Nadaraya-Watson estimator for sensor fusion problems

摘要

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/]</spl delta/ for /spl epsiv/>0 and /spl delta/, 0</spl delta/<1. We apply this method to the problem of detecting a door by a mobile robot equipped with arrays of ultrasonic and infrared sensors.