System: Instructions: Compose a comprehensive reply to the query using the search results given. Cite each reference using [Page Number] notation (every result has this number at the beginning).Citation should be done at the end of each sentence. If the search results mention multiple subjects with the same name, create separate answers for each. Only include information found in the results and don't add any additional information. Make sure the answer is correct and don't output false content. Ignore outlier search results which has nothing to do with the question. You can give additional information using you own knowledge but you have to start by the tag [Own knowledge]. If you any caracters should be printed as mathematical formula put it in between brackets. Only answer what is asked. The answer should be short and concise. Answer step-by-step.

User: search results:

[Page no. 19] "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 P(A|B) = P (B|A)P (A) P (B) where A and B indicate two events in event space. GMM aims at describing the sample from {x1, x2, · · · , xn} in a component of GMM as P(xj) = K � i=1 πiN(xj; µi, Σi), (19) where N(xj; µi, Σi) is the probability density of the ith component belonging to a component with mean µi and variance Σi. πi is the weight with a constraint that �K i=1 πi = 1, 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 trajec- tory 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 [9]. 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 [68]. Bayesian inference classiﬁes new coming data, and the classiﬁed ones update the probability functions of Bayesian inference. For samples {x1, x2, · · · , xn}, the probability of the corresponding labels y1:n is p(y1:n|x1:n). 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 p(y1:n|x1:n) by sampling p(yi|y−i, x1:n) iteratively. According to Bayes’ theorem, p(yi|y−i, x1:n) is represented as p(yi|y−i, x1:n) ∝ p(xi|yi)p(yi|y−i) where p(xi|yi) is the likelihood and p(yi|y−i) is the marginal distribution."

[Page no. 2] "A survey on trajectory clustering analysis Jiang Biana, Dayong Tianb,∗, Yuanyan Tangc, Dacheng Taod aCentre for Artiﬁcial Intelligence, FEIT, University of Technology Sydney, Australia bSchool of Electronics and Information, Northwestern Polytechnical University, China cFaculty of Science and Technology, University of Macau, China dUBTECH Sydney AI Centre, FEIT, University of Sydney, Australia Abstract 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 behav- ior 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 di- mensions. This paper provides a holistic understanding and deep insight into trajectory clustering, and presents a comprehensive analysis of representative methods and promising future directions. Keywords: trajectory clustering, object movement, pattern recognition, survey, review 1. Introduction With the development of tracking and surveillance devices, tremendous num- bers of object trajectory data were collected, which makes extracting useful information imperative and challenging. Trajectory clustering is an eﬃcient ∗Corresponding author Email addresses: <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS> (Jiang Bian), <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS> (Dayong Tian), <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS> (Yuanyan Tang), <PRESIDIO_ANONYMIZED_EMAIL_ADDRESS> (Dacheng Tao) Preprint submitted to Journal of LATEX Templates July 18, 2024 arXiv:1802.06971v1 [cs.CV] 20 Feb 2018  method to analyze trajectory data and it has been applied in pattern recogni- tion, 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 [1], traﬃc monitoring [2][3][4], activity understanding [5][6][7], abnormal detection [8][9][10][11], 3 dimensional reconstruction[12], weather forecasting [13] and ge- ography [14]. Trajectory"

[Page no. 9] "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 8  similarity without considering lengths and time ordering [67][68]. W(A, B) = min f 1 n n � i=1 ||ai − bf(i)||2, (6) where A has n points and B has m points, all mappings f : [1, n] → [1, m] should satisfy the requirements that f(1) = 1, f(n) = m and f(i) ≤ f(j), for all 1 ≤ i ≤ j ≤ n. 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 LCSSϵ,δ(A, B) is written as LCSSϵ,δ(A, B) =                          0, if A or B is empty 1 + LCSSϵ,δ(Head(A), Head(B)), if ||aN − bM|| < ϵ and |N − M| < δ max(LCSSϵ,δ(Head(A), B), LCSSϵ,δ(A, Head(B))), otherwise, (7) where Head(A) indicates ﬁrst N −1 points belonging to A and Head(B) denotes ﬁrst M − 1 points of B. Finally, D(A, B) = 1 − LCSSϵ,δ(A,B) max(N,M) . In [40][69][70], more other distance types are proposed to consider more properties such as angle distance, center distance and parallel distance, which are deﬁned as dangle(Li, Lj) =    ||Lj|| × sin(θ), 0o ≤ θ ≤ 90o ||Lj||, 90o ≤ θ ≤ 180o, (8) where θ is the smaller intersecting angle between Li and Lj. dcenter(Li, Lj) = ||centeri − centerj||, (9) where dcenter(Li, Lj) is the Euclidean distance between center points of Li and Lj. 9"

[Page no. 14] "B) = 1 NA � ai∈A ||(xa i − xb ψ(i), ya i − yb ψ(i) + γd(βa i , βb ψ(i)))||, (11) where ψ(i) = arg minj∈B ||(xa i − xb j, ya i − yb j)|| and the minimum distance value is counted as the distance between A and B. NA is the total number of points belonging to A, d(βa i , βb ψ(i)) indicates the dissimilarity of A and B, and γ is weight parameter. In the ﬁne-clustering stage, the model aims at distinguishing distortions by considering directed similarity SA→B and conﬁdence CA→B SA→B = � ai∈A c(ai, bψ(i))s(ai, bψ(i)) � ai∈A c(ai, bψ(i)) , (12) CA→B = � ai∈A c(ai, bψ(i))2 � ai∈A c(ai, bψ(i)) , (13) where c(ai, bψ(i)) = exp( −||(xa i −xb ψ(i),ya i −yb ψ(i))|| σ1 ) and s(ai, bψ(i)) = exp(−d(βa i ,βb ψ(i)) σ2 . Furthermore, a similar hierarchical framework is explored to group videos by constructing the trajectories of video [92] as an unordered tree, and a ker- nel method recognizes videos by clustering the trees. In addition, Hierarchical Clustering models also recognize actions from video in [93] and [94]. For two trajectories in video, a = {xa 1, xa 2, · · · , xa ta} and b = {xb 1, xb 2, · · · , xb tb}, the dis- tance is computed as follows, d(a, b) = max t∈[τ1,τ2] dspatial[t] · 1 τ2 − τ1 τ2 � t=τ1 dvelocity[t], (14) where dspatial[t] is the positional distance at time stamp t, and dvelocity[t] is the similarity measurement of velocity. An aﬃnity matrix w(a, b) = exp(−d(a, b)) is calculated and trajectories are clustered by greedy agglomerative hierarchical models [93][94]. The clusters are overlapped because of similar parts, so every trajectory is weighted and optimized to classify"

[Page no. 22] "diﬀerences of trajectory data and requires a huge number of data to train it. Though the su- pervised methods obtain the classiﬁers by observing a number of training data, 21  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 super- vised 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 [4][8][9][132]. Detected anomaly trajectory data are used to recalculate the rep- resentation of anomaly trajectory clusters in [132]. Trajectory data of video are modeled as the combination of normal and abnormal patterns, and probabilistic terms characterize the patterns in [9]. From this modeling, the terms can be updated by the detected inquiry trajectory. In order to detect abnormal trajec- tories faster in complex scene, low-rank approximation is employed to describe trajectory data and the new detected abnormal ones update the threshold in [8]. Inspired by Hierarchical Frameworks, trajectories and the clusters are rep- resented as a tree where children nodes indicate trajectories and roots de- note the representations of the clusters in [133], [134] and [135]. A new clus- ter is created if no clusters close to the inquiry trajectory. Trajectory T is constructed as"

Query: What are the semi supervised methods described in this survey ?
Answer:
Assistant: