Metrics on the space of sets of trajectories are important for scientists in the field of computer vision, machine learning, robotics, and general artificial intelligence. However, existing notions of closeness between sets of trajectories are either mathematically inconsistent or of limited practical use. In this paper, we outline the limitations in the current mathematically-consistent metrics, which are based on OSPA ( 2008); and the inconsistencies in the heuristic notions of closeness used in practice, whose main ideas are common to the CLEAR MOT measures (Keni and Rainer 2008) widely used in computer vision. In two steps, we then propose a new intuitive metric between sets of trajectories and address these limitations. First, we explain a solution that leads to a metric that is hard to compute. Then we modify this formulation to obtain a metric that is easy to compute while keeping the useful properties of the previous metric. Our notion of closeness is the first demonstrating the following three features: the metric 1) can be quickly computed, 2) incorporates confusion of trajectories' identity in an optimal way, and 3) is a metric in the mathematical sense.
It is worth mentioning a few works that differ from the main stream in this aspect. One of them is @cite_1 , where the authors define a similarity measure based on comparing the occurrence of special discrete events in @math and @math , and another is @cite_30 , where the authors propose an information theoretic measure of similarity between sets of trajectories. Finally, in @cite_7 the authors propose a similarity measured based on hidden Markov models that does not assume that the temporal sampling rates of the trajectories are equal.
@cite_30 @cite_1 @cite_7
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A new class of distances appropriate for measuring similarity relations between sequences, say one type of similarity per distance, is studied. We propose a new "normalized information distance," based on the noncomputable notion of Kolmogorov complexity, and show that it is in this class and it minorizes every computable distance in the class (that is, it is universal in that it discovers all computable similarities). We demonstrate that it is a metric and call it the similarity metric . This theory forms the foundation for a new practical tool. To evidence generality and robustness, we give two distinctive applications in widely divergent areas using standard compression programs like gzip and GenCompress. First, we compare whole mitochondrial genomes and infer their evolutionary history. This results in a first completely automatic computed whole mitochondrial phylogeny tree. Secondly, we fully automatically compute the language tree of 52 different languages. In Semantic Textual Similarity (STS), systems rate the degree of semantic equivalence, on a graded scale from 0 to 5, with 5 being the most similar. This year we set up two tasks: (i) a core task (CORE), and (ii) a typed-similarity task (TYPED). CORE is similar in set up to SemEval STS 2012 task with pairs of sentences from sources related to those of 2012, yet different in genre from the 2012 set, namely, this year we included newswire headlines, machine translation evaluation datasets and multiple lexical resource glossed sets. TYPED, on the other hand, is novel and tries to characterize why two items are deemed similar, using cultural heritage items which are described with metadata such as title, author or description. Several types of similarity have been defined, including similar author, similar time period or similar location. The annotation for both tasks leverages crowdsourcing, with relative high interannotator correlation, ranging from 62 to 87 . The CORE task attracted 34 participants with 89 runs, and the TYPED task attracted 6 teams with 14 runs. The authors propose models for both temporal and spatial locality of reference in streams of requests arriving at Web servers. They show that simple models based on document popularity alone are insufficient for capturing either temporal or spatial locality. Instead, they rely on an equivalent, but numerical, representation of a reference stream: a stack distance trace. They show that temporal locality can be characterized by the marginal distribution of the stack distance trace, and propose models for typical distributions and compare their cache performance to the traces. They also show that spatial locality in a reference stream can be characterized using the notion of self-similarity. Self-similarity describes long-range correlations in the data set, which is a property that previous researchers have found hard to incorporate into synthetic reference strings. They show that stack distance strings appear to be strongly self-similar, and provide measurements of the degree of self-similarity in the traces. Finally, they discuss methods for generating synthetic Web traces that exhibit the properties of temporal and spatial locality measured in the data. We investigate techniques for analysis and retrieval of object trajectories in two or three dimensional space. Such data usually contain a large amount of noise, that has made previously used metrics fail. Therefore, we formalize non-metric similarity functions based on the longest common subsequence (LCSS), which are very robust to noise and furthermore provide an intuitive notion of similarity between trajectories by giving more weight to similar portions of the sequences. Stretching of sequences in time is allowed, as well as global translation of the sequences in space. Efficient approximate algorithms that compute these similarity measures are also provided. We compare these new methods to the widely used Euclidean and time warping distance functions (for real and synthetic data) and show the superiority of our approach, especially in the strong presence of noise. We prove a weaker version of the triangle inequality and employ it in an indexing structure to answer nearest neighbor queries. Finally, we present experimental results that validate the accuracy and efficiency of our approach.