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 Xqq[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 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 f′ satisfies P[I(f′) -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.