Influential spatial facility prediction over large scale cyber-physical vehicles in smart city

3.1 Partition of region

Given N facilities, calculating the nearest distance for one object requires N calculations. The computation overhead increases with the number of both objects and facilities. Assume we have N facilities and M objects, Θ(M N) distance calculations are needed in order to get influence values for all facilities. The computation overhead is exacerbated if vehicles are moving and update locations fastly. Obviously, it is unnecessary to calculate the distance for every pair of facility f and object p at every time stamp.

More specifically, we partition a geographic region into equal-size grids and provide an index for each grid. Since different facilities have different scope area, the number of grids covering different facilities varies. If a grid is inside the scope of a facility, the value that denotes the influence of this grid on the facility is stored. This value is simply the total number of objects inside this grid. With this approach, we do not need to calculate the distances to access all N facilities to discover the shortest one (whose time complexity is Θ(N)); instead, mapping the index to the value incurs much less computation overhead (whose time complexity is (Θ(1)).

Let us use an example (Fig. 2) to walk through this grid-based approach. Two facilities f ₁ and f ₂ are in the region in Fig. 2 a. Each has an associated scope and each has multiple objects around. The cell influence view (Fig. 2 b) shows that every grid cell has a nearest facility, and the objects view Fig. 2 c shows objects in every cell. We can overlay these two views and get all facilities’ influence very quickly (Fig. 2 d).

Although the grid-based approach seems simple, two practical issues need to be addressed. First, ambiguous grid may lead to inaccuracy in calculation of facility influence: some grids are at the intersection of ranges of more than one facility. Second, the choice of the grid size significantly affects the overhead of computing facility influence. We next address these two practical issues.

3.2 Ambiguous grid

The following theorem provides a method to classify a cell being unique or ambiguous.

where NF(g) denotes the nearest facility of grid cell g, H(p) outputs the cell that p is mapped to, and Pr(·) is the probability.

Since every object can be mapped to a cell, we separate the objects set \(\mathcal {P}\) into two subsets: one is unique cell set \(\mathcal {P'}\) and the other is ambiguous cell set \(\mathcal {P''}\). If the mapped cell g is a unique cell, then Pr(NF(g)=f|g=H(p))=1; when the mapped cell g is an ambiguous cell, we design three methods to compute Pr(NF(g)=f|g=H(p)).

Centering is the simplest scheme as it uses center point to represent a grid and finding the nearest facility of a point is easy. However, it is too coarse-grained because it increases the inaccuracy if one object does not lie in the same region as the center. If locations mapped to a grid follow uniform distribution, sampling and area schemes are recommended. The area scheme is a generalization of the sampling scheme and can use the information of Voronoi diagram [19] generated by facility set \(\mathcal {F}\).

A natural question now arises: what is the difference between a facility’s actual influence and approximate influence? If we assume objects follow uniform distribution, then we have the following theorem.

Theorem 2 shows that if l is small, the approximate influence of a facility is very close to its actual influence. We can use \(\hat I_{\mathcal {P}}(\,f)\) to approximately estimate \(I_{\mathcal {P}}(\,f)\) if the grid size is not too big. In practical, we also shall not set the grid size too small. We demonstrate the reason below.

3.3 Grid size

Grid size l has an impact to facility influence on three aspects: memory size requirements, computing overhead, and accuracy. We next analyze these three factors and discuss how to determine a suitable grid size.

Memory Intuitively, smaller grid size makes a larger grid count in a certain geographic region. For instance, assume Beijing has an area of S (nearly to 16,800 km²), the grid count is equal to S/l ² where l is the grid length. If one grid only stores one value, i.e., facility influence (assume 32-bit), then the total memory needed is equal to (S/l ²)∗32 bits. We generally do not set grid size too small considering memory usage.

Computing overhead. Given N facilities and M objects, each object p falls into one grid, so overhead of computing facility influence is Θ(1) for this object-facility pair. The total computing overhead is Θ(M), which is irrelative with grid size.

Accuracy If we set l larger, we have fewer unique cells and more ambiguous cells, which will decrease the accuracy of facility influence. The principle is not to set l too large. Empirically, we set l to be average distance of adjacent facilities. In the evaluation section, we would discuss the grid size impacting over facility influence.

3.4 Facility influence computation

After the entire geographic region is divided into cells, we can get a mapping between cells and facilities. Given an object’s location, the grid index indicates which cell the object should be mapped. The corresponding nearest facility and its probability can be retrieved. Therefore, it is unnecessary to compute distance between all facilities and the object to determine the nearest facility. Our iterative process works as shown in Algorithm 1. When an object’s location is updated (i.e., changed), we use grid index to get a new grid cell ID. If the current cell ID is not equal to the previous one, we extract the new nearest facility ID which is stored in the grid data structure. We then compare the current facility ID with the previous one. If they are equal, we just skip and do nothing. Otherwise, we plus facility influence value into current facility and minus it from the previous one. Then, we calculate the facility influence using (3).

Analysis: The time complexity of Algorithm 1 is O(N), where N is the number of objects. Lines 2 and 3 describe that the grid cell ID is obtained from object’s location using grid index, so the calculation is constant time. Lines 5 and 6 retrieve data from grid data structure, which also takes constant time. The upper bound of algorithm is determined by the the number of objects.

4.1 Construction of Markov model

We have previously used a grid partition-based approach to calculate facility influence. Now, we utilize the same grid representation as in the previous section to predict a vehicle’s future location. Partitioned grids can be converted to a grid graph, where each node is one grid and an edge exists between two nodes implying that the two grids are adjacent.

There are two advantages using a grid partition-based approach. First, a grid representation is appropriate for facility influence prediction. Second, a grid-based approach can avoid the data sparsity problem when predicting future location. When a trajectory consists of a sequence of locations, it can be represented using a sequence of grids. Therefore, we will have lots of trajectory data in grid index to represent each trajectory. For predicting future locations with historical trajectories, lots of training data is needed for our TMM model.

Each grid cell is viewed as a state and the Markov model uses these states to build its state transition matrix. In the matrix, the transition probability between grid cells (i.e., states) is trained by the historical step counts, i.e., the number of transition steps from one cell to another is used to derive the transition probability between these two cells. To further understand the processing of Markov model for trajectories, we use an example to illustrate it. In Fig. 4, the region is partitioned into 3×3 grids. Trajectory T ₁ is a sequence of grid cells: g ₁,g ₄,g ₅ (Fig. 4 a). Each grid is a state in Markov model, so T ₁ is represented as a sequence of states over time (Fig. 4 b).

The prediction of objects’ future locations consists of two phases: a training phase where the historical trajectories are mined offline to obtain transition matrix and an online prediction phase where the historical trajectory of a given object is analyzed and its future locations are predicted.

where |S _i¬| is the total number of travels from g _i to all neighboring cells.

For each pair of adjacent nodes in the grid graph, we compute the transition probabilities offline using Eq. (4). These probabilities are stored as entries of a two-dimensional matrix where one dimension corresponds to the node of current state and the other dimension corresponds to the next state. We denote the matrix and its entries by the transition matrix M and M _ij , respectively.

4.2 Prediction of future locations

Prediction of object’s future locations essentially provides a set of cells that the object will reside along with a probability. We next use an example (Fig. 5) to illustrate how to predict future location using TMM model. Suppose one object p currently stays at grid cell g ₅. Next time, it has four possible movements: g ₂, g ₄, g ₆, and g ₈ (Fig. 5 a). If its trajectory is unknown, the transition probability (in the transition probability matrix of Markov model) is directly used as the probability of it going to each of the four cells. If its trajectory is known, we can use Bayesian theorem to calculate the probabilities (Fig. 5 b), where prior probability is the transition probability and the likelihood probability is how the trajectory and the historical data is similar to each other.

where L _i→k denotes l ₁ distance (step count of the shortest path) from g _i to g _k and L _de,i→k denotes the maximal steps where an object may run more paths from g _i to g _k . L _de,i→k is often set to be 0.2L _i→k .

4.3 Prediction of future facility influence

Facility influence is determined by objects around them. If we can predict these objects’ future locations, then we can predict facility influences in the future. This raises several interesting challenges. One challenge is how to select a proper time step size. Objects have different speed and direction, e.g., taking a taxi requires a larger time step size compared with walking. Mobility is also affected by surrounding environments, such as traffic congestion or speed limitation. It is hard to predict when objects will arrive at the predicted place even though we make a successfully prediction. The other challenge is that the prediction is related to the size of grid cell. If the cell is too small, object may cross several cells in one time step, resulting in data sparsity problem for the training phase of the Markov model construction; if the cell is too large, object will stay in one cell for several time steps, resulting in inaccuracy to facility influence. In our experiments, we have tried multiple cell size from 100 to 1000 m and empirically selected an appropriate value.

The speed of an object is related to the object’s previous speed. If we can get these two pieces of information as well as the speed, we can determine the grid cells the object will arrive. To be specific, suppose travel time is t, the speed is v, so the distance from current location is s=v t. Then, we can get a set of possible cells C within distance s from current location. If cell g in C is next to current cell, we can directly use Eq. (7) to compute the probability. After getting all possible cells’ probability, we can use the highest one as the final prediction result for that object. Algorithm 2 summarizes how prediction of facility influence is done.

5.1 Experimental methodology

To validate the effectiveness and efficiency of proposed algorithms, we take taxis as moving objects for facility influence computation. We use a large real-world dataset which consists of 1-month moving trajectories of taxis in Beijing. It contains more than 27,000 taxis over one billion trajectories. Facilities are overpasses, subway stations, tourism destinations, shopping malls, etc, which are extracted from over 1000 Beijing POIs [7]. Each taxi uploaded its location to the database server every 30 s. Figure 6 a shows the coverage of taxis in 1 day. We can see that taxi trajectories almost cover all central urban area of Beijing. Figure 6 b is a typical taxi’s trajectory in a day.

We ran our proposed algorithm on a desktop machine with Intel Core I5-3380 2.90GHz dual core CPU. The data is stored in SQL server database. There are many data structures to construct grid index, such as hash table, hash array, or matrix. We use hash array to index the grids. Each grid use 100 bits to store two values: facility IDs and the corresponding influence value. We train our trajectory-based Markov model with built-in toolkit [21]. We first separate every taxi trajectory into sub-trajectories which starts in one grid cell and ends in another. Then, we split the sub-trajectory database to two parts, one for training Markov transition matrix and the other for testing the effectiveness of our algorithm. We then compute and save the Markov transition matrix M using Eq. (4). It takes several hours to train matrix M using the massive trajectories. We then compute M ²,M ³,⋯ and save them on a hard disk. When doing online prediction, we put all the matrices into memory, avoiding slow matrix multiplication operations.

5.2 Grid partition

Figure 7 depicts the running time to different grid partitions. The running time increases along with the increase in the number of facilities from 100 to 500, in Fig. 7 a. This is because grid partition needs to calculate every facility’s scope and then map it into grid cells, larger facility count leads to higher computing overhead. Figure 7 b depicts the impact of grid size on running time. It confirms our intuition that smaller grid size causes higher computing overhead in grid partition. The smaller the grid size, the higher the grid count in a fixed geographic region.

In brief, if grid size l is too small, the index requires larger memory space and higher overhead. We propose to set l from 100 to 500. The facility count depends on specific application scene, which affects the computing overhead.

5.3 Facility influence computing

We evaluate Algorithm 1 for calculating the facility influence. Figure 8 a depicts how taxi count affects the response time of facility influence calculation. In these experiments, we set grid size l=100 and facility count m=300. The calculation only takes 100 ms when the taxi count is close to 10,000. The response time increases very slowly with the increase of taxi count n. This is because the calculation maps location to grid index, which has a constant complexity Θ(1). Figure 8 b shows how facility count affects response time. In these experiments, grid size is l=100 and taxi count n=10,000. Our approach remains very efficient and the response time is under 0.5 s. These results have confirmed that using grid is suitable for calculating facility influence in real time.

We next evaluate the effectiveness of dealing with ambiguous grids. We basically measure the accuracy of associating objects with facility. We consider the ground truth being the closest facility in Euclidean distance. In these experiments, facility count m=300 and taxi count n=10,000. Figure 9 depicts the results using the three methods we proposed: centering (a), sampling (b), and area (c). If grid size is too large, there are large ambiguous grids at the intersection of multiple facilities. The accuracy decreases with the increase of grid size. When the grid length becomes 1000 m, the accuracy of all three methods is unacceptable with the worst case close to 56 %. When the grid size is small (i.e., close to 100 meters), the accuracy of all three methods stays high (i.e., close to 96 %.) Combining the results of grid partition shown in Fig. 7 and Table 2, large grid size indicates low memory usage and low overhead. We suggest to set l between 100 and 500 m according to different surroundings (various facility types). Fig. 9 also depicts the accuracy with different k values, i.e., calculating top k most influential facilities. If k equals facility count m, it basically calculates influence for all facilities. Here, we measure different k values impacting facility influence calculation (total count is 300), including 10, 50, 80, and 100. There is no difference in accuracy regardless of the k value.

5.4 Prediction accuracy

We firstly evaluate location prediction accuracy of our TMM. Two factors affect the accuracy: the trajectory length and the number of steps taken. Step indicates that the location is transformed from one grid cell to another cell (includes jumping circle). Figure 10 depicts the impact of trajectory length on location prediction accuracy (assume future step is 1). When the trajectory length is long (10 steps in this case), the accuracy is close to 85 %. When the number of historical steps equals 1, the accuracy is below 70 %. This is reasonable since more historical data will lead to higher accuracy in prediction. The cumulative distribution function (CDF), as shown Fig. 10 b, depicts that the accuracy is higher than 80 % on average.

In order to verify the effectiveness of our TMM for future step count, we compare it with the naive MM which is considered as baseline. MM is a common methods in location prediction with normal transition, which directly uses the transition probability of current state to neighbors. The trajectory length used for TMM and MM ranges from 3 to 10.

Figure 11 a depicts impact of future steps on the prediction accuracy. The accuracy of TMM is much higher than MM in every steps, which indicates that historical trajectories can to a large extent dominate future locations. Figure 11 b shows the CDF of prediction accuracy of the two models: for 80 % of the time, the prediction accuracy in TMM is lower than 80 %. While the prediction accuracy in MM is lower than 40 %, which is significantly lower than TMM.

For the trajectory length, we suggest to use a longer one so that the prediction accuracy is higher. For large future step count, the prediction accuracy seems low from our experiments. However, it does not affect the facility influence prediction. This is because when the taxi is moving within the scope of a facility, prediction error in future location cannot cause the error in facility influence prediction. The error in influence prediction only occurs when the taxi moves from the scope of one facility to another, which case error occasionally occurs in future step prediction.

We next evaluate the effectiveness of predicting facility influence. We set the parameters l=500,m=500,n=30,000; the historical steps is bigger than 4 and we only consider the situation in 5 future steps. The ground-truth of our test uses nearest distance to calculate the facility influence. We use the sampling method to deal with ambiguous grids. Figure 12 a depicts the accuracy of dealing with ambiguous grids and also future influence prediction. We can see that our facility influence prediction accuracy is close to the accuracy of location prediction, i.e., above 80 % in all k values tested. Our facility influence calculation accuracy has better accuracy than prediction by nearly more than 10 %, and this difference is caused by the error in future location prediction. Figure 12 b shows the CDF of facility influence prediction accuracy. Our solution has achieved an acceptable accuracy for predicting facility influence.

We observe that although location prediction can introduce uncertainty which then affects the final influence prediction, as long as the predicted grid is not far away from the real location and they have the same nearest facility, this uncertainty has weak impact on influence prediction.