1) Sage-Husa adaptive filtering.

i. Basic principle. The Sage-Husa adaptive filtering performs real-time estimation of the current observation noise by the recursive method. In the estimation process, the updated weight is used to modify the observation noise variance in real-time. The model error can be reduced, the divergence of the filter can be suppressed, and the filter’s accuracy can be improved by the real-time accurate estimation of the variance of observation noise. First, consider the following random linear offline system (see Eq (1)): (Eq 1)

In Eq (1), X_t is the system state vector at time t; A_t,t−1 is the state transition matrix from time t-1 to time t; C_t,t−1 is the control-input matrix from time t-1 to time t, A_t,t−1 and C_t,t−1 can both change as time t changes; Y_t is the observation vector at time t, H_t is the observation matrix at time t; w_t−1 is the process noise at time t-1, and v_t is the observation noise at time t, which are independent of each other, and both are normal white noise sequences with time-varying mean and covariance matrix. Among them, process noise and observation noise conform to the following statistical characteristics (see Eq (2)): (Eq 2)

Where the Sage-Husa adaptive algorithm can be described as Eq (3): (Eq 3)

(Where can be obtained by recursion of time-varying noise estimator (see Eq (4)): (Eq 4)

Where, d_t is the update rate and represents the update rate of noise parameters.

It can be seen from the above equations that the Sage-Husa adaptive algorithm estimates and corrects the statistical characteristics of process noise and observation noise in real-time while using GPS measurement data for recursive filtering to achieve the purpose of adaptive filtering. The method is simple in principle and good in real-time, so it has been widely applied in many fields. According to the above algorithm, it is easy to implement the computer program, and the autoregressive operation can be continued.

ii. Basic steps. When using Sage-Husa adaptive filtering to de-noise GPS positioning data, the basic steps are as follows:

1. Firstly, the state equation and the observation equation are established, and then the observation vector is determined.
2. The prior data were used to calibrate and test the model, and then the model noise covariance w_t−1 and v_t was determined.
3. The model was validated with measured data.

We used Sage-Husa adaptive filtering algorithm to process and analyze the data collected by the mobile phone GPS (as shown in Fig 1(a) is the distribution map of original GPS data; Fig 1(b) is the distribution map of GPS data after Sage-Husa adaptive filtering. According to the experimental analysis, the significant data fluctuation can be effectively eliminated by Sage-Husa adaptive filtering, and the mean error of longitude and latitude is controlled at 5.2m.

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Fig 1. GPS data distribution before and after SAGE-HUSA adaptive filtering.

https://doi.org/10.1371/journal.pone.0259472.g001

2) DBSCAN algorithm.

Density-based Spatial Clustering of Applications with Noise (DBSCAN) algorithm is a representative density-based Clustering analysis algorithm, which can find arbitrary shapes of clusters and effectively shield interference from noise data.

i. Working principle. DBSCAN algorithm uses the high-density connectivity of clusters to find clusters of arbitrary shape quickly. Its core idea is that for each object constituting the cluster, the number of objects in its Eps neighborhood must not be less than a given value MinPts. The following are some definitions involved in the basic idea of density-based clustering:

1. Eps: the distance set by the user is Eps. the object is in the area with itself as the dot and Eps as the radius called the Eps neighborhood of the object.
2. Core object: an object is a core object if the number of data points in its Eps neighborhood is greater than or equal to the given value MinPts.
3. Direct density reachable: in sample set D, p is the core object, q is in the Eps neighborhood of p, and object q is directly density reachable from object p.
4. Density Reachable: in sample set D, a series of sample points P₁, P₂, P_n,…, P₁ = p, P _n = q, P_i∈D, 1≤ i ≤ n, if object P_i is direct density reachable from the P_i-1, then object p is density reachable from object q, this relationship is asymmetric.
5. Density connected: If there is an object O in object set D, objects p and q are density-connected if object O is density-reachable to objects p and q. This relationship is symmetric.
6. Noise: a density-based cluster is a collection of the largest density-connected objects based on density-reachability. Objects that are not contained in any cluster are called noise.

In the practical application of urban site location selection, Eps can represent the distance from the residence or workplace to the stops of users who choose to take urban public transport. MinPts can define the minimum number of covered passengers required to meet the stop setting conditions. Density-connected points represent all users served by any stop. The density reachability relationship is the basis for the DBSCAN algorithm to divide the travel demand of the same passengers. The density connected point represents the passenger travel cluster that can be divided into one cluster.

ii. Steps in the DBSCAN algorithm. Based on the above definition, the DBSCAN algorithm uses a method to find density reachable objects for clustering analysis. The correctness of the DBSCAN algorithm is guaranteed because a cluster is equivalent to the set O consisting of all the density connected points objects of any core object in the cluster.

DBSCAN finds all density-reachable objects contained in each cluster by iteratively looking for all directly density-reachable objects. The specific methods are as follows:

Input: database D, Eps, Minpts (given minimum number);

Output: cluster.

1. Check the unchecked object p in the database, and if p has not been processed (grouped into a cluster or marked as noise), check its Eps neighborhood N_Eps(p), if the number of objects contained is not less than MinPts, a new cluster C is created, and add all points of N_Eps(p) to C;
2. For all unchecked objects q in C, check its Eps neighborhood N_Eps(q), if N_Eps(q) contains at least MinPts objects, then objects of N_Eps(q) that do not belong to any cluster are added to C;
3. Repeat step (2) to continue checking for unchecked objects in C until no new objects are added to the current cluster C;
4. Repeat steps (1) through (3) until all objects fall into a cluster or are marked as noise.

The density reachable objects are obtained by continuously executing region queries to achieve a region query and return all objects in the specified region.

iii. Disadvantages of the DBSCAN algorithm. DBSCAN needs to manually set two parameters, Eps and Minpts, which are set by the user and remain the same during the run. When Eps is constant and Minpts changes, Minpts is too small to treat the noise in the data as cluster data, and Minpts is too big to treat low-density data as noise. The result is that the clustering accuracy is not high, and the clustering effect is not reasonable without providing a sound decision basis and wisdom support. The clustering effect depends on the initial parameters greatly. It is an effective way to improve the clustering effect to determine the reasonable initial parameters.

DBSCAN algorithm has two significant disadvantages:

1. It is sensitive to the input parameter Eps, and only clusters with similar density can be found, which affects the quality of clustering results. In practice, it is challenging to determine before the algorithm is run.
2. It is challenging to find clusters with significant density differences. Because Eps and MinPts that determine the density threshold are globally unique, DBSCAN can only find clusters with approximate densities.

3) Improved DBSCAN algorithm.

i. Clustering evaluation index based on the duty cycle. Usually, we choose the clustering evaluation index to determine the quality of clustering results, also known as clustering validity analysis. A good clustering partition should have the following characteristics: the samples in different clusters should be as diverse as possible, and the samples in the same cluster should be as similar as possible.

It is found that the factors affecting the clustering results include not only the degree of clustering and the boundary distance between clusters but also the number of trajectory points in the cluster. Because the traditional evaluation index only considers the cohesion degree, clustering distance, and other coefficients, trajectory clustering has limitations. In the evaluation of clustering cohesion degree, the inner density of clusters is not considered, and the relationship between the number of internal clusters and the size of the cluster is ignored. In irregular clustering, the influence degree of a single variable is often too large, and the clustering result usually stays at the boundary point, which cannot achieve the optimal selection of parameters.

Aiming at the problem that the existing evaluation index is unsuitable for location point clustering, this paper proposes a novel validity index based on the internal and external duty cycle index (IEDCI). The equation is as follows: (Eq 5)

According to Eq (5), the internal and external duty cycle involves three regions S_i,, S_j and S_i+j (as shown in Fig 2), which S_i and S_j are the regions enclosed by the outermost points in class i and j. S_i+j represents the region bounded by the outermost point of the two clusters merged. Duty cycle balanced relationship between the distance within the cluster and the distance between the clusters to solve the inappropriate situation of the single point being grouped or all points being grouped. The area is a two-dimensional standard, which can evaluate the degree of dispersion of two clusters, to effectively avoid the possible linear extreme distance of some points in two clusters.

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Fig 2. Duty cycle coefficient.

https://doi.org/10.1371/journal.pone.0259472.g002

After defining the internal and external duty cycle concept, an evaluation index IEDCI (Internal and External duty cycle Index) is proposed based on the internal and external duty cycle. Eq (6) is as follows: (Eq 6)

Where, n_i、n_j represents the number of points in the i_th and j_th cluster, and k represents the number of the current clusters. represents the maximum value of the ratio of any two different clustering sets, and F(k) is the result of the evaluation index based on the duty cycle. The smaller the value is, the better the classification result is. F(k) differences in the number of clusters may lead to different results. The clustering parameters have the best effect (cluster threshold MinPts and neighborhood radius Eps) when the value of F(k) is the smallest.

A validity evaluation index based on clustering points and the clustering duty cycle is proposed, which is used to evaluate the clustering results generated by different input parameters and determine the current optimal input parameters according to the previous feedback to find the optimal input parameters and clustering results. When processing the clustering results, the Silhouette Coefficient and the Davius-Bouldin index (DBI) only consider the relationship between the cohesion degree and the clustering distance and do not fully consider the influence of the clustering points in a single clustering result on the overall clustering effect. Therefore, the clustering evaluation proposed is better than the above evaluation indexes. The specific experimental results are shown in Tables 6 and 7.

ii. A parameter-free DBSCAN algorithm. The DBSCAN algorithm requires two critical parameters, MinPts, and Eps. However, when selecting these two clustering parameters, the DBSCAN algorithm depends on users’ practical experience. The improved DBSCAN clustering algorithm proposed clusters the data and automatically determines the input parameters: the input parameters of the evaluation index are the clustering results generated during the clustering process, and then the evaluation results are obtained. The specific algorithm is shown in Table 1.

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Table 1. Improved DBSCAN clustering algorithm.

https://doi.org/10.1371/journal.pone.0259472.t001

1. The input parameters
   1. ■ D is the current input data set. D₁ (x₁, y₁) represents x and y of the plane coordinates in the set.
   2. ■ MaxEps is the maximum distance between two plane coordinate points, which can be flexibly determined according to practical meaning.
   3. ■ MinEps is the minimum distance between two plane coordinate points, which can be flexibly determined according to practical meaning.
   4. ■ E is the distance between any two points in the set, ranging from MinEps to MaxEps.
   5. ■ MaxNum sets the upper limit of the clustering threshold because if the number of clusters is too large, the data set may not cluster effectively.
   6. ■ MinNum sets the lower limit of the cluster threshold. If the number of clusters is too small, it may lead to too many clusters, and even a point becomes a cluster without a calculated value.
   7. ■ M determines the optimal quantity threshold for a cluster, which ranges between MaxNum and MinNum.
2. Output parameters
   1. ■ ResultC is the clustering result. Different input parameters can be used to get different clustering results.
   2. ■ MinIEDCI is the minimum duty cycle and is initially set to infinity.
   3. ■ BestEps is the best value for E, initially set to 0.
   4. ■ Best MinPts is the best value for M, with an initial value of 0.

Different input parameters will produce different clustering results. The algorithm gives a range of input parameters, traverses all parameter values within the field, and generates clustering results to prevent some parameters from being lost. The optimal evaluation value can be obtained by evaluating and calculating the clustering results, and the optimal input parameters can be calculated based on the backpropagation method. The algorithm process is described as follows:

1. STEP 1 Building the input parameter range

In different application scenarios, the value of the optimal clustering input parameter fluctuates within a specific range. Since the range of input parameters determines the efficiency of algorithm execution and the possibility of finding the optimal value, it is essential to establish a suitable range of input parameters before algorithm execution. Too many clustering times, the data set may not form effective clustering; Too few clustering times and too scattered clustering are not practical. In addition, the distance between cluster points will affect the compactness within the cluster. If the distance measure is too large, the clusters are too discrete to distinguish different clusters effectively. If the distance measure is too small, the clustering distance is too close, producing too many trivial and worthless clustering results. Therefore, at the early stage of clustering, the maximum and minimum values of Eps and MinPts should be determined to construct the effective range of clustering parameters.

1. STEP 2 Generating clustering results

Cyclic density clustering was carried out to complete the clustering calculation of all travelers’ trajectory points for six months using the neighborhood radius range in Step 1 as the input parameter. All the clustering results (result C) were saved.

1. STEP 3 Evaluating the clustering results

Evaluation indexes such as Silhouette Coefficient, DBI (David-Bouldin index), and IEDCI (Internal and External Duty Cycle Index) were used to evaluate the clustering results. The best clustering parameters BestEps and BestMinPts were saved in the evaluation index.

1. STEP 4 Obtaining the optimal clustering results

The best clustering result was calculated, taking BestEps and BestMinPts in Step 3 as input parameters. The clustering result is the clustering of the actual activity trajectory of the travelers, which is all the possible travel areas of the public transport travelers in the follow-up research.

1) Acquisition of travelers’ trajectory data.

In this study, the trajectory points of individual users’ smartphones were collected as the original data set. The GPS positioning information was mainly obtained using the integrated positioning sensor in the experimental mobile phone for recording and storage. At present, Baidu Map, Tencent Map, and Google Map all provide map-based API development applications. With the authorization of smartphone users, the collection of smartphone users’ trajectory points can be realized by using the APP of mobile phones. According to the setting of the positioning software, the user’s trajectory point information is collected every 10 seconds. In an ideal state, the number of trajectory points collected in a day should be 8640. However, the number of recorded trajectory points varies under different positioning strategies due to the differences in hardware performance, operating systems, carrier networks, and signal stability of location sensors of users’ smartphones.

2) Preprocessing of travelers’ trajectory data.

It is necessary to preprocess the collected original trajectory point data to meet the data format requirements of the clustering algorithm. In this paper, the GPS coordinate data processed by Sage-Husa adaptive filtering is preprocessed again, including data cleaning, data centralization, and standardization. Data cleaning aims to eliminate null values in the original data set, remove or correct incorrect data, and prepare for the accurate discovery of clustering patterns. Data centralization and standardization is since all variables in the collected data often use different units of measurement, and the difference between the data is significant. The data with a large absolute value can overwhelm a small absolute value in the clustering process. The latter cannot reflect the characteristics of the existing system.

The accuracy of the GPS is easily affected by environmental factors. Therefore, taking location L₁ (downtown) and location L₂(remote valley), the GPS integrated device is used to output positioning data every 5s for a fixed location. The 500 continuously acquired data are composed of the original data set algorithm test.

The data distribution after Sage-Husa adaptive filtering, the mean error of longitude and latitude of L₁ is 4.6 m. The mean error of longitude and latitude of L₂ is 6.3m (as shown in Table 2). After calculation, the proposed algorithm is close to the actual coordinate and meets the GPS positioning error of 3–5 m.

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Table 2. Comparison of positioning errors between L₁ and L₂.

https://doi.org/10.1371/journal.pone.0259472.t002

1) 2D synthetic data sets.

2D synthetic data sets included random numbers generated for a computer simulation. Each data set has 1,200 points, each represented as a coordinate and divided into a cluster. These data sets are Distinct Cluster, Fuzzy cluster, Halo cluster, and non-cluster (as shown in Fig 3). In these data sets, the structure of the clear cluster and fuzzy cluster are convex, the design of the halo cluster is annular, and the form of non-cluster is splash.

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Fig 3. 2-D synthetic data sets.

https://doi.org/10.1371/journal.pone.0259472.g003

2) Case data set.

The case data used in this paper are from the ebus APP, which is about urban public transport mobile software. Travelers can use ebus to query public transportation information such as nearby stations, line transfers, real-time arrival prediction. During the query process, the user’s location information is synchronously recorded in the ebus APP. This paper used the location information data of 500 volunteers in Jinhua city over the past six months (From January 2021 to June 2021 (see S2 File)). A total of 500. TXT files were obtained, each representing each traveler’s location information during the six months. The trajectory data of each traveler is represented by the x, y coordinates of the trajectory points. In addition, since the data set represents the real traveler’s trajectory points, the data structure is varied compared to the 2D synthetic data sets, including linear, annular, convex, and splash. The case data set is shown in Table 3, where UID is the unique identifier of the user SIM card, LNG is the longitude of the current user position, LAT is the dimension of the current user position, and UP_TIME is the coordinate upload time (see Table 1 in S1 File).

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Table 3. The travel data structure of urban public transport in Jinhua city.

https://doi.org/10.1371/journal.pone.0259472.t003

To verify the reliability and accuracy of the Sage-Husa-based adaptive filtering algorithm proposed, the GPS data collected is de-noised and compared with the original GPS data without denoising. Table 4 is the comparison and analysis results of the positioning errors of the algorithm in different directions and dimensions.

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Table 4. Comparison of positioning accuracy based on Sage-Husa adaptive filtering algorithm.

https://doi.org/10.1371/journal.pone.0259472.t004

The case shows that the proposed algorithm effectively controls the error range and further improves the positioning accuracy of GPS. Positioning accuracy eastward and northward are improved by 17.34% and 15.24%, respectively, and the positioning accuracy in 2D and 3D dimensions is enhanced by 14.25% and 18.17%, respectively. After using this algorithm, the number of noise points is reduced obviously, and the data error is controlled in a smaller range and concentrated to the actual coordinate points.

1) 2D synthetic data sets.

This paper uses compactness and separation to evaluate the clustering results of four 2-D synthetic data sets. Table 6 shows the compactness evaluation results of the three evaluation indexes. According to the results, IEDCI has a better evaluation value for the distinct cluster, fuzzy cluster, and non-cluster data set. Table 7 shows the separation evaluation results of the three evaluation indexes. It can be seen from the results that IEDCI has a better evaluation value for clear cluster and non-cluster data sets.

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Table 6. Compactness evaluation results of different performance indexes.

https://doi.org/10.1371/journal.pone.0259472.t006

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Table 7. Separation evaluation results of different performance indexes.

https://doi.org/10.1371/journal.pone.0259472.t007

2) Case data set.

Case data sets are to be used to evaluate the performance of the algorithm. The entire evaluation process is as follows.

1. (1) Optimal input selection: The algorithm in Table 1 was performed using the three evaluation indexes of Silhouette Coefficient, DBI, and IEDCI. After traversing all possible values in the parameter range, the algorithm can obtain the optimal input parameters corresponding to the three evaluation indexes, as shown in Table 8. We use a three-dimensional surface diagram to illustrate the process of getting the optimal input parameters (see Fig 5).

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Fig 5. 3-Dimensional surfaces with different performance indexes.

https://doi.org/10.1371/journal.pone.0259472.g005

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Table 8. Optimal values of MinPts and Eps.

https://doi.org/10.1371/journal.pone.0259472.t008

In the figure, the axis x represents all possible values of Eps, the coordinate y represents all possible values of MinPts, and the coordinate z represents the corresponding values. The values of Eps and MinPts are the best when the value is the smallest. After parameter selection within the same input range, the optimal parameter value generated by the Silhouette Coefficient will be generated at the boundary point, and both DBI and IEDCI will obtain the optimal parameter value within the range, minimize the value of the evaluation index.

1. (2) Clustering results: Three different clustering results are generated using the optimal input values of the three evaluation indexes. As shown from Fig 6, for the clustering points within the same range, the results generated by the evaluation index of the Silhouette Coefficient gather the discrete points in the red ellipse into a cluster. However, from the actual situation of the traveler’s trajectory, the clustering results are unsatisfactory because travelers have too many activities. In the DBI clustering results, the red ellipse is divided into two parts. Similarly, the distance from point A to point B in Fig 6(b) is far beyond the range of human activities (500 meters). In this paper’s clustering results of the algorithm, the range of traveler’s movement is smaller than the radius of the resident’s trajectory. Therefore, this algorithm performs well in practical applications.

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Fig 6. Clustering results of different performance indicators.

https://doi.org/10.1371/journal.pone.0259472.g006

1. (3) Clustering evaluation: The compactness and separation degree of the generated clustering results were evaluated, and the evaluation results are shown in Table 9. Based on fully considering the effects of clustering density and clustering spacing, the results obtained by the proposed method have bigger separation and smaller compactness, which is more consistent with the actual situation of human activities in trajectory clustering.

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Table 9. Evaluation results of separation and compactness.

https://doi.org/10.1371/journal.pone.0259472.t009