Time Series Motif Discovery (TSMD) is defined as searching for patterns that are previously unknown and appear with a given frequency in time series. Another problem strongly related with TSMD is Word Segmentation. This problem has received much attention from the community that studies early language acquisition in babies and toddlers. The development of biologically plausible models for word segmentation could greatly advance this field. Therefore, in this article, we propose the Variable Input Length Map (VILMAP) for Motif Discovery and Word Segmentation. The model is based on the Self-Organizing Maps and can identify Motifs with different lengths in time series. In our experiments, we show that VILMAP presents good results in finding Motifs in a standard Motif discovery dataset and can avoid catastrophic forgetting when trained with datasets with increasing values of input size. We also show that VILMAP achieves results similar or superior to other methods in the literature developed for the task of word segmentation.
Defining @math as a threshold that establishes a minimum allowed similarity or the maximum distance allowed between two occurrences of a Motif, there are two major definitions for the problem of Time Series Motif Discovery (TSMD), according to Mueen (2014) @cite_6 :
@cite_6
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Let @math be a tree space (or tree network) represented by a weighted tree with @math vertices, and @math be a set of @math stochastic points in @math , each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the @math -threshold probability and the expectation of the closest-pair distance of a realization of @math . We propose the first algorithm to compute the @math -threshold probability in @math time for any given threshold @math , which immediately results in an @math -time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a @math -approximation for the expected closest-pair distance in @math time, by arguing that the expected closest-pair distance can be approximated via @math threshold probability queries. For the latter, we study the @math most-likely nearest-neighbor search ( @math -LNN) via a notion called @math most-likely Voronoi Diagram ( @math -LVD). We show that the size of the @math -LVD @math of @math on @math is bounded by @math if the existence probabilities of the points in @math are constant-far from 0. Furthermore, we establish an @math average-case upper bound for the size of @math , by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers @math -LNN queries in @math time using average-case @math space, and worst-case @math space if the existence probabilities are constant-far from 0. Finally, we also give an @math -time algorithm to construct the LVD data structure. We consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math . A natural time-dependent similarity measure for two trajectories is their average distance at corresponding times. We give algorithms for computing the most similar subtrajectories under this measure, assuming the two trajectories are given as two polygonal, possibly self-intersecting lines with time stamps. For the case when a minimum duration of the subtrajectories is specified and the subtrajectories must start at corresponding times, we give a linear-time algorithm. The algorithm is based on a result of independent interest: We present a linear-time algorithm to find, for a piece-wise monotone function, an interval of at least a given length that has minimum average value. In the case that the subtrajectories may start at non-corresponding times, it appears difficult to give exact algorithms, even if the duration of the subtrajectories is fixed. For this case, we give (1+@e)-approximation algorithms, for both fixed duration and when only a minimum duration is specified. The study and comparison of sequences of characters from a finite alphabet is relevant to various areas of science, notably molecular biology. The measurement of sequence similarity involves the consideration of the different possible sequence alignments in order to find an optimal one for which the “distance” between sequences is minimum. By associating a path in a lattice to each alignment, a geometric insight can be brought into the problem of finding an optimal alignment. This problem can then be solved by applying a dynamic programming algorithm. However, the computational effort grows rapidly with the number N of sequences to be compared @math , where l is the mean length of the sequences to be compared).It is proved here that knowledge of the measure of an arbitrarily chosen alignment can be used in combination with information from the pairwise alignments to considerably restrict the size of the region of the lattice in consideration. This reduction implies fewer computations and less memory ...