GeneralFunctionFittingwithRadialBasisFunctionNetwork.pptVIP

General Function Fitting with Radial Basis Function Network CS570 Term Project 3? ???, ??? * * Contents * Radial Basis Function Network * Least Square fitting * Function fitting Program Product * Conclusion Discussion Radial Basis Function h ( x) = ? ((x-c)T R-1 (x-c)) where R is metric, (x-c)T R-1(x-c) is the distance between the input x and the center c in the metric defined by R. Gaussian: ? (z) = e –z Multiquadric: ? (z) = (1+z) Inverse Multiquadric: ? (z) = (1+z) Cauchy: ? (z) = (1+z)-1 Euclidean metric R = r2I, for some radius ? -? Gaussian: r2 (x-c)2 h(x) = exp Multiquadric: r r2 + (x-c)2 h(x) = Inverse multiquadric: h(x) = r r2 + (x-c)2 Cauchy: h(x) = r r2 + (x-c)2 Radial Basis Function Radial Basis Function Network Each of n components of the input vector x feeds forward to m basis functions whose outputs are linearly combined with weights {wj}m into the network output f(x) h1(x) x1 xi xn hj(x) hm(x) f (x) … … … … w1 wj wm j=1 f (x) = j=1 m wj hj (x) S = i=1 p ( yi - f (xi))2 C = i=1 p ( yi - f (xi))2 + j=1 m ?j wj 2 where the {?j} m are regularization parameters. j=1 where {(xi, yi)}p is training set I=1 , sum-square-error , cost function Linear Model Function Fitting  with Least Squares Aw = HT y H = h1 (x1) h2 (x1) hm(x1) … h1 (x2) h2 (x2) hm(x2) … h1 (xp) h2 (xp) hm(xp) … … … … , where H, the design matrix, is A-1 = (HTH + ? )-1 , where ? = … … 0 … … … … ?1 0 0 ?2 0 ?m 0 0 w = A-1HT y normal equation y = y2 y1 … yp w = w2 w1 … wm vector of training set outputs vector of weights which minimizes the cost function