Patent Publication Number: US-2022221287-A1

Title: Moving number estimating device, moving number estimating method, and moving number estimating program

Description:
TECHNICAL FIELD 
     The present disclosure relates to a human mobility estimation device, a human mobility estimation method, and a human mobility estimation program. 
     BACKGROUND ART 
     Position information about humans obtained from sources such as GPS may be provided as temporal area population data in cases where individuals cannot be tracked because of privacy concerns. Here, temporal area population data refers to information about the number of people in each area for each timestep, which is the length of a certain time interval. Areas are assumed to be obtained by dividing a geographical space into a grid, for example. There exists a need to estimate the human mobility across areas and across timesteps from such temporal area population data. 
     In the related art, a framework (collective graphical model) that estimates individual probability models from aggregated data is used to estimate movement probability and human mobility across areas from demographic information. 
     CITATION LIST 
     Non-Patent Literature 
     
         
         Non-Patent Literature 1: D. R. Sheldon and T. G. Dietterich. Collective Graphical Models. In Proceedings of the 24th International Conference on Neural Information Processing Systems, pp. 1161-1169, 2011 
         Non-Patent Literature 2: T. Iwata, H. Shimizu, F. Naya, and N. Ueda. Estimating People Flow from Spatiotemporal Population Data via Collective Graphical Mixture Models. ACM Transactions on Spatial Algorithms and Systems, Vol. 3, No. 1, pp. 1-18, May 2017 
       
    
     SUMMARY OF THE INVENTION 
     Technical Problem 
     In the movement of people in the real world, certain relationships exist between the distance between areas and the human mobility between the areas, such as that people more readily move to nearby areas and less readily move to distance areas. However, the related art does not consider the influence that the distance between areas exerts on the ease of movement while accounting for traffic information, and has a limitation in that the estimation accuracy is lowered when applied to real-world data. In addition, there is also a problem of simply using a distance such as the Lp distance in a grid or the simple Euclidean distance between two points as the distance between areas. The reason for this problem is that in the real world, the ease of movement between areas for people is not determined solely by the simple physical distance. 
     In particular, information indicating what transport facilities and means of transportation exist between areas is highly relevant to the ease of movement between areas. For example, even in the case where the Euclidean distance between an area A and an area B is equal to the Euclidean distance between the area A and an area C, the circumstances will be different if a railway line exists between the area A and the area B whereas only a narrow street exists between the area A and the area C. The number of people moving from A to B will be greater than the number of people moving from A to C. A method that uses simple distances is incapable of accounting for the properties and relationships of such movement routes, and therefore the problem of lowered accuracy will arise when such a method is applied to real-world data. 
     The technology of the disclosure has been devised in light of the above points, and an object thereof is to provide a human mobility estimation device, a human mobility estimation method, and a human mobility estimation program capable of estimating human mobility between areas accurately while accounting for traffic information. 
     Means for Solving the Problem 
     A first aspect of the present disclosure is a human mobility estimation device comprising: an intersection determination unit that makes an intersection determination of whether or not a traffic route included in map information intersects a boundary between areas on a basis of the traffic route and an area shape between areas that are adjacent in the map information, the intersection determination being made for each pair of adjacent areas; a movement cost graph construction unit that constructs, on a basis of a weight determined for the boundary in a case where the intersection determination is made that no intersection exists, a movement cost graph expressing a movement cost for each area that accounts for the weight; a shortest distance calculation unit that calculates a corrected shortest distance between areas on a basis of the movement cost graph; and a human mobility estimation unit that estimates a human mobility between areas for each timestep, the timestep being the length of a predetermined time interval, according to a predetermined function for estimating the human mobility between areas on a basis of the corrected shortest distance between areas and a population of each of the areas at different predetermined times. 
     A second aspect of the present disclosure is a human mobility estimation method executed by a computer, the method comprising: making an intersection determination of whether or not a traffic route included in map information intersects a boundary between areas on a basis of the traffic route and an area shape between areas that are adjacent in the map information, the intersection determination being made for each pair of adjacent areas; constructing, on a basis of a weight determined for the boundary in a case where the intersection determination is made that no intersection exists, a movement cost graph expressing a movement cost for each area that accounts for the weight; calculating a corrected shortest distance between areas on a basis of the movement cost graph; and estimating a human mobility between areas for each timestep, the timestep being the length of a predetermined time interval, according to a predetermined function for estimating the human mobility between areas on a basis of the corrected shortest distance between areas and a population of each of the areas at different predetermined times. 
     A third aspect of the present disclosure causes a computer to execute a process comprising: making an intersection determination of whether or not a traffic route included in map information intersects a boundary between areas on a basis of the traffic route and an area shape between areas that are adjacent in the map information, the intersection determination being made for each pair of adjacent areas; constructing, on a basis of a weight determined for the boundary in a case where the intersection determination is made that no intersection exists, a movement cost graph expressing a movement cost for each area that accounts for the weight; calculating a corrected shortest distance between areas on a basis of the movement cost graph; and estimating a human mobility between areas for each timestep, the timestep being the length of a predetermined time interval, according to a predetermined function for estimating the human mobility between areas on a basis of the corrected shortest distance between areas and a population of each of the areas at different predetermined times. 
     Effects of the Invention 
     According to the human mobility estimation device, the human mobility estimation method, and the human mobility estimation program of the present invention, an effect of estimating human mobility between areas accurately while accounting for traffic information is obtained. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a diagram illustrating a representation of the case of outputting human mobility between areas from the numbers of people in each area of a grid at different time periods. 
         FIG. 2  is a block diagram illustrating a configuration of a human mobility estimation device according to an embodiment. 
         FIG. 3  is a block diagram illustrating a hardware configuration of the human mobility estimation device. 
         FIG. 4  is a diagram illustrating an example of data stored in a temporal area population data storage DB. 
         FIG. 5  is a diagram illustrating an example of data stored in a temporal area population data storage DB. 
         FIG. 6  is a conceptual diagram illustrating an example of the curve of a traffic route. 
         FIG. 7  is a diagram illustrating an example of intersection determination. 
         FIG. 8  is a diagram illustrating an example of constructing a movement cost graph and calculating a shortest distance. 
         FIG. 9  is a diagram illustrating an example of data stored in an area-to-area distance storage DB. 
         FIG. 10  is a diagram illustrating an example of data saved in a human mobility storage DB. 
         FIG. 11  is a diagram illustrating an example of an algorithm used by an estimation method. 
         FIG. 12  is a flowchart illustrating the flow of a human mobility estimation process by a human mobility estimation device. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Hereinafter, an example of an embodiment of the disclosed technology will be described with reference to the drawings. Note that in the drawings, structural elements and portions that are identical or substantially the same are denoted with the same reference signs. Also, the dimensional ratios in the diagrams are exaggerated for convenience, and may be different from actual ratios in some cases. 
     The premise of the technology according to the present disclosure is illustrated in  FIG. 1 .  FIG. 1  is a diagram illustrating a representation of the case of outputting human mobility between areas from the numbers of people in each area of a grid at different time periods. As illustrated in  FIG. 1 , the human mobility between areas is estimated from the populations of the areas. 
     The technology of the present disclosure is capable of obtaining an estimation result of the human mobility that accounts for the “ease of movement” between areas by incorporating into the estimation distances that account for map information in the above premise. As a result, a more accurate human mobility estimation result can be obtained. 
     Hereinafter, a configuration of the present embodiment will be described. 
       FIG. 2  is a block diagram illustrating a configuration of a human mobility estimation device according to the present embodiment. 
     As illustrated in  FIG. 2 , a human mobility estimation device  100  includes an operation unit  110 , a map information storage DB  120 , a temporal area population data storage DB  122 , an area-to-area distance storage DB  124 , a human mobility storage DB  126 , an intersection determination unit  130 , a movement cost graph construction unit  132 , a shortest distance calculation unit  134 , a human mobility estimation unit  136 , and an output unit  138 . 
       FIG. 3  is a block diagram illustrating a hardware configuration of the human mobility estimation device  100 . 
     As illustrated in  FIG. 3 , the human mobility estimation device  100  includes a central processing unit (CPU)  11 , read-only memory (ROM)  12 , random access memory (RAM)  13 , storage  14 , an input unit  15 , a display unit  16 , and a communication interface (I/F)  17 . The components are communicably interconnected through a bus  19 . 
     The CPU  11  is a central processing unit that executes various programs and controls each unit. In other words, the CPU  11  reads out a program from the ROM  12  or the storage  14 , and executes the program by using the RAM  13  as a work area. The CPU  11  follows a program stored in the ROM  12  or the storage  14  to control each of the above components and perform various computational processes. In the present embodiment, a human mobility estimation program is stored in the ROM  12  or the storage  14 . 
     The ROM  12  stores various programs and various data. The RAM  13  stores programs or data temporarily as a work area. The storage  14  includes a hard disk drive (HDD) or a solid-state drive (SSD), and stores various programs, including an operating system, and various data. 
     The input unit  15  includes a pointing device, such as a mouse, and a keyboard, and is used to perform various inputs. 
     The display unit  16  is a liquid crystal display for example, and displays various information. The display unit  16  may also adopt a touch panel configuration and function as the input unit  15 . 
     The communication interface  17  is an interface for communicating with other equipment such as a terminal, and a standard such as Ethernet (registered trademark), FDDI, or Wi-Fi (registered trademark) is used, for example. 
     Next, each functional component of the human mobility estimation device  100  will be described. Each functional component is achieved by causing the human mobility estimation program stored in the ROM  12  or the storage  14  to be read out, loaded into the RAM  13 , and executed by the CPU  11 . 
     The operation unit  110  receives various operations with respect to data in the map information storage DB  120 . The various operations refer to operations of registering, modifying, and deleting map information recorded in the map information storage DB  120 . The operation unit  110  receives various operations with respect to data in the temporal area population data storage DB  122 . The various operations refer to operations of registering, modifying, and deleting temporal area population data recorded in the temporal area population data storage DB  122 . Additionally, the operation unit  110  receives an instruction to execute the estimation of human mobility. 
     The map information storage DB  120  stores map information including target areas. The map information includes information about features such as railway lines, highways, and bus routes. As an example, OpenStreetMap data converted into a database format such as PostGIS or the like is conceivable as the map information. 
     The temporal area population data storage DB  122  stores population data for each area at different time periods. Additionally, the temporal area population data storage DB  122  also stores area shape data for each area. The temporal area population data storage DB  122  stores the shapes of areas in the shapefile format as an example. A shapefile is a map data file containing shape information and attribute information.  FIGS. 4 and 5  are diagrams illustrating examples of data stored in the temporal area population data storage DB  122 . 
     The intersection determination unit  130  reads out map information from the map information storage DB  120  and area shape data from the temporal area population data storage DB  122 , and for each pair of areas adjacent to each other, makes an intersection determination of whether or not a traffic route intersects the boundary between the areas. A traffic route herein refers to a structure which is expressible as the shape of a curve in a map space, such as a railway line or a highway for example, and also along the curve of which many people may conceivably move. 
     The movement cost graph construction unit  132  constructs a movement cost graph that expresses a movement cost for each area, on the basis of the intersection determination regarding an area boundary and a traffic route performed by the intersection determination unit  130  and a weight α determined for the boundary in the case of determining that an area boundary and a traffic route do not intersect. The movement cost graph is a graph that treats each area as a vertex and accounts for weights by drawing weighted undirected edges between the vertices representing adjacent areas. Here, when drawing an edge between an area i and an area j, the movement cost graph construction unit  132  assigns the weight  1  if the boundary between the area i and the area j intersects with a traffic route, and assigns the weight α (where α&gt;1) if no intersection exists. The weight α is a parameter that controls how much to consider the existence of a traffic route, and the larger the value, the more importance is placed on the existence of the traffic route. In other words, the weight α is determined according to the type of traffic route that exists near the area. Depending on the type of traffic route, in the case where an area having a highly influential route exists nearby, such as a railway line for example, a relatively larger weight α may be determined. On the other hand, in the case of a bus route or the like, the movement speed is slow compared to a railway line, and therefore a relatively smaller weight α may be determined. 
     The shortest distance calculation unit  134  calculates the corrected shortest distance between areas on the basis of the movement cost graph, and saves the calculated shortest distance in the area-to-area distance storage DB  124 . By performing a shortest route calculation on the movement cost graph constructed by the movement cost graph construction unit  132 , the shortest distance calculation unit  134  can calculate the shortest distance between areas corrected to account for the map information. For example, in the case of wanting to obtain the distance between the area i and the area j, the shortest distance calculation unit  134  calculates the shortest distance between the vertex representing i and the vertex representing j on the movement cost graph. On this occasion, calculating the shortest distance between all of the areas requires obtaining the shortest distance between all of the vertices in the movement cost graph. This can be achieved with a computational complexity of O(N 3 ) using the Warshall-Floyd algorithm taking N to be the number of areas, or with a computational complexity of O(N 2  log N) by Dijkstra&#39;s algorithm treating each vertex as the starting point. The computational complexity of Dijkstra&#39;s algorithm uses the property that the number of edges is O(N) because the movement cost graph is a planar graph. 
     Hereinafter,  FIGS. 6 to 8  will be used to describe an example of a method of calculating the shortest distance between areas while accounting for map information by the series of functions of the intersection determination unit  130 , the movement cost graph construction unit  132 , and the shortest distance calculation unit  134 .  FIG. 6  is a conceptual diagram illustrating an example of the curve of a traffic route.  FIG. 7  is a diagram illustrating an example of an intersection determination.  FIG. 8  is a diagram illustrating an example of constructing the movement cost graph and calculating the shortest distance. 
     As in  FIG. 6 , consider a case in which areas exist in a grid and a railway line exists as a curve, and at this time it is desirable to obtain the distance from the area marked with the star to the other areas. As illustrated in  FIG. 7 , first, for each pair of adjacent areas, the intersection determination unit  130  determines whether or not the boundary line between areas crosses a traffic route, which is a railway line in this example. The dashed lines are boundaries determined to have an intersection, while the solid lines are boundaries determined not to have an intersection. Next, on the basis of the intersection determination, the movement cost graph construction unit  132  draws edges to construct a graph (left side of  FIG. 8 ). In this example, α=2. Finally, the shortest distance calculation unit  134  obtains the shortest distance from the vertex representing the area marked with the star (right side of  FIG. 8 ). By adopting such a method, the shortest distance between areas where the railway line exists is calculated to be smaller than the shortest distance between areas where the railway line does not exist, and distances that account for the traffic route are obtained. In this way, the shortest distance between areas corrected to account for traffic information is calculated. 
     The area-to-area distance storage DB  124  stores the distances between areas calculated above.  FIG. 9  is a diagram illustrating an example of data stored in the area-to-area distance storage DB  124 . 
     The human mobility estimation unit  136  reads out the temporal area population data from the temporal area population data storage DB  122  and the corrected shortest distance between areas from the area-to-area distance storage DB  124 . Subsequently, the human mobility estimation unit  136  estimates the human mobility between areas for each timestep according to a function for estimating the human mobility between areas on the basis of the corrected shortest distance between areas and the population of each area at different times. The human mobility estimation unit  136  saves an estimation result of the human mobility between areas for each timestep in the human mobility storage DB  126 . The human mobility estimation unit  136  uses a time interval saved in the temporal area population data storage DB  122  as the time interval of the timestep. In the present embodiment, the time interval is set to one hour. 
       FIG. 10  is a diagram illustrating an example of data saved in the human mobility storage DB  126 . 
     The output unit  138  reads out and outputs the estimated human mobility between areas for each timestep from the human mobility storage DB  126 . 
     The above describes each component of the human mobility estimation device  100 . 
     Next, the basic principles of a specific estimation method of the human mobility estimation unit  136  will be described. To describe the functions for estimating human mobility between areas, a notation like the following is defined with respect to the signs used in the estimation. 
     Namely, V is the set of all areas, and T is the maximum value of the timestep. In other words, the timesteps are t=0, . . . , T−1. Also, G=(V, E) is an undirected graph expressing adjacency relationships between the areas. 
     Also, Γ i  is the set of movement candidate areas from the area i. 
     Also, N ti  is the population in the area i at the time t. Here, t is taken to be t=0, . . . , T−1, and i is taken to be i∈V. 
     Also, M tij  is the number of people who move from the area i to the area j during the period from the time t to the time t+1. Here, t is taken to be t=0, . . . , T−2, while i and j are taken to be i, j∈V. 
     The above is the symbol notation. 
     Hereinafter, an objective function for estimation will be defined. First, let θ 3  be the movement probability from the area i to the area j. It is postulated that the human mobility M ti ={M tij |j∈V} from the area i at the time t will be generated according to the probability in Expression (1) below, using the movement probability 
       θ i ={θ ij   |j∈Γ   i }
 
     from i. 
     
       
         
           
             
               
                 
                   
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     Taking the logarithm of Expression (2) gives the following Expression (3). 
     
       
         
           
             
               
                 
                   
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     The above uses Stirling&#39;s approximation 
       log  n!≈n  log  n−n    
     for the intermediate transformation. Also, the portion that does not depend on the desired variable to estimate is omitted as a constant. 
     Additionally, the following Expressions (4) and (5) hold as constraints expressing a law of conservation of the number of people. 
     
       
         
           
             
               
                 
                   
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     Here, it is postulated that the movement probability θ ij  can be approximated from the three elements of the ease of departure from each area, the ease of people congregating to each area, and the influence that distance exerts on the probability of movement. As an example, it is postulated that θ ij  can be written in the form of the following Expression (6a). 
     
       
         
           
             
               
                 
                   
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                                   s 
                                   j 
                                 
                                 · 
                                 
                                   exp 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         - 
                                         β 
                                       
                                       · 
                                       
                                         d 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             i 
                                             , 
                                             j 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     ∈ 
                                     
                                       
                                         Γ 
                                         i 
                                       
                                       ⁢ 
                                       \ 
                                       ⁢ 
                                       
                                         { 
                                         i 
                                         } 
                                       
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     s 
                                     k 
                                   
                                   · 
                                   
                                     exp 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           - 
                                           β 
                                         
                                         · 
                                         
                                           d 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               i 
                                               , 
                                               k 
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             ( 
                             
                               
                                 j 
                                 ≠ 
                                 i 
                               
                               , 
                               
                                 j 
                                 ∈ 
                                 
                                   Γ 
                                   i 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             ( 
                             otherwise 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     In the above, π and s are given by Expression (6b) below. Note that k is a constant for setting the total for the movement candidate areas to 1. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       6 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     π 
                     = 
                     
                       { 
                       
                         
                           π 
                           i 
                         
                         | 
                         
                           i 
                           ∈ 
                           V 
                         
                       
                       } 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     s 
                     = 
                     
                       { 
                       
                         
                           s 
                           i 
                         
                         | 
                         
                           i 
                           ∈ 
                           V 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   
                     6 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     Note that π i  is a value expressing the ease of departure from the area i, and satisfies 0≤π i ≤1. Also, s i  is a score expressing the ease of people congregating to the area i, and satisfies s 1 ≥0. Note that there is a degree of freedom with regard to the constant multiple of s i . Also, d(i, j) is the distance between the area i and the area j read out from the area-to-area distance storage DB  124 . In other words, in d(i, j), the corrected shortest distance between areas is used. Also, β is a parameter expressing the influence that distance exerts on the probability of movement, and satisfies β≥0. 
     Substituting Expression (6a) into the logarithmic likelihood according to Expression (3) gives the following Expression (7). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       7 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     log 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       P 
                       ⁡ 
                       
                         ( 
                         
                           
                             M 
                             | 
                             N 
                           
                           , 
                           π 
                           , 
                           s 
                           , 
                           β 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           t 
                           = 
                           0 
                         
                         
                           T 
                           - 
                           s 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             V 
                           
                         
                         ⁢ 
                         
                           
                             log 
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   π 
                                   i 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             M 
                             tii 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           t 
                           = 
                           0 
                         
                         
                           T 
                           - 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             V 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               j 
                               ∈ 
                               
                                 
                                   Γ 
                                   i 
                                 
                                 ⁢ 
                                 \ 
                                 ⁢ 
                                 
                                   { 
                                   i 
                                   } 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               { 
                               
                                 
                                   log 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     π 
                                     i 
                                   
                                 
                                 + 
                                 
                                   log 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     s 
                                     j 
                                   
                                 
                                 - 
                                 
                                   β 
                                   · 
                                   
                                     d 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         i 
                                         , 
                                         j 
                                       
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   log 
                                   ( 
                                   
                                     
                                       ∑ 
                                       
                                         k 
                                         ∈ 
                                         
                                           
                                             Γ 
                                             i 
                                           
                                           ⁢ 
                                           \ 
                                           ⁢ 
                                           
                                             { 
                                             i 
                                             } 
                                           
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         s 
                                         k 
                                       
                                       ⁢ 
                                       
                                         exp 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               - 
                                               β 
                                             
                                             · 
                                             
                                               d 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   i 
                                                   , 
                                                   k 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               } 
                             
                             ⁢ 
                             
                               M 
                               tij 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           t 
                           = 
                           0 
                         
                         
                           T 
                           - 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             V 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               j 
                               ∈ 
                               
                                 Γ 
                                 i 
                               
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 M 
                                 tij 
                               
                               - 
                               
                                 
                                   M 
                                   tij 
                                 
                                 ⁢ 
                                 log 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   M 
                                   tij 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     + 
                     
                       const 
                       . 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     The logarithmic likelihood function of Expression (7) is used to estimate M, π, s, and β according to a maximum likelihood estimation. The optimization problem to solve is expressed by the following Expressions (8a) to (8d). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       8 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     maximize 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ℒ 
                       ⁡ 
                       
                         ( 
                         
                           M 
                           , 
                           π 
                           , 
                           s 
                           , 
                           β 
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     8 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       subject 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       to 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         N 
                         ti 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           j 
                           ∈ 
                           
                             Γ 
                             i 
                           
                         
                       
                       ⁢ 
                       
                         
                           M 
                           tij 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               t 
                               = 
                               0 
                             
                             , 
                             1 
                             , 
                             … 
                             ⁢ 
                             
                                 
                             
                             , 
                             
                               T 
                               - 
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     8 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     N 
                     
                       
                         t 
                         + 
                         1 
                       
                       , 
                       i 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         j 
                         ∈ 
                         
                           Γ 
                           i 
                         
                       
                     
                     ⁢ 
                     
                       
                         M 
                         tji 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             t 
                             = 
                             0 
                           
                           , 
                           2 
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           
                             T 
                             - 
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     8 
                     ⁢ 
                     c 
                   
                   ) 
                 
               
             
             
               
                 
                   M 
                   ≥ 
                   0 
                 
               
               
                 
                   ( 
                   
                     8 
                     ⁢ 
                     d 
                   
                   ) 
                 
               
             
           
         
       
     
     Also, let the following Expression (9) and (10) hold. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       9 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ℒ 
                     ⁡ 
                     
                       ( 
                       
                         M 
                         , 
                         π 
                         , 
                         s 
                         , 
                         β 
                       
                       ) 
                     
                   
                   := 
                   
                     
                       ∑ 
                       
                         t 
                         = 
                         0 
                       
                       
                         T 
                         - 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           ∈ 
                           V 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               Γ 
                               i 
                             
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               log 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 ij 
                               
                               ⁢ 
                               
                                 M 
                                 tij 
                               
                             
                             + 
                             
                               M 
                               tij 
                             
                             - 
                             
                               
                                 M 
                                 tij 
                               
                               ⁢ 
                               log 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 M 
                                 tij 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
             
               
                 
                   = 
                   
                     
                       
                         ∑ 
                         
                           t 
                           = 
                           0 
                         
                         
                           T 
                           - 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             V 
                           
                         
                         ⁢ 
                         
                           
                             log 
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   π 
                                   i 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             M 
                             tii 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           t 
                           = 
                           0 
                         
                         
                           T 
                           - 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             V 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               j 
                               ∈ 
                               
                                 
                                   Γ 
                                   i 
                                 
                                 ⁢ 
                                 \ 
                                 ⁢ 
                                 
                                   { 
                                   i 
                                   } 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               { 
                               
                                 
                                   log 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     π 
                                     i 
                                   
                                 
                                 + 
                                 
                                   log 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     s 
                                     j 
                                   
                                 
                                 - 
                                 
                                   β 
                                   · 
                                   
                                     d 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         i 
                                         , 
                                         j 
                                       
                                       ) 
                                     
                                   
                                 
                                 - 
                                 
                                   log 
                                   ( 
                                   
                                     
                                       ∑ 
                                       
                                         k 
                                         ∈ 
                                         
                                           
                                             Γ 
                                             i 
                                           
                                           ⁢ 
                                           \ 
                                           ⁢ 
                                           
                                             { 
                                             i 
                                             } 
                                           
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         s 
                                         k 
                                       
                                       ⁢ 
                                       
                                         exp 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               - 
                                               β 
                                             
                                             · 
                                             
                                               d 
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   i 
                                                   , 
                                                   k 
                                                 
                                                 ) 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                               } 
                             
                             ⁢ 
                             
                               M 
                               tij 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           t 
                           = 
                           0 
                         
                         
                           T 
                           - 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             V 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               j 
                               ∈ 
                               
                                 Γ 
                                 i 
                               
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 M 
                                 tij 
                               
                               - 
                               
                                 
                                   M 
                                   tij 
                                 
                                 ⁢ 
                                 log 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   M 
                                   tij 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Although the expressions are solvable in this form, consider solving the optimization problem in the form of Expression (11) below to also account for factors such as the case where noise is present in the observation. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       10 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       ℒ 
                       ′ 
                     
                     ⁡ 
                     
                       ( 
                       
                         M 
                         , 
                         π 
                         , 
                         s 
                         , 
                         β 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ℒ 
                       ⁡ 
                       
                         ( 
                         
                           M 
                           , 
                           π 
                           , 
                           s 
                           , 
                           β 
                         
                         ) 
                       
                     
                     - 
                     
                       
                         λ 
                         2 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             t 
                             = 
                             0 
                           
                           
                             T 
                             - 
                             2 
                           
                         
                         ⁢ 
                         
                           
                              
                             
                               
                                 N 
                                 ti 
                               
                               - 
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     ∈ 
                                     
                                       Γ 
                                       i 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   M 
                                   tij 
                                 
                               
                             
                              
                           
                           2 
                         
                       
                     
                     - 
                     
                       
                         λ 
                         2 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             t 
                             = 
                             0 
                           
                           
                             T 
                             - 
                             2 
                           
                         
                         ⁢ 
                         
                           
                              
                             
                               
                                 N 
                                 
                                   
                                     t 
                                     + 
                                     1 
                                   
                                   , 
                                   i 
                                 
                               
                               - 
                               
                                 
                                   ∑ 
                                   
                                     j 
                                     ∈ 
                                     
                                       Γ 
                                       i 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   M 
                                   tji 
                                 
                               
                             
                              
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The above is subject to the constraints expressed by the following Expressions (12a) and (12b). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       11 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     maximize 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         L 
                         ′ 
                       
                       ⁡ 
                       
                         ( 
                         
                           M 
                           , 
                           π 
                           , 
                           s 
                           , 
                           β 
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     12 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     subject 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     to 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     M 
                   
                   ≥ 
                   0 
                 
               
               
                 
                   ( 
                   
                     12 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     Also, λ is a parameter that controls how strongly the constraint is obeyed. 
     Expression (11) above is a function for estimating the human mobility between areas. 
     Next, the optimization of M will be described. Because the objective function L′ is concave for M, a convex optimization method such as the L-BFGS-B algorithm (see Reference Literature 1) can be used to obtain a globally optimal solution.
     [Reference Literature 1] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, A Limited Memory Algorithm for Bound Constrained Optimization., SIAM Journal on Scientic Computing, vol. 16, pp. 1190-1208, 1995.   

     Next, the optimization of π will be described. Rearranging the objective function L′ in terms of π and β gives the following Expression (13a). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       12 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ℒ 
                     ′ 
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         ∈ 
                         V 
                       
                     
                     ⁢ 
                     
                       [ 
                       
                         
                           
                             log 
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   π 
                                   i 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   t 
                                   = 
                                   0 
                                 
                                 
                                   T 
                                   - 
                                   s 
                                 
                               
                               ⁢ 
                               
                                 M 
                                 tii 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           log 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               π 
                               i 
                             
                             · 
                             
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     t 
                                     = 
                                     0 
                                   
                                   
                                     T 
                                     - 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ∑ 
                                     
                                       j 
                                       ∈ 
                                       
                                         
                                           Γ 
                                           i 
                                         
                                         ⁢ 
                                         \ 
                                         ⁢ 
                                         
                                           { 
                                           i 
                                           } 
                                         
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     M 
                                     tij 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   
                     13 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     In the above, the portion not dependent on π and β is omitted. The value π* that would maximize the expression can be denoted by a closed form like the following Expression (13b) by Lagrange&#39;s method of undetermined multipliers. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       13 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     π 
                     i 
                     * 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           t 
                           = 
                           0 
                         
                         
                           T 
                           - 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               
                                 Γ 
                                 i 
                               
                               ⁢ 
                               \ 
                               ⁢ 
                               
                                 { 
                                 i 
                                 } 
                               
                             
                           
                         
                         ⁢ 
                         
                           M 
                           tij 
                         
                       
                     
                     
                       
                         ∑ 
                         
                           t 
                           = 
                           0 
                         
                         
                           T 
                           - 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               Γ 
                               i 
                             
                           
                         
                         ⁢ 
                         
                           M 
                           tij 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     13 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     Next, the optimization of s and β will be described. Rearranging the objective function L′ in terms of s and β gives the following Expression (14a). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     14 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ℒ 
                     ′ 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           ∈ 
                           V 
                         
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             
                               A 
                               i 
                             
                             ⁢ 
                             log 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               s 
                               i 
                             
                           
                           - 
                           
                             
                               B 
                               i 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               log 
                               ( 
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     ∈ 
                                     
                                       
                                         Γ 
                                         i 
                                       
                                       ∖ 
                                       
                                         { 
                                         i 
                                         } 
                                       
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     s 
                                     k 
                                   
                                   ⁢ 
                                   
                                     exp 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           - 
                                           β 
                                         
                                         · 
                                         
                                           d 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               i 
                                               , 
                                               k 
                                             
                                             ) 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     - 
                     
                       β 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       D 
                     
                   
                 
               
               
                 
                   ( 
                   
                     14 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     In the above, A i  and D are given by the following Expression (14b). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     15 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       A 
                       i 
                     
                     := 
                     
                       
                         
                           ∑ 
                           
                             t 
                             = 
                             0 
                           
                         
                         
                           T 
                           - 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               
                                 Γ 
                                 i 
                               
                               ∖ 
                               
                                 { 
                                 i 
                                 } 
                               
                             
                           
                         
                         ⁢ 
                         
                           M 
                           tji 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       B 
                       i 
                     
                     := 
                     
                       
                         
                           ∑ 
                           
                             t 
                             = 
                             0 
                           
                         
                         
                           T 
                           - 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             ∈ 
                             
                               
                                 Γ 
                                 i 
                               
                               ∖ 
                               
                                 { 
                                 i 
                                 } 
                               
                             
                           
                         
                         ⁢ 
                         
                           M 
                           tij 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     D 
                     := 
                     
                       
                         
                           ∑ 
                           
                             t 
                             = 
                             0 
                           
                         
                         
                           T 
                           - 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             ∈ 
                             V 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               j 
                               ∈ 
                               
                                 
                                   Γ 
                                   i 
                                 
                                 ∖ 
                                 
                                   { 
                                   i 
                                   } 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               d 
                               ⁡ 
                               
                                 ( 
                                 
                                   i 
                                   , 
                                   j 
                                 
                                 ) 
                               
                             
                             · 
                             
                               M 
                               tij 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     14 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     In the above, the portion not dependent on π and β is omitted. For simplicity, the right side is denoted f(s, β). 
     To maximize f(s, β), a framework referred to as the Minorization-Maximization algorithm (hereinafter, the MM algorithm; see Reference Literature 2) is used. The MM algorithm is a technique of generating a candidate point sequence in cases where directly maximizing a function is difficult, and works by iteratively solving a maximization problem of an approximation function for the lower bound.
     [Reference Literature 2] D. R. Hunter. MM algorithms for generalized Bradley-Terry models. The Annals of Statistics, Vol. 32, No. 1, pp. 384-406, Feb. 2003.   

     A specific method of applying the MM algorithm will be described. The following Expression (15) holds for x, y&gt;0. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     16 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       - 
                       log 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     x 
                   
                   ≥ 
                   
                     1 
                     - 
                     
                       log 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       y 
                     
                     - 
                     
                       x 
                       y 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     In the above, x i  and y i  are given by the following Expression (16). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     17 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       x 
                       i 
                     
                     = 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             
                               Γ 
                               i 
                             
                             ∖ 
                             
                               { 
                               i 
                               } 
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           s 
                           k 
                         
                         ⁢ 
                         
                           exp 
                           ⁡ 
                           
                             ( 
                             
                               
                                 - 
                                 β 
                               
                               · 
                               
                                 d 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     i 
                                     , 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       y 
                       i 
                     
                     = 
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             
                               Γ 
                               i 
                             
                             ∖ 
                             
                               { 
                               i 
                               } 
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           s 
                           k 
                           
                             ( 
                             u 
                             ) 
                           
                         
                         ⁢ 
                         
                           exp 
                           ⁡ 
                           
                             ( 
                             
                               
                                 - 
                                 
                                   β 
                                   
                                     ( 
                                     u 
                                     ) 
                                   
                                 
                               
                               · 
                               
                                 d 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     i 
                                     , 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Expression (15) is applied to i∈V to obtain the following Expression (17). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     18 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     f 
                     ⁡ 
                     
                       ( 
                       
                         s 
                         , 
                         β 
                       
                       ) 
                     
                   
                   ≥ 
                   
                     
                       
                         ∑ 
                         
                           i 
                           ∈ 
                           V 
                         
                       
                       ⁢ 
                       
                         [ 
                         
                           
                             
                               A 
                               i 
                             
                             ⁢ 
                             log 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               s 
                               i 
                             
                           
                           - 
                           
                             
                               C 
                               i 
                               
                                 ( 
                                 u 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   ∈ 
                                   
                                     
                                       Γ 
                                       i 
                                     
                                     ∖ 
                                     
                                       { 
                                       i 
                                       } 
                                     
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   s 
                                   k 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   exp 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         - 
                                         β 
                                       
                                       · 
                                       
                                         d 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             i 
                                             , 
                                             k 
                                           
                                           ) 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     - 
                     βD 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     In the above, C i   (u)  is expressed by the following Expression (18). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     19 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     C 
                     i 
                     
                       ( 
                       u 
                       ) 
                     
                   
                   := 
                   
                     
                       B 
                       i 
                     
                     
                       
                         ∑ 
                         
                           k 
                           ∈ 
                           
                             
                               Γ 
                               i 
                             
                             ∖ 
                             
                               { 
                               i 
                               } 
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           s 
                           k 
                           
                             ( 
                             u 
                             ) 
                           
                         
                         ⁢ 
                         
                           exp 
                           ⁡ 
                           
                             ( 
                             
                               
                                 - 
                                 
                                   β 
                                   
                                     ( 
                                     u 
                                     ) 
                                   
                                 
                               
                               · 
                               
                                 d 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     i 
                                     , 
                                     k 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     If the right side of Expression (17) is denoted f (u) (s, β), the properties of the following Expressions (19) and (20) hold. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       20 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     f 
                     ⁡ 
                     
                       ( 
                       
                         
                           s 
                           
                             ( 
                             u 
                             ) 
                           
                         
                         , 
                         
                           β 
                           
                             ( 
                             u 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       f 
                       
                         ( 
                         u 
                         ) 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           s 
                           
                             ( 
                             u 
                             ) 
                           
                         
                         , 
                         
                           β 
                           
                             ( 
                             u 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     f 
                     ⁡ 
                     
                       ( 
                       
                         s 
                         , 
                         β 
                       
                       ) 
                     
                   
                   ≥ 
                   
                     
                       
                         f 
                         
                           ( 
                           u 
                           ) 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           s 
                           , 
                           β 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           ∀ 
                           s 
                         
                         , 
                         β 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     Using the above, f(s, β) is maximized by Algorithm 1 in  FIG. 11 . In Algorithm 1, the objective function f(s, β) increases monotonically, as demonstrated by Expression (21a). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       21 
                     
                     ] 
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           f 
                           ⁢ 
                           
                             ( 
                             
                               
                                 s 
                                 
                                   ( 
                                   
                                     u 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                               , 
                               
                                 β 
                                 
                                   ( 
                                   
                                     u 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         ≥ 
                       
                     
                     
                       
                         
                           f 
                           
                             ( 
                             u 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               s 
                               
                                 ( 
                                 
                                   u 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                             , 
                             
                               β 
                               
                                 ( 
                                 
                                   u 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         ( 
                         
                           ∵ 
                           
                             ( 
                             20 
                             ) 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       ≥ 
                     
                     
                       
                         
                           
                             f 
                             
                               ( 
                               u 
                               ) 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 s 
                                 
                                   ( 
                                   
                                     u 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                               , 
                               
                                 β 
                                 
                                   ( 
                                   u 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         ≥ 
                         
                           
                             f 
                             
                               ( 
                               u 
                               ) 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 s 
                                 
                                   ( 
                                   u 
                                   ) 
                                 
                               
                               , 
                               
                                 β 
                                 
                                   ( 
                                   u 
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                           
                       
                     
                   
                   
                     
                       = 
                     
                     
                       
                         
                           f 
                           
                             ( 
                             u 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               s 
                               
                                 ( 
                                 u 
                                 ) 
                               
                             
                             , 
                             
                               β 
                               
                                 ( 
                                 u 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       
                         ( 
                         
                           ∵ 
                           
                             ( 
                             19 
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     21 
                     ⁢ 
                     a 
                   
                   ) 
                 
               
             
           
         
       
     
     Next, the formula for updating s and β in Algorithm 1 will be derived. First, consider a substitution like the following for s (u+1) . 
     
       
         
           
             
               s 
               
                 ( 
                 
                   u 
                   + 
                   1 
                 
                 ) 
               
             
             ← 
             
               
                 
                   arg 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   max 
                 
                 s 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   f 
                   
                     ( 
                     u 
                     ) 
                   
                 
                 ⁡ 
                 
                   ( 
                   
                     s 
                     , 
                     
                       β 
                       
                         ( 
                         u 
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Because the following Expression (21b) holds, s (u+1)  can be obtained in a closed form. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     22 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       ∂ 
                       
                         
                           f 
                           
                             ( 
                             u 
                             ) 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             s 
                             , 
                             
                               β 
                               
                                 ( 
                                 u 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                     
                       ∂ 
                       
                         s 
                         i 
                       
                     
                   
                   = 
                   
                     
                       0 
                       ⇔ 
                       
                         s 
                         i 
                       
                     
                     = 
                     
                       
                         A 
                         i 
                       
                       
                         
                           ∑ 
                           
                             k 
                             ∈ 
                             
                               
                                 Γ 
                                 i 
                               
                               ∖ 
                               
                                 { 
                                 i 
                                 } 
                               
                             
                           
                         
                         ⁢ 
                         
                           
                             C 
                             k 
                             
                               ( 
                               u 
                               ) 
                             
                           
                           ⁢ 
                           
                             exp 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   
                                     β 
                                     
                                       ( 
                                       u 
                                       ) 
                                     
                                   
                                 
                                 · 
                                 
                                   d 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       k 
                                       , 
                                       i 
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     21 
                     ⁢ 
                     b 
                   
                   ) 
                 
               
             
           
         
       
     
     Next, consider a substitution like the following for β (u+1) . 
     
       
         
           
             
               s 
               
                 ( 
                 
                   u 
                   + 
                   1 
                 
                 ) 
               
             
             ← 
             
               
                 
                   arg 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   max 
                 
                 s 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   f 
                   
                     ( 
                     u 
                     ) 
                   
                 
                 ⁡ 
                 
                   ( 
                   
                     s 
                     , 
                     
                       β 
                       
                         ( 
                         u 
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     By a simple calculation, the following can be confirmed for ∀β∈R (where R is the set of real numbers). 
     
       
         
           
             
               
                 
                   ∂ 
                   2 
                 
                 ⁢ 
                 
                   
                     f 
                     
                       ( 
                       u 
                       ) 
                     
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         ɛ 
                         
                           ( 
                           
                             u 
                             + 
                             1 
                           
                           ) 
                         
                       
                       , 
                       β 
                     
                     ) 
                   
                 
               
               
                 ∂ 
                 
                   β 
                   2 
                 
               
             
             &lt; 
             0 
           
         
       
     
     In other words, f(u) is a concave function with respect to β. Consequently, to obtain β (u+1) , it is sufficient to solve a concave, single-variable function maximization problem with respect to β, and β (u+1)  can be obtained efficiently by a technique such as the golden-section search or Newton&#39;s method. 
     The above is a description of the basic principles of the estimation method. 
     Next, the operations of the human mobility estimation device  100  will be described. 
       FIG. 12  is a flowchart illustrating the flow of the human mobility estimation process by the human mobility estimation device  100 . The human mobility estimation process is performed by causing the human mobility estimation program to be read out from the ROM  12  or the storage  14 , loaded into the RAM  13 , and executed by the CPU  11 . The human mobility estimation device  100  executes the following process upon receiving an instruction to estimate the human mobility through the operation unit  110 . Note that data is preregistered in the map information storage DB  120  and the temporal area population data storage DB  122 . 
     In step S 100 , the CPU  11  acts as the intersection determination unit  130  to read out map information from the map information storage DB  120  and area shape data from the temporal area population data storage DB  122 . 
     In step S 102 , the CPU  11  acts as the intersection determination unit  130  to make the intersection determination of whether or not a traffic route intersects a boundary between areas, the determination being made for each pair of adjacent areas. 
     In step S 104 , the CPU  11  acts as the movement cost graph construction unit  132  to construct a movement cost graph expressing the movement cost for each area on the basis of the intersection determination made by the intersection determination unit  130  and the weight X determined for a boundary in the case where the intersection determination is made that no intersection exists. 
     In step S 106 , the CPU  11  acts as the shortest distance calculation unit  134  to calculate the corrected shortest distance between areas on the basis of the movement cost graph, and saves the calculated shortest distance in the area-to-area distance storage DB  124 . 
     In step S 108 , the CPU  11  acts as the human mobility estimation unit  136  to read out temporal area population data from the temporal area population data storage DB  122  and the corrected shortest distance between areas from the area-to-area distance storage DB  124 . 
     In step S 110 , the CPU  11  acts as the human mobility estimation unit  136  to estimate the human mobility between areas for each timestep according to the function in Expression (11) above, on the basis of the corrected shortest distance between areas and the population of each area at different times. 
     In step S 112 , the CPU  11  acts as the output unit  138  to output the human mobility between areas for each timestep estimated in step S 110  as an estimation result. 
     According to the human mobility estimation device  100  of the present embodiment as described above, human mobility between areas can be estimated accurately while accounting for traffic information. 
     Note that the human mobility estimation executed by the CPU loading software (a program) in the foregoing embodiment may also be executed by any of various types of processors other than a CPU. The processor in this case may be an programmable logic device (PLD) whose circuit configuration is changeable after fabrication, such as a field-programmable gate array (FPGA), or a special-purpose electric circuit that acts as a processor having a circuit configuration specially designed to execute a specific process, such as an application-specific integrated circuit (ASIC), for example. In addition, the human mobility estimation may also be executed by a single processor from among these various types, or by a combination of two or more processors of the same or different types (such as a combination of multiple FPGAs, or a CPU and an FPGA, for example). Furthermore, the hardware structure of these various types of processors is more specifically an electric circuit combining circuit elements such as semiconductor elements. 
     Additionally, the foregoing embodiment describes an aspect in which the human mobility estimation program is stored (installed) in advance in the storage  14 , but is not limited thereto. The program may also be provided by being stored on a non-transitory storage medium such as Compact Disc-Read-Only Memory (CD-ROM), Digital Versatile Disc-Read-Only Memory (DVD-ROM), or Universal Serial Bus (USB) memory. Furthermore, the program may also be downloaded from an external device over a network. 
     The following supplement is additionally disclosed regarding the foregoing embodiment. 
     (Supplementary Item 1) 
     A human mobility estimation device comprising: 
     a memory; and 
     at least one processor connected to the memory, wherein 
     the processor is configured to
         make an intersection determination of whether or not a traffic route included in map information intersects a boundary between areas on a basis of the traffic route and an area shape between areas that are adjacent in the map information, the intersection determination being made for each pair of adjacent areas,   construct, on a basis of a weight determined for the boundary in a case where the intersection determination is made that no intersection exists, a movement cost graph expressing a movement cost for each area that accounts for the weight,   calculate a corrected shortest distance between areas on a basis of the movement cost graph, and   estimate a human mobility between areas for each timestep, the timestep being the length of a predetermined time interval, according to a predetermined function for estimating the human mobility between areas on a basis of the corrected shortest distance between areas and a population of each of the areas at different predetermined times.       

     (Supplementary Item 2) 
     A non-transitory storage medium storing a human mobility estimation program that causes a computer to execute a process comprising: 
     making an intersection determination of whether or not a traffic route included in map information intersects a boundary between areas on a basis of the traffic route and an area shape between areas that are adjacent in the map information, the intersection determination being made for each pair of adjacent areas; 
     constructing, on a basis of a weight determined for the boundary in a case where the intersection determination is made that no intersection exists, a movement cost graph expressing a movement cost for each area that accounts for the weight; 
     calculating a corrected shortest distance between areas on a basis of the movement cost graph; and 
     estimating a human mobility between areas for each timestep, the timestep being the length of a predetermined time interval, according to a predetermined function for estimating the human mobility between areas on a basis of the corrected shortest distance between areas and a population of each of the areas at different predetermined times. 
     REFERENCE SIGNS LIST 
     
         
         
           
               100  human mobility estimation device 
               110  operation unit 
               120  map information storage DB 
               122  temporal area population data storage DB 
               124  area-to-area distance storage DB 
               126  human mobility storage DB 
               130  intersection determination unit 
               132  movement cost graph construction unit 
               134  shortest distance calculation unit 
               136  human mobility estimation unit 
               138  output unit