Name
Title | ||
|---|---|---|
Convergence order in trajectory estimation by piecewise-cubics and exponential parameterization |
Abstract | ||
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This paper discusses the problem of estimating the trajectory of the unknown curve gamma from the sequence of m + 1 interpolation points Q(m) = {gamma(t(i))}(i=0)(m) in arbitrary Euclidean space E-n. The respective knots T-m = {t(i)}(i=0)(m) (in ascending order) are assumed to be unknown. Such Q(m) is coined reduced data. In our setting, a piecewise-cubic Lagrange interpolation (gamma) over cap (3) : [0, (T) over cap] -> E-n is applied to fit Q(m). Here, the missing knots T-m are replaced by their estimates (T) over cap (m) = {t(i)}(i=0)(m) in accordance with the exponential parameterization. The latter is controlled by a single parameter lambda is an element of [0, 1]. This work analyzes the intrinsic asymptotics in approximating gamma by (gamma) over cap (3) based on the exponential parameterization and Q(m). The multiple goals are achieved. Firstly, the existing result established for lambda = 1 (i.e. for the cumulative chord parameterization) is extended to the remaining cases of lambda is an element of [0, 1) and more-or-less uniformly sampled Q(m). As demonstrated herein, a quartic convergence order alpha(1) = 4 in trajectory estimation drops discontinuously to the linear one alpha(lambda) = 1, for all lambda is an element of [0, 1). Secondly, the asymptotics derived in this paper is also analytically proved to be sharp with the aid of illustrative examples. Thirdly, the latter is verified in affirmative upon conducting numerical testing. Next, the necessity of more-or-less uniformity imposed on Q(m) is shown to be indispensable. In addition, several sufficient conditions for (gamma) over cap (3) to be reparameterizable to the domain of gamma are formulated. Lastly, the motivation for using the exponential parameterization with lambda is an element of [0, 1) is also outlined. |
Year | DOI | Venue |
|---|---|---|
2019 | 10.3846/mma.2019.006 | MATHEMATICAL MODELLING AND ANALYSIS |
Keywords | DocType | Volume |
interpolation,reduced data,convergence order and sharpness | Journal | 24 |
Issue | ISSN | Citations |
1 | 1392-6292 | 0 |
PageRank | References | Authors |
0.34 | 2 | 2 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Magdalena Wilkołazka | 1 | 0 | 0.34 |
| Ryszard Kozera | 2 | 163 | 26.54 |