End-member modeling of compositional data: Numerical-statistical algorithms for solving the explicit mixing problem

• Published in: Mathematical Geology Vol. 29; no. 4; pp. 503 - 549
• Main Author: Weltje, Gert Jan
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01.05.1997; Springer Nature B.V
• Subjects: Algorithms; Constraint modelling; Earth sciences; Earth, ocean, space; Exact sciences and technology; Geochemistry; Iterative methods; Mathematical models; Mineralogy; Mixing; Mixing processes; Modelling; Numerical schemes; Parameter estimation; Partitioning; Petrology; Physical properties of sedimentary rocks; Robustness (mathematics); Sedimentary rocks; Sedimentology; Statistical analysis; Studies; algorithms; models; mixing; quartz; least-squares; feldspar; sandstone; linear models; Earth; sedimentary rocks; composition; mathematical methods; silicates; geometry; framework silicates; clastic rocks
• ISSN: 0882-8121; 1874-8961; 1874-8953; 1573-8868
• DOI: 10.1007/BF02775085

Linear mixing models of compositional data have been developed in various branches of the earth sciences (e.g., geochemistry, petrology, mineralogy, sedimentology) for the purpose of summarizing variation among a series of observations in terms of proportional contributions of (theoretical) end members. Methods of parameter estimation range from relatively straightforward normative partitioning by (nonnegative) least squares, to more sophisticated bilinear inversion techniques. Solving the bilinear mixing problem involves the estimation of both mixing proportions and end-member compositions from the data. Normative partitioning, also known as linear unmixing, thus can be regarded as a special situation of bilinear unmixing with (supposedly) known end members. Previous attempts to model linear mixing processes are reviewed briefly, and a new iterative strategy for solving the bilinear problem is developed. This end-member modeling algorithm is more robust and has better convergence properties than previously proposed numerical schemes. The bilinear unmixing solution is intrinsically nonunique, unless additional constraints on the model parameters are introduced. In situations where no a priori knowledge is available, the concept of an “ optimal ” solution may be used. This concept is based on the trade-off between mathematical and geological feasibility, two seemingly contradictory but equally desirable requirements of the unmixing solution.