Least squares fitting Gaussian type curve

Application of gradient descent method to the sedimentary grain-size distribution fitting

“…Numerical methods for minimizing functional ϕ (θ ) defined by Problem 2.1 require an initial approximation θ (0) ∈ Θ which is as good as possible. Many initial value techniques have been developed (see [10][11][12][13][14][15]18,19]) and the choice, however, is arbitrary. In this section we determine the initial value θ (0) by linearizing the nonlinear Problem 2.1, which is called the linearization method, via the following two steps.…”

“…Some theorems about the existence of the LSE for some special function can be found, e.g., in [10][11][12][13][14][15][16][17]. In [10] Jukić and Scitovski have given the existence theorem of LS fitting Gaussian type curve.…”

“…In [10] Jukić and Scitovski have given the existence theorem of LS fitting Gaussian type curve. Actually, the existence theorem of the LSE for the sum of several normal functions, which are indeed a series of special Gaussian functions, could be readily proved with the given condition in this paper in the similar way as Jukić and Scitovski did in [10], and the details will be given in Section 3.…”

“…One essential suggestion is the Gauss-Newton (GN) method, which has been applied to nonlinear data fitting in many studies (see, e.g., [10,13,18,20]). In addition Hölmstrom and Petersson also presented some other methods such as the quasi-Newton method, Levenberg-Marquardt (LM) method, etc in [18].…”

“…To answer the second question, some authors have given valuable proposals (see [10][11][12][13][14][15]18,19]). In [18] Hölmstrom and Petersson used a modified Proney method for the initial approximation for a sum of exponential functions; Jukić and Marković et al [10][11][12][13][14][15] have proposed several practical initial value techniques for some special functions; Hasdorff [19] derived an expression for initial approximation from gradient. In this paper we will determine the initial approximation by linearizing the nonlinear LS problem.…”