In the light of regularized dynamic time warping kernels, this paper re-considers the concept of time elastic centroid for a setof time series. We derive a new algorithm based on a probabilistic interpretation of kernel alignment matrices. This algorithm expressesthe averaging process in terms of a stochastic alignment automata. It uses an iterative agglomerative heuristic method for averagingthe aligned samples, while also averaging the times of occurrence of the aligned samples. By comparing classification accuracies for45 heterogeneous time series datasets obtained by first nearest centroid medoid classifiers we show that: i) centroid-basedapproaches significantly outperform medoid-based approaches, ii) for the considered datasets, our algorithm that combines averagingin the sample space and along the time axes, emerges as the most significantly robust model for time-elastic averaging with apromising noise reduction capability. We also demonstrate its benefit in an isolated gesture recognition experiment and its ability tosignificantly reduce the size of training instance sets. Finally we highlight its denoising capability using demonstrative synthetic data:we show that it is possible to retrieve, from few noisy instances, a signal whose components are scattered in a wide spectral band.
A single alignment path is required to calculate the time elastic centroid of a pair of time series (Def. ). However, multiple path alignments need to be considered to evaluate the centroid of a larger set of time series. Multiple alignments have been widely studied in bioinformatics @cite_16 , and it has been shown that determining the optimal alignment of a set of sequences under the sum of all pairs (SP) score scheme is a NP-complete problem @cite_6 @cite_29 . The time and space complexity of this problem is @math , where @math is the number of sequences in the set and @math is the length of the sequences when using dynamic programming to search for an optimal solution @cite_1 . This latter result applies to the estimation of the time elastic centroid of a set of @math time series with respect to the DTW measure. Since the search for an optimal solution becomes rapidly intractable with increasing @math , sub-optimal heuristic solutions have been subsequently proposed, most of them falling into one of the following three categories.
@cite_29 @cite_16 @cite_1 @cite_6
2586337417 2167747244 2136266535 2150696241
Dynamic Time Warping (DTW) is probably the most popular distance measure for time series data, because it captures flexible similarities under time distortions. However, DTW has long been suffering from the pathological alignment problem, and most existing solutions, which essentially impose rigid constraints on the warping path, are likely to miss the correct alignments. A crucial observation on pathological alignment is that it always leads to an abnormally large number of links between two sequences. Based on this new observation, we propose a novel variant of DTW called LDTW, which limits the total number of links during the optimization process of DTW. LDTW not only oppresses the pathological alignment effectively, but also allows more flexibilities when measuring similarities. It is a softer constraint because we still let the optimization process of DTW decide how many links to allocate to each data point and where to put these links. In this paper, we introduce the motivation and algorithm of LDTW and we conduct a nearest neighbor classification experiment on UCR time series archive to show its performance. In this paper, we consider the problem of estimating the spatiotemporal alignment between N unsynchronized video sequences of the same dynamic 3D scene, captured from distinct viewpoints. Unlike most existing methods, which work for N = 2 and rely on a computationally intensive search in the space of temporal alignments, we present a novel approach that reduces the problem for general N to the robust estimation of a single line in RN. This line captures all temporal relations between the sequences and can be computed without any prior knowledge of these relations. Considering that the spatial alignment is captured by the parameters of fundamental matrices, an iterative algorithm is used to refine simultaneously the parameters representing the temporal and spatial relations between the sequences. Experimental results with real-world and synthetic sequences show that our method can accurately align the videos even when they have large misalignments (e.g., hundreds of frames), when the problem is seemingly ambiguous (e.g., scenes with roughly periodic motion), and when accurate manual alignment is difficult (e.g., due to slow-moving objects). We consider the joint optimization of beamformers and linear receivers in a MIMO interference network. Each transmitter transmits a single beam corresponding to a rank-one precoder. When the number of users K is greater than the number of antennas at each terminal N, the maximum degrees of freedom is achieved via spatial interference alignment. Interference alignment is feasible for up to K = 2N−1 users, in which case there is a finite number of solutions to the alignment conditions. This number of solutions increases rapidly with N, and the solutions depend only on the cross-channel coefficients (i. e., they are independent of the direct channels). To maximize the achievable sum rate at high SNRs we therefore wish to select an aligned solution which is best matched to the direct channels. We evaluate the performance of this scheme for large K and N, assuming that the solution is the best out of a random subset of aligned solutions. We then compare numerically this performance with the performance of previously proposed numerical (e. g., forward-backward) techniques for optimizing beams, and a new technique which tracks the local optimum as the SNR is incrementally increased, similar to a homotopy method for improving convergence properties. We observe that the incremental technique typically achieves better performance than the previously proposed methods. Alignment of time series is an important problem to solve in many scientific disciplines. In particular, temporal alignment of two or more subjects performing similar activities is a challenging problem due to the large temporal scale difference between human actions as well as the inter intra subject variability. In this paper we present canonical time warping (CTW), an extension of canonical correlation analysis (CCA) for spatio-temporal alignment of human motion between two subjects. CTW extends previous work on CCA in two ways: (i) it combines CCA with dynamic time warping (DTW), and (ii) it extends CCA by allowing local spatial deformations. We show CTW's effectiveness in three experiments: alignment of synthetic data, alignment of motion capture data of two subjects performing similar actions, and alignment of similar facial expressions made by two people. Our results demonstrate that CTW provides both visually and qualitatively better alignment than state-of-the-art techniques based on DTW.