hrvatski

Jedna metoda procjene parametara u smislu minimizacije sume L_p ortogonalnih udaljenosti

Points T_i(x_i, y_i), \, i=1, ..., m, are given in the plane. Optimal parameters b, c of a nonlinear function-model x\mapsto f(x ; b, c) should be estimated, such that the sum of orthogonal distances L_p (p>= 1) from points T_i, \, i=1, ..., m, to the graph of function f is minimal, i.e. function F(b, c)=\sum\limits_{i=1}^m d_p(T(x_\pi, f(x_\pi ; b, c)), T_i), should be minimized, where x_\pi=\arg \min\limits_xd_p(T(x, f(x ; b, c)), T_i). The problem will be considered for the linear, the exponential and the logistic function, respectively, for the most important cases: p=1, 2, \infty. If (x_i, y_i), \, i=1, ..., m, are considered to be some experimental or empirical data, then we deal here with the estimation of parameters of a generally nonlinear function in the sense of L_p orthogonal deviations, which is widely used in applied research. Since the problem in question is the problem of nondifferentiable minimization of nonlinear function F, it will be solved by using the Nelder-Mead Downhill Simplex Method, and for the calculation of L_p distances from points T_i to the graph of function f the method of one-dimensional minimization will be used, which will be elaborated in the paper by I.Soldo and K.Sabo, that will be also presented at this conference. All programs and subprograms will be done by using the {\em Mathematica} software system. We will thereby use graphic options and animation of the iterative process. Software that will be created for this purpose can be used for designing illustrative examples used in the teaching procress, but also for practical applications in various fields, such as agriculture, medicine, economics, biology, etc.

procjena parametara; ortogonalna regresija

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