Nadaraya-Watson estimator for sensor fusion problems

Proceedings of International Conference on Robotics and Automation Pub Date : 1997-04-20 DOI:10.1109/ROBOT.1997.619268

N. Rao

引用次数: 16

Abstract

In a system of N sensors, the sensor S/sub j/, j=1,2...,N, outputs Y/sup j/spl isin//[0, 1], according to an unknown probability density p/sub j/(Y/sup j|/X), corresponding to input X/spl isin/[0, 1]. A training n-sample (X/sub 1/,Y/sub 1/), (X/sub 2/,Y/sub 2/), ..., (X/sub n/,Y/sub n/) is given where Y/sub i/=(Y/sub i//sup 1,/Y/sub i//sup 2,/...,Y/sub i//sup N/) such that Y/sub i//sup j /is the output of S/sub j/ in response to input X/sub i/. The problem is to estimate a fusion rule f:[0,1]/sup N//spl rarr/[0,1], based on the sample, such that the expected square error, I(f), is minimized over a family of functions /spl Fscr/ with uniformly bounded modulus of smoothness. Let f* minimize I(.) over /spl Fscr/; f* cannot be computed since the underlying densities are unknown. We estimate the sample size sufficient to ensure that Nadaraya-Watson estimator f/spl circ/ satisfies P[I(f/spl circ/)-I(f*)>/spl epsiv/]0 and /spl delta/, 0

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传感器融合问题的Nadaraya-Watson估计

在有N个传感器的系统中，传感器S/sub j/， j=1,2，…，N，根据未知概率密度p/sub j/(Y/sup j|/X)，对应输入X/spl isin/[0,1]，输出Y/sup j/spl isin//[0,1]。培训n个抽样(X /子1 / Y /订阅1 /),(X /子2 / Y / sub 2 /),…(X / an / Y / an /)是考虑到Y / (i / = (Y /子我/ /吃晚饭1,/ Y /订阅/ /一口2,/……，Y/下标i//sup N/)，使得Y/下标i//sup j/是S/下标j/响应输入X/下标i/的输出。问题是基于样本估计融合规则f:[0,1]/sup N//spl rarr/[0,1]，使得期望平方误差I(f)在光滑模数一致有界的函数族/spl Fscr/上最小化。令f*最小化I(.) / /spl Fscr/;由于底层密度未知，因此无法计算F *。我们估计的样本量足以保证nadarya - watson估计量f/spl circ/满足P[I(f/spl circ/)-I(f*)>/spl epsiv/]0和/spl delta/， 0

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Proceedings of International Conference on Robotics and Automation

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