Application of gradient descent method to the sedimentary grain-size distribution fitting. (English) Zbl 1173.86310

Summary: Existence of a least squares solution for a sum of several weighted normal functions is proved. The gradient descent (GD) method is used to fit the measured data (i.e. the laser grain-size distribution of the sediments) with a sum of three weighted lognormal functions. The numerical results indicate that the GD method is not only easy to operate but also could effectively optimize the parameters of the fitting function with the error decreasing steadily. Meanwhile the overall fitting results are satisfactory. As a new way of data fitting, the GD method could also be used to solve other optimization problems.


86A32 Geostatistics
65D10 Numerical smoothing, curve fitting
62J02 General nonlinear regression
90C31 Sensitivity, stability, parametric optimization
62P12 Applications of statistics to environmental and related topics
93E24 Least squares and related methods for stochastic control systems
Full Text: DOI


[1] Sun, D. H.; Bloemendal, J.; Rea, D. K.; Vandenberghe, J.; Jiang, F. C.; An, Z. S.; Su, R. X., Grain-size distribution function of polymodal sediments in hydraulic and aeolian environments, and numerical partitioning of the sedimentary components, Sedimentary Geology, 152, 263-277 (2002)
[2] Sun, D. H.; Su, R. X.; Bloemendal, J.; Lu, H. Y., Grain-size and accumulation rate records from Late Cenozoic aeolian sequences in northern China: Implications for variations in the East Asian winter monsoon and westerly atmospheric circulation, Palaeogeography, Palaeoclimatology, Palaeoecology, 264, 39-53 (2008)
[3] Sun, D. H.; Bloemendal, J.; Rea, D. K.; An, Z. S.; Vandenberghee, J.; Lu, H. Y.; Su, R. X.; Liu, T. S., Bimodal grain-size distribution of Chinese loess, and its palaeoclimatic implications, Catena, 55, 325-340 (2004)
[4] Sun, D. H.; An, Z. S.; Su, R. X.; Lu, H. Y.; Sun, Y. B., Eolian sedimentary records for the evolution of monsoon and westerly circulations of northern China in the last 2.6 Ma, Science in China (Series D), 46, 10, 1049-1059 (2003)
[5] Sun, D. H., Monsoon and westerly circulation changes recorded in the late Cenozoic aeolian sequences of Northern China, Global and Planetary Change, 41, 63-80 (2004)
[6] Dearing, J. A., Sedimentary indicators of lake-level changes in the humid temperate zone: A critical review, Journal of Paleolimnology, 18, 1, 1-14 (1997)
[7] Nocedal, J.; Wright, S. J., Numerical Optimization (2006), Springer · Zbl 1104.65059
[8] Fletcher, R., Practical Methods of Optimization, vol. 1 (1981), John Wiley & Sons · Zbl 0474.65043
[9] Gill, P. E.; Murray, W.; Wright, M. H., Practical Optimization (1981), Academic Press: Academic Press London · Zbl 0503.90062
[10] Jukić, D.; Scitovski, R., Least squares fitting Gaussian type curve, Applied Mathematics and Computation, 167, 286-298 (2005) · Zbl 1080.65011
[11] Marković, D.; Jukić, D.; Benšić, M., Nonlinear weighted least squares estimation of a three-parameter Weibull density with a nonparametric start, Journal of Computational and Applied Mathematics (2008)
[12] Jukić, D.; Kralik, G.; Scitovski, R., Least squares fitting Gompertz curve, Journal of Computational and Applied Mathematics, 169, 359-375 (2004) · Zbl 1054.65009
[14] Jukić, D.; Benšić, M.; Scitovski, R., On the existence of the nonlinear weighted least squares estimate for a three-parameter Weibull distribution, Computational Statistics and Data Analysis, 52, 4502-4511 (2008) · Zbl 1452.62485
[15] Jukić, D.; Scitovski, R., Existence of optimal solution for exponential model by least squares, Journal of Computational and Applied Mathematics, 78, 317-328 (1997) · Zbl 0872.65120
[16] Jukić, D.; Scitovski, R.; Späth, H., Partial linearization of one class of the nonlinear total least squares problem by using the inverse model function, Computing, 62, 163-178 (1999) · Zbl 0934.65014
[17] Jukić, D.; Sabo, K.; Scitovski, R., Total least squares fitting Michaelis-Menten enzyme kinetic model function, Journal of Computational and Applied Mathematics, 201, 230-246 (2007) · Zbl 1106.92027
[18] Hölmstrom, K.; Petersson, J., A review of the parameter estimation problem of fitting positive exponential sums to empirical data, Applied Mathematics and Computation, 126, 31-61 (2002) · Zbl 1023.65009
[19] Hasdorff, L., Gradient Optimization and Nonlinear Control (1976), Wiley: Wiley New York
[20] Atieg, A.; Watson, G. A., A class of methods for fitting a curve or surface to data by minimizing the sum of squares of orthogonal distances, Journal of Computational and Applied Mathematics, 158, 277-296 (2003) · Zbl 1034.65007
[21] Böckmann, C., Curve fitting and identification of physical spectra, Journal of Computational and Applied Mathematics, 70, 207-224 (1996) · Zbl 0853.65150
[22] Nyarko, E. K.; Scitovski, R., Solving the parameter identification problem of mathematical models using genetic algorithms, Applied Mathematics and Computation, 153, 651-658 (2004) · Zbl 1048.65075
[23] Ahn, S. J.; Rauh, W.; Warnecke, H.-J., Least-squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola, and parabola, Pattern Recognition, 34, 2283-2303 (2001) · Zbl 0991.68127
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