Fitting monotonic polynomials to data
The regression model \(Y=P_k(x)+\varepsilon=\sum_j^k b_j x^j+\varepsilon\) is considered, where \(P_k\) is a monotonically increasing polynomial (i.e. \(P'_k(x)\geq 0\)). To fit this model to the data \((Y_i,x_i)\), \(i=1,\dots,n\), the author uses the generalized least squares criterion \(S(b)=(Y-Xb)^T\Gamma(Y-Xb)\), where \(Y=(Y_1,\dots,Y_n)\), \(X=((x_i)^j)_{i=1,\dots ,n, \;j=1,\dots, k}\), and \(\Gamma\) is a general symmetric matrix. Then the problem is to minimize \(S(b)\) subject to \(P'_k(x)\geq 0\) for all real \(x\). This problem is reduced to the problem of minimization of \(S(b)\) subject to constraints \(h(x_m^*)b=0\), \(m=1,\dots, M\), where \(h(x)=(0,1,2x,\dots, kx^{k-1})\), and \(X^*=\{x_1^*,\dots,x_M^*\}\) is the set of points at which the constraint \(P'_k(x) =0\) is active. The author proposes to use the iterative primal-dual active set algorithm for construction of \(X^*\) and minimization of \(S(b)\). Examples for the data simulated by the formula \(Y=4x(x-2)^2(x+0.5)^2(x^2+2)+\varepsilon\) and for car (fuel consumption; engine power output) data are presented. The application of the algorithm for the transformation of data to normality and a modification of the algorithm for the \(L_1\) criterion (instead of the least squares) are discussed.
scientific article; zbMATH DE number 1409678