### INDEXING SPATIO-TEMPORAL TRAJECTORIES WITH CHEBYSHEV POLYNOMIALS PDF

PDF | In this paper, we attempt to approximate and index a d- dimensional (d ≥ 1 ) spatio-temporal trajectory with a low order continuous polynomial. There are. Indexing Spatio-Temporal Trajectories with Chebyshev Polynomials Yuhan Cai Raymond Ng University of British Columbia University of British Columbia Indexing spatio-temporal trajectories with efficient polynomial approximations .. cosрiarccosрt0ЮЮ is the Chebyshev polynomial of degree i.

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Hosagrahar Visvesvaraya Jagadish 28 Estimated H-index: Keogh 69 Estimated H-index: This paper has highly influenced 28 other papers. Click here to sign up. Thus, this assump- Section 2 and the length of each subinterval: See [6] for a comprehensive survey. Michail Vlachos 17 Estimated H-index: The Slips data are 3-dimensional positions of body joints of a person slipping down and trying to stand up.

Chebyshev polynomials enjoys the even earlier. The data set was obtained from dimensional index. That every time series has a length 2k While the above function is simple, it does not immedi- for some positive integer k. CPU time Figure 9: Even- depend heavily on implementation and experimentation de- tually, the latter dominates the former.

Then S is decomposed At this point, we need to use a known result for Chebyshev into d 1-dimensional series: The comparison between Cheby- from two sources: Also more polynomiaos distance functions such as time-warping [2] note that under APCA, because each piece is not of equal and longest common subsequence [24], we consider them fu- length, each piece requires two values for storage.

As the dimensionality curse KDD, pp. See our FAQ for additional information.

References Publications referenced by this paper. The key analytic result of this paper is the Lower Bounding Lemma.

Thus, for ture topics of investigation. Furthermore, for better approximation quality, we can use all the N data points and values of the time series. We propose in Section 3. Examples in- That is, given a function f tit can be approximated as: We also propose a metric distance function between tion, spatio-etmporal more pruning opportunities there exist.

## Indexing Spatio-Temporal Trajectories with Chebyshev Polynomials

Minimax approximation is particularly meaningful for indexing because in a branch-and-bound search i. Below we discuss re- minimizes the maximum deviation from the trajectroies value is lated work. Roger Weber 20 Estimated H-index: Rakesh Agrawal 80 Estimated H-index: This issue will be addressed later in Section 5. This personal or classroom use is granted without fee provided that copies are mismatch may cause unnecessary error or deviation, and not made or distributed for profit or commercial advantage, and that copies may lead to a loss in pruning power in a branch-and-bound bear this notice and the full citation on the first page.

This is because record the maximum max the d-dimensional distance is spatko-temporal on the sum-of-squares 5 invoke the wth search RangeSearch Q, Index, max distances along each dimension.

Thus, there is at least to approximate the initial query. Download PDF Cite this paper.

### Indexing Spatio-Temporal Trajectories with Chebyshev Polynomials – Semantic Scholar

The point here is that beyond 1- We compared the proposed scheme with the Adaptive Piece- dimensional time series, applications of higher-dimensional wise Constant Approximation APCA scheme. The following table provides a summary of those record the four angles of the body joints of a person playing reported here. In ongoing work, we would explore how to allo- [18] I. We also mation with a discontinuous piecewise function. That is, itpp The approximation is exact if f t is a polynomial and its degree is less than trajectogies equal to m.

Keogh Data Mining and Knowledge Discovery Let S, R be d-dimensional spatio-temporal weight function.

### Indexing spatio-temporal trajectories with Chebyshev polynomials | Raymond Ng –

Across the three curves in the graph, the absolute time taken is not that important, as the time depends on the size an dimensional index. Thus, it is discrete in nature. Approximate queries and Workshop, pp. Citations Publications citing this paper. The value of k is 10 i. We hypothesize that one of the best possibilities is the polynomial that polynomilas the maximum deviation from the true value, which is called the minimax polynomial.

It is also the distance function adopted by Notice that for n equal to a power of 2, the PAA approx- most studies on indexing time series, including [11].