Daily Human Mobility: A Reproduction Model and Insights from the Energy Concept

Abstract

Human movements have raised broad attention, and many models have been developed to reproduce them. However, most studies focus on reproducing the statistical properties of human mobility, such as the travel distance and the visiting frequency. In this paper, a two-step Markov Chain model is proposed to generate daily human movements, and spatial and spatiotemporal attributes of reproduced mobility are examined. In the first step, people’s statuses in the next time slot are conditioned on their previous travel patterns; and in the second step, individual location in such a slot is probabilistically determined based on his/her status. Our model successfully reproduces the spatial and spatiotemporal characteristics of human daily movements, and the result indicates that people’s future statuses can be inferred based on travel patterns they made, regardless of exactly where they have traveled, and when trips happen. We also revisit the energy concept, and show that the energy expenditure is stable over years. This idea is further used to predict the proportion of long-distance trips for each year, which gives insights into the probabilities of statuses in the next time slot. Finally, we interpret the constant energy expenditure as the constant ‘cost’ over years.

1. Introduction, Background, and Our Dataset

Understanding the nature of human mobility is important in many fields, such as urban planning, transportation management, geography, and epidemiology. It has raised broad attention in the last decade. Several statistical properties of human mobility, such as the displacement, mean square displacement, the radius of gyration, frequency of visiting different locations, etc., are carefully examined, and the distributions of some of them can be well-approximated [1,2,3,4,5,6,7]. At the same time, models have been developed for the description and reproduction of human mobility. Pioneer models are mostly centered on the population level, including the famous gravity model [8], the intervening opportunities model [9], and, more recently, the radiation model [10], the universal opportunity model [11], and the visitation law [12]. Models at the individual level are also under rapid development, which covers great variations from the classical continuous-time random-walk model to the currently popular preferential-return model (EPR) and many of its extensions [2,13,14,15,16].

However, most studies concerning reproduction models focus on reproducing statistical properties of human mobility, for example, displacement and the radius of gyration. Few of them compare the generated data with the real dataset about when and where trips happen, and about where exactly people have been to in what order. In this research, one-day human movement is studied. We propose a model of reproducing mobility for people with various individual attributes. The spatial characteristics (i.e., where people have been, and in what order locations were visited) and spatiotemporal characteristics (i.e., when trips happened for each type of traveling) of reproduced movements are examined, and these characteristics are well-reproduced by our method. To predict people’s future travel behaviors from current knowledge, we apply the energy concept, and attempt to clarify the energy law behind human mobility. Finally, a simple model is proposed to predict the proportion of long-distance trips given the average percentage of time spent on each travel mode.

This paper is composed of five sections (Figure 1). In the following parts of this section, we present an overview of the state of the art on human mobility, including several statistical rules and popular reproduction models (1.1). Our datasets and the preprocessing are also introduced (1.2). In Section 2, we propose our method of reproducing human movements, and evaluate its performance. The model is introduced in Section 2.1. There are two important elements in the framework of reproduction: the time-varying transition matrix, and the next-status probability. The first element is widely employed in transportation studies, and is briefly introduced in Section 2.1.1. For the second element, two terms, types of mobility sequences and types of current mobility sequences, are introduced in Section 2.1.2. The next-status probability is defined using the two terms in Section 2.1.3. In Section 2.1.4, we present the reproduction framework using the two elements. In Section 2.2, we examine spatial (Section 2.2.1) and spatiotemporal (Section 2.2.2) characteristics of reproduced movements. In Section 3, to predict future human mobility using the proposed model, we revisit the energy concept, and study how it may inform us about the next-status probability over time. Section 3.1 introduces the energy law and some discussions on it. Section 3.2 examines the law using our datasets. In Section 3.3, the proportion of long-distance trips is estimated based on the constant energy distribution over years, and constant energy expenditure is further interpreted as constant ‘cost’ in Section 3.4. Section 4 provides a discussion, and Section 5 draws conclusions of the paper.

1.1. Literature Review

Human motion is a popular topic. The statistical properties are well-examined in previous studies, and several popular models to reproduce and/or predict human mobility have been developed. In this sub-section, a brief review is given about statistical characteristics and such models.

1.1.1. Statistical Rules

A large body of research focuses on the statistical rules about spatiotemporal characteristics of human mobility. It was shown that the distribution of displacement can be fitted by a truncated power-law model, as well as the radius of gyration [1,2,3,4]. The frequency of visiting different locations follows a power-law distribution at the collective level, and a truncated power-law scaling at the individual level. Such non-uniformity suggests the non-Markovian character of human mobility [17]. Using the concept of motif, researchers noticed that only a limited number of major networks are present in daily mobility [5], and algorithms were developed to extract temporal motifs from subsequences of daily trips [6]. Besides, it was reported that seasonality explains about 17% of the intrapersonal monthly variations in the size of activity space [7]. Many researchers focus on other physical properties of human movements. It was reported that the apparent speed of trips increases with travel distance following a power-law function [18]. Marchetti developed the idea of travel-time budget, but later rereports suggest that the stability of travel time varies by area [19]. Kölbl and Helbing applied the energy concept to daily trips, and noticed that the distribution of human energy consumption rate follows a canonical-like distribution [20]. However, this assertion is doubted by later research [21]. The energy concept will be discussed in detail in Section 3. In addition, extensive discussions have been made regarding the impact of key factors in urban planning, such as land use [22] and residential neighborhood characteristics [23,24,25], on travel behaviors. Several propositions commonly made by urban planners have not yet been fully examined, and our ability to predict the impacts of corresponding policies on travel behaviors remains limited [26].

1.1.2. Reproduction Models

Grounded on the above findings, many models have been developed to reproduce or predict human mobility in spatial and/or temporal dimensions. Song et al., explored the predictability in human dynamics, and found a 93% potential predictability in user mobility [27]. Schneider et al., proposed a simple ‘coin flipping’ model to generate the overall behaviors of people [5]. Song et al., introduced two generic mechanisms, exploration and preferential return (EPR), to build a generative model [2]. The model successfully reproduces several of the statistical characteristics of human mobility, such as the number of visited locations, mean square displacement, etc. Based on their research, many studies have been conducted that improve the model in various directions, adding other factors such as the collective relevance of new locations (d-EPR) [13], the recency of location visits (recency-EPR) [14], limited-memory of exploration and preferential return (memory-EPR) [15], social contacts (STSEPR) [16], etc. Jiang et al., proposed a time-varying Markov Chain model, namely TimeGeo, which captures not only the circadian rhythm of daily trips with a global variable, but the likelihood of conducting activity with three individual-specific parameters [28]. Pappalardo and Simini proposed a method of generating trajectories in a framework named DITRAS in two steps: the generation of mobility diary, and the generation of trajectory [29]. Toole et al., proposed GeoSim to reproduce human mobility, which also takes into consideration social contacts [30]. In addition, Wu et al., developed a model where travel demands are generated first, and human movements are driven by such demands [31]. Besides, Ying et al., proposed a model to predict the future location of users based on users’ previous location features, and the method showed good performance [32]. In recent years, machine learning and neuron networks were employed to predict users’ movements, taking into account user-specific and contextual variables [33,34,35], and many of them achieved excellent performance compared with traditional models.

1.2. Dataset, Mobility Sequences

In this sub-section, the datasets that we use are introduced, and sequences indicating the spatial and temporal properties of individual mobilities are built.

1.2.1. Person Trip Survey Data

Person Trip survey has been conducted every ten years by the Ministry of Land, Infrastructure, and Tourism of Japan in major urban areas. It is conducted to households, and collects information about their travels on a given day. ‘Trips’ defined in Person Trip survey data (hereafter PT data) are illustrated in Figure 2, and details are presented in

Table 1. Details about Person Trip survey data.

Item	Content
Regions subject to survey	Tokyo, Kanagawa, Saitama, Chiba, and Southern Ibaraki prefectures
Survey time and day	24 hrs on weekdays in October 1968, 1978, 1988, 1998, and 2008, excluding Monday and Friday
Object of survey	Persons over the age of 5 living in the above regions
Sampling	Random sampling based on census data
Valid data	272,230, 588,343, 667,918, 883,012, and 594,314 samples for 1968, 1978, 1988, 1998, 2008, respectively
Content of data	Personal attributes, place and time of departure and arrival, the purpose of trip, etc.

. It is a one-day dataset, and place and time of the departure and arrival, travel purpose, and means of trip, as well as personal attributes (age, gender, occupation, car ownership, etc.), are included in the dataset. In our research, the survey covers an area of a circle with a radius of 70 km centered on the Tokyo Railway station [36]. The day starts from 3:00 AM at night, and ends at 3:00 AM the next day. We further filter out samples whose trip information, such as trip purpose, travel time, and location, is missing. After data preprocessing, 272,230 samples, 588,343 samples, 667,918 samples, 883,012 samples, and 594,314 samples were obtained for year 1968, 1978, 1988, 1998, 2008, respectively.

1.2.2. Mobility Sequences

Using PT data, individual locations at any time of the day can be inferred. We divide a day into 30-min slots (48 slots for a day). For every person, there is a sequence of 48 slots, where each slot is assigned with his/her location during that 30-min interval (Figure 3, mobility sequence hereafter). If a person is taking a trip during the 30-min interval, the location after the trip will be assigned to the time slot. The spatial resolution is the administrative region. Figure 4 shows all the regions (hereafter Region 1, Region 2, …) in the Tokyo Metropolitan area in the survey, and we will use the term throughout the paper. Accordingly, the number of mobility sequences is the same as the number of samples for each year.

2. The Model for Reproduction

By considering the temporal similarity of individual travel behaviors, we cluster people into four groups using hierarchical clustering algorithms (

Table 2. Four groups of people.

Group ID	Attributes
Group 1	Household wives/husbands, the unemployed, and farmers
Group 2	Workers and college students with ages greater than 14
Group 3	High school students between 15–19 years old
Group 4	Children between 5–14 years old

). Since this paper mainly introduces the reproduction method and the energy concept, the detail of the algorithm is out of focus. For each year, people are classified into four groups based on their ages and occupations, and the reproduction model is validated for each group of people in each year. To reproduce mobility sequences that are spatially and temporally similar to the original mobility sequences, we develop a two-step time-varying Markov Chain model. The model is detailed in Section 2.1, and in Section 2.2, the performance is evaluated spatially and spatiotemporally.

2.1. Method of Reproduction

There are two ingredients for the reproduction model for each year: the time-varying transition matrix and the next-status probability. In Section 2.1.1, the first ingredient is introduced, and reasons why it is not enough are given. Section 2.1.2 and Section 2.1.3 define the second one. Section 2.1.4 presents the steps of reproduction using the two elements.

2.1.1. The Time-Varying Location Transition Matrix

The time-varying location transition probability matrix, $P\left(t\right)$, is a matrix indicating the probability for people to have a trip between two regions at time $t$. The entry in the $i$-th row and the $j$-th column, $P^{i,j}\left(t\right)$, is the probability of a person in $Region i$ beginning a trip to $Region j$ at time $t$ (Figure 5). For each group of people, transition probabilities can be aggregated from the mobility sequences. Moreover, the probability of being at each region at the time $t=1$ can be calculated. $\pi \left(i\right)$ stands for the probability of being at $Region i$ when $t=1$.

In simple scenarios, human mobility can be reproduced using such matrices in a Markov process, where the next location is probabilistically chosen from $P\left(t\right)$ at time $t$. The next location only depends on the current location. This is the so-called time-varying Markov Chain model. However, the mobility sequences generated using a simple Markov process are remarkably different from real human mobility, and properties of human movements cannot be reproduced. For example, in our dataset, most people go back home at night. In a Markov process, the next location only depends on the current location, and thus, people ‘forget’ their start location (home), and randomly select their destination at night. In the following two subsections, another element is introduced that memorizes their previous types of locations.

2.1.2. Types of Mobility Sequences and Types of Current Mobility Sequences

There are various types of mobility sequences ($M_a$ for person $a$), such as ‘A’ type, ‘A-B-A’ type, ‘A-B-C-A’ type, etc. (Figure 6a). ‘A’ stands for the region that shows up first in the mobility sequence; ‘B’ stands for the second appearing region; and ‘C’ indicates the third one, etc. ‘A-B-A’ type implies that the person is in the same region at the start and the end of the day, but travels to another region during the day; ‘A’ type indicates that the person stays in a region, and makes no trip the whole day; ‘A-B-A-A’ type means the person travels once more at last (maybe at the end of the day) in region ‘A’ compared with the ‘A-B-A’ type, etc. In Figure 6a, for person $a$, s/he traveled from $Region 1$ to $Region 2$, and later from $Region 2$ back to $Region 1$, and the type of mobility sequence is ‘A-B-A’. ‘A’ stands for $Region 1$ and ‘B’ stands for $Region 2$. For person $b$, ‘A’ stands for $Region 3$ and ‘B’ stands for $Region 2$. Both person $a$ and person $b$ have the ‘A-B-A’ type of mobility sequences, but the regions traveled, and the times of trips are different. For any person, the type of current mobility sequence ($M_a\left(t\right), t=1, 2, \dots , 48$) is the type of mobility sequence the person has until the current time (Figure 6b). In Figure 6b, the person has made no trip until 6:00, and thus, the type of current mobility sequence at 6:00 is ‘A-’. S/he had made only one trip from $Region 1$ to $Region 2$ before 12:00, and thus, the type of current mobility sequence at 12:00 is ‘A-B-’. Similarly, at 20:00, the type of current mobility sequence is ‘A-B-A-’. $M_a$ and $M_a\left(t\right)$ are both sequences of letters, but $M_a$ is the type of mobility for a whole day, whereas $M_a\left(t\right)$ is that for a given time. They only suggest the type of mobility sequence of a person, but indicate nothing about where those regions are, and at exactly what time each trip happened. For different individuals, even if they have the same type of current mobility sequence at the same time, the regions visited, and the time of trips may vary greatly.

2.1.3. Next-Status Probability

We assume that people who have the same types of current mobility sequence are likely to have similar patterns of mobility in the next time slot. Grounded on this idea, the mobilities of these people are predicted in the same way. For people with any type of current mobility sequence at time $t$, $M\left(t\right)$, there can be three types of statuses at time $t+1$ (Figure 7):

• S1. Staying in the current region and making no trip, with the probability denoted by $P_t(S1|M)$. In the case of Figure 7, if the person does not move, the location at $t+1$ is $Region 1$. At $t+1$, the type of current mobility sequence is $M\left(t+1\right)=M\left(t\right)$, (i.e., ‘A-A-B-C-A-’). The probability of such a case is denoted by $P_t(S1|M=A-A-B-C-A-)$.
• S2. Having a trip to a region visited. The probability is denoted by $P_t(S2|M)$. It can be a collection of probabilities if the person has multiple visits until time $t$. $Sum(P_t(S2|M))$ is the sum of probabilities in the collection. In the case of Figure 7, the person may have a trip to $Region 1$ (A), $Region 2$ (B), or $Region 3$ (C), and $M\left(t+1\right)$ can be ‘A-A-B-C-A-A-’, ‘A-A-B-C-A-B-’, or ‘A-A-B-C-A-C-’ accordingly. Having a trip to $Region 1$ indicates a trip inside $Region 1$. The probabilities of these cases are denoted by $P_t(S2=A|M=A-A-B-C-A-)$, $P_t(S2=B|M=A-A-B-C-A-)$, and $P_t(S2=C|M=A-A-B-C-A-)$. $P_t(S2|M=A-A-B-C-A-)$ = $\{P_t(S2=A|M=A-A-B-C-A-), P_t(S2=B|M=A-A-B-C-A-), P_t(S2=C|M=A-A-B-C-A-)\}$.
• S3. Traveling to an area that the person has never been to. The probability of such a case is denoted by $P_t(S3|M)$ (= 1 − $P_t(S1|M)$ − $Sum(P_t(S2|M))$). In the case of Figure 7, the person travels to an unvisited region ($Region other$), and $M\left(t+1\right)=‘A-A-B-C-A-D-’$. The probability of this case is denoted by $P_t(S3|M=A-A-B-C-A-)$.

For any type of current mobility sequence, the probabilities of being at the three types of statuses at the next time slot can be aggregated from the mobility sequences for each group of people. We call such probabilities the next-status probabilities hereafter.

2.1.4. Method of Reproducing Human Mobility

For each group of people, two elements are calculated: the time-varying transition matrix, $P\left(t\right)$ (including the probability of being at each region at the start time, $\pi \left(i\right)$), and the next-status probabilities, $P_t(S1|M)$, $P_t(S2|M)$, and $P_t(S3|M)$. In our framework of reproduction, the second element is used to choose individual status in the next time slot conditioned on the type of current mobility sequence (see Section 2.1.3), whereas the first element is used to assign an individual a location if the status at the next time slot is S3 (i.e., traveling to an area that the person has never been to, see Section 2.1.3). Based on the framework, we do the following steps (Figure 8):

• Step 1. At the time $t=1$, individual location is randomly chosen where the probability for $Region i$ to be chosen is $\pi \left(i\right)$. The type of current mobility sequence of any person is set to be ‘A-’, and ‘A’ stands for the first location of the person.
• Step 2. When $1\le t\le 47$, there can be three types of statuses at time $t+1$: staying in the current region and making no trip ($S1$), having a trip to a previously visited region ($S2$), and traveling to an unvisited region ($S3$). The probabilities of these statuses are $P_t(S1|M)$, $P_t(S2|M)$, and $P_t(S3|M)$, accordingly.
• Step 3. When the next status is $S1$ or $S2$, the location of the person in the next time slot is the current location or one of the previously visited locations. When the next status is $S3$, the next location (denoted by $l\left(t+1\right)$) is chosen from unvisited regions. The probability of choosing an unvisited region is given by:

  $$P_a^{i,j}\left(t\right)=\frac{P^{i,j}\left(t\right)}{{{\displaystyle \sum }}_k{P}^{i,k}\left(t\right)}$$

  (1)

  where $P_a^{i,j}\left(t\right)$ is the probability for person $a$ in $Region i$ to choose $Region j$ at time $t$; $Region i$ is the region the person in at time $t$; $Region j$ and $Region k$ are unvisited regions; $P^{i,j}\left(t\right)$ is the entry in the $i$-th row and the $j$-th column of transition matrix $P\left(t\right)$.
• Step 4. By looping Step 2 and Step 3 until $t=47$, mobility sequences are generated.

2.2. Results and Evaluation

Human mobilities are complicated, and there are many types of current mobility sequences (i.e., $M\left(t\right)$). When applying the method introduced in Section 2.1 for each group, only the types of current mobility sequences that cover more than 0.1% of the population are focused. The next-status probabilities of these types are calculated. For other types of current mobility sequences, the next location is chosen from the time-varying location transition matrix $P\left(t\right)$ in a Markov process. After reproducing mobility sequences for each group, we evaluate the spatial (Section 2.2.1) and spatiotemporal (Section 2.2.2) similarity between the generated and the real datasets. The evaluation for 2008 is shown in the following sub-sections. For other years, see Appendix A.

2.2.1. Spatial Evaluation

In our research, spatial characteristics of mobility sequence are what regions are traveled by people in what order. Here, we define a region sequence as a sequence of regions (e.g., $Region 2\to Region 3\to Region 2$). For any person from the original or the generated dataset, his/her mobility sequence belongs to a region sequence by discarding its temporal information. For example, if there are two persons who have the first trip from $Region 2$ to $Region 3$ at 9:00, and the second trip from $Region 3$ to $Region 2$ at 17:00, then we have two mobility sequences belonging to region sequence $Region 2\to Region 3\to Region 2$, and the count of such a region sequence increases by two. Different from the type of mobility sequence introduced in Section 2.1.2, region sequence claims what regions are visited.

We evaluate the spatial similarity by comparing the counts of region sequences from the original dataset and the reproduced dataset (Figure 9 and Figure A1, Figure A2 and Figure A3, one dot for one region sequence). To clearly present our results, for each group of people (see Table 2), the comparison is shown in three figures (comparisons for ‘A’ type, ‘A-A-A’ and ‘A-B-A’ types, and other types). In Figure 9, each dot represents one region sequence. The x-value is the number of mobility sequences belonging to such a region sequence in the original dataset, and the y-value is the corresponding count in the reproduced dataset. If the x-value and the y-value are close (i.e., the dot is close to the diagonal line drawn from the origin to the top right), the numbers of people with such a region sequence in the two datasets are close, where we would say that the spatial attributes are well-reproduced. Results show that, for Group 1 and Group 4, the spatial attributes of reproduced mobility sequences are similar to those of the real mobility sequences; for Group 2 and Group 3, the ‘A’ type is well-reproduced spatially, whereas the region sequences of other types with large counts are generated fewer than real counts. This may be owing to the differences between next-status probabilities for frequent region sequences (with large counts) and the next-status probabilities for less frequent region sequences. However, we treat all these region sequences as the same type of current mobility sequences in our model. Overall, our method can reproduce the spatial attributes well. The results suggest that people who have the same type of mobility sequences are likely to have the same future status (‘A’, ‘B’, etc.), no matter when they had each trip and where they have visited.

2.2.2. Spatiotemporal Evaluation

In this section, we evaluate the spatiotemporal characteristics of reproduced datasets. The spatiotemporal similarity between the original and the reproduced datasets is high only when two conditions are met: the counts of region sequences are similar (Section 2.2.1); and the trips for each region sequence happen at similar times between the two datasets. Mobility sequences with more than two trips are very complicated. We compare the similarity only for ‘A-B-A’ and ‘A-A-A’ types of mobility sequences with two trips, which cover large proportions of people in the research (about 50%, see Figure 10). For each group in each year, we do the following steps for the original and reproduced mobility sequences:

• Step 1. For any target person $a$, we get his/her region sequence (defined in Section 2.2.1), and the times of the two trips, $T_a=\left(T_a^1,T_a^2\right)$.
• Step 2. For people ($a, b, \dots$) whose mobility sequences belong to the same region sequence $R$, we have a collection of times of the two trips of these people, $T^R=\left\{\left(T_a^1, T_a^2\right),\left(T_b^1,T_b^2\right), \dots \right\}$.
• Step 3. The collection of times of trips, $T^R$, is plotted on a 2-D space where the x-axis is the time of the first trip, and the y-axis is the time of the second trip; one point for one mobility sequence that is defined in 1.2.2 (Figure 11).
• Step 4. The 2-D space is cut into 3-h-by-3-h cells, and the number of points in each cell is counted (also Figure 11).

For region sequence $R$ (defined in Section 2.2.1), $O_R^{i,j}$ is the number of points in the cell in the $i$-th row and the $j$-th column for the original dataset (Figure 11a), and $G_R^{i,j}$ is the number of points in the same cell for the reproduced dataset (Figure 11b). $O_R^{i,j}$ indicates the number of people in the original dataset whose mobility sequences belong to $R$, and whose first trip happened in the $i$-th time interval and the second trip happened in the $j$-th time interval. $G_R^{i,j}$ indicates the same in the generated dataset. If $O_R^{i,j}$ and $G_R^{i,j}$ are close for any $i$, $j$, then, the temporal attributes of region sequence $R$ are well-reproduced. For each group of people, $O_R^{i,j}$ and $G_R^{i,j}$ for all $R$s are shown in one plot (Figure 12 and Figure A4, Figure A5 and Figure A6). We can notice that for Group 1 and Group 4, the spatiotemporal attributes are well-reproduced, whereas the numbers of points in favored time-cells (with large counts) are underestimated for the other two groups. The inaccuracy could be caused by differences between the next-status probabilities for various region sequences, the same reason as is introduced in Section 2.1.1. Overall, given the two strict, but essential, conditions, the spatiotemporal attributes are well-reproduced. The results suggest that the probability of having a trip at a certain time can be approximated only using the information of the type of current mobility sequences, regardless of previous locations and times of trips.

For region sequences with $n$ trips, we need an $n$-dimensional space to plot the times of trips, and, as a result, cells are also $n$-dimensional. However, the number of cells increases exponentially as $n$ grows, and the count of points in each cell is too few for comparison. Thus, only the ‘A-B-A’ and ‘A-A-A’ types of mobility sequences are of concern.

3. The Next-Status Probability and Energy Law

After reproducing human mobility for 1978–2008, we want to reproduce people’s mobility for any time. There are two important ingredients for the reproduction in our method: the time-varying location-transition matrix, $P\left(t\right)$, and the next-status probabilities, ($P_t(S1|M)$, $P_t(S2|M)$, and $P_t(S3|M)$). The former one can be estimated using the OD matrix, and there is a huge amount of literature regarding its uses, models, synthetic indicators, and so on [37]. It may also be estimated from many types of data that are inexpensive, such as the call detail records (CDR). However, the later element, the next-status probabilities, is not easily available. To obtain it straightforwardly, it is necessary to track a large number of individuals’ daily movements with high temporal resolution. However, this can hardly be done inexpensively.

It is further noticed that the distribution of the number of trips over decades remains stable (Figure 13), despite the development in transportation, and changes in the urban population. With the evolution of transportation, people may travel further over years. In terms of the next-status probabilities, one of the main differences over time is that, when trips happen, there is a higher proportion of people whose next statuses are different from the current status. In other words, when a trip happens, people are more likely to travel to a different region (a different status) as the average travel distance increases. To deduce people’s travel behaviors given a specific time, or predict them in the future, we have to capture elements that are constant or change structurally over years. Providing that the stability of travel-time budget varies by area [38] (in our case, it grows slowly by year), we turn to the travel energy, which is poorly investigated in previous studies, but turned out to be more stable over years in our study area.

In this section, the energy concept is employed to get knowledge about how next-status probabilities (i.e., $P_t(S1|M)$, $P_t(S2|M)$, and $P_t(S3|M)$) may change over the years. We revisit the energy law, and apply it to predict the proportion of cross-region trips. Cross-region trips are trips from one region to another, for example, from $Region 1$ to $Region 2$. When status-transition happens, the proportion of cross-region trips implies if the next status remains the same as the current one. It roughly indicates the proportion of long-distance trips. However, it should also be noted that the proportion is dependent on the average size of regions (the larger the size, the lower the proportion). Such a proportion is a meaningful indicator of trip length only when the boundaries of regions are fixed.

3.1. The Energy Law

Human movement consumes energy from the human body, and one would reasonably study statistics about energy for human mobility. Using data collected from the UK, Kölbl and Helbing examined travelers who have only one travel mode apart from walking (single-mode traveler hereafter) [20]. It was noticed the average daily journey time for a given travel mode is inversely proportional to the energy consumption rate of the mode. They further showed that the scaled daily travel energy follows a canonical-like distribution, regardless of travel modes. However, in a later report, this conjecture is doubted. By applying the energy per minute of travel calculated from [20] (

Table 3. Average energy consumption per minute for different travel modes. (a) Values used by Kölbl and Helbing. (b) Values applied in this study.

(a) Values Used by Kölbl and Helbing
Travel mode	Train	Car (driver)	Bus	Car (passenger)	Bicycle	Walk
Energy (kJ/min)	4.0	8.2	9.2	10.4	14.6	15.4
(b) Values Applied in this Study
Travel mode	Train	Car	Bus	Taxi	Bicycle	Walk	Auto_b 1 (1968, 1978)	Auto_b1 1 (1988–2008)	Auto_b2 1 (1988–2008)
Energy (kJ/min)	3.2	6.4	7.2	11.9	17.1	18.7	11.88	14.7	9.6

¹ Auto_b, auto_b1, and auto_b2 are motorized bikes.

a), Hubert and Toint examined the daily average energy consumption by travel mode using data collected from Belgium, France, and Great Britain [21]. From their study, the average energy consumption varies by travel mode and by area, which does not agree well with the theory. They additionally present three reasons why the conjecture is not convincing:

• The calculated energy consumption per minute for car passengers is higher than car drivers, which is counter-intuitive.
• Travelers with more than two travel modes show higher energy expenditure compared with single-mode travelers.
• The average expenditure of 615 kJ/day used by the former research is surprisingly small for an average person who consumes 250 kJ/h.

In the following sub-sections, we revisit this conjecture using the aforementioned PT data collected from 1968 to 2008, to infer how the cross-region trips may change over years, as well as to make an effort to answer the three questions.

3.2. Energy Expenditure for Single-Mode Travelers and the Whole Population

People who spend on one travel mode for more than 90% of their daily travel time are regarded as single-mode travelers in this study. These people are classified by their travel modes into the following classes: walk, bicycle, taxi (in comparison with car passengers in previous studies), car, bus, train, and motorized bike (auto_b). For 1988, 1998, and 2008, motorized bikes are further divided into two types by their engine sizes: auto_b1 (<50 cc) and auto_b2 (≥50 cc).

The travel time for single-mode travelers is examined first. Figure 14a shows the change of average travel time for each class of single-mode travelers over years on a logarithmic scale. It can be concluded that the stability of travel time is not guaranteed, which could be because of the rapid development of transportation, and the changes in urban structure over decades. Given the unstable travel time, if the energy per unit time for each travel mode is assumed to be constant over years, the energy expenditure for each class of single-mode traveler is not stable. This conclusion does not agree with observations in [20]. However, this does not imply the energy expenditure is unstable. Figure 14b shows the proportion of each class of single-mode travelers in the whole population. The y-axis is on a logarithmic scale, so we can investigate the shares of both frequent and infrequent modes by year. With the development of transportation, people switch their travel modes. Consequently, even if the energy expenditures for each class of single-mode travelers changes, the distribution of energy expenditure for the whole population may be stable.

With an estimated average travel-energy budget of $\tilde{E}\approx$615 kJ per day for single-mode travelers, we estimate the average energy consumptions per minute for travel modes given by:

$$C_j=\frac{1}{Y}{{\displaystyle \sum }}_y\frac{\tilde{E}{n}_j^y}{{{\displaystyle \sum }}_i^{n_j^y}{t}_i}$$

(2)

where $C_j$ is the average energy expenditure per unit time of travel mode $j$; $n_j^y$ is the number of single-mode travelers using travel mode $j$ in year $y$; $t_i$ is the travel time for one of such travelers; $Y$ is the number of years. The energy expenditure is not estimated for each year, but the average over years, because it is assumed that energy expenditures for activities (e.g., walking, driving) do not vary significantly by time. Calculated energy expenditures for modes are shown in Table 3b. Using the calculated values, the daily energy for any travelers $i$ is defined as:

$$E_i={{\displaystyle \sum }}_j{C}_j{t}_i\left(j\right)$$

(3)

where $E_i$ is the daily energy expenditure; $t_i\left(j\right)$ is the time that the person spent on travel mode $j$.

The distribution of daily energy expenditure for each year is estimated (Figure 15 for the whole population, and Appendix B for four groups of people in Table 2). Despite the differences in the change of the shares of transportation modes in the Tokyo Metropolitan area, the overall energy consumption remains stable over five decades.

3.3. Estimating the Proportion of Cross-Region Trips for Each Year

Given the stable number of trips (Figure 13), and the stable energy expenditure over years (Figure 15), the average energy consumption for a single trip should be stable, as it equals the average energy consumption divided by the average number of trips. Figure 16 shows that for cross-region trips, in-region trips, and for all trips.

The proportion of cross-region trips (denoted by $P_c$), ultimately, is about travel distance. Given fixed region boundaries and population distribution in the area, the further the travel distance, the higher the proportion. It can be concluded that as the average travel distance $d\to \infty$, $P_c\to 1$, and if $d\to 0$, $P_c\to 0$. With the average speed of travel mode estimated in the following table (

Table 4. Average travel speed by mode.

Mode	Walk	Bike	Train	Car	Bus	Taxi	Auto_b 1
Speed (km/h)	3.2	15	50	30	10	18.3	17

¹ Auto_b is the motorized bike. Due to the lack of valid resources, auto_b1 and auto_b2 mentioned in Table 3 are not differentiated, but treated as auto_b.

), we calculate the average travel distance for datasets from 1978–2008 (data from 1968 is excluded because the survey area is smaller than other years, and regions are also smaller). The proportion of cross-region trips over years can also be obtained. Figure 17 shows their relationship, and there is approximately a linear correlation (formula shown in the same figure) when the average travel distance is within the range. This can be used to predict the proportion of cross-region trips given the travel distance. However, the complete relationship should be non-linear, as when $d\to \infty$, $P_c\to 1$.

The average time spent on travel modes for one trip for each year is plotted in Figure 18a. It can be noticed that the time spent on foot is decreasing, whereas that on the train is gradually increasing. With the switch of travel modes, the energy remains stable, whereas the travel time is increasing constantly. Knowing the average proportion of time spent on each mode for a single trip ($p_j\left(y\right)$ for mode $j$ in year $y$, see Figure 18b), we have:

$${{\displaystyle \sum }}_{j}T\left(y\right)\times p_j\left(y\right)\times C_j=\overline{C}$$

(4)

where $T\left(y\right)$ is the average time spent on a trip in year $y$; $C_j$ is the average energy consumption per unit time for travel mode $j$; $\overline{C}$ is the energy consumption for an average trip (≈278.79 kJ). From Equation (4), we have:

$$T\left(y\right)=\frac{\overline{C}}{{{\displaystyle \sum }}_j{p}_j\left(y\right)\times C_j}$$

(5)

The average travel distance for one trip in year $y$ is given by:

$$D\left(y\right)={{\displaystyle \sum }}_{j}T\left(y\right)p_j\left(y\right)v_j=\frac{\overline{C}}{{{\displaystyle \sum }}_j{p}_j\left(y\right)\times C_j}{{\displaystyle \sum }}_j{p}_j\left(y\right)v_j$$

(6)

where $v_j$ is the speed of travel mode roughly estimated in Table 4. With $D\left(y\right)$, the proportion of cross-region trips can be estimated using the linear regression in Figure 17. The estimated results are shown in Figure 19. Unsurprisingly, the estimation is good. Using the proportion of cross-region trips, the change of sequence types of mobility may be inferred. For example, the sum of proportions of mobility sequences with two trips (mostly ‘A-B-A’ and ‘A-A-A’ types of sequences) is stable, and with the increase of cross-region trips, the proportion of the ‘A-B-A’ type will increase, whereas that of ‘A-A-A’ will decrease. However, because of the complexity of human movements, the model of predicting each mobility type by year is yet to be developed.

3.4. A Further Interpretation of Energy Law

Let’s focus on the scaled daily energy expenditure of individuals (Figure 20):

$$r_i=\frac{E_i}{\overline{E}}$$

(7)

where $\overline{E}\approx$798.9 kJ is the average daily energy consumption over years. Following Equations (2) and (3), we have:

$$r_i=\frac{{{\displaystyle \sum }}_j{C}_j{t}_i\left(j\right)}{\overline{E}}=\frac{1}{\overline{E}}\times {{\displaystyle \sum }}_j\left(\frac{1}{Y}{{\displaystyle \sum }}_y\frac{\tilde{E}{n}_j^y}{{{\displaystyle \sum }}_s^{n_j^y}{t}_s}\right)t_i\left(j\right)=\frac{\tilde{E}}{\overline{E}}\times {{\displaystyle \sum }}_j\left(\frac{1}{Y}{{\displaystyle \sum }}_y\frac{n_j^y}{{{\displaystyle \sum }}_s^{n_j^y}{t}_s}\right)t_i\left(j\right)$$

(8)

By the definition of $\overline{E}$, we have:

$$\frac{\tilde{E}}{\overline{E}}=\frac{1}{\frac{1}{Y}\left[{{\displaystyle \sum }}_y\left({{\displaystyle \sum }}_k^{n^y}{{\displaystyle \sum }}_j\left(\frac{1}{Y}{{\displaystyle \sum }}_y\frac{n_j^y}{{{\displaystyle \sum }}_s^{n_j^y}{t}_s}\right)t_k\left(j\right)\right)/n^y\right]}$$

(9)

where $n^y$ is the number of people in year $y$. From Equation (9), $\frac{\tilde{E}}{\overline{E}}$ is independent of how $\tilde{E}$ is set initially, because it is a combination of travel times. Since $\overline{E}$ is about 798.9 kJ, and $\tilde{E}$ was set to be 615 kJ, following Equation (8), we have:

$$r_i\approx 0.77\times {{\displaystyle \sum }}_j\left(\frac{1}{Y}{{\displaystyle \sum }}_y\frac{n_j^y}{{{\displaystyle \sum }}_s^{n_j^y}{t}_s}\right)t_i\left(j\right)$$

(10)

How $\tilde{E}$ is set initially does not affect the distribution of $r_i$. The calculated energy, $C_j$, might be interpreted as the average ‘cost’ per minute of travel mode, which is reversely proportional to the average travel time of single-mode travelers. In other words, the higher the cost per minute, the less time single-mode travelers spend on it. The energy law claims the stability of such daily travel costs. In addition, when $\tilde{E}$ is set to be 615 kJ, the cost agrees with the energy consumption of physical activities of the corresponding travel mode. This may imply that the energy spent on travel mode is an important factor when people consider the ‘cost’ of travel mode. Given the above hypothesis, the three questions from [21] may be answered:

• The calculated energy consumption per minute for car passengers is higher than car drivers, which is counter-intuitive.
• Travelers with more than two travel modes show higher energy expenditure compared with single-mode travelers.
• The average expenditure of 615 kJ/day is surprisingly small for an average person who consumes 250 kJ/h.

For the first question, the energy expenditure of car passengers is measured from people taking taxis. Since taking a taxi is expensive, the cost is high, accordingly. For the second question, the energy expenditure of single-mode travelers is about 0.77 times as much as the average (including single-mode travelers and multi-mode travelers). For the third question, the setting of $\tilde{E}$ has no effect on the distribution of scaled daily energy expenditure (Equation (10)).

In addition, from our dataset, the distribution of energy expenditure over years seems to be more stable than the distribution of travel time, which increases over years. However, in the same year, for regions with different distances from the city center, the energy expenditure shows more differences than travel time. To verify the stability of energy consumption for the same area over years, further validations are needed.

4. Discussion

From the experiment in Section 2, we notice that people’s future status and mobility can be probabilistically determined based on their current type of mobility sequences (e.g., ‘A-B-’), regardless of exactly where people have traveled, and when trips happen. This may be because most people in a group follow general activity-travel patterns, no matter where they live, or the times for trips. For example, for workers with the ‘A-B-’ type of current mobility sequence, ‘A’ is likely to be the home location, and ‘B’ is likely to be the workplace location, and thus, having an ‘A-B-’ type in the afternoon suggests that they are likely to go back to ‘A’ (home location) from 5:30 PM to 8:00 PM. The pattern is generally true no matter where ‘A’ and ‘B’ are for workers, or when the first trip happened. Given a current mobility sequence for a group of people, the interpretation of each letter (type of location and activity) could be made, and the next-status probabilities imply the type of the next location/activity given the current interpretation. The model also shows some limitations. The region sequences of types other than the ‘A’ type with large counts are reproduced fewer than real counts (Figure 9 and Figure 12). There are differences between next-status probabilities for frequent region sequences (with large counts) and the next-status probabilities for less frequent region sequences. However, we treat all of these region sequences as the same type of current mobility sequences in the reproduction model. Secondly, for people with less frequent current mobility sequences, their next locations are chosen based on the time-varying location transition matrix. Travel activities of these people can be remarkably different from the real dataset, but the difference is not significantly shown in Figure 9 and Figure 12 because of their small counts. However, their travel activities dominate the tails of distributions of travel distance, travel time, etc.

To further get insights into how the next-status probabilities change in the future or various scenarios, we need to capture related factors that are constant or change structurally over years. It is observed that the distribution of energy expenditure in the whole area is stable over years. However, different from [20], it is not stable for a specific travel mode in our case. We argue that the constant energy budget can be interpreted as the constant travel cost over years, where the cost per minute for a travel mode is believed to be constant over years, and is inversely proportional to the average travel time of its single-mode travelers. Based on the constant distribution of energy (cost) expenditure, the proportion of cross-region trips is predicted given the travel-mode share. This suggests that we may predict the status-transition probability given the development of public transportation for various scenarios. However, the precise correlations between travel behaviors and contextual changes, including infrastructure, built environment, and lifestyle, are still unknown, and further attention will be given to such topics.

The existing literature has demonstrated many methods of reproducing human mobility using various data sources, and most of their evaluations center on general statistics. In this study, we examined the reproduction results specifically about where people visited in what order, and when trips happen for each region sequence. The reproduced dataset is a good representation of reality, and can be used in a wide variety of fields. For example, in recent years, researchers simulated the spread of COVID-19 by employing the agent-based disease transmission model that is data-hungry [39], and our synthetic data can be applied for such tasks. Also, our method is a tool to infer the spatial and temporal attributes of human mobility from knowledge about aggregated statistics. In this study, the proportion of cross-region trips is also modeled using the concept of energy, though the bridge between it and the proportion of types of mobility sequences is yet to be built. The energy concept in travel behaviors needs more validation. In addition, COVID-19 may cause some long-lasting effects, including disruptions to habitual behaviors, changes in attitude [40], etc. Differences in next-status probabilities can be expected if our model is applied in a post-pandemic future.

5. Conclusions

In this paper, we first propose a model of reproducing human mobility based on the idea that people who have the same type of current mobility sequences are likely to behave in similar patterns in the future. In the framework of the model, individual status in the next time slot is chosen based on the type of current mobility sequence. If the next status is traveling to an unvisited region, the location would be selected using the time-varying location-transition matrix in a Markov process. The method can successfully reproduce the spatiotemporal characteristics of human mobility. The results suggest that people’s future status and mobility can be inferred from the current type of mobility sequence.

In the preceding sections, the energy law is revisited and applied to model the proportion of cross-region trips. We show that the travel mode use is not stable over years, and thus, the distribution of energy expenditures of each class of single-mode travelers should vary. This conclusion is different from the observations in [20], which may owe to the development of transportation, and mode-switch of people over decades in our study. At the population level, the distribution of energy expenditure is stable. Using such a conclusion, the share of cross-region trips is modeled in two steps. Given the average travel-time shares of modes for a trip, the travel time for each mode is calculated, and the average travel distance is estimated in the first step. In the second step, using the correlation between travel distance and cross-region trips, the proportion of cross-region trips is estimated. Finally, we suggest that the stable energy expenditure might be interpreted as a stable ‘cost’ of daily trips. The average ‘cost’ per minute of a travel mode is inversely proportional to the average time spent by single-mode travelers using such a mode. However, differences in the ‘cost’ are shown in the city center and suburb areas. More evidence from other cities is necessary about if the energy expenditure is stable for the same city over years.