Nearest neighbor pattern classification

The case of n unity-variance random variables x1, XZ,. * *, x, governed by the joint probability density w(xl, xz, * * * x,) is considered, where the density depends on the (normalized) cross-covariances pii = E[(xi jzi)(xi li)]. It is shown that the condition holds for an “arbitrary” function f(xl, x2, * * * , x,) of n variables if and only if the underlying density w(xl, XZ, * * * , x,) is the usual n-dimensional Gaussian density for correlated random variables. This result establishes a generalized form of Price’s theorem in which: 1) the relevant condition (*) subsumes Price’s original condition; 2) the proof is accomplished without appeal to Laplace integral expansions; and 3) conditions referring to derivatives with respect to diagonal terms pii are avoided, so that the unity variance assumption can be retained. Manuscript received February 10, 1966; revised May 2, 1966. The author is with the Ordnance Research Laboratory, Pennsylvania State University, State College, Pa. RICE’S THEOREM and its various extensions ([l]-[4]) have had great utility in the determination of output correlations between zero-memory nonlinearities subjected to jointly Gaussian inputs. In its original form, the theorem considered n jointly normal random variables, x1, x2, . . . x,, with respective means 21, LE2, . . . ) Z, and nth-order joint probability density, P(z 1, .x2, . 1 . , :r,,) = (27p ply,, y . exp { -a F F ;;;, ~ (2,. 2,)(:r, 5,) I , (1) where IM,l is the determinant’ of J1,, = [p,,], Pr-. = E[(sr 5$.)(x, ,2J] = xvx, &IL:, is the correlation coefficient of x, and x,, and Ail,, is the cofactor of p.? in ilf,. From [l], the theorem statement is as follows: “Let there be n zero-memory nonlinear devices specified by the input-output relationship f<(x), i = 1, 2, . 1 . , n. Let each xi be the single input to a corresponding fi(x) Authorized licensed use limited to: IEEE Xplore. Downloaded on December 22, 2008 at 14:33 from IEEE Xplore. Restrictions apply.