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2. HEURISTICS AND 11IE METHOD OF RECURRENT BACK-PROPAGAnON Suppose that we observe a realization of a sequence {Z,}={Z,: t= O. I} of ran-dom vectors. where z,=(y,. X [) T (with T denoting the transposition o~ rator). Y, is (for simplicity) a scalaf, and X, is avx 1 veCtOr, veINs {I, 2}. We interpret Y, as a target value at time t, and Xt as a vector of input variables influencing Yt and generated by nature. Xt may contain lagged values of Y,(eg Y,-lt Y,-2t...) as well as lagged values of other variables. For con-venience, we assume tluoUghout that the first element of X,(ie X, I) is always equal to one. LetXt & (XO,..., X,) denote the hiStory of the X process from time zerotbrough time t.(Similarly, for any sequence {ar}, at.(ao,..., ar).) Suppose we are interested in approximating E (Y, IX'). the conditional expectation of Y, given X'. by a parametric function of X'. so that f,: m: v (t+ l) x e-+ IR (say) defines a family of approximations f,(X'. 8) as 8 ranges over the parameter space ec mI, s elN, say.