Nadaraya-Watson estimator for sensor fusion problems

In a system of N sensors, the sensor S_j, j=1,2...,N, outputs Y^j∈[0, 1], according to an unknown probability density p_j(Y^j|X), corresponding to input X∈[0, 1]. A training n-sample (X₁,Y₁), (X₂,Y₂), ..., (X_n,Y_n) is given where Y_i=(Y_i^1,Y_i^2,...,Y _i^N) such that Y_i^j is the output of S_j in response to input X_i. The problem is to estimate a fusion rule f:[0,1]^N→[0,1], based on the sample, such that the expected square error, I(f), is minimized over a family of functions ℱ with uniformly bounded modulus of smoothness. Let f* minimize I(.) over ℱ; f* cannot be computed since the underlying densities are unknown. We estimate the sample size sufficient to ensure that Nadaraya-Watson estimator fˆ satisfies P[I(fˆ)-I(f*)>ε]<δ for ε>0 and δ, 0<δ<1. We apply this method to the problem of detecting a door by a mobile robot equipped with arrays of ultrasonic and infrared sensors