Pattern Ensembling for Spatial Trajectory Reconstruction
Shivam Pathak, Mingyi He, Sergey Malinchik, Stanislav Sobolevsky
PPattern Ensembling for Spatial TrajectoryReconstruction
Shivam Pathak [email protected] Mingyi He [email protected] Sergey Malinchik [email protected] Stanislav Sobolevsky [email protected] Center for Urban Science and ProgressNew York UniversityNew York, NY, 11201 Lockheed Martin Advanced Technology LabLockheed Martin CorporationCherry Hill, NJ, 08002
Abstract
Digital sensing provides unprecedented opportunity to assess and understand mo-bility. However incompleteness, missing information, possible inaccuracies andtemporal heterogeneity in the geolocation data can undermine its applicability. Asmobility patterns are often repeated, we propose a method to use similar trajectorypatterns from local vicinity and probabilistically ensemble them to robustly recon-struct missing or unreliable observations. We evaluate the proposed approach incomparison with traditional functional trajectory interpolation using a case of seavessel trajectory data provided by The Automatic Identification System (AIS). Byeffectively leveraging the similarities in real-world trajectories, our pattern ensem-bling method helps reconstructing missing trajectory segments of extended lengthand complex geometry. It can be used for locating mobile objects when temporaryunobserved as well as for creating an evenly sampled trajectory interpolation usefulfor further trajectory mining.
The recent advances in location acquisition technologies and the widespread use of digital deviceshas helped curating a variety of spatial trajectories data, representing the mobility patterns of variousobjects, such as people, vehicles or vessels. A large body of papers demonstrates inferences andmodeling of human mobility based on partial information provided by opportunistic digital sensing,such as anonymized mobile phone data [1–3], GPS traces [4–6], Bluetooth [7], credit card transactions[8] as well as social media data [9, 10] along with successful urban applications of such inferences.However incompleteness, missing information and possible inaccuracies in such data often undermineits applicability. Furthermore, trajectory mining can benefit from having regular representations ofthe trajectory, including having location observations at even time intervals, which is often not thecase for the real-world trajectory data. In order to address this problem the present paper proposes anapproach for inferring missing locations and the entire trajectory segments based on available partialobservations.We will illustrate the approach on one of the examples of mobility data - The Automatic IdentificationSystem (AIS), being a rich source of spatial data for sea vessels which also serves as a representativeillustration of the above issues. The AIS is an automatic vessel tracking system used by Vessel Traffic
Preprint. Under review. a r X i v : . [ phy s i c s . d a t a - a n ] J a n ervices(VTS) for maritime navigation and safety. The AIS uses transponders on the ship to transmittime-stamped vessel positional information to nearby ships and VTS to identify and locate vessels. Italso collects dynamic data related to ships movements such as speed, time, and positional coordinates.The dynamic data of AIS yields the trajectories of vessels, and this large dataset collectively generatedby thousands of ships in the sea is critical for understanding vessel navigational behaviors. In the pastfew years, the AIS trajectory datasets are under focus for both research and practical applications.The major applications of AIS Data are: (a) analyzing maritime traffic patterns [11, 12] and collisionavoidance [13–15], (b) vessel behavior study[16, 17] and anomaly detection [18, 19], (c) knowledgeextraction[20, 13, 21, 22]and pattern mining [23, 24], and (d) trajectory clustering[25, 22] andpredictions [26–28].Although the AIS data is useful for various purposes, its incompleteness or the presence of a largenumber of missing segments in its trajectories subdues its applicability. These missing segmentsare created either due to vessel inactivity, device failure, or inclement weather conditions at thepoint of data collection. In the past few years, this problem has led to the emergence of a newstream of studies for Trajectory Reconstruction, and several approaches are proposed to resolvethis issue. Jie et al. [29] use bilateral filtering on the dynamic information of AIS for smootheningthe trajectories and performing interpolations. [30] focuses on improving the general applicabilityof AIS data by improving the quality of spatial and temporal data. [31] segments the trajectoriesbased on their shape as line, curve, or arc and uses linear fitting for interpolation. In general, allthese methods linear in nature and thus are only able to reconstruct easily inferable or relativelylinear missing segments. Several researchers have also attempted to use Neural Network (NN), [32]utilizes LSTM, and proposed a Neural Network-Based automatic reconstruction of Missing VesselTrajectory Data. Although NN based methods generalize well for missing segments of short lengths,they fail to reconstruct segments of longer lengths. Also, these methods miss an opportunity to utilizepatterns from trajectories of other objects present in the local vicinity of the target trajectory withthe missing segments. Lastly, all the traditional methods, as well as NN-based methods, produce asingle inference and do not account for its uncertainty. While the probabilistic inferences of missingobservations incorporating uncertainty could be particularly useful within probabilistic pattern miningframeworks.Mobility patterns are often repeated within the same or across different users trajectories. While mo-bility of objects (people, vehicles, ships) is often constrained of follows a number of (known/unknown)common rules. For ships, they often follow specific routes while navigating in the sea; thus, forany given trajectory, there may exist similar trajectories of the same or other vessels traversing thesame route. These similar trajectories can be used to reconstruct missing gaps present in the giventrajectory.We base our method on this principle and propose to impute the missing segment in the target trajectoryby extracting and ensembling patterns from similar trajectories present in the spatial vicinity. Weachieve this objective in three steps. (a) First, we select candidate trajectories in close proximityusing box-similarity approach (b) Second, we calculate relevance weights for candidate patternsusing similarities between the known portions of the target trajectory and segments correspondingto it within the candidate trajectories. (c) Third, we align all these candidate patterns and ensemblethem using relevance weighting to generate the inference for the missing pattern.We illustrate the proposed method applying it to AIS data and evaluate its performance to reconstructmissing segments of variable time and distance gaps in comparison with traditional functionalinterpolation methods, used as a baseline. The attributes of AIS data can be broadly categorised into three categories [33] : Static Data (vesselname and identity), Dynamic Data (position, heading, speed etc.), and Voyage Data (Draught, CargoDescription, Destination etc.). Last et al. [26] provides a comprehensive analysis and understandingof Automatic Identification System (AIS) data. To conduct this study we use the raw dynamic AISData [34] collected by the United States Coast Guard and provided by The Bureau of Ocean EnergyManagement (BOEM) and the National Oceanic and Atmospheric Administration (NOAA). Thechunk of AIS data we used corresponds to U.S. and international waters of UTM Zone 18 [34]. This2hunk of data contains AIS data and continuous positional logs of 7209 vessels for the month of July2017. Overall, the data has 35 million positional points.The raw AIS data is the continuous positional logs of vessels for the entire considered period; thus,it is noisy. It includes the cases of vessels being idle and yet transmitting their position as wellas significant gaps where the positional points are missing. For this study, we clean these naturalinconsistencies present in the AIS data and obtain clean segments of the trajectory that can be treatedas consistent movements.Figure 1: Distributions of Time Differences and Speeds between consecutive points of raw AIStrajectories.In order to eliminate points corresponding to the vessel’s idle period, we use the distribution of vesselestimated speed between any two consecutive points. This distribution is shown in figure 1. Thedistribution is bimodal with a clear cut-off at 1km/hrs. Whereas the distribution of speed above1km/hr follows the normal navigational behavior, speeds below it are unnaturally low and are likelyto represent stationary periods. Thus, if in any section of the trajectory the speed between subsequentpoints is lesser than 1km/hr continuously for a long period, we mark the section as idle and split thetrajectory at the beginning and end of this section.Next, we consider the distribution of time differences between the consecutive points. This distributionpeaks at 60,180, and 300 seconds. These time marks correspond to the frequency generally at whichthe AIS devices in the vessels transmit the data. Thus, if the time difference between any set of twoconsecutive points is higher than two or more times of the typical interval, we consider a missingsegment between these points. Again to eliminate this missing gap, we split the trajectories at thesepoints. While cleaning, we also found that 5% of points in our dataset had an adjacent missingsegment at one of their ends.After forming the trajectories by eliminating noisy points, we performed cleaning at the trajectorylevel to eliminate noisy trajectories. Again, we use distributions of Time, Speed, and Distance atthe trajectory level to execute this cleaning. Ultimately, we obtained 81K clean trajectories with 8.1million positional points. The statistics corresponding to these clean trajectories could be found intable 1 and table 2.To develop an evaluation framework for testing the methods discussed in the next section, we create asample set of 600 trajectories and induce synthetic missing segments in them. While creating thissample set, we controlled the time, distance and curvature of both the trajectories and the missingsegments to have equal representation of linear/non-linear shapes and short/long lengths of themissing segments. 3able 1: Distribution of attributes between consecutive points of clean trajectoriesDistribution of Attributes for Points (8183942 total)Time(Seconds) Distance(Kms) Speed(Kms/hr)mean 75.30 0.39 18.91std 19.75 0.52 40.1125% 66.00 0.26 12.8850% 70.00 0.35 16.5975% 80.00 0.47 22.94Table 2: Distribution of attributes at trajectory level for clean trajectoriesDistribution of Attributes for Trajectories (81223 total)Time(Seconds) Distance(Kms) Speed(Kms/hr) Number of Pointsmean 7586.71 39.37 19.60 100.76std 14553.14 81.60 9.35 202.6125% 1774.50 8.45 13.06 23.0050% 2981.00 15.66 16.47 39.0075% 6199.00 32.25 22.89 78.00
Trajectory can be considered as a sequence of n points with temporal attribute timestamp (t) andpositional attributes Latitude (LAT) and Longitude (LON). When we compare two trajectories, therecan be a difference between the number of points in them. This difference originates from two factors:(a) difference in time frequency at which the two trajectories were sampled or (b) difference in thespeed of the vessels. Illustration of these factors is shown in figure 2.To work with pairs of Trajectories, we need to find methods for (a) aligning the points and (b)calculating similarities between the sequences of location observations. There are multiple methodsfor calculating similarities between the two trajectories, the most used metrics are based on Euclideandistance[35], LCSS (Longest common subsequence) [36], Hausdorff distance, Frechet distance, andDTW [37].Distance measures like Euclidean and Manhattan assume prior alignment and calculate distancesbetween paired sets of points matching the two trajectories. The mean or sum of these distances can beused as the final similarity measure. The requirement of prior alignment makes it difficult to use thesemethods with real-world trajectories. Another method, Longest Common Sub-Sequence (LCSS),considers only the distances between point pairs that have a distance below a certain threshold.The limitation of the threshold restricts the applicability of this method when the trajectories arediverging. Hausdorff and Frechet distances are based on calculating the maximum of minimumdistances between the sets of points in the different trajectories. Frechet distance is a modificationof Hausdorff and is more suitable for trajectory similarity as it also considers time sequence whilealigning the sets of points i.e., any point in trajectory A will not be paired up with any of the points intrajectory B which were already paired up back in time.Dynamic Time Warping (DTW) is a widely used method for finding similarity between temporalsequences of varying speeds. It matches every point in either of the trajectories with one or morepoints in the other and finds the most optimal match. DTW uses distance matrix and dynamicprogramming in a time-efficient way to return the most optimal match. Previously, Li et. al [38] haveused DTW to find similarity between the trajectories, for developing their dimensionality reductionbased multi-step clustering method. DTW fulfills both the match and similarity measure requirementsof our study. 4igure 2: Illustration of similar trajectories with different number of points originating due to thedifference in (a) sampling frequency and (b) speed of vessel Earlier in this paper, we defined trajectory as a sequence of points. Often inclement weather conditionsor AIS device inactivity or failure leads to a missing set of points from the trajectories. Our study’sobjective is to infer a sequence of points that can most appropriately reconstruct the unobservedsegment of the original trajectory.A straightforward linear solution to this problem is to use the very last known points before andafter the missing segment and interpolate a straight line between them. Hereafter, we will refer tothis method as linear interpolation. A modification on top of the linear interpolation could be to usethe last sets of points leading and trailing the missing segment and fit a polynomial function to useits predictions for reconstruction. In this study, we use both of these methods as our baseline forcomparison. Similar to [32], we use cubic function as the baseline polynomial function, and hereafterwe will refer to this as cubic interpolation. The number of known points to use for fitting cubicfunction is a hyper-parameter for this approach. Using a small number of points can lead to highvariance, and using the entire set of known points may create high bias. Thus, we experimented withthe different number of known points for fitting the cubic spline. The cubic interpolation performedbest when using 15 known points on both ends of the missing segment, and we use this version forperformance comparison with our proposed method.In both of the above methods, we are using information present within the target trajectory having themissing segment. An alternative approach could be to also use the similar trajectories present in thespatial vicinity to provide likely patterns for reconstructing the observations gap. An ensemble ofthese patterns using a set of suitable relevance weights could lead to improved predictive performance.Our proposed method is based on this idea and performs in three stages: (a) Filter/select candidatetrajectories, (b) Calculate relevance weights, (c) Align and ensemble patterns.5igure 3: The Three Stages of Pattern EnsemblingOne way to select similar trajectories is to compare the known portions of the target trajectory withevery other available trajectory, calculate the similarities between them, and pick the top ones havingthe highest similarity score. The number of similarity score calculations required here will be equalto the number of reference trajectories and will grow with the size of the reference set, thus becomingcomputationally expensive. A much faster way could be to use spatial box constraints around theknown points and filter only those trajectories which are traversing through them. To keep our methodcomparable with the cubic baseline, we use points belonging to 15-time steps before and after themissing segment and place a bounding box spatial constraints with sides of length 1Km on top ofthem. The selection of these points and the type of spatial constraints can be further improved foroptimal performance.After selecting the set of candidate trajectories, we calculate their relevance weights based on theirrelevance to the target trajectory and presumed ability to reconstruct the observations gap. To getthese weights, we calculate the similarity between the known portions of the target trajectory and thesegments corresponding to it in the candidate trajectories. Here, we use the DTW similarity score. Togenerate relevance weights for these candidate patterns with the mass summing up to 1, we use thesoftmax function from the multinomial classification on the set of similarity scores and redistributethe relevance weight amongst all the candidates.In the last step, we align and ensemble the candidate patterns using their relevance weights to predictthe final sequence for reconstructing the missing observations. Again, we use DTW for aligningpoints from different candidate segments. Note, DTW can only align two sets of points at a time,whereas we can have more than two candidate patterns for reconstruction. Thus, we start by pickingtwo patterns with maximum relevance weights, align them using DTW, and average each pair ofpoints based on the candidate’s weights to create an ensembled pattern. This new ensembled patternhas a weight equivalent to the sum of the relevance weights of the candidate patterns it ensembled.Next, we pick the third candidate pattern and its weight, ensemble it with the already assembledpattern, and iteratively repeat this until all the candidate patterns are covered. One can assume thatwe can directly use the candidate pattern with the highest weight to reconstruct the missing gap.This method could work well in general cases, but due to its over-reliance on a single pattern, it6an fail when the candidate patterns are not clearly relevant. In general, ensembling averages outinconsistencies in individual candidates and leads to increased inferential performance.
Figure 4: Comparison of fit accuracy of Pattern Ensembling and Functional Interpolation with respectto time and distance of missing segments.We evaluate the performance of the method for reconstructing the segments using the sample setcreated earlier with the induced missing segments. First, we align the prediction and the actualmissing segment, next we calculate the geospatial distances between the paired sets of points in Kmsand after that take their mean to get an average error of the missing segment inference. We conduct aquantitative study to evaluate the effect of increasing time/distance length of the missing segmentson the different interpolation methods’ performance. Moreover, we conduct a qualitative study tounderstand the role and impact of the the shape of trajectories or missing segments on the segmentreconstruction performance.The total of 111 samples out of 600 generated lack any suitable candidate trajectories in their spatialvicinity; thus, pattern ensembling does not generate any inference for those cases. The availabilityof a candidate trajectory is dependent on the tightness of spatial constraints used for filtering. Thenumber of those cases could be reduced by optimizing the filtering strategy used in the first stage.The mean error of the fit over the entire sample set for linear, cubic and the pattern ensemblinginterpolations is 0.478km, 0.403km, and 0.197km respectively. The significantly lower mean errorfor pattern ensemble interpolation highlights its superior performance over the other methods. Tobetter understand this improved performance, we sort and bucket the samples based on the time anddistance of their missing segment, and plot the mean errors with respect to those as shown on theFigure 4. The plot clearly demonstrates that the fit error for linear and cubic interpolation increaseswith increased time and distance of the missing segment, whereas for the pattern ensembling this errorremains consistently low. Functional interpolation methods assume that the trajectory follows certainanalytic shape; while this assumption might not misrepresent reality when the length of the missingsegment is relatively low, however when the length increases, for real trajectories, this assumptionstarts breaking down and large errors of fit accumulate. On the contrary, the pattern ensemblingmethod draws the patterns from other real-world trajectories and thus efficiently reconstructs thecomplex shapes of the real-world trajectories, undiscoverable for the functional interpolation. Thisfeature enables the pattern ensembling method to keep its predictions close to the original shapeseven for the longer missing segments.Our qualitative study of cases with complex shapes of trajectories reveals three more benefits ofpattern ensembling over the functional interpolation methods. Figure 5 illustrates these cases onspecific examples. First, pattern ensembling is robust against curvature and complexities of naturaltrajectories and provides reconstructions with minimal error, whereas the other methods fail to do so.Second, the baseline methods often fail to generalize, and the error accumulates to a high numberwhen the missing segments are long. Third, the baseline interpolation methods overfit the known dataand are prone to incurring a massive off-shift in their prediction whereas, in our developed method,the probabilistic ensemble of different patterns helps to avoid overfitting and generalizes well.7igure 5: Cases with more complex shapes and geometries illustrating comparison of Cubic andPattern Ensembling interpolations
In this paper, we consider the problem of missing segments in spatial trajectories data. As mobilitypatterns are repetitive in nature and often reproduced by the same or different users, we proposeda new pattern ensembling method to reconstruct these missing segments. It effectively utilizessimilar candidate patterns from local vicinity and probabilistically ensembles them to produce thereconstructed segment.The three step approach to filter candidates, calculate their probabilistic relevance weights, andensemble the inferences was successful in reconstructing missing segments in AIS trajectories. Theempirical results demonstrate the superior performance of the proposed method for robustly recon-structing the missing segments compared to functional interpolation. The advantage of the patternensembling method is particularly noticeable while reconstructing segments of higher length/durationand more complex geometry.The approach can help recovering missing locations of mobile objects, such as people, vehicles orvessels and aid trajectory mining by providing evenly sampled trajectory representations.
The authors thank the US National Geospatial Intelligence Agency for supporting this work andfurther thank Christopher Farah for stimulating discussions and useful feedback.
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