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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Multi-fractional order estimator ： ウィキペディア英語版 Multi-fractional order estimator The multi-fractional order estimator (MFOE)〔Bell, J. W., Simple Disambiguation Of Orthogonal Projection In Kalman’s filter Derivation, Proceedings of the International Conference on Radar Systems, Glasgow, UK. October, 2012.〕〔Bell, J. W., A Simple Kalman Filter Alternative: The Multi-Fractional Order Estimator, IET-RSN, Vol. 7, Issue 8, October 2013.〕 is a straightforward, practical, and flexible alternative to the Kalman filter (KF)〔Kalman, R. E., A New Approach to Linear Filtering and Prediction Problems, Journal of Basic Engineering, Vol. 82D, Mar. 1960.〕〔Sorenson, H. W., Least-squares estimation: Gauss to Kalman, IEEE Spectrum, July, 1970.〕 for tracking targets.〔Radar tracker〕 The MFOE is focused strictly on simple and pragmatic fundamentals along with the integrity of mathematical modeling. Like the KF, the MFOE is based on the least squares method (LSM) invented by Gauss〔〔〔 and the orthogonality principle at the center of Kalman's derivation.〔〔〔〔 Optimized, the MFOE yields better accuracy than the KF and subsequent algorithms such as the extended KF〔Burkhardt, R., et.al., Titan Systems Corporation Atlantic Aerospace Division; Shipboard IRST Processing with Enhanced Discrimination Capability; Sponsor: Naval Surface Warfare Center, Dahlgren, VA; Contract #: N00178-98-C-3020; September 19, 2000 (p. 41).〕 and the interacting multiple model (IMM).〔Blom, H. A. P., An efficient filter for abruptly changing systems, in Proceedings of the 23rd IEEE Conference on Decision and Control Las Vegas, NV, Dec. 1984, 656-658.〕〔Blom, H. A. P., and Bar-Shalom, Y., The interacting multiple model algorithm for systems with Markovian switching coefficients, IEEE Trans. Autom. Control, 1988, 33, pp. 780–783〕〔Bar-Shalom, Y. and Li, X. R., Estimation and Tracking : Principles, Techniques, and Software Artech House Radar Library, Boston, 1993.〕〔Mazor, E., Averbuch, A., Bar-Shalom, Y., Dayan, J., Interacting Multiple Model Methods in Target Tracking: A Survey; IEEE T-AES, Jan 1998〕 The MFOE is an expanded form of the LSM, which effectively includes the KF〔〔〔 and ordinary least squares (OLS)〔()〕 as subsets (special cases). OLS is revolutionized in〔 for application in econometrics. The MFOE also intersects with signal processing, estimation theory, economics, finance, statistics, and the method of moments. The MFOE offers two major advances: (1) minimizing the mean squared error (MSE) with fractions of estimated coefficients (useful in target tracking)〔〔 and (2) describing the effect of deterministic OLS processing of statistical inputs (of value in econometrics)〔 ==Description== Consider equally time spaced noisy measurement samples of a target trajectory described by〔〔 $y\_n\; =\; \backslash sum\_\; ^J\; c\_j\; n^\; +\; \backslash eta\_n=\; x\_n\; +\; \backslash eta\_n$ where ''n'' represents both the time samples and the index; the polynomial describing the trajectory is of degree J-1; and $\backslash eta\_n$ is zero mean, stationary, white noise (not necessarily Gaussian) with variance $\backslash sigma\_n\; ^2$. Estimating x(t) at time $\backslash tau$ with the MFOE is described by $\backslash hat\; x\; (\backslash tau)\; =\; \backslash sum\_^\; y\_n\; w\_n\; (\backslash tau)$ where the hat (^) denotes an estimate, ''N'' is the number of samples in the data window, $\backslash tau$ is the time of the desired estimate, and the data weights are $w\_n\; (\backslash tau)\; =\; \backslash sum\; \_m\; U\_\; T\_m\; (\backslash tau)\; f\_m$ The $U\_$ are orthogonal polynomial coefficient estimators. $T\_(\backslash tau)$ (a function detailed in〔〔) projects the estimate of the polynomial coefficient $c\_m$ to the desired estimation time $\backslash tau$. The MFOE parameter 0≤$f\_m$≤1 can apply a fraction of the projected coefficient estimate. The combined terms $U\_T\_m$ effectively constitute a novel set of expansion functions with coefficients $f\_m$. The MFOE can be optimized at time $\backslash tau$ as a function of the $f\_m$s for given measurement noise, target dynamics, and non-recursive sliding data window size, ''N''. However, for all $f\_m=1$, the MFOE reduces and is equivalent to the KF in the absence of process noise, and to the standard polynomial LSM. As in the case of coefficients in conventional series expansions, the $f\_m$s typically decrease monotonically as higher order terms are included to match complex target trajectories. For example, in〔 the $f\_m$s monotonically decreased in the MFOE from $f\_1=1$ to $f\_5\; \backslash gtrsim\; 0$ , where $f\_m\; =\; 0$ for m ≧ 6. The MFOE in〔 consisted of five point, 5th order processing of composite real (but altered for declassification) cruise missile data. A window of only 5 data points provided excellent maneuver following; whereas, 5th order processing included fractions of higher order terms to better approximate the complex maneuvering target trajectory. The MFOE overcomes the long-ago rejection of terms higher than 3rd order because, taken at full value (i.e., $f\_s=1$), estimator variances increase exponentially with linear order increases. (This is elucidated below in the section "Application of the FOE".) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Multi-fractional order estimator」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.