Nadaraya-Watson estimator for sensor fusion
Nageswara S. V. Rao · Optical Engineering · 1997
In a system of N sensors, the sensor S j , j = 1; 2 : : : ; N , outputs Y (j) 2 [0; 1], according to an unknown probability density p j (Y (j) jX), corresponding to input X 2 [0; 1]. A training n-sample (X 1 ; Y 1 ), (X 2 ; Y 2 ), : : :, (X n ; Y n ) is given where Y i = (Y (1) i ; Y (2) i ; : : : ; Y (N) i ) such that Y (j) i is the output of S j in response to input X i . The problem is to estimate a fusion rule f : [0; 1] N 7! [0; 1], based on the sample, such that the expected square error I(f) = Z [X \\Gamma f(Y )] 2 p(Y jX)p(X)dY (1) dY (2) : : : dY (N) dX is minimized over a family of functions F with uniformly bounded modulus of smoothness, where Y = (Y (1) ; Y (2) ; : : : ; Y (N) ). 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) \\Gamma I(f ) ? ffl] ! ffi for any ffl ? 0 and ffi, 0 ! ffi ! 1...