On the convergence rate of the unscented transformation
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Nonlinear state-space models driven by differential equations have been widely used in science. Their statistical inference generally requires computing the mean and covariance matrix of some nonlinear function of the state variables, which can be done in several ways. For example, such computations may be approximately done by Monte Carlo, which is rather computationally expensive. Linear approximation by the first-order Taylor expansion is a fast alternative. However, the approximation error becomes non-negligible with strongly nonlinear functions. Unscented transformation was proposed to overcome these difficulties, but it lacks theoretical justification. In this paper, we derive some theoretical properties of the unscented transformation and contrast it with the method of linear approximation. Particularly, we derive the convergence rate of the unscented transformation.
- Ahn, K. W., Chan, K. S. (2011). Approximate conditional least squares estimation of a nonlinear state-space model via unscented Kalman filter. Technical report# 405 Department of Statistics and Actuarial Science, University of Iowa.
- Boyce, W. E., DiPrima, R. C. (2004). Elementary differential equations and boundary value problems (8th ed.). New Jersey: Wiley.
- Diekmann, O., Heesterbeek, J. A. P. (2000). Mathematical epidemiology of infectious diseases: Model building, analysis and interpretation. West Sussex: Wiley
- Foster, J., Richards, F. B. (1991). The Gibbs phenomenon for piecewise-linear approximation. The American Mathematical Monthly, 98, 47–49.
- Holmquist, B. (1988). Moments and cumulants of the multivariate normal distribution. Stochastic Analysis and Applications, 6, 273–278.
- Isserlis, L. (1918). On a formula for the product-moment coefficient of any order of a noraml frequency distribution in any number of variables. Biometrika, 12, 134–139.
- Julier, S. J., Uhlmann, J. K. (1997). A new extension of the Kalman filter to nonlinear systems. Proceedings of AeroSense: The 11th International symposium on aerospace/defense sensing, simulation and controls. Orlando, Florida, Vol Multi Sensor Fusion, Tracking and Resource Management II.
- Julier, S. J., Uhlmann, J. K. (2004). Unscented filtering and nonlinear estimation. IEEE Review, 92, 401–422.
- Milstein, G. N., Tretyakov, M. V. (2004). Stochastic numerics for mathematical physics. Berlin, Heidelberg: Springer.
- Shim, H. T., Park, C. H. (2005). The survey of Gibbs phenomenon from Fourier series to hybrid sampling series. Journal of Applied Mathematics and Computing, 17, 719–736.
- Simon, D. (2006). Optimal state estimation. New Jersey: Wiley.
- Triantafyllopoulos, K. (2003). On the central moments of the multidimensional gaussian distribution. The Mathematical Scientist, 28, 125–128.
- Wan, E. A., van der Merwe, R. (2000). The unscented Kalman filter for nonlinear estimation. Adaptive systems for signal processing, communications, and control symposium (AS-SPCC) The IEEE, Oct 2000 Lake Louise, Alberta, pp. 153–158.
- On the convergence rate of the unscented transformation
Annals of the Institute of Statistical Mathematics
Volume 65, Issue 5 , pp 889-912
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- Unscented transformation
- Nonlinear transformation
- Monte Carlo
- Linear approximation
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