Abstract. End-member mixing analysis (EMMA) is a method of interpreting stream water chemistry variations and is widely used for chemical hydrograph separation. It is based on the assumption that stream water is a conservative mixture of varying contributions from well-characterized source solutions (end-members). These end-members are typically identified by collecting samples of potential end-member source waters from within the watershed and comparing these to the observations. Here we introduce a complementary data-driven method (convex hull end-member mixing analysis – CHEMMA) to infer the end-member compositions and their associated uncertainties from the stream water observations alone. The method involves two steps. The first uses convex hull nonnegative matrix factorization (CH-NMF) to infer possible end-member compositions by searching for a simplex that optimally encloses the stream water observations. The second step uses constrained K-means clustering (COP-KMEANS) to classify the results from repeated applications of CH-NMF and analyzes the uncertainty associated with the algorithm. In an example application utilizing the 1986 to 1988 Panola Mountain Research Watershed dataset, CHEMMA is able to robustly reproduce the three field-measured end-members found in previous research using only the stream water chemical observations. CHEMMA also suggests that a fourth and a fifth end-member can be (less robustly) identified. We examine uncertainties in end-member identification arising from non-uniqueness, which is related to the data structure, of the CH-NMF solutions, and from the number of samples using both real and synthetic data. The results suggest that the mixing space can be identified robustly when the dataset includes samples that contain extremely small contributions of one end-member, i.e., samples containing extremely large contributions from one end-member are not necessary but do reduce uncertainty about the end-member composition.
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Structure Assisted NMF Methods for Separation of Degenerate Mixture Data with Application to NMR Spectroscopy
In this paper, we develop structure assisted nonnegative matrix factorization
(NMF) methods for blind source separation of degenerate data. The motivation
originates from nuclear magnetic resonance (NMR) spectroscopy, where a multiple mixture NMR spectra are recorded to identify chemical compounds with similar
structures. Consider the linear mixing model (LMM), we aim to identify the chemical compounds involved when the mixing process is known to be nearly singular. We first consider a class of data with dominant interval(s) (DI) where each of
source signals has dominant peaks over others. Besides, a nearly singular mixing process produces degenerate mixtures. The DI condition implies clustering
structures in the data points. Hence, the estimation of the mixing matrix could be
achieved by data clustering. Due to the presence of the noise and the degeneracy
of the data, a small deviation in the estimation may introduce errors in the output.
To resolve this problem and improve robustness of the separation, methods are
developed in two aspects. One is to find better estimation of the mixing matrix by
allowing a constrained perturbation to the clustering output, and it can be achieved
by a quadratic programming. The other is to seek sparse source signals by exploiting the DI condition, and it solves an 1 optimization. If no source information is
available, we propose to adopt the nonnegative matrix factorization approach by
incorporating the matrix structure (parallel columns of the mixing matrix) into the
cost function and develop multiplicative iteration rules for the numerical solutions.
We present experimental results of NMR data to show the performance and reliability of the method in the applications arising in NMR spectroscopy.
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- Award ID(s):
- 1924548
- NSF-PAR ID:
- 10339326
- Date Published:
- Journal Name:
- International journal of mathematics and computation
- Volume:
- 33
- Issue:
- 1
- ISSN:
- 0974-570X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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