.1016/S1361-9209(98)00001-7. Di Lorenzo, G., Reades, J., Calabrese, F., & Ratti, C. (2012). Predicting personal mobility with individual and group travel histories. Environment and Planning B: Planning and Design, 39(5), 838–857http://doi.org/10.1068/b37147.

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