Mobility Pattern of Taxi Passengers at Intra-Urban Scale: Empirical Study of Three Cities

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1 Journal of Systems Science and Information Dec., 2017, Vol. 5, No. 6, pp DOI: /JSSI Mobility Pattern of Taxi Passengers at Intra-Urban Scale: Empirical Study of Three Cities Mengqiao XU Faculty of Management and Economics, Dalian University of Technology, Dalian , China stephanie1996@sina.com Ling ZHANG Faculty of Management and Economics, Dalian University of Technology, Dalian , China @qq.com Wen LI Faculty of Management and Economics, Dalian University of Technology, Dalian , China @qq.com Haoxiang XIA Faculty of Management and Economics, Dalian University of Technology, Dalian , China hxxia@dlut.edu.cn Abstract The study of human mobility patterns is of both theoretical and practical values in many aspects. For long-distance travel, a few research endeavors have shown that the displacements of human travels follow a power-law distribution. However, controversies remain regarding the issue of the scaling laws of human mobility in intra-urban areas. In this work, we focus on the mobility pattern of taxi passengers by examining five datasets of three metropolitans. Through statistical analysis, we find that the distribution with a power-law tail can best approximate both the displacement and the duration time of taxi trips in all the examined cities. The universality of the scaling laws of human mobility is subsequently discussed, in view of the analysis of the data. The consistency of the statistical properties of the selected datasets that cover different cities and study periods suggests that, the identified pattern of taxi-based intra-urban travels seems to be ubiquitous over cities and time periods. Keywords human mobility pattern; taxi travel; displacements; duration time 1 Introduction The study of human mobility patterns is of great importance in many aspects, such as control of epidemic spreading [1], tourism management [2], transport prediction and planning [3], emergency management [4] and city structure [5]. However, mainly due to the limitations in Received July 10, 2017, accepted September 25, 2017 Supported by the National Natural Science Foundation of China ( , , ) Recommended by the 18th International Symposium on Knowledge and Systems Sciences (KSS2017) which was held in Bangkok during November 17 19, 2017

2 538 XU M Q, ZHANG L, LI W, et al. collecting and analyzing the large-scale mobility data, the mobility or travel patterns of massive populations in different geographical scales had not been studied well until recently. In the last decade, with the increasing availability of different types of large-scale locationbased data of massive populations, as well as the rapid prominence of data-driven computational social science [6], our understandings of human mobility patterns have been greatly deepened. This issue has also attracted great attention of scientists in the areas of non-linear physics and complex systems science. One well-noted work was performed by Brockmann, et al. [7]. By tracing the circulation of bank notes in the United States of America, they identified that the travelling distances of banknote carriers followed a power-law distribution, indicating that the carriers travels were alike to Lvy flights with attenuation of dispersal. A two-parameter continuous-time random walk model was then developed to reproduce the observed travelling pattern in their work. González, et al. [8] studied the trajectories of 100,000 mobile-phone users, finding that the distribution of displacements over all users was well approximated by a truncated power-law. González, et al. s work was subsequently extended by Song, et al. [9], who developed an individual-mobility model based on the idea of preferential return to account for the scaling law observed in mobile-phone trajectories. Another interesting model was proposed by Han, et al. [10], and they found that the power-law of travel distances can be lained by the hierarchical structure of transport systems. The aforementioned series of studies are fascinating. However, datasets used in these studies are with limited accuracy to depict human mobility in smaller geographical scales. For the dataset of banknote circulation, one recorded travel of a banknote does not precisely reflect the movement of any single person, since the banknote may change hands several times during the period of this recorded banknote travel. Especially, it would be difficult to trace the short-distance travels of human beings through the banknote records, e.g., within the travel range of less than 10 km as shown in Brockmann, et al. s own paper. The inaccuracy of the mobile-phone dataset lies in its shortage of tracing short-distance travels, since the mobilephone positioning records depend on the locations of communication company s base stations and the travels within the range of one single base-station cannot be traced properly. Consequently, a question can be raised whether the human travels in shorter-distances, especially within the range of intra-urban areas, statistically follow similar mobility patterns as observed in the longer-distance (i.e., from around ten kilometers to thousands of kilometers) datasets. Comparing with the datasets of banknotes and mobile phones, the data of Global-Positioning- System (GPS) trajectories of vehicles can reflect the intra-urban travels more directly and with finer granularity. Therefore, the availability of the GPS trajectory data has stimulated various studies on the mobility patterns in intra-urban areas in recent years. Based on the GPS data generated by fifty taxicabs during a six-month period in Sweden, Jiang, et al. [11] found that the travel distances by taxi passengers followed a two-phase power-law distribution. The power-law was also supported by Yao and Lin [12], according to their analysis of taxi trajectories in a South China city. In comparison, various other studies on the datasets of private cars [13] and taxis [14 16] showed the onential distributions of travel distances. A distribution of travel distances was suggested by Veloso and Phithakkitnukoon [17], who investigated the taxi data in the period of five months in Lisbon, Portugal. Roth, et al. [18]

3 Mobility Pattern of Taxi Passengers at Intra-Urban Scale: Empirical Study of Three Cities 539 analyzed the dataset of individuals subway travels in London during one week, finding that the travel distances approximated a negatively binomial distribution. The previous researches indicate that the intra-urban travels do not simply follow the same power-law of longer-distance travels. But the distributions of intra-urban travels fitted in different researches are contradictory. Such situation demands more thorough investigations based on intra-urban mobility data. A great proportion of previous studies are based on taxi trajectories. As taxi plays an important role in city transportation, the taxi trajectory data have been widely used to study different transport problems, e.g. mining of the attractive areas [19], evaluation of road network [20], route discovery [21], mining of the cabdrivers drive patterns [22 24], modeling of passenger-searching strategies [25], and transportation planning [26]. These researches also stimulate us to investigate the intra-urban mobility pattern with the taxi trajectory data. Thus, in this paper we examine the travel distributions of taxi passengers by analyzing five datasets of the three metropolitans, New York City and two cities in China. The inconsistent results obtained in previous researches imply that the human mobility patterns may be inherently intricate. Therefore, in this work we limit our research scope in order to improve the accuracy of results. On one hand, we restrict the travel mode to the single means of taxi-taking. On the other hand, the travel distances are limited to the range of a single city, more specifically within the metropolitan areas as the examined cities are all with more than one million in population, although these cities are significantly different in their geographical characteristics and traffic situations. The remainder of this paper is organized as follows. In Section 2, the datasets to be used are described and the method of data cleaning is introduced. We describe the analysis method in Section 3. In Section 4, two statistical metrics of taxi passengers are respectively analyzed, namely the distribution of travel displacements, and the distribution of duration time of travels. Finally, the results are discussed and the whole paper is concluded in Section 5. 2 Data Description and Cleaning In order to lore human mobility patterns in the intra-urban areas more precisely, we use five datasets of GPS trajectories of taxis in three cities, namely New York City and two cities in China. In this work, we anonymize the two Chinese cities as requested by the data owners. The dataset D 1 was collected from 7,804 taxis within the urban areas in a North China City A, from Jan. 1st to May. 28th, This dataset is from the Department of Road Transport of City A, covering the trajectory data of most taxicabs of that city, i.e., 7804 out of about 8900 taxicabs. The datasets D 2, D 3 and D 4 are all from the open datasets by New York City (NYC) Taxi and Limousine Commission, which can be publicly accessed at the website 1 The dataset D 2 covers the trajectories of more than 13,000 yellow taxicabs in November and December of Both D 3 and D 4 are from NYC green taxicabs in whole year of 2014 (D 3 ) and 2015 (D 4 ), respectively. The dataset (D 5 ) records the GPS trajectories of 7700 taxis in the city of a South China City B. We use these five datasets from three cities in order to examine whether the taxi-based 1 record datas.html

4 540 XU M Q, ZHANG L, LI W, et al. mobility pattern is stable in terms of geographical position, service areas and time scale. First, we aim to examine whether the intra-urban mobility pattern is common in different cities. For this reason, we try to select datasets with geographical diversity. We choose one city in US and two cities in China. For Chinese cities, we furthermore choose one in North China and another in South. Second, we hope to examine whether the travel modes are similar in different service areas of a same city and under different distance-scales. Therefore, we use both Yellow Taxis and Green Taxis of New York City. In NYC, Yellow taxis can pick up passengers everywhere in the city, and according to an analysis of trips using GPS by the Taxi and Limousine Commission, 95% of yellow taxi pick-ups occurred in Manhattan below 96th Street and at JFK and LaGuardia airports. This resulted in significantly lower access to legal taxi rides for people in outer boroughs. Green Taxis were then introduced to pick up street-hail passengers to fill in the gap. Green Taxis cannot provide hail service to passengers below West 110th Street and East 96th Street, or at the two NYC airports. We use both Yellow and Green Taxis to examine the taxi-based trips follow the similar statistical distributions although they run in different regions of the same metropolitan area, and the distances of service coverage are different. Third, we concern whether time matters for the mobility pattern. Thus, we choose datasets with different time scales. For the dataset of North China City A, we use the trajectory of 5 months; for the dataset of South China City B, the trajectory data cover 15 days; in the NYC yellow taxis dataset, the data cover 60days; in two NYC green taxi datasets, each covers the data of one whole year (2014 and 2015 respectively). Furthermore, we divided the NYC green taxis data into two datasets, so as to examine whether the same mobility pattern occur in different years. In all the five datasets, the following information is contained, i.e., vehicle ID (anonymized in D 2, D 3 & D 4 ), pick-up time, drop-off time, travel distance, pick-up longitude, pick-up latitude, drop-off longitude and drop-off latitude. Two main statistical metrics are extracted from the records, i.e., the passenger s trip displacement, the passenger s trip duration time. The basic information of all the five datasets is summarized in Table 1. Table 1 Basic information of the five datasets Dataset D 1 D 2 D 3 D 4 D 5 Dataset Name North China City A NYC yellow NYC green NYC green South China city B Effective Days # of Taxicabs 7804 >13000 >6000 > # of Trips per day 193k 485k 43k 60k 140k Avg.Displacement (km) Avg.Duration (min) To note: As the vehicleids are protected in NYC datasets, we do not have the exact number of taxicabs. Before analyzing these datasets, data cleaning is processed to eliminate the noise data. First, records with unusually short displacements (i.e., less than 0.5 km) are removed, as walk

5 Mobility Pattern of Taxi Passengers at Intra-Urban Scale: Empirical Study of Three Cities 541 is the more common choice for a less-than-0.5 km trip. Hence, we consider those recorded less-than-0.5 km trips as error data generated by incorrect GPS traces. Second, we discard the trips that are farther than 100 km (50 km for NYC datasets), as the accounts of such long-distance travels are very few in the datasets. Finally, the trips with less-than-one-minute and more-than-3-hours in duration are also ignored, since such trips are unusual for an average taxi passenger. 3 Method 3.1 Cullen and Frey Graph In this work, we use descriptive statistics to select candidate theoretic distributions or models to fit the trajectory data. In descriptive statistical analysis, kurtosis and skewness are two widely used coefficients. Kurtosis is a measure to depict the tailedness of a distribution in comparison with the normal distribution for which the kurtosis is equal to 3. Distributions with kurtosis greater than 3 are called leptokurtic, while distributions with kurtosis less than 3 are called platykurtic. Skewness measures the level of asymmetry of a distribution on its mean. If the skewness of a distribution is positive, the part of its probability density function that is on the right of the mean is fatter than the left part. On the contrary, the negative skewness of a distribution reveals that the left part of the density function is fatter than the right part. Cullen and Frey graph kurtosis Observation of Displacements on D 1 Theoretical distributions normal uniform onential logistic beta (Weibull is close to and ) square of skewness Figure 1 Cullen and Frey graph

6 542 XU M Q, ZHANG L, LI W, et al. Combining the two coefficients, Cullen and Frey [29] used the skewness-kurtosis plot (i.e., Cullen and Frey Graph) to depict the skewness and kurtosis of the commonly used distributions, as shown in Figure 1. For normal, uniform, onential and logistic distributions, the values of skewness and kurtosis are unique. Thus, each of those distributions is represented as a single point on the plot. Gamma and distributions are represented as lines, while the Beta distribution is represented by a shaded area in the plot. Table 2 The skewness and kurtosis of the two statistics in the five datasets Statistic Dataset Skewness Kurtosis D Displacement D D D D D Duration D D D D In addition, we choose the kurtosis and skewness of displacements in dataset D 1 as the representative to illustrate how these values help us to choose candidate distributions. As shown in Figure 1, the solid circle observation is the kurtosis and the square of skewness of displacements on dataset D 1, indicating that the displacements of D 1 may be fitted well by distribution. Moreover, we calculate the values of kurtosis and skewness of the records of passenger displacement, passenger duration time in the five datasets, as shown in Table 2. For all the datasets, all the values of kurtosis are greater than 3, while all the values of skewness are positive, indicating that these data series would probably be well-fit by some right-skewed and leptokurtic distribution. This indicates the normal, uniform and onential distributions may not fit the data well. Therefore, we select three major right-skewed and leptokurtic distributions as candidates to fit the data, namely, Weibull, and distributions. Besides, as some previous researches have illustrated the good-fit of the onential distribution and the power-law distribution, these two type of distributions are also taken into account in the following analysis. 3.2 Selection of Candidate Distributions As mentioned above,, Weibull, Gamma, onential, and power-law distributions are selected as candidates to fit our datasets. Their probability density functions are shown below. The probability density function of distribution is defined by Equation (1). P(x) = 1 ( xσ 2π (ln x ) µ)2 2σ 2, (1)

7 Mobility Pattern of Taxi Passengers at Intra-Urban Scale: Empirical Study of Three Cities 543 where µ is the mean and σ is the standard deviation. The Weibull distribution is defined by the probability density function of Equation (2). P(x) = α β ( x ) α 1 ( ( x ) α ), (2) β β where α > 0 is the shape parameter, and β > 0 is the scale parameter. The probability density function of Gamma distribution is defined by Equation (3). P(x) = βα Γ(α) xα 1 ( βx ), (3) where α and β, respectively, are shape and rate parameters. The probability density function of onential distribution is defined by Equation (4). P(x) = ( λx ), (4) where λ is the rate parameter. The probability density function shown in Equation (5) is used to define the power-law distribution. P(x) = α 1 x min ( x x min ) α, (5) where α 1 x min is the normalizing constant and α is the power parameter. To find out which candidate distributions fit the data best, two commonly-used fitting criteria are Akaike Information Criterion (AIC) [30] and Bayesian Information Criterion (BIC) [31]. In this paper, Akaike information criterion or AIC is used as the criterion for measuring the goodness-of-fit between the theoretical distribution (i.e., the model) and empirical data. The overall procedure of model selection is comprised of three steps: a. Calculating the model parameters. In the paper, the parameters of candidate models are computed by Maximum Likelihood Estimation (MLE) [32]. b. Calculating the AIC score of each model. For the candidate model i ( i {1, 2, 3, 4, 5} ), the corresponding AIC score is computed by AIC i = 2 logl i +2K i, where K i represents the number of parameters of model i, and L i is the likelihood function. c. Selecting the best-fit model. AIC weights (w i ) [33] denotes the relative likelihoods of the model, and we use it as the criterion to select the best-fitting model. Let AIC min = min AIC i and i = AIC i AIC min, and the AIC weights can be calculated by W i = ( i/2) m j=1 ( j/2). The candidate model with largest AIC weight is the one fitting the actual data best. 4 Results and Analyses 4.1 Displacement Distribution By utilizing the previous datasets, we first examine the distribution of displacements of travels. Figure 2 illustrates the box plots of the displacements extracted from the five datasets, where the black dots represent the mean values of displacements, and black triangle points

8 544 XU M Q, ZHANG L, LI W, et al. denote the most frequent displacements. In Figure 2, we can firstly find that in all plots the mean values are greater than the medians, indicating that all data series are comprised of abundant of short-distance travels and relatively fewer long-distance travels. Among the five displacement data series, the average travel distance of taxi trips in NYC is approximately 3.0 km while that in North China City A is 6.3 km and that in South China City B is 4.2 km. The median displacement value of trips in NYC is about 2.0 km, while that in North China City A is 5.0 km and that in South China City B is 3.2 km. Moreover, the most frequent displacement in NYC is 1.0 km, which is also shorter than that in North China City A (2.9 km) and in South China City B (1.7 km). The most frequent displacements in all three cities are shorter than the corresponding mean and median values of displacements. What s more, the shorter mean, median and most-frequent displacements in NYC than those in the two Chinese cities may possibly reflect the faster pace of life in NYC. Displacement,r (km) North China City A NYC yellow NYC green 2014 NYC green 2015 South China City B Datasets Figure 2 The box plots of displacements in the five datasets The previous plots reveal that the distributions of displacement are skewed in all the five datasets. Subsequently we examine the fitness of the five candidate theoretic distributions to the actual data, as shown in Figure 3. Figure 3 illustrates the fittings of four datasets, namely, D 1 (North China City A), D 2 (New York Yellow), D 4 (New York Green 2015) and D 5 (South China City B). As it is very similar with that of D 4, the fitting result of dataset D 3 is not illustrated in Figure 3 to ease the layout of the plots. In Figure 3, the actual data points are shown as black empty circles, while the fitting data generated by the, Weibull, and onential distributions are respectively presented as red solid, blue dashed, green dotted and purple dot-dash curves. By comparing the actual data points with the theoretical curves, we can intuitively estimate that the and onential distribution fit the actual data better than the other two distributions. As for the power-law distribution, our observation is that

9 Mobility Pattern of Taxi Passengers at Intra-Urban Scale: Empirical Study of Three Cities 545 this type of distribution does not generate good fit to the full-scoped data in general. However, the actual displacements that are greater than 18 km can be fitted well by the power-law, as illustrated by the orange two-dash line in Figure 3. For such long-distance displacements, the power-law fit is apparently better than the other four types of distributions, which is consistent with the result shown in Brockmann, et al [7] (a) North China City A (b) NYC yellow (c) NYC green (d) South China City B Figure 3 Fittings of the distributions of displacement at log-log scale Furthermore, we calculate the AIC weights of the five candidate models. The AIC weights of model are 1 in all datasets, while the AIC weights of the other four models are zero. This indicates that all the datasets are fitted best by the. The previous analysis indicates that the displacements in all five datasets can be best fitted by the distributions with a power-law tail for the long-distance travels. It is worth noting that the taxi trips that are within 18 km respectively cover 97.4%, 98.5%, 99.95%, 99.98%, 98.5% of the recorded trips in the five datasets. To more specifically show the fitting results of, Weibull, and onential distributions to the displacements within the range of 18 km, we omit the data records of long-distance travels. As shown in Figure 4, the advantage of the distribution over the other three candidate distributions becomes more apparent. In a word, the distribution of short-distance trips (less than 18 kilometers) can be best fitted by the while long-distance trips follow a power-law distribution.

10 546 XU M Q, ZHANG L, LI W, et al (a) North China City A (b) NYC yellow (c) NYC green (d) South China City B Figure 4 Fittings of the distributions of displacement at log-log scale Table 3 Parameters for the distributions to fit the displacement data Datasets MLE for parameters (with 95% CI bounds) Mean (µ in Equation (1)) Standard Deviation (σ in Equation (1)) D (1.5727,1.5733) (0.7578,0.7582) D (0.7154,0.7162) (0.8498,0.8503) D (0.7835,0.7843) (0.8069,0.8075) D (0.7453,0.7451) (0.8054,0.8060) In the previous fittings of the actual data records, the parameters of models are computed by the Maximum Likelihood Estimation (MLE) method. Table 3 lists the adopted parameters of the distributions to fit the displacements in the five datasets, and the 95% confidence bounds of parameters are listed in the brackets. In order to further illustrate the goodness of fit, the Probability-Probability (P-P) plot [34] of the, Weibull, and onential models are drawn in Figure 5, where theoretical probabilities are on the horizontal axis whilst empirical probabilities being on the vertical. When the theoretical probabilities and empirical probabilities are mostly the same, the fitting curve is closer to the diagonal. The P-P

11 Mobility Pattern of Taxi Passengers at Intra-Urban Scale: Empirical Study of Three Cities 547 plots in Figure 5 reveal that the displacements in the all examined datasets are best fitted by the in the four theoretical models. To this end, the result in this work differs from that in a few earlier contributions where the onential was claimed as the best approximation [15, 16]. P-P plot P-P plot Empirical probabilities Empirical probabilities Theoretical probabilities Theoretical probabilities (a) North China City A (b) NYC yellow P-P plot P-P plot Empirical probabilities Empirical probabilities Theoretical probabilities Theoretical probabilities (c) NYC green 2015 (d) South China City B Figure 5 P-P plots of the four candidate models Furthermore, the tail parts in the plots of Figure 3, i.e., the trip displacements that are greater than 18km, are specifically examined. As shown in Figure 6, the greater-than-18km displacements can best fitted by the power-law in all the datasets, while the fittings of other theoretical distributions are unsatisfactory. The power parameters of the fitting power-law distributions (i.e., α in Equation (5)) are 4.14, 11.36, 8.96, 7.74, and 5.61 respectively for the five datasets. This reveals that the greater-than-18km displacement are schematically similar in different cities (i.e., the power-law), but with different parameters from city to city. In New York City, we observe steep decay of the amount of long-distance taxi trips, while the decay is much gentler in the two Chinese cities, indicating that the tail of long-distance trip may relate with the city s structure, traffic conditions and even the effects of social and economic factors. In some cities, the tails seem to be lighter than the fitted to the entire dataset, while in some others, the tails are heavier. To sum up, the previous investigation reveals that the trips of taxi passengers are comprised of abundant of short-distance trips and relatively fewer

12 548 XU M Q, ZHANG L, LI W, et al. long-distance one. In all the examined datasets, we find the displacements of taxi passenger trips can be best fitted by the distribution with a power-law tail (a) North China City A (b) NYC yellow (c) NYC green (d) South China City B Figure 6 Fittings of the distributions of displacement beyond 18 km at log-log scale 4.2 Duration Distribution Second, we examine the distributions of passenger duration time in the five datasets. The duration time represents the time consumed from the origin to the destination in a passenger s trip. Figure 7 illustrates the box plots of duration time, in which the black dots represent the mean values and the black triangles denote the peak values. In Figure 7 we can find that the durations of the five datasets are similar in average, as in all the datasets the mean values are about 15.0 minutes and the median values are about 12.0 minutes. Comparing with South China City B and North China City A, this partly reflects the severer traffic congestion in NYC, since the average displacements are shorter in NYC datasets under similar average duration time. Figure 8 plots the fittings of the five theoretical distributions (i.e.,, Weibull,, onential and power-law) to the actual duration data series at the log-log scale. Again, we skip illustrating the fitting result of the Dataset of NYC Green 2014 to ease the layout of Figure 8. Similar with the case of displacement distribution, the model fits the duration data best in all the five datasets. The goodness-of-fit of the are also validated by the AIC weights, which is as same as the result of displacements. The AIC weights of model are 1 in all datasets, while the weights of the other four models are zero. As same as in the fitting of

13 Mobility Pattern of Taxi Passengers at Intra-Urban Scale: Empirical Study of Three Cities 549 displacement data, the parameters of the theoretical distributions are computed by MLE. The parameters of the adopted distribution to fit the duration data with 95% CI are listed in Table 4. Different from the case of displacement distribution, the fitting parameters are quite similar in different cities. This is consistent with the results of Figure 7. Runtime,t (min) Dalian NYC yellow NYC green 2014 NYC green 2015 Nanjing Datasets Figure 7 The box plots of duration time in the five datasets Prob(t) Prob(t) Duration Time,t (min) (a) North China City A Duration Time,t (min) (b) NYC yellow Prob(t) Prob(t) Duration Time,t (min) (c) NYC green Duration Time,t (min) (d) South China City B Figure 8 Fittings of the distributions of duration time at log-log scale

14 550 XU M Q, ZHANG L, LI W, et al. Table 4 Parameters for the distributions to fit duration data with 95% CI Datasets MLE for parameters (with 95% CI bounds) Mean Standard Deviation D (2.4765,2.4770) (06912,0.6915) D (2.4266,2.4273) (0.7197,0.7191) D (2.3470,2.3476) (0.7022,0.7026) D (2.5812,2.5830) (0.7792,0.7804) In a word, the previous analysis shows that comparing with Weibull,, onential and power-law models, the records of durations of trip in all five datasets can be fitted best by the model. More accurately, for the trips that elapse more than 100 minutes, the power-law fits best. Thus, the overall duration records in each dataset are fitted best by the with a power-law tail. 5 Comparison with Related Work By examining five datasets in three cities, in this paper we find that the displacements and the durations of taxi passengers, can be approximated best by the distribution with a power-law tail. As distribution is essentially a continuous distribution of a random variable whose logarithm is normally distributed, a possible lanation for the generation of the distribution of taxi-travels is as follows. If a passenger s travel behavior is regarded as a result of two sequential decision-making processes, both of which are actually random, first, one has to select the direction to move, and then one travels a certain distance to reach the destination. Assuming that passengers travel behaviors are independent of each other and both of the two decision-making processes addressed above follow normal distributions, we should get a distribution of travel distance. Because in the present case the distribution turns a variant of random variables, e.g. travel decisions and travel distances of the population, into a sum. The present results can be further discussed by comparing with the related work. Our investigation firstly reveals that the power-law distribution, which has often been observed in long-distance travels, does not provide a good fit to the taxi-based travels within the range of intra-urban areas. It is interesting to note that in the well-noted contributions such as Brockmann, et al. [7] and González, et al. [8], the power-law fits the actual data well for the travels that are across the distances from 10 1 km to around 10 3 km, while the fit is less satisfactory for the travels that are shorter than 10 km. By contrast, the main finding of this work is that the perfectly fits both the taxi passengers displacements and durations for the travels that are shorter than 18 km in distance and 60 minutes in time, while the fitting becomes unsatisfactory for the taxi-based travels that are farther than 18 kilometers and longer than 100 minutes. Putting the results of these endeavors together, our conjecture is that human mobility may not follow a simple universal scaling-rule. The should be a good candidate to fit the trips that are in the intra-urban areas, while long-distance trips

15 Mobility Pattern of Taxi Passengers at Intra-Urban Scale: Empirical Study of Three Cities 551 probably follow the power-law. For intra-urban travels that are usually shorter than 10 kilometers, some research also stand on the power-law fitting. For example, Yao, et al. [12] suggested the power-law fitting of travel distances by analyzing the one-day taxi GPS trajectories of one company in one South China city. According to the prior analysis, we argue that the power-law does not fit the intra-urban taxi-based trips well. Especially, the power-law model does not account for the increasing taxitaking probability with the travel distance in the range of short-distance travels. Furthermore, in Yao, et al. s [12] work, the power-law distribution of displacement was lained by a Maxwell- Boltzmann model, in which the human movements were analogous to the random movements of gas molecules. This analogy is questionable as human travels are usually purposeful and a great fraction of intra-urban movements are the routine travels, e.g., between residential quarters and business and commercial centers (i.e., the working places and shopping areas). The aggregation of living, working, and shopping and recreational areas in a city often causes a great proportion of travels are converged to some specific distances. This phenomenon is not lained well by a power-law distribution. Intuitively, the fitting is more reasonable than the power-law fitting for the taxi-based trips, since taxi is not the most preferable option for average travelers in their extremely short and long travels. Thus, it would be reasonable to estimate that a most-frequent distance (and duration time) can be observed, while such characteristic value is missing in a power-law distribution. The research method used in this paper is also similar with those in Wang, et al. s [16] and Liang, et al. s [14] contributions. The distributions of the durations of occupied trips identified in this work agree with the corresponding results in Wang, et al. [16]. However, for the distributions of displacements, our result is different from theirs, as well as Liang, et al. s [14], as both the previous studies have claimed the onential law of displacement distributions. We believe the provides better fitting than the onential according to three evidences. The first evidence is from the analysis based on Cullen and Frey Graph as shown in Figure 1. We find that the data series under investigation would probably be well-fitted by the rightskewed and leptokurtic distributions. The kurtosis and skewness features of the investigated datasets can almost exclude the onential fitting. Secondly, the onential fitting does not pass the Akaike tests in all the datasets, including the datasets used in Wang, et al. s paper. This is mainly due to the unsatisfactory fitting of the onential to the short-distance trips. As shown in Figures 3 and 4 of this paper, the fit is apparently better than the onential fit for short-distance trips, although the longer-distance trips can be well-fitted by both theoretical distributions. In taxi-based urban transportation, the short-distance trips are non-negligible. In our datasets, the trips that are shorter than 3km respectively occupy 26%, 71.5%, 65%, 67% and 47% of the all trips in the five datasets. It is worth noting that the trips that are shorter than one kilometer is omitted in Wang, et al. s work, as well as in Liang, et al. s [14]. This may partly lain their flaws. Thirdly, the better fitting of the is more clearly shown in the probability-probability (PP) plots in Figure 5, where the fitting is best and the onential fitting performs poorest comparing with other theoretical models. Such PP-plot-based analysis is another key difference between our paper and Wang, et al. s [16]. Especially in the first quarter (i.e., the

16 552 XU M Q, ZHANG L, LI W, et al. left-lower parts of the plots) the theoretical probabilities of the onential approximations do not fit the empirical data well. Besides the previously-noted power-law and onential, various other researches have also suggested other theoretical distributions to fit the displacements and time durations for intraurban human travels. Csáji, et al. s [27] declared a distribution of travel distances of the mobile-phone users in Portugal. This work provides some support on their result with denser and more accurate spatiotemporal data. Veloso and Phithakkitnukoo [17] suggested the distribution by examining the taxi data in Lisbon. They claimed that the onential can fit the trips that is longer than around 3 km. As distribution is morphologically similar with the distribution claimed in this work. The result in this work is actually quite similar with theirs. Roth, et al. [18] claimed a negatively binomial distribution according to the investigation of London subway data. The displacement distributions shown in this work are schematically different from that in Roth, et al. s [18] work, since the travel distances in their London subway dataset obviously follow a left-skewed distribution, as shown in Figure 2 of their paper. Such difference indicates that the means of transport has remarkable influence on the travel pattern. People are more likely to ride subway to travel a long distance; when travel distance is shorter, the probability of taking-a-taxi becomes higher. 6 Conclusion In this work, our focus is to examine the pattern of taxi-based intra-urban travels. Under such restriction, we find the distribution with a power-law tail can best fit the displacements and durations of occupied trips. This result may reflect the general mobility pattern of the taxi passengers in intra-urban areas. First, the probability for taking taxi to travel a very-short distance and duration is low. This may be lained by the fact that people are unlikely to take a taxi when the destination is easy to go on foot. Second, the likeliness for an average passenger to take taxi would steeply increase after the travel distance exceeds the limit of walkable distance. Then the highest probability for passengers to take taxis is reached at a range of distances that are relatively short (e.g., around 1.0 km in New York City, 1.7 km in South China City B, and 2.9 km in North China City A as shown in Figure 2). Third, the probability gently decays for the taxi trips with longer distance and duration time. When the travel distance is extremely long (e.g., across the boundary of the city), the probability of taxi-taking becomes very low again, which is the same as in the case of extremely short travel distance. The previous phenomena of steep increase and relatively gentle decrease of the probability of taxi trips may possibly ascribe to the combinatorial effect of two factors, the demand of intra-urban human travel itself and the passengers preference to select taxi as the transport mode. For the general traffic demand, a great proportion of traffics are routinized between passengers often-visited places (e.g., residential places, the working places, and shopping centers). Furthermore, a great proportion of such routinized trips are across relatively short distances. For example, one shopper tends to select a shopping center that is close to her living place. On the other hand, leveraging the time efficiency and economic cost, the motive for taking-taxi to cross a relatively short distance is usually higher than to cross a long distance. However, the

17 Mobility Pattern of Taxi Passengers at Intra-Urban Scale: Empirical Study of Three Cities 553 decay of taxi-taking demand is gentle in the reasonable distances. For the travels that cross very long distances, people are more likely to select the substitutional means of transport such as train, subway, and bus; and the amount of taxi trips greatly drops. According to the analysis of this work, our conclusion is twofold. On one hand, we contend that the taxi-based intra-urban travels can usually be well fitted by the with a power-law tail. Especially for the taxi-based travels that are shorter than 18 kilometers and 60 minutes, the fit is better than the other widely-discussed fits such as the power-law and the onential. Furthermore, it is worth noting that we purposefully select datasets that are from different geographical places and with different time scales. The power-law-truncated model fits all the datasets well, indicating that the identified pattern of taxi-based intra-urban travel does not depend on time and region. On the other hand, the stability of the taxi-based intra-urban travel pattern does not mean that our results can easily be generalized to reach a universal scaling law of human mobility. Instead, the apparent inconsistences between our work and some other earlier contributions implies that the mobility pattern may enormously depend on multiple factors, especially the spatiotemporal scales and the means of transport. From the aspect of spatiotemporal scales, it seems that mobility patterns of travels that cross long distances and take long time are different from those of shorter travels, in accordance with the comparison between our work and Brockman, et al. s [7] and González, et al. s [8]. There is probably no simple statistical rule to lain the human mobility pattern in all spatiotemporal scales. The different results obtained in our work and Roth, et al. s [18] demonstrate the influence of the means of transport on the corresponding pattern of human travels. Within the intraurban areas, the subway-based travels tend to follow a left-skewed distribution, while taxi-based travels tend to be characterized by the right-skewness. This inconsistency gives rise to a stillunanswered question, whether there is a general pattern of mobility if taking all means of transport into consideration. If so, what is it? As regards the future work, we are going to extend the current research in three directions. First, in the current research, we only take the displacements and durations of taxi-based trips into account; and more accurate geographical and time information of the travels is neglected. More thorough spatiotemporal mining of taxi travel records will be carried out, in order to obtain results that are more accurate. For example, an implication of this work is that the distinction of functional quarters in cities may be a vital cause for the aggregation of taxi-taking trips around some particular distances. We are to examine this by giving further analysis upon the basis of origin-destination data mining [28]. Another direction of future research is to cover different means of transport. In the current work, we only consider the travels by taxi due to lacking the data of other means of transport. Next, we are to seek the available data of other means of public transport, especially the travel data of smart transport cards in different cities, and to study the mobility patterns of the passengers who use multiple means of public transport. Finally, based on the current study, we will try to develop dynamic models to lain the identified the pattern of taxi-based intra-urban travels.

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