End-member modeling of compositional data: Numerical-statistical algorithms for solving the explicit mixing problem |
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| Authors: | Gert Jan Weltje |
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| Institution: | (1) Geology Department, Faculty of Earth Sciences, Utrecht University, P.O. Box 80021, 3508 TA Utrecht, The Netherlands |
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| Abstract: | 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. |
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| Keywords: | inversion compositional data linear mixing model constrained weighted least squares |
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