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Indexing spatiotemporal trajectories with Chebyshev polynomials Cai, Yuhan

Abstract

In this thesis, we investigate the subject of indexing large collections of spatiotemporal trajectories for similarity matching. Our proposed technique is to first mitigate the dimensionality curse problem by approximating each trajectory with a low order polynomial-like curve, and then incorporate a multidimensional index into the reduced space of polynomial coefficients. There are many possible ways to choose the polynomial, including Fourier transforms, splines, non-linear regressions, etc. Some of these possibilities have indeed been studied before. We hypothesize that one of the best approaches is the polynomial that minimizes the maximum deviation from the true value, which is called the minimax polynomial. Minimax approximation is particularly meaningful for indexing because in a branch-and-bound search (i.e., for finding nearest neighbours), the smaller the maximum deviation, the more pruning opportunities there exist. In general, among all the polynomials of the same degree, the optimal minimax polynomial is very hard to compute. However, it has been shown that the Chebyshev approximation is almost identical to the optimal minimax polynomial, and is easy to compute [32]. Thus, we shall explore how to use the Chebyshev polynomials as a basis for approximating and indexing d-dimensional (d≥1) trajectories. , The key analytic result of this thesis is the Lower Bounding Lemma. That is, we show that the Euclidean distance between two d-dimensional trajectories is lower bounded by the weighted Euclidean distance between the two vectors of Chebyshev coefficients. This lemma is not trivial to show, and it ensures that indexing with Chebyshev coefficients admits no false negatives. To complement the analytic result, we conduct comprehensive experimental evaluation with real' and generated 1-dimensional to 4-dimensional data sets. We compare the proposed scheme with the Adaptive Piecewise Constant Approximation (APCA) scheme. Our preliminary results indicate that in all situations we test, Chebyshev indexing dominates APCA in pruning power, I/O and CPU costs.

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