Space Time Path Data Warehouse Mining based on Simplicial

Space Time Path Data Warehouse Mining based on Simplicial Complex Analysis Azedine Boulmakoul*, Lamia Karim** University of Hassan II-Mohammedia, Faculty of Science and Technology, Computer Science department BP 146 Mohammedia 20650, Mohammedia, Morocco *[email protected] m **lkarim.lkarim@g mail.co m Abstract. Current exponential growth in rich spatio-temporal data generated by positioning technologies has led to unprecedented excitement about loc ation based services. In this paper, we propose a general space time path data warehouse and the application of the Q-analysis developed by R. H. Atkin on it for extraction of implicit and potentially useful information. We propose to analyze the space time path data warehouse of a supermarket, as a case study, to highlight the usefulness of our data warehouse and mining approach that applies q-analysis method and uses topological characteristics, such as connectivenness, and eccentricity. Another objective of the proposed case study is to analyze spatio-temporal knowledge about how customers are using the supermarket’s space in a time interval. This analyze allows the supermarket respo nsible to best succeed in extracting actionable knowledge from these data, fin ding the optimal set of eye-catchers regions and also the eccentric one. The use of this knowledge permits to reorganize product disposition in the space in o rder to profit and maximize cross -selling.

1 Introduction Many interesting location based services are being developed based on trajectory analysis. For example, in a traffic management system, traffic jams may be determined by mining movement patterns of groups of cars. This allows managing trajectories, attaching whatever semantics the application requires, and developing robust and efficient methods to aggregate a set of trajectories that may support complex analysis. With the advances in mobile communication and positioning technology, large amounts of moving objects data from various types of devices, such as GPS equipped mobile phones o r vehicles with navigational equipment, has been collected. From these devices, movements of objects are collected in the form of trajectories in databases and then useful information for high quality Location Based Services (LBS) is extracted, such as traffic flow control or location-aware advertising. In supermarkets’ field, the retail uses a set of very different merchandising techniques to increase the gross profit margin through sales and cost reduction. This requires improving the efficiency of operation and providing attractive services for customers. The goal of the merchandising techniques is to get the maximum of clients in maximum regions to maximize sales and facilitate the movement. But since theses merchandising techniques are most of time probabilistic, and a supermarket, as good as it is, can lose customers and sales if its regions / products are mismatched, badly organized, or its products are often out of stock;

Space Time Path Data Warehouse Analysis based on the Concept of Simplicial Complex

The supermarket’s responsible needs to evaluate these merchandising techniques. Using qanalysis method to analyze the space-time path of supermarket’s customers, retailers could determine the regions where most of the stores sales activities occur and where customers tend to stay for a long time, then they can decide where to display products and how to organize an efficient storage environment. A more efficient storage environment can provide convenient services for customers and thus increases sales. The rest of the paper is organized as follows. Section 2 discusses the related wo rk on trajectories data mining and knowledge discovery. Next in Section 3 we present the proposed space time path data warehouse conceptual star schema using hierarchy and MultiDimER notation. Then in Section 4, we review mathematical foundations of q-analysis method which are the foundations of the core subjects of the paper. In section 5, we illustrate the use of Q-analysis on the space time path data warehouse to extract implicit and potentially useful information with a case study in a supermarket field. Finally, the last section summarizes the results obtained in this paper and discusses directions for future work.

2 Related Works Several works for trajectory data mining and knowledge discovery have focused on the geometrical properties of trajectories, without considering the moving object’s activities and the background semantic information. However, useful information, needed by different applications fields, may only be extracted from trajectory data if their moving object’s activities and the background semantic information are considered (Alvares, 2007). In the literature, there is only a few works that consider semantic information of trajectories. In (Alvares, 2007), an intersection based approach is presented to integrate trajectory sample point s with geographic information, with the algorithm SMoT. Ashbrook and Starner (2003) use the well-know clustering algorithm K-means to find important places in trajectories. The problem of K-means is that the number of clusters must be given a priori, what in our problem is unknown. Zhou (2007) uses information obtained from GPS to classify personal gazetteers. The gazetteers correspond to the most important places of a person, such as home, work, supermarket, etc. A set of trajectories is processed by the DJ-Cluster algorithm in order to find the baseline places. DJ-Cluster is a density-based algorithm similar to DBSCAN that works on the notion of connectivity between neighborhoods. Howe ver, DJ-Cluster does not consider the temporal dimension. GDBSCAN (Sander et al., 1998) is another extension of DBSCAN, developed for clustering non-spatial attributes. ST-DBSCAN (Birant et al.,2007) was developed to consider both spatial and temporal aspects, but it treats them sep arately in the evaluation function. In our case, we use the spatio-temporal data together to consider activities and semantic information. Several works for trajectory clustering have been developed to find similar sub trajectories or dense regions as (Giannotti et al., 2007), (Lee et al., 2007) and (Nanni et al., 2006). Moving object’s trajectories have been studied by several research communities in the context of several locations based fields. Several methods have been proposed for trajectories data analysis, including the use of spatio-temporal databases and data mining techniques. They can be classified according to the following groups: (i) generation of trajectory according to the geometrical properties of trajectories (Laube et al., 2005); (ii) extraction of clusters of sample points from trajectories, basically using time and space to determine trajectories located in dense regions, trajectories with similar shapes or distances, and traject ories that

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move between regions during the same time interval (Giannotti et al., 2007); (iii) analysis of trajectories from a semantic point of view, trying to add context information (Spaccapietra et al., 2008) et (Rocha et al., 2010); (iv) development of architectures, ETL (Extraction, Tran sformation and Loading) processes, data models and languages for h elping in the construction of trajectory data warehouses (TDWs). In Yavas et al. (2005), a mining algorithm for predicting user movements in a Personal Communication Systems network was proposed, which defines the mobility pattern as a sequence of cells in the network and mines frequent paths based on sequential pattern mining. Similar to the work in (Tsoukatos et al., 2001), which employs grid-based spatial decompositions for discretizing data uses a predefined cell network for represent ing spatial properties in data. Mamoulis et al. address the problem of mining sequential patterns from spatio temporal data by considering the patterns as th e form of trajectory segments (Mamoulis et al., 2004). They first decompose the original trajectories into segments then, group them according to their shape and closeness. They introduce a spatio-temporal pattern mining algorithm, which finds frequent sequential patterns based on a tree stru cture and an Apriori paradigm. Moreover (Mamoulis et al., 2004) needs a lot of complex computations for sorting and merging segments repeatedly and it degrades overall processing performance. Giannotti et al. (2007) describes movement patterns in both spatial and temporal contexts, based on RoI (Region of Interest). They first identify RoIs, which are mostly visited regions, then, find frequent patterns from sequences of regions of interest.

3 Space Time Path Data Warehouse Space time path presents individual activities under different constraints in a space time context; it models a trajectory as sequence of activities. The Unified Moving Object Trajectories’ Meta-model (Boulmakoul et al., 2012) describes a general meta-model that could be used by different application domains; it can also use an object approach and integrates previous trajectories models described in literature. Using the space-time event ontology, the meta-model models Space according to OGC Spatial Data Model (OGC, 2008), Observation domain of trajectory, according to OGC Sensor Meta Model and OGC Feature Type, Phys ical and virtual activities between the beginning and the end of Space Time Path (Shaw, 2011), sensors used for collecting moving object’s traces and Movement patterns using composite Region of Interest. Data resulting from different spatio-temporal sensors are gathered for decision making purposes such as a new hypermarket implementation. Multidimensional modeling is the foundation of data warehouses (Song et al., 2001). It is characterized by two primitives which are Facts and Dimensions. Those latter are used to construct the star schema, the snowflake schema or the constellation schema. Malinowski et al. (2006) defined a MultiDimER schema as a finite set of dimensions and fact relationships. It allows represen ting facts with measures as well as the different kinds of hierarchies. Using hierarchies, in the proposed schema, allows to benefit from roll-up and drill-down operations. In the Figure 1, we present the proposed conceptual space time path warehousing star schema including hierarchy. Thanks to redundancy, we can provide horizontally-scalable systems as distribution of data across multiple machines is easy and does not cause problems. Additionally, space time path data warehouse schema presents trajectories in both spatial and te mporal contexts based on Region of Interest and activities. The spatial neighborhood is presented using the following dimensions: Point of Interest dimension characterized by longitude,

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Space Time Path Data Warehouse Analysis based on the Concept of Simplicial Complex

latitude and name. A hierarchy of spatial neighborhood is used for roll-up and drill-down operations. Area of interests has a shape and a surface. City region of interests to present a city as a region of interests, it contains about the city region of interest name, su rface, first language and religion. In other applications domain, region of in terests could be presented as a Voronoi network of interest or a modal network.

Semantic Move

Semantic Begin

Moving Object

id_semantic_move t_begin_move t_end_move

id_semantic_begin spatial_point_begin semantic_type adress_begin t_begin

id_MO fist_name last_name age adress tel email mobile_sensor_type

Semantic object T oponyme

Semantic Stop id_semantic_stop spatial_point semantic_type adress_stop t_begin_stop t_end_stop

Semantic End T ime period id_Period begin_Period end_Period

id_semantic_end spatial_point semantic_type adress_end t_end

Space T ime Event id_SpaceT imeEvent type_SpaceT imeEvent

Hour id_Hour timestamp

Space T ime Path H



Spatial Object

Space Time Path2 average_duration minimum_duration Day maximum_duration average_distance name average_speed number_of_stops number_of_moves most_frequent_RoI Month most_frequent_activity name count users count trajectories shape

Year



name

Point of Interest id_PointofInterest name_PointofInterest longitude latitude



id_Spatial_Object spatial_reference geometry

Line

Point

id_Line points shape_line

id_Point longitude latitude

Polygon

Activity id_activity t_begin_activity t_en_activity duration_activity

id_Polygon name_Polygon shape_Polygon lines Drive id_vehicule duration_drive

Area of Interest id_AreaOfInterest name_AreaOfInterest shape_AreaOfInterest surface__AreaOfInterest

Shopping Call

id_Shopping type_Shopping



id_call call_to call_from destination_code cost_call

City Region of Interst id_CityRoI name_CityRoI surface_CityRoI first_language religion_CityRoI



Modal Network id_Modal_Network name_Modal_Network

FIG. 1 – Space time path warehousing conceptual star schema using hierarchy and Mult iDimER notation.

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4 Mathematical Foundations Q-analysis was originally developed by Atkin (1974, 1977), as an approach for studying the structural characteristics of social systems in which two sets of indicators, features, or characteristics are related to each other. Subsequently, Q-analysis has been applied in such diverse areas as chess urban planning, geographical studies, spatial databases, and water distribution. The technique of Q-analysis provides indices such as connectivity level, eccentricity and complexity.

4.1 Simplicial complexes We first consider a finite set V={v i , i:1..k} and a collection K of its subsets. We denote any one of these subsets by the symbol σ p when it contains (p+1) elements of V, and we call such a subset p-simplex. If σq is q-simplex defined by a (q+1) subsets of σ q , we say that σq is a face of σp and we write σq < σp. The relation < is a partial ordering of the collection K. The collection K is called a simplicial complex if and only if : (1) Each singleton set { σi } is a member of K, each being a σ 0. (2) Whenever σp ∈ K and σq < σp then σq ∈ K The set V is called the vertex set of the complex K, each p -simplex is said to be of dimension p; the largest integer n for which σ n ∈ K is called the dimension of K, and written dim K.

4.2 Algebraic combinatorial topology Let us define some basic concepts in combinatorial topology. Let  a binary relation between finites sets Y and X, where X= {x1 ,…,xn }, Y= {y i ,…,y m}, if then exists at least one y i ∈ Y, such that a (p+1) subset of X is -related to it, we call that (p+1) subsets of X a p-simplex. We write a p-simplex σp is a (p+1) subset of X: σp = . Any subset of σp, say a q-simplex, is said to be a face of σ p , we write σq≤ σp . The collection of simplicies is called Simplicial Complex K, More precisely we shall denote the co mplex by Ky(X, ).

4.3 A concept of q-nearness We introduce a notion of q-nearness between Simplicies and then we use it to analyze the proximity of chains of connection. - Definition: if two simplicies σr and σs share a q-face, that is to say σr ∩ σs = σq, we shall say that they are q-near. - Remarque: if σr and σs are q-near, they are also be t-near, for t= 0,1,…,q.

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Space Time Path Data Warehouse Analysis based on the Concept of Simplicial Complex

4.4 Chains of q-connection in K Given two simplicies σr, σs in K, we shall say they are q-connected, if there exist a finite sequence of simplicies { σi ; i=1..N} such that (i) σr is q-near σ1 (ii) σs is q-near σh (iii) σi is q-near σi+1 ; i=1, …, (h-1) We use the notation [σr, σs ] for a chain of connection between σ r and σs . The process of identifying that pieces of K which are q-connected, for all values of q ranging from 0 to N, constitutes a sequence of partitioning of simplicies of K, defined by: (σp , σr) q if and only if σp is q-connected to σr. This q is reflexive, symmetric and transitive and is therefore an equivalence relation. The quotient set K/q denotes the equivalence classes, under q . We denote the cardinality of K/q by Qq . When we analyze K by finding all the values of Q0 ,Q1 ,Q2 ,…,QN where N = dimK , we say that we have performed a Q-analysis on K.

4.5 Eccentricity For a given a simplicial family in Q-analysis approach, eccentricity (ecc) of a simplex has been devised to measure the extent to which the simplex is integrated into the simplicial family (Hocking et al., 1988). The formula for calculating eccentricity of a simplex is given by (equation 1) (Atkin, 1974): (1) where “ ” is called top-q and denotes the dimension of simplex σ and “ ” is called bottomq and is the largest number of vertices minus 1 that σ shares with any simplex. A simplex is eccentric when it is badly embedded within the complex. (Jiang et al, 2006) suggests another measure of eccentricity called ecc’ (equation 2) 2  i qi σi ecc'(σ) = q max (q max +1) (2) Where q i each q-level where σ appears, σi is the number of elements in σi ’s equivalence class at level q i and q max the maximum level of the complex. In the proposed case study (figure 2) we found the following results. The difference between ecc and ecc’ is that ecc depends on the other simplicies and takes values in interval of [0, ∞].

5 Case Study In this case study, we apply q-analysis devised by R. H. Atkin on the space time path data warehouse of a supermarket. Our objective is to analyze spatio-temporal knowledge about how customers are using the supermarket space in a time interval. This analyze allows the supermarket responsible to best succeed in extracting actionable knowledge from these data, finding the optimal set of eye-catchers regions and also the eccentric one. The use of this knowledge permits to reorganize product’s disposition in the space in order to profit and

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maximize cross-selling. The supermarket space time path network between 15 spatial regions and 10 trajectories of 10 volunteers’ customers is shown in figure 2. We partition the supermarket space in a voronoi network rather than grids representation because the spatial stru cture of a supermarket is dynamic. The regions are Dairy, Frozen Food, Drinks, Snacks, Canned Foods, Bakery, Chilled Food, Household, Merchandise, Cleaning Products, Pets, World Cuisine, Fruit & Veg, Sea Food, Meat noted R1 , R2, R3, R4, R5, R6, R7, R8, R9, R10, R11, R12, R13, R14, R15 and 10 trajectories of 10 volunteers customers are noted T1 , T2, T3, T4, T5, T6, T7, T8, T9, T10. T2

T4 T1

T10

R1 R3 R2

R4

R5

T8

R10

T5

R6

R7

R9 T7

R8

T6

R15 R11

R13

R14

R12 T3

T9

FIG. 2 – The supermarket space time path network. To do this, we marked all points of interests of the studied supermarket and then we partitioned the space of the supermarket as a network of regions. The Voronoi cell of a point p ∈ S, defined Vp, is the set of points x ∈ R that are closer to p than to any other point in S. The union of the Voronoi cells of all generating points p ∈ S forms the Voronoi diagram of S, defined VD(S). The different regions/cells are shown on the map and recorded in the dat abase in the form of geometric objects. In the other hand, to track the spatio-temporal positions of the supermarket’s customers, we have equipped a set of passengers / moving objects with positioning devices that are used to send, every 30 seconds, information about the trajectory of each mobile. The structure of each message sent is as follows: moving_object_id; latitude; longitude; date; time; observ ation. A collector process that use asynchronous sockets with a pool is developed to collect the received spatio-temporal customers’ messages and then construct the space time path data warehouse. We evaluate in the present case study the supermarket’s merchandising technique from spatial perspective. Otherwise, we aim to see if all regions constituting the supermarket is connected (can converge into the same cluster group), detect regions of the same class and also regions / products isolated. Certainly we can know through sales and stock store analysis, products with a low sales rate, but we cannot detect exactly where the nature of the product that is not interesting or it has deposited in an isolated area of the store. Hence there is need of analyzing the shopping paths of supermarket's customers.

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Space Time Path Data Warehouse Analysis based on the Concept of Simplicial Complex

Let T be a set of trajectories and R a set of regions. D is a database of trajectories with region presentation TR, where each TR has a unique identifier (trid) and contains a set of regions. Based on the space time path data warehouse, we represent our trajectories/customer to region/place relationship in map form (Fig. 2) for analytical purposes, as the incidence matrix Λ=(T x R), see Table 1. Each element a ij in Λ is only one of the two values 1 and 0, with 1 (respectively 0) indicating the presence (respectively absence) of the i th customer in the jth region. Hence this matrix is an incidence matrix. A simplex, which is a basic object in the study of algebraic topology, is attached to each column of the inc idence matrix, and then the collection of these simplexes forms a simplicial family.   T  1  T2  T  3  T4  T5 T  6  T7  T8 T  9  T10 

R1 0 1 0 0 0

0 0 0 0 0

R2

R3

0

0 0 0 0

1 0 0 1 0 0

0 0 0 0

0 0 0 0

1

R4 1 0

R5

0 0 0 0

0 0 0 0 0 0

0 0 0 0

0 0 0 1

R6 0 0 1 0 1 0 0 0 0 0

R7 0 1 0 0 1 0 0 0 0 0

R8 0 0 0 0 0 0 0 0 0 1

R9 0 0 0 1 0 0 0 1 0 0

R10 1 0 0 0 0 0 0 1 0 0

R11 R12 R13 R14 R15  0 0 0 0 0  0 0 0 0 0  1 1 1 1 0   0 0 1 1 0  0 0 0 0 0  1 0 0 0 0   0 0 0 1 1  0 0 0 1 1  0 1 1 1 1  0 0 0 0 0 

TAB. 1 – Incidence matrix for analyzing Supermarket’ Space Time Path (Regions).   R  1  R2 R  3  R4  R5 R  6 R T Λ×Λ -Ω =  7  R8 R  9  R10  R11   R12  R13 R  14  R15 

R1 0

R2 0 0

R3   1

R4   0 0

R5     0

R6      1

R7 0 0    0 1

R8     0   0

R9   0      1

R10 R11 R12 R13 R14 R15                0   0 0    0               0 0 0 0                  0   0 1 0  1    0 0  1 0 0 0   1 1 1 0  2 2 0  4 2   2 

TAB. 2 – Shared face Matrix for analyzing Supermarket’ Space Time Path (Regions). At q = 4 we At q = 3 we At q = 2 we At q = 1 we At q = 0 we

have Q4 = 1; have Q3 = 1; have Q2 = 1; have Q1 = 6; have Q0 = 1;

{R14 } {R14 } { R13 , R14 , R15 } { R3 } { R6 } { R7 } { R10 } { R11 }{ R9 , R12 R13 , R14 , R15 } all ; { R1 , R2 R3 , R4 , R5 , R6 R7 , R8 , R9 , R10 R11 , R12 , R13 , R14 , R15 }

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4321 0

Q = {111 6 1 }

TAB. 3 – Summary of results for analyzing Supermarket’ Space Time Path (Regions). Overall, the results of Q-analysis of the complex in table were interesting enough. They showed that in the present case study, there are the following equivalence classes at the different dimensional levels of q=0, q=1, q=2, q=3 and q=4. Each equivalence class is enclosed in the curly brackets (TAB. 3). The sign “-” in the matrix stands for -1, and shows that there 4321 0

is disconnected components (Table 2).The structure vector Q = {111 6 1 } indicates the number of q-connected components at each level; thus, Q1 = 6 means that there are 6 components connected at the 4th level. At levels 4, 3, 2, and 0, all the elements are in the same complex. To find out how the individual simplices -or subsystems-are integrated into the complex, Equation (1) defining eccentricity is used, results are shown in the following figure. σ0(R1) 1

σ2(R15) σ4(R14) σ2(R13)

σ0(R2)

0,8

Ecc σ1(R3)

0,6 0,4

σ0(R4)

0,2 0

σ1(R12)

σ0(R5)

σ1(R11)

σ1(R6)

σ1(R10)

σ1(R7) σ1(R9)

σ0(R8)

FIG. 3 – The supermarket space time path network. Thus the complex does not appear to be very homogeneous. In part icular, ecc(R3 )= ecc(R6 )= ecc(R7 )= ecc(R10 )= ecc(R11 )= 1 means that regions R3 ,R6 ,R7 , R10,R11 are not well integrated with the other regions. In this case study, the q-analysis reveals that there is a problem in the present merchandising technique and there is a necessity to review and reorganize the space in order to provide convenient services for customers and thus increases sales.

6 Conclusion Choosing the right location is crucial for the success of every company because location decisions determine the external market conditions and the internal scope for action. In the present paper, we proposed a general space time path data warehouse and instantiated it for supermarket field to analyze individual space time paths and get implicit and potentially useful information for predicting or evaluating supermarkets merchandising strategies. A supermarket space time path has been analyzed and the structure of this system has been scrutinized by means of q-analysis. A set of indices has been defined from the q-analysis like the structure vector Q(i) indicating the connectivity level between regions, the eccentricity, measuring the level of integration of a given region into the supermarket. Using the first incidence matrix (table 1), we can analyze customers spatio-temporal behavior by calculating

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Space Time Path Data Warehouse Analysis based on the Concept of Simplicial Complex shared face matrix for analyzing customers’ trajectories (T*T). By means of these indices, the q-analysis method enables to examine the merchandising strategy of the supermarket, identify eccentric regions that are not well integrated into the network, and hence to pinpoint elements of the system that may require attention. In the future, we are interested in exploring further useful information using others q-analysis indices and also social network analysis.

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Résumé.

La croissance exponentielle des données spatio-temporelles générées par les technologies de positionnement est un élément important de l’ambition des services basés sur la localisation. Dans le présent article, nous proposons l'application de la méthode Q-analyse développée par RH Atkin sur l’entrepôt de données des chemins espace -temps pour extraire les informations implicites et potentiellement utiles. Nous proposons d'analyser l’entrepôt de données des chemins espace-temps d'un supermarché, comme une étude de cas, de mettre en évidence l'importance de l'application de la méthode q -analyse et l'utilisation des caractéristiques topologiques, comme connectivenness, et l'excentricité. Notre objectif est d'analyser les connaissances spatio-temporelles sur la façon dont les clients utilisent l'espace du supermarché dans un intervalle de temps. Cette analyse permet aux respo nsables du supermarché de réussir l'extraction de connaissances actionnables à partir de ces données, trouver l'e nsemble optimal des régions accroches et aussi l'excentrique. L'utilisation de cette connaissance permet de réorganiser la disposition des produits dans l'espace afin de tirer profit et de maximiser les ventes croisées.

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