Discriminant analysis in the presence of interferences: Combined application of target factor analysis and a Bayesian soft-classifier
Graphical abstract
Highlights
► A novel background-independent soft classification method is described. ► Target factor analysis is combined with Bayesian decision theory. ► Target factors are taken from a library of target analytes. ► Trace metal transfers to complex backgrounds are analyzed by LIBS. ► The method is shown to be both conservative and accurate.
Introduction
An important problem in chemical analysis is the assignment of object membership among a set of defined classes. A detailed discussion of modern classification methods has been given in several texts [1], [2]. Current literature provides examples of the use of discriminant analysis in forensic evidence evaluation [3], [4], [5], [6], homeland security applications [7], [8], and disease diagnosis [9], [10], [11], among other applications.
Most classification methods are not particularly well adapted to classifying an unknown in the presence of a significant background signal that is independent of the analyte class and highly variable from sample to sample. Classification procedures that have been optimized using class examples without including interferences in the examples, may fail to perform well on samples that contain impurities. However, recent results have shown some success in performing discriminant analysis in the presence of interfering factors. For example, >97% correct classification of the maximum temperature reached by burning soils has been accomplished by partial least squares discriminant analysis (PLS-DA) in the presence of up to 20% ash interference [12]. It is often the case that the number of analyte classes is small in comparison to the number of potential interfering species and background signatures, thereby complicating the task of securing training sets that contain a representative set of possible interferences.
This paper examines an approach to analyte classification in the presence of a complex and undetermined interference. The method explored here makes use of target factor analysis (TFA), as described by Malinowski [13], to determine the presence of an analyte in a mixture without requiring that every component of the mixture be identified. The method introduced here differs from a previously reported use of TFA for discriminant analysis (also referred to as procrustes discriminant analysis), which has been shown to be mathematically equivalent to PLS-DA [14], [15]. In contrast, the method discussed here relies on testing a library of analyte spectra from representative classes to identify the class or classes that contribute to the data set. The data sets examined in this paper are taken from spectra obtained by laser-induced breakdown spectroscopy (LIBS) from steel plates that had been penetrated by 9 mm bullets with either copper jackets (CJ), metal alloy jackets (MJ) or non-jacketed lead (NJ) projectiles. The same bullets were applied in trace quantities to porcelain tiles and metal plates as one and two analyte (mixed) samples. The bullet jacketing materials constitute the analyte classes. The objective is to test the TFA method to identify the class of bullet that penetrated a steel plate or was applied to porcelain or steel based on the posterior probabilities calculated from the LIBS spectra collected at the edges of the holes and in the application areas where residues of jacket materials were deposited. The LIBS spectra reflect contributions from the steel or porcelain substrates and the bullet's jacket material. The class of bullet with the highest probability of contributing to the spectral set is identified by the posterior probabilities. Similarly, significant posterior probabilities for the more than one class of analyte can indicate the presence of multiple classes. The method also defines criteria that must be met before a posterior probability is calculated for a given class. Consequently, if none of the analyte classes meet the criteria, the sample may be evaluated as not containing any of the analyte classes, and the method may truly serve as a soft classifier.
Section snippets
Instrumentation
The LIBS instrument used in this research (LIBS2000+, Ocean Optics, Dunedin, FL, USA) was equipped with a Q-switched Nd:YAG nanosecond pulsed laser (CFR200, Big Sky Lasers, Bozeman, MT, USA). Fundamental (1064 nm) laser output (pulse width of 9 ns and 63 mJ pulse−1) was used to generate laser-induced plasmas. The spectrometer delay for spectral collection was optimized at 2.5 μs, giving maximum signal intensity and signal-to-noise ratio and an integration time of 1 ms. The plasma emission between 200
Results and discussion
Fig. 1 shows LIBS spectra from each class (CJ, MJ and NJ) in the library, and Fig. 2 shows LIBS spectra of the steel and porcelain substrates. The two strong peaks at 324.75 nm and 327.40 nm in the CJ spectra (Fig. 1a) are attributed to Cu I transitions. The spectrum of the MJ projectile (Fig. 1b) also contains emissions attributed to the Cu I transitions, along with emissions at 361.94, 341.48 and 352.45 nm that are attributed to Ni. The representative spectrum from the NJ class projectile (Fig. 1
Conclusions
The combined TFA and Bayesian soft classifier are intended to produce posterior probabilities that a sample contains a representative analyte from one of the classes in a library. The method may also indicate that a system does not contain any of the analytes represented in the library. The resulting posterior probabilities are intended to aid the analyst in classifying the sample in the presence of an unknown background signal. The method is designed to work in cases where spectra of multiple
Acknowledgements
This work was supported under award number W911NF0610446 from the U.S. Army Research Office and award number 2004-IJ-CX-K031 from the Office of Justice Programs, National Institute of Justice, Department of Justice. Points of view in this document are those of the authors and do not necessarily represent the official position of the U.S. Army or U.S. Department of Justice. The work was done jointly at the Center for Research and Education in Optics and Lasers and the National Center for
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