An algorithm for nonlinear L1 curve-fitting based on the smooth approximation

Abstract

In this paper the author proposes a simple and efficient algorithm for L 1 nonlinear curve-fitting. It is based on the smooth approximation in arbitrarily small neighborhoods of the points of discontinuous differentiability caused by the absolute value operator. Such a smooth approximation can be minimized using efficient algorithms in L 2 norm. The approximate function in the arbitrarily small neighborhood 2 β t is taken as u t 2 + β t 2) 2β t , u t = y t − f( x t , θ), t =1, 2,…, n (the number of observations). In order to form β t , we give β max and β min, (0 ≤ β min ≤ β max), and let β t = β max for ¦u t¦ > β max , β t = ¦u t¦ for β min ≤ ¦u t¦ ≤ β max , and β t = β min for ¦u t¦ < β min . The author proves that the convergence is independent of the β max but β min. The convergence is also proved when Levenberg-Marquardt's iterative procedure is employed. β min is a tolerance error, and can be set an arbitrary small value in all iterations. It is easy to see that our algorithm will reduce to Tishler-Zang's algorithm (1982) if β 1 = β 2 = … = β n , and will be equivalent to the method of least squares with weighted 1 β t if β max = max¦u t¦ . One numerical example shows that the present algorithm is efficient.