ABlindSourceSeparationMethodforNearly ...

1 [ ] 7 Oct 2011A Blind Source Separation Method for NearlyDegenerate Mixtures and Its applications to NMRS pectroscopyYuanchang Sun and Jack Xin AbstractIn this paper, we develop a novel blind source separation (BSS) method fornonnegative and correlated data, particularly for the nearly degenerate motivation lies in nuclear magnetic resonance (NMR) spectroscopy , wherea multiple mixture NMR spectra are recorded to identify chemical compoundswith similar structures (degeneracy).There have been a number of successful approaches for solving BSS problemsby exploiting the nature of source signals. For instance, independent componentanalysis (ICA) is used to separate statistically independent (orthogonal) sourcesignals. However, signal orthogonality is not guaranteed in many real-worldproblems. This new BSS method developed here deals with nonorthogonal sig-nals. The independence assumption is replaced by a condition which requiresdominant interval(s) (DI) from each of source signals over others.

2 Additionally,the mixing matrix is assumed to be nearly singular. The method first estimatesthe mixing matrix by exploiting geometry in data clustering. Due to the de-generacy of the data, a small deviation in the estimation mayintroduce errors(spurious peaks of negative values in most cases) in the output. To resolvethis challenging problem and improve robustness of the separation, methodsare developed in two aspects. One technique is to find a betterestimation ofthe mixing matrix by allowing a constrained perturbation tothe clustering out-put, and it can be achieved by a quadratic programming. The other is to seeksparse source signals by exploiting the DI condition, and itsolves an 1opti-mization. We present numerical results of NMR data to show the performanceand reliability of the method in the applications arising inNMR spectroscopy . Department of Mathematics, University of California at Irvine, Irvine, CA 92697, IntroductionBlind source separation (BSS) is a major area of research in signal and image pro-cessing.

3 It aims at recovering source signals from their mixtures without detailedknowledge of the mixing process. applications of BSS include signal analysis andprocessing of speech, image, and biomedical signals, especially, signal extraction, en-hancement, denoising, model reduction and classification problems[8]. The goal ofthis paper is to study new BSS methods for nearly degenerate dataarising from Nu-clear Magnetic Resonance (NMR) spectroscopy . The BSS problem isdefined by thefollowing matrix modelX=A S ,withAij 0, Sij 0,( )whereX Rm p, A Rm n, S Rn p. Rows ofXrepresents the measured mixedsignals, rows ofSare the source signals. TheX, Sare sampled functions of anacquisition variable which may be time, frequency, position, wavenumber, etc, de-pending on the underlying physical process. Hence there arepsamples in the mea-surements. The objective of BSS is to solve forAandSgivenX. In the con-text of nmr spectroscopy , the mixing coefficients are not typically measured.

4 Thisis where BSS techniques become useful. The problem is also known as nonnega-tive matrix factorization (NMF [18]). Similar to factorizing a compositenumber(48 = 6 8 = 8 6 = 4 12 = 12 4 = 2 24 = 24 2 = 3 16 = 16 3), there arepermutation and scaling ambiguities in solutions to BSS. For any permutation matrixPand invertible diagonal matrix , (AP , 1P 1S) is another pair equivalent tothe solution (A, S), sinceX=AS= (AP )( 1P 1S).( )Various BSS methods have been proposed relying onprioriknowledge of sourcesignals such as spatio-temporal decorrelation, statistical independence, sparseness,nonnegativity, etc, [7, 8, 12, 16, 19, 20, 21, 25, 30, 31, 32, 33]. Recently there havebeen considerable interests for solving nonnegative BSS problems,which emerge incomputer tomography, biomedical image processing, nmr spectroscopy [2, 3, 14, 17,18, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35]. This work is originated from analyticchemistry, in particular, nmr spectroscopy .

5 applications include identification oforganic compounds, metabolic fingerprinting, disease diagnosis, and drug design. Aschemical mixtures abound in human organs, for example, blood, urine, and metabo-lites in brain and muscles. Each compound has a unique spectral fingerprint definedby the number, intensity and locations of its NMR peaks. In drug design, structuralinformation must be isolated from spectra that also contain the target molecule, sideproducts, and different spectra come from Fourier transform of NMR measurement of ab-sorbance of radio frequency radiation by receptive nuclear spins of the same mixturesample at different time segments when exposed to high magnetic fields. The NMRspectra are nonnegative. Besides, NMR spectra of different chemical compounds areusually not independent, especially as compounds (component molecules) have similarfunctional groups, the peaks overlap in the composite NMR spectra making it difficultto identify the compounds involved.

6 ICA-type approaches recover independent source1signals and thus are unable to separate NMR source spectra. New methods need tobe invented to handle this class of data. Recently nonnegative BSS has been attractedconsiderable attention in nmr spectroscopy [1, 17, 25, 28, 29, 31,32, 33, 34, 36, 37].For example, Naanaa and Nuzillard (NN) proposed a nonnegative BSSmethod in[25] based on a strict local sparseness assumption of the source signals. The NN as-sumption (NNA) requires the source signals to be strictly non-overlapping at somelocations of acquisition variable ( , frequency). In other words,each source sig-nal must have a stand-alone peak where other sources are strictly zero there. Such astrict sparseness condition leads to a dramatic mathematical simplification of a generalnonnegative matrix factorization problem ( ) which is non-convex. Geometricallyspeaking, the problem of finding the mixing matrixAreduces to the identificationof a minimal cone containing the columns of mixture matrixX.

7 The latter can beachieved by linear programming. In fact, NN s sparseness assumption and the geo-metric construction of columns ofAwere known in the 1990 s [2, 35] in the problemof blind hyper-spectral unmixing, where the same mathematical model ( ) is analogue of NN s assumption is called pixel purity assumption [6]. The resultinggeometric (cone) method is the so called N-findr [35], and is now a benchmark inhyperspectral unmixing. NN s method can be viewed as an applicationof N-findr toNMR data. It is possible that measured NMR data may not strictly satisfy NN ssparseness conditions, which introduces spurious peaks in the results. Postprocessingmethods have been developed to address the resulting errors. Such a study has beenperformed recently in case of (over)-determined mixtures [30] where it is found thatlarger peaks in the signals are more reliable and can be used to minimize errors dueto lack of strict this paper, we consider how to separate the data if NN assumption is not satis-fied.

8 We are concerned with the regime where source signals do not have stand-alonepeaks yet one source signal dominates others over certain intervals of acquisition vari-able. In other words, a dominant interval(s) condition (DI) is required for sourcesignals. This is a reasonable condition for many NMR spectra. For example, the DIcondition holds well in the NMR data which motivated us. The data is producedby the so-called DOSY (diffusion ordered spectroscopy ) experiment where a physicalsample of mixed chemical compounds in solvent (water) is tries todistinguish the chemicals based on variation in their diffusion rates. However, DOSY fails to separate them if the compounds have similar chemical functional groups ( ,they have similar diffusion rates). In this application, the diffusion rates of the chem-icals serve as the mixing coefficients. This presents an additional mathematical chal-lenge due to the near singularity of the mixing matrix.

9 Separating these degeneratedata is intractable to the convex cone methods, thus we are prompted to develop newapproaches. Examination the DI condition reveals a great deal about the geometry ofthe mixtures. Actually, the scattered plot of columns ofXmust contain several clus-ters of points, and these clusters are centered at columns ofA. Hence, the problem offindingAboils down to the identification of the clusters, and it can be accomplishedby data clustering, for example, K-means. Although the data clustering in generalproduces a fairly good estimate of the mixing matrix, its output deviates from thetrue solution due to the presence of the noise, initial guess of the clustering algorithm,and so on. In the case of nearly singular mixing matrix, a small perturbation canlead to large errors in the source recovery ( , spurious peaks).To overcome this2difficulty and improve robustness of the separation, we propose two different meth-ods.

10 One is to find a better estimation of mixing matrix by allowing a constrainedperturbation to the clustering output, and it is achieved by a quadratic intention is to move the estimation closer to the true solution. The other is toseek sparse source signals by exploiting the DI condition. An 1optimization problemis formulated for recovering the source paper is outlined as follows; In section 2, we shall review the essentials ofNN approach, then we propose a new condition on the source signalsmotivated byNMR spectroscopy data. In section 3, we introduce the method. In section 4, wefurther illustrate our method with numerical examples including the processing ofan experimental DOSY NMR data set. Section 5 is the conclusion. We shall use thefollowing notations throughout the paper. The notationAjstands for thej-th columnof matrixA,Sjfor thej-th column of matrixS,Xjthej-th column of thej-th rows of matrixSandX, or thej-th source and mixture, work was partially supported by NSF-ADT grant DMS-0911277 and NSFgrant DMS-0712881.