A survey on trajectory clustering analysis

02/20/2018 ∙ by Jiang Bian, et al. ∙ The University of Sydney University of Macau 0

This paper comprehensively surveys the development of trajectory clustering. Considering the critical role of trajectory data mining in modern intelligent systems for surveillance security, abnormal behavior detection, crowd behavior analysis, and traffic control, trajectory clustering has attracted growing attention. Existing trajectory clustering methods can be grouped into three categories: unsupervised, supervised and semi-supervised algorithms. In spite of achieving a certain level of development, trajectory clustering is limited in its success by complex conditions such as application scenarios and data dimensions. This paper provides a holistic understanding and deep insight into trajectory clustering, and presents a comprehensive analysis of representative methods and promising future directions.

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1 Introduction

With the development of tracking and surveillance devices, tremendous numbers of object trajectory data were collected, which makes extracting useful information imperative and challenging. Trajectory clustering is an efficient method to analyze trajectory data and it has been applied in pattern recognition, data analysis and machine learning, etc. Furthermore, trajectory clustering aims to gain space, time or even potential information inside trajectory data, so it is ubiquitous in some application fields such as object motion prediction

chen2010searching , traffic monitoring atev2006learning li2007traffic gurung2014traffic , activity understanding bashir2007object wang2016globally yao2017learning , abnormal detection wang2014detection yuan2017statistical zhao2018gestalt chaker2017social , 3 dimensional reconstructionkumar2017spatio , weather forecasting ferreira2013vector and geography mo2015application .
Trajectory data are recorded in different formats according to device types, object movements and even purposes. For instance, GPS tracking devices generate a trajectory by tracking object movement as , which is a consecutive sequence of points in geographical space, and denotes a combination of coordinates and time stamp like , as shown in Fig.1. In some specific circumstances, other properties relevant to object movement are added, such as velocity, direction or acceleration. As shown in Fig.2, for image data, a sequence of pixels in consecutive frames forms up a trajectory, which is similar to optical flow cai2016framework wang2016robust .

Figure 1: Trajectory generated by GPS tracking devices
Figure 2: Trajectory generated from camera device

In order to measure similarities among different types of trajectory data, data representation, feature extraction and distance metric selection are critical preliminary works of trajectory clustering. For example, trajectories can be represented as a vector and downsampled to an unified length, so Euclidean distance is used

nanni2006time . Trajectories also can be treated as samples of a probablistic distribution. Hence, Bhattacharyya Distance li2006coarse is used to measure the distance between two distributions.
According to the availability of labeled data, trajectory clustering methods are divided in three categories: unsupervised, supervised, semi-supervised. Unsupervised models aim at clustering data without human experts supervision or labeled data. An inference function has been drawn by analyzing unlabeled data sets ester1996density ferreira2013vector wang2006learning xiang2008spectral . Supervised models are learned prior to trajectory clustering. Generally, labeled data are used to learn a function mapping data to their labels, i.e. clusters. The clusters of unlabeled data are predicted by this function, then gao2007efficient yuan2017statistical wu2016harnessing cho2014classifying . Labeling data need a heavy burden of manual works by human experts. It is unfeasible for large data sets. Semi-supervised compromises the previous two types of models. It is trained by labeled data and tuned by unlabeled data gurung2014traffic wang2014detection yuan2017statistical .
The rest of this paper is organized as follows. Preliminary works are introduced in Sect.2, and the models based on unsupervised algorithms are described in Sect.3. A description of the models under supervised algorithms are presented in Sect.4. Sect.5 discusses some models based on semi-supervised algorithms. Finally, promising future directions are given in Sect.6, and conclusions are made in Sect.7.

2 Preliminaries

2.1 Trajectory Clustering Preparation

In some clustering models zhang2009learning sillito2008semi naftel2006motion hu2013incremental , trajectory data are required to be set as a unified length so that they could be measured. However, as shown in Fig.3, for two arbitrary trajectories, their lengths maybe largely different from each other. Therefore, representing trajectories in an unified length with little loss of information is a major preliminary work of these models. This procedure is called clustering preparation.

Figure 3: Trajectory data with varying lengths

2.1.1 Trajectory Transformation Algorithms

For some methods, original data are represented in other space with a same length. For instance, trajectory data are projected into a subspace hu2015exploiting

. Linear transformation algorithm aims at representing trajectory as a combination of basis trajectories

akhter2011trajectory . Curve fitting is another method to approximate trajectories by a parameterized quadratic curve zhang2009learning . In order to distinguish similar curves, the direction of the last trajectory point is chosen as an additional parameter. In sillito2008semi , trajectory data are approximated by a uniform cubic B-spline curve, so that a representation capable of encoding both the shape and the spatiotemporal profile of trajectory data is obtained. In addition, the lengths of trajectories are added to distinguish the trajectories with similar shapes. According to the fact that trajectory data contain a lot kinds of positional information, such as coordinates, speed and directions, vector fields are employed to represent trajectory data ferreira2013vector . Vector fields give trajectory a smooth streamline and induce a notion of similarity of trajectories. Principle Component Analysis (PCA) is a statistical procedure to compute a set of linearly uncorrelated variables called principle components by orthogonal transformation. To avoid partially extracted information, a number of organized segmentations substitute for the corresponding trajectory in bashir2007object and bashir2007real

. The time ordering data are transformed and represented in frequency domain by Discrete Fourier Transformation (DFT), so a trajectory can be represented as a fixed length vector comprised of Fourier coefficients in

naftel2006motion and hu2013incremental . In jiang2012trajectory , the interaction of trajectories are encoded and set as elements of codebook, so camera motion is ignored and the model’s robustness is improved.

2.1.2 Re-sampling Methods

Re-sampling methods choose trajectory points by sampling rule to unify trajectory lengths. Trajectory data are segmented as sub-trajectories, and all of them are re-sampled to a fixed length so that sub-trajectories are aligned as matrix bashir2007real . In a complex scene such as hand writing data set, Equidistant sampling fixes the problem that two same characters are recorded in different temporal sequence because of different writing speeds schenk2006novel . Since re-sampled trajectory points are discontinuous, it is critical that normalization should be involved after re-sampling liou2011trajectory . It has been widely acknowledged that re-sampling method causes information loss piotto2009syntactic . Therefore, sparsity regularization is used in wang2009evaluation ochs2014segmentation elhamifar2009sparse cappe2007overview .

2.1.3 Trajectory Substitute

Sub-trajectories hold partial and hidden information of original trajectory data hu2013incremental lee2007trajectory , so they are put together and describe trajectory more flexible. For instance, the latent motion rule beneath hurricane trajectories is figured out and a certain hurricane trend chart is printed by analyzing sub-trajectories of past hurricane trajectories in ferreira2013vector . Sub-trajectories also lead to simplified trajectories which represent trajectory data as some smaller, less complex primitives suitable for storage and retrieval purposes anagnostopoulos2006global . In yan2010hybrid , sub-trajectories are generated by well-defined policies based on facility performance, time range or distance range. In bashir2007object and bashir2007real , trajectory is segmented at the so-called significant changing points at which direction or speed changes dramatically. Curvature describes direction information, and it could be extracted if a trajectory is treated as a curve by connecting consecutive trajectory points. Curvatures are computed by transforming 3-dimensional position coordinates of points into spherical system and quantized as up, down, left, right faria20093d , then a trajectory is segmented at the points where curvature changes. In addition, Minimum Description Length (MDL) principle traces the sub-trajectories holding primary movement of trajectory data by minimizing the differences between sub-trajectories and the corresponding trajectories in lee2007trajectory . Minimum Bounding Rectangles (MBR) is proposed to separate trajectories under occlusion and optimize the inter-object separability in anagnostopoulos2006global . It optimizes the bounding rectangles containing sub-trajectories to ensure that the distance between two rectangles are closer than the distance of trajectories.
Some specific regions of surveillance area hold special semantic information and attract more attention so Regional Segmenting method is implemented. The whole scene is split into several regions and boundaries of the regions segment trajectories zhang2007trajectory . As independent motion pattern, sub-trajectories characterize more information while original trajectory presents limited information.

2.1.4 Points of Interest

Some specific regions of surveillance area hold special semantic information. Thus, the points inside the special regions are used to represent trajectory or scene in tuytelaars2008local and all these points are called Points of Interest (POI). The points outside the regions are ignored because they are short of useful information. For instance, activity analysis is a key part in surveillance application to seek low-level situational awareness by understanding and characterizing behaviors of objects in the scene morris2008survey , so it is critical to extract POI in the special regions. In topographical map, POI inside the special regions are represented as a single node. For example, two types of POI are introduced in morris2008survey where the first one is the points in entry/exit zones and the second one is the points at the scene landmarks that objects intend to approach, move away or stay for a long time. Except for the special areas, points are represented by a node if their speed are less than a threshold in brandle2006track and morris2011trajectory . The importance of points can be measured and high-scored ones are selected in zhou2013dense . For video data, POI are obtained by Pyramid Representation wang2011action

. In addition, optical flow is another popular implementation by estimating trajectory motion in

wang2016robust and fragkiadaki2015learning .

2.1.5 Scale-invariant Features

In image frames, more robust and representative features are needed rather than only positional information of trajectory points in jiang2012trajectory and wang2011action . In wang2011action , histograms of oriented gradients (HOG) and histograms of optical flow (HOF) features are used to describe static appearance information and local motion information of trajectories, respectively. HOG feature computes orientation information to keep scale-invariant property of tracking point and it is fast to implement jiang2012trajectory klaser2008spatio laptev2008learning willems2008efficient matikainen2009trajectons matikainen2010representing . Furthermore, Scale-invariant Feature Transform (SIFT) descriptor represents image patch around tracking point uemura2008feature sun2009hierarchical wang2008video vig2012space wang2013dense , and computes scale and orientation information of image patches to localize tracking object in consecutive frames. In uemura2008feature , Kanade Lucas Tomasi (KLT) tracker is used to find trajectory points and SIFT is applied to represent them. In wang2008video , Difference-of-Gaussian (DOG) detector is used to detecting trajectory points instead of KLT in uemura2008feature .

2.2 Common Distance Measurements

Essentially, trajectory are allocated into cohesive groups according to their mutual similarities. An appropriate metric is necessary morris2009learning zhang2006comparison atev2010clustering .
Euclidean Distance: Euclidean distance requires that lengths of trajectories should be unified and the distances between the corresponding trajectories points should be summed up,

(1)

where and indicate the th point of trajectory on Cartesian coordinate. is the total number of points. In nanni2006time , Euclidean distance is used to measure the contemporary instantiations of trajectories.
Hausdorff Distance: Hausdorff distance measures the similarities by considering how close every point of one trajectory to some points of the other one, and it measures trajectories and without unifying the lengths in liu2014entropy chen2011clustering ,

(2)
(3)

Bhattacharyya Distance

: Bhattacharyya distance measures how closely of two probability distributions. In

li2006coarse , it is employed to measures similarities of quantized directions of points,

(4)

where and it is used to measure the separability of and . and are quantized directions.
Frechet distance: Frechet distance measures similarity between two curves by taking into account location and time ordering. After obtaining the curve approximations of trajectories and , their curves map unit interval into metric space , and a re-parameterization is added to make sure cannot be backtracked. Frechet distance is defined as

(5)

where is distance function of , , are continuous and non-decreasing re-parameterization.
Dynamic Time Warping (DTW) Distance: DTW is a sequence alignment method to find an optimal matching between two trajectories and measure the similarity without considering lengths and time ordering shao2016integral bautista2016gesture .

(6)

where has points and has points, all mappings should satisfy the requirements that and .
Longest Common Subsequence (LCSS) Distance: LCSS aims at finding the longest common subsequence in all sequences, and the length of the longest subsequence could be the similarity between two arbitrary trajectories with different lengths. The distance is written as

(7)

where indicates first points belonging to and denotes first points of . Finally, .
In lee2007trajectory lee2008traclass li2010incremental , more other distance types are proposed to consider more properties such as angle distance, center distance and parallel distance, which are defined as

(8)

where is the smaller intersecting angle between and .

(9)

where is the Euclidean distance between center points of and .

(10)

where is the Euclidean distances of to and is that of to . and are the projection points of and onto respectively.
Distance metrics are used in much more fields relating to trajectories clustering, e.g., density clustering lee2007trajectory lee2008traclass palma2008clustering bu2009efficient andrienko2009interactive . It is critical to choose an optimal distance according to the scene. For instance, LCSS distance is proved to provide outperforming performance without concerning trajectories length morris2009learning . Hausdorff distance aims at finding the minimum distance between two trajectories and ignore time-order in data. A comparison of distance is listed in Table I,

Measurement types Unifying lengths Computational complexity
Euclidean distance Yes
Hausdorff distance No
Bhattacharyya distance Yes
Frechet distance No
LCSS distance No
DTW distance No
other distance types No
Table 1: Summary of common distance measurements

3 Unsupervised Algorithms of Trajectory Clustering

Unsupervised algorithms infer a function to describe internal relationships between unlabeled data. Clustering is the method to draw this hidden structure, and some models relating to trajectory clustering are reviewed such as Densely Clustering models, Hierarchical Clustering models and Spectral Clustering models.


3.1 Densely Clustering Models

Figure 4: DBSCAN
Figure 5: DBSCAN for trajectory clustering

Given the centroids, the closely points are packed together and this procedure is called densely clustering. Inspired by this idea, Density-based spatial clustering of applications with noise (DBSCAN) which has been widely applied to trajectory clustering is proposed in ester1996density . In DBCSAN, point is chosen as the core point and distance threshold is given in advance. The points inside circle of which the radius is and the center is are called directly reachable to . Furthermore, points are reachable to if there is a path that is directly reachable to and each is directly reachable jeung2008discovery wang2013action

. Other points are the outliers. Thus, the distance metric and the core parts selection are important. For solving the problem that DBSCAN cannot cluster the trajectories with large differences in densities

lee2007trajectory lee2008traclass khaing2014efficient , all trajectories are partitioned and substituted by sub-trajectories, then sub-trajectories are clustered and all clusters are grouped at the last step. However, different from measuring distance by Euclidean distance in lee2007trajectory , the distance is measured by a combination of angle distance, center distance, parallel distance with equal weight in lee2008traclass and khaing2014efficient

. The core trajectories are computed from the clusters and used for classifying new coming trajectory in

lee2008traclass , zheng2015trajectory , deng2011trajectory and zheng2010geolife , e.g., all trajectories points belonging to same cluster are averaged as a new point at each time, and all averaged points form the representations of clusters lee2008traclass . In an adaptive multi-kernel-based method, shrunk clusters represent all groups by considering the attributes including positions, speeds and points, which retains much more discriminative messages in xu2015unsupervised .

Besides DBSCAN, there are some other models belonging to Densely Clustering models cluster trajectory data. K-means clusters trajectories by searching centroids of clusters repeatedly

ferreira2013vector morris2011trajectory suzuki2007learning galluccio2012graph hu2006system melo2006detection

. For improving the performance of K-means, Expectation Maximization (EM) algorithm is implemented to solve optimization problem iteratively

zhou2007detecting . Due to the issues such as data imprecision and complexity of large data sets, a trajectory may belong to multiple clusters so EM is used to classify them kim2008real . Fuzzy C-Means (FCM) algorithm employs parameters to measure the level of cluster fuzziness for each trajectory, called fuzzifier. The algorithm searches correct direction in each iteration for cluster trajectories pelekis2009clustering pelekis2011clustering simonnet2017multi .

3.2 Hierarchical Clustering Models

Hierarchical Clustering models help to understand trajectory by multiple features, so this tree-type construction is proper to implement. Hierarchical Clustering models generally fall into two clustering types, Agglomerative and Divisive. As shown in Fig.6, two hierarchical types are also known as “bottom-up” and “top-down” approaches.

(a) Agglomerative clustering
(b) Divisive clustering
Figure 6: Hierarchical clustering models

In Agglomerative frameworks, trajectories are grouped and the similar clusters are merged by searching their common properties. Optimal classifications are obtained by repeating representation computation and clusters merging until meeting the requirements. Inspired by this idea, Agglomerative clustering models were explored in zheng2009mining to mine the locations that users are interested, Hypertext Induced Topic Search (HITS) model is proposed to achieve this goal and movement tracks of users are recorded as trajectories. Top interesting trajectory clusters are obtained iteratively and the most popular locations are generated.
Different from Agglomerative, Divisive frameworks cluster trajectory data into groups and split them recursively to reach the requirements. Following this framework, trajectory data are characterized by direction feature and clustered by Dominant-set embedded Bhattacharyya distance in initial clustering stage li2006coarse . In each cluster, trajectories are split further except for the ones holding similar positions. Because of the good performance of iterative models, Test-and-Divide (TAD) model is proposed zheng2013discovery . It is a Divisive framework detecting all the closed trajectories firstly and splitting them recursively. More attributes of trajectory points are considered to improve the performance in wang2006learning . For instance, trajectory where . It is comprised of 2-dimensional position and an additional attribute such as velocity or object size. In the coarse clustering step, the distance measurement between trajectory and its nearest observation trajectory are shown as follows,

(11)

where and the minimum distance value is counted as the distance between and . is the total number of points belonging to , indicates the dissimilarity of and , and is weight parameter. In the fine-clustering stage, the model aims at distinguishing distortions by considering directed similarity and confidence

(12)
(13)

where and .
Furthermore, a similar hierarchical framework is explored to group videos by constructing the trajectories of video gaidon2014activity as an unordered tree, and a kernel method recognizes videos by clustering the trees. In addition, Hierarchical Clustering models also recognize actions from video in raptis2012discovering and tabatabaei2011ganc . For two trajectories in video, and , the distance is computed as follows,

(14)

where is the positional distance at time stamp , and

is the similarity measurement of velocity. An affinity matrix

is calculated and trajectories are clustered by greedy agglomerative hierarchical models raptis2012discovering tabatabaei2011ganc . The clusters are overlapped because of similar parts, so every trajectory is weighted and optimized to classify in ni2015motion . Since one motion object may generate several trajectories, it is critical to employ as much features as possible to ensure object recognition, and a multi-layer classifier is invented in li2006coarse antonini2006counting .

3.3 Spectral Clustering Models

Trajectory data can be represented as a matrix called affinity matrix, and the relationships between them are extracted as the elements of matrix. The top eigenvectors form clusters with distinctive gaps between them which can be readily used to separate data into different groups xiang2008spectral . In addition, affinity matrix characterizes videos turchini2015understanding and represents the relationships. In hu2007semantic , affinity matrix is constructed as follows,

(15)

where , and indicates the th point of trajectory . Considering different lengths of trajectories, some novel models are explored to construct affinity matrix brunskill2007topological brox2010object and it is constructed as

(16)

where and are points, and indicates scale invariance computed by the median of the nearest neighbors. In order to increase the separation of points belonging to different groups, SVD decomposition is used to construct the affinity matrix lauer2009spectral . In addition, a novel distance method is explored to compute trajectories and atev2006learning so that spatial distinction can be considered.

(17)
(18)

where is the directed Hausdorff distance, indicates the value of and denotes the subset of points which the ones matching to the point in trajectory .
For clustering high dimensional trajectory data by Spectral Clustering models, several novel methods are explored in zhang2016red , hong2017high and chen2009spectral . For example, a new similarity metric captures causal relationships between time series in hong2017high and a mixture of affinity subspaces is applied to approximate trajectory in chen2009spectral . Trajectory data are represented by considering covariance features of trajectories in ergezer2016anomaly , so it avoids considering different lengths of trajectory data. Spectral clustering works with multiple-instance learning frameworks to achieve human action recognition in yi2016human .
Spectral Clustering models are derived from Graph Theory in which an undirected graph represents the relationships and constructs a symmetric adjacency matrix presenting them brouwer2011spectra . By constructing a graph, both explicit and implicit intentions inside trajectory data are mined chen2015action . The graph is cut into sub-graphs to classify trajectories, and each sub-graph represents its own cluster zhang2009learning lin2012integrating . Hierarchical layers search sub-clusters in each cluster by treating trajectories points as graph nodes and this procedure is called Hierarchical graph partitioning guo2010graph . For considering more variables, a novel measurement function comprised of the entropy rate of a random walk on a graph is presented in liu2014entropy . From the idea that an undirected graph can be represented as an adjacent matrix, a directed graph also can be involved li2007traffic . Trajectory Binary Partition Tree (BPT) represents video in palou2013hierarchical by representing trajectories as nodes so the edges indicate relationships between a pair of trajectories, and graph cut method groups trajectory data. Because of the robustness of composite feature descriptors, the descriptors including Speeded Up Robust Features (SURF) and Maximally Stable Extremal Regions (MSER) are employed in lin2012integrating . An object creates several trajectories if different parts of the object are tracked, so a model is invented to describe trajectories by feature patches liu2009trajectory . The edges are computed by geometric distance and appearance distance. Hausdorff distance is utilized to measure the similarities and set as weights of edges in junejo2008euclidean . Since the great performance of PageRank, it is used to score the edges in cho2010authority , too.

3.4 Discussion

Densely Clustering models classify trajectories by distance metrics mostly, which may result in classifying trajectory data by spatial information. Hierarchical Clustering models fix this problem by considering more attributes in each level. However, this operation cost much more time in computation. Spectral Clustering models compute internal relationships by analyzing the affinity matrix, and it saves much more computational resource by processing all trajectory data together. But keuper2015motion mentions that Spectral Clustering models have their own limitation that they are well defined only for the non-negative affinities between trajectories. Furthermore, that trajectory lengths are required to be unified is another issue of applying Spectral Clustering models.

4 Supervised Algorithms of Trajectory Clustering

Supervised algorithms aims at learning a function which determines the labels of testing data after analyzing labeled training data. Therefore, supervised algorithms outperform on scene understanding and utilization of ground truth, and the algorithms could save much more computation resource. In some supervised algorithms, trajectory data are classified by unsupervised algorithms and the representations of clusters are obtained to classify new inquiry trajectories. For example, in Densely Clustering models, the representations can be computed from the grouped training trajectory data and new coming trajectories are clustered quickly in

bashir2007object and pelekis2009clustering . Trajectory data are classified and organized in a tree-construction and new coming trajectories are clustered by searching the tree in gurung2014traffic and pelekis2017temporal .

4.1 Nearest Neighbor Algorithms

Nearest Neighbor algorithms, such as -Nearest Neighbor algorithm (-NN), are finding a voting system to determine the category of a new coming entity and all data are kept in the same feature space. In trajectory clustering, the distances from an inquiry trajectory to all labeled trajectory data are computed, and the label of the inquiry trajectory is voted by its nearest neighbors. Shown in Fig.7, the inquiry trajectory is assigned as blue cluster if and assigned as red one if .

Figure 7: -NN for trajectory clustering. Inquiry trajectory is the green one, the labeled data are the red and the blue ones which means two clusters.

In the implementation, it is important to choose a suitable distance metric according to the scenario, occlusion, trajectory data sources and feature types. Therefore, trajectory data are represented by MBR and classified by -NN in gao2007efficient . It avoids occlusion and increasing inter-object separability. Furthermore, trajectory data are represented in Riemannian manifold devanne20153 so their shapes can be modeled and compared by using an elastic metric. For accessing -NN faster, fast nearest neighbor (fastNN) algorithm organizes trajectory data in an Octree poularakis2016low . With the increasing inquiry trajectories, the trends of trajectory data in a fixed period are required instead of general representation, so a circumstance that dynamically searches the nearest neighbors in a fixed period or the ones belonging to some specific types is considered in frentzos2007algorithms

. Support Vector Machine (SVM) is trained to generate the hypervolume, and the inquiry trajectory is determined as outliers if it falls outside the hypervolume

piciarelli2008trajectory . Structural Support Vector Machine (Structural SVM) is explored to detect social groups in crowds in solera2016socially . Furthermore, SVM works with Graph Theory to cluster trajectories in singh2017graph .

4.2 Statistical Models

Statistical models exploit a set of probability distributions to represent the data generating process such as Gaussian Mixture model (GMM) and Bayesian inference. GMM usually combines with EM algorithm to train each component, and Bayesian inference obtains a set of probability functions which determine the categories of inquiry trajectory data. Bayes’ theorem is critical for Bayesian inference and written as

where and indicate two events in event space.
GMM aims at describing the sample from in a component of GMM as

(19)

where is the probability density of the th component belonging to a component with mean

and variance

. is the weight with a constraint that , and they can be computed according to event frequency. Generally, EM algorithm iteratively optimizes the parameters of GMM, but Maximum Likelihood algorithm is implemented instead of EM if labeled trajectory data are available in training stage. For example, video events are treated as a linear combination of a set of event patterns, and two probabilistic terms are proposed to characterize video events in yuan2017statistical . Furthermore, the abnormal patterns are scored by summarizing the probabilities of trajectory data of the corresponding video. GMM models the variance caused by the environmental factors and embedded into DTW to recognize gestures bautista2016gesture .
Bayesian inference classifies new coming data, and the classified ones update the probability functions of Bayesian inference. For samples , the probability of the corresponding labels is

. Derived from Markov Chain Monte Carlo (MCMC) algorithm, the distribution of variables can be approximated by a joint distribution, so Gibbs sampling is used to approximate

by sampling iteratively. According to Bayes’ theorem, is represented as where is the likelihood and is the marginal distribution. In Dirichlet Process (DP) model which is one of the Bayesian inference frameworks, is formulated as where is scale parameter and is base measure in sample space. The clusters can be parameterized for classifying new inquiry data, e.g., Dirichlet Process Mixture model (DPMM) is used to represent all clusters with parameterized indexes in hu2013incremental . Finally, the new inquiry trajectory is classified by a trained DPMM as where is the likelihood and

is the prior probability. In order to learn coupled spatial and temporal patterns, Hierarchical Dirichlet Process (HDP) algorithm is applied in

wang2016globally . Bayesian model is used to segment object by classifying trajectories, so that human motion is also detected devanne2017motion .

4.3 Neural Network

Neural network is an artificial system simulating the biological neural network in animal brains. The network is constructed by a number of mutually connected neurons, and each neuron is represented as a real number. Neural networks can represent data such as deep generative model. It is trained to represent multivariate time series if trajectory data are generated as a vector yuan2016deep , and a deep fully-connected Neural Network with weight decay and sparsity constraint transfers trajectory data from different viewpoints to a fixed viewpoint in compact representation rahmani2018learning .

Figure 8: CNN is one of the classical models of Neural Network and widely used in images classification

In most cases, Neural Network is used to classify data. It can be viewed as a mathematical function where is the observation and

indicates the corresponding label. For example, Convolutional Neural Network (CNN, or ConvNet) consists of multiple layers including convolutional, pooling and fully connected layers. That layout tolerates the variations of the input data, avoids overfitting problem and distinguishes data as similar as Multilayer perceptron (MLP). CNN has been proved efficient in clustering issue of computer vision. As Fig.

8 shown, CNN is comprised by two convolutional layers, two pooling layers, two fully connected layers and one output layer which acts as an image classifier. CNN is employed for trajectory clustering in wu2016harnessing and cho2014classifying . Furthermore, CNN also ranks the trajectory clustering results in fragkiadaki2015learning

. A flexible deep CNN called Deep Event Network (DevNet) is trained by ImageNet data set, and the trained DevNet is tuned to extract generic image-level features of trajectory data in

gan2015devnet . In order to figure out the differences between image classification and multimedia event detection, DevNet fine tunes parameters by a specific data set, and backward passing is employed to identify pixels in consecutive frames to recount events. Deep Neural Network (DNN) is another Neural Network which learns a more compact and powerful representation of trajectories hasan2015continuous . Furthermore, DNN keeps the structural relationships between trajectories in shi2015learning , and mines the relationship between multiple features including spatiotemporal features, audio features and inter-class relationship to classify videos in wu2014exploring and jiang2015exploiting

. Self-Organizing Map learns the similarities between trajectories in a 2-dimensional grid and each element of the grid indicates a specific prototype in

naftel2006motion and schreck2009visual . In training steps, each training trajectory is trying to find the most suitable prototype in network, and adjust the neighbors of the matched one accordingly.

4.4 Discussion

Nearest Neighbor algorithm only considers the spatial relationships between a pair of trajectory data but ignores local characters. Statistical model makes up for this imperfection by combining them in a mixture model or inferring the relationships in Bayesian models. Neural Network considers the differences of trajectory data and requires a huge number of data to train it. Though the supervised methods obtain the classifiers by observing a number of training data, overfitting problem may happen when the model overreacts training data.

5 Semi-supervised Algorithms of Trajectory Clustering

Semi-supervised algorithms fall between unsupervised algorithms and supervised algorithms. The algorithms make use of a small number of labeled data and continuous inquiry data to complete tasks. The model is trained by labeled data firstly, then inquiry data are kept sending to the trained model to make sure that it can be updated to outperform the previous model. Semi-supervised procedure needs only a small cost in terms of human classification efforts. This procedure not only avoids overfitting problem, but also is more accurate than the unsupervised ones.
Therefore, some semi-supervised algorithms are invented from unsupervised or supervised algorithms. For example, trajectory data are classified firstly and the new inquiry ones are clustered to update the classifier automatically gurung2014traffic wang2014detection yuan2017statistical laxhammar2014online . Detected anomaly trajectory data are used to recalculate the representation of anomaly trajectory clusters in laxhammar2014online . Trajectory data of video are modeled as the combination of normal and abnormal patterns, and probabilistic terms characterize the patterns in yuan2017statistical . From this modeling, the terms can be updated by the detected inquiry trajectory. In order to detect abnormal trajectories faster in complex scene, low-rank approximation is employed to describe trajectory data and the new detected abnormal ones update the threshold in wang2014detection .
Inspired by Hierarchical Frameworks, trajectories and the clusters are represented as a tree where children nodes indicate trajectories and roots denote the representations of the clusters in piciarelli2006line , kulic2008incremental and li2013effective . A new cluster is created if no clusters close to the inquiry trajectory. Trajectory is constructed as a vector of 2 dimensional coordinates where . A representation of cluster is computed as , where and is an approximation of the local variance of the cluster at time . The inquiry trajectory is assigned to the nearest cluster and the corresponding cluster should be updated by the new one. For the nearest cluster point to the point of trajectory , is updated as following

(20)

where is the update rate between 0 and 1.
Considering the fact that Bayesian model is derived from Bayes’ theorem, the parameters are optimized by sampling training data, and it is feasible to update the model by classified new inquiry data hu2013incremental . Furthermore, in order to add new trajectory data, the previous samples and the new ones are sampled by Gibbs Sampling as

(21)

where is trajectory data, indicate the known states of the previous samples, and . denote the states of new inquiry trajectory data except for the th one. From Bayes’ theorem, the cluster process is rewritten as .

is estimated by the previous samples and it is assumed to be Gaussian distribution. The only issue need to be fixed is carrying out Gibbs Sampling on

to compute .

6 Promising Future Directions and Tasks

In trajectory clustering, we saw that how to measure trajectories with different lengths is important. Thus, a representation method or feature descriptor is essential for trajectory clustering. In recent years, transforming trajectory data into other space are paid more attention, such as DFT which keeping data information and unifying lengths of trajectory data hu2013incremental . For other preparation works, re-sampling is efficient for sparse scene wang2009evaluation , but it limits the robustness of model. Curve approximation fits the movement of trajectory zhang2009learning sillito2008semi . Hence, trajectory data preparation may be a promising and helpful direction.
Recently, Densely Clustering models have achieved great progress in trajectory clustering. In particular, novel distance metrics have been proposed to measure trajectory data according to different properties. Furthermore, for the trajectory data with large difference in density, grid construction is employed to improve the performance uncu2006gridbscan . Besides grid-based DBCSAN, sub-trajectories are acted as the substitute for trajectory in lee2007trajectory , lee2008traclass and khaing2014efficient .
Though Spectral Clustering models and Graph method share a similar idea, they are intrinsically different. Spectral Clustering models are easy to implement and have no restriction on data dimensions, but the models require non-negative affinities and this limitation restricts the performance and the application. Therefore, a suitable affinity matrix construction method is needed. Furthermore, it is critical to determine scale value when the affinity matrix is being computed, because it determines the clustering is fail or not. Thus, Spectral Clustering models need to handle the problem of constructing affinity matrix.
In supervised algorithms, a large number of training data are required to obtain an efficient model. However, such as in Neural Network, there may have overfitting problem and some special steps are needed like pooling layers in CNN. In addition, it should be noticed that a meaningful distance metric is essential for Nearest Neighbor algorithms.

7 Conclusion

In this paper, we reviewed the methods of trajectory clustering. They are classified into three categories: unsupervised, supervised and semi-supervised algorithms. Unsupervised algorithms can be grouped into three sub-categories: Densely Clustering models, Hierarchical Clustering models and Spectral Clustering models. Supervised algorithms are divided into Nearest Neighbor algorithms, statistical models and Neural Network. By means of a comprehensive analysis, we found that unsupervised algorithms have the disadvantages of high computation cost and heavy memory load, although there is no training data requirement and human experts supervising. Semi-supervised algorithms combine the advantages of both previous algorithms, and may result in more efficient methods. Finally, we proposed several promising future directions and tasks, and this paper could help readers to gain a thorough understanding of trajectory clustering.

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