Anal. Chem. 1998, 70, 5108-5110
Efficient Computation of Net Analyte Signal Vector in Inverse Multivariate Calibration Models Nicolaas (Klaas) M. Faber
Netherlands Forensic Science Institute, Volmerlaan 17, 2288 GD Rijswijk, The Netherlands
The net analyte signal vector has been defined by Lorber as the part of a mixture spectrum that is unique for the analyte of interest; i.e., it is orthogonal to the spectra of the interferences. It plays a key role in the development of multivariate analytical figures of merit. Applications have been reported that imply its utility for spectroscopic wavelength selection as well as calibration method comparison. Currently available methods for computing the net analyte signal vector in inverse multivariate calibration models are based on the evaluation of projection matrices. Due to the size of these matrices (p × p, with p the number of wavelengths) the computation may be highly memory- and time-consuming. This paper shows that the net analyte signal vector can be obtained in a highly efficient manner by a suitable scaling of the regression vector. Computing the scaling factor only requires the evaluation of an inner product (p multiplications and additions). The mathematical form of the newly derived expression is discussed, and the generalization to multiway calibration models is briefly outlined. The goal of multivariate calibration is to construct a predictive model. Generally, two modes of calibration are possible. In the classical model, the instrument responses are expressed as a function of analyte concentration, whereas in the inverse model, the role of the variables is reversed. Although the classical model reflects the causal relationship between predictor and predicted variables, the inverse model is usually preferred in practice since it allows one to construct a model for individual substituents. The interferences are implicitly modeled whereas in the classical model one has to explicitly estimate parameters describing all interferences. This paper is concerned with the inverse model. The predictive ability of an inverse calibration model is often assessed exclusively by calculating the root-mean-square error of prediction (RMSEP) for an independent test set. This procedure is not entirely satisfactory for the following reasons. First, depending on the number of test samples, RMSEP estimates may be highly variable. Second, since squared prediction errors enter the calculation, RMSEP estimates are not robust against outliers. Thus, chance effects are inherent to the common validation procedure and it is desirable to have additional criteria for judging model performance. This could, for example, lead to a better motivated choice of model dimensionality. The selection of optimum model dimensionality is problematic if the plot of model dimensionality versus RMSEP does not exhibit a clear minimum. 5108 Analytical Chemistry, Vol. 70, No. 23, December 1, 1998
Figure 1. Geometrical representation of NAS vector. The NAS vector is orthogonal to the space spanned by the spectra of the interferences.
In the univariate context, analytical figures of merit such as sensitivity, signal-to-noise ratio, and limit of detection are widely used for characterizing model performance. Their multivariate analogues have been derived by Lorber.1 A key role in the development of multivariate analytical figures of merit is played by the net analyte signal (NAS) vector. It has been defined by Lorber as the part of a mixture spectrum that is useful for model building. The NAS vector should therefore be unique for the analyte of interest, and this requirement implies that it must be orthogonal to the spectra of the interferences. Figure 1 is a geometrical representation of the orthogonality property of the NAS vector. It is seen that, in principle, only the space spanned by the interferences’ spectra is important, not their individual spectra. Algebraically, the NAS vector follows from an orthogonal projection as
r* ) Pr
(1)
where r* (p × 1) is the NAS vector, P (p × p) is an orthogonal projection matrix, r (p × 1) is the mixture spectrum, and p is the number of wavelengths. The original theory of Lorber was developed within the classical model formulation where the pure spectra of the interferences can be used to compute the projection matrix P. However, in applications of the inverse model, the pure spectra are not known in general. Hence an alternative procedure is needed. Recently, two methods have been introduced for computing the NAS vector for inverse multivariate calibration models.2,3 (1) Lorber, A. Anal. Chem. 1986, 58, 1167-1172. (2) Lorber, A.; Faber, N. M.; Kowalski, B. R. Anal. Chem. 1997, 69, 16201626. (3) Xu, L.; Schechter, I. Anal. Chem. 1997, 69, 3722-3730. 10.1021/ac980319q CCC: $15.00
© 1998 American Chemical Society Published on Web 10/20/1998
(The method developed by Wentzell et al.4 implies knowledge of the pure spectra and is therefore not generally applicable.) These methods have in common that the full projection matrix P is evaluated. Due to the size of this matrix (p × p), the computation may be highly memory- and time-consuming (e.g., measuring at 1000 wavelengths is not exceptional in near-infrared applications). This paper shows how calculation of the projection matrix can be circumvented. Xu and Schechter5 have developed a NAS vector-based error indicator function that can be used to find the optimum wavelength region for spectroscopic calibration. Moreover, the NAS vector has been reported to give insight into the results of a calibration method comparison study.4 Owing to the large number of potential applications, the subject of NAS vector calculation should be of considerable interest to the analytical chemist. ALTERNATIVE METHOD FOR NAS VECTOR COMPUTATION The starting point for the derivation of the alternative expression for the NAS vector is the model equation. In inverse calibration, the relationship between analyte concentration, c, and instrument response vector, r, is given by
c ) btr +
(2)
where b (p × 1) is the regression vector (model parameters), is a residual (zero mean random variable), and the superscripted t denotes vector transposition. In applications, the data are often centered with respect to the mean of the training samples. Usually, mean centering of analyte concentration is made explicit in the notation by adding the symbol for mean analyte concentration, jc, to the right-hand side of eq 2 whereas mean centering of r is implied. In this paper, optional mean centering is also implied in the notation for analyte concentration for convenience. Sa´nchez and Kowalski6 identified the regression vector as the contravariant vector, which is an important concept in tensor algebra. Translated to the inverse calibration problem, this result implies that the regression vector is orthogonal to the spectra of the interferences. (If the spectra of the interferences vary, due to the orthogonality constraint the regression vector for the analyte contravaries.) This is consistent with the idea that interferences are implicitly modeled in inverse calibration. It is clear that the inner product in eq 2 will only give contributions from the part of the mixture spectrum, r, which is orthogonal to the spectra of the interferences; this part has been identified above as the NAS vector. Consequently, eq 2 can be rewritten by expanding the mixture spectrum in two mutually orthogonal contributions as
c ) bt(r* + r⊥) + ) btr* +
(4)
and an alternative expression for the NAS vector follows as
r* ) E[c](bt)+ ) (E[c]/btb)b
(5)
where E[‚] symbolizes the expected value and the superscripted + denotes the Moore-Penrose pseudoinverse. In practice, the true and expected values on the far right-hand side of eq 5 have to be replaced by measured values (e.g., reference value for analyte concentration for training or test samples), estimates (regression vector), or predictions (analyte concentration for unknown samples). The evaluation of eq 5 amounts to taking the inner product of the regression vector estimate with itself (p multiplications and additions). The computational effort involved compares favorably with the previously mentioned methods for calculating the NAS vector.2,3 The new method has been tested on practical data sets (e.g., near-infrared spectra), and the differences between various NAS vector estimates were well within the uncertainty limits derived using the method of error propagation (see Appendix I in ref 7). DISCUSSION The classical model and the inverse model are also known as K-matrix and P-matrix methods, respectively, where the prefixes K and P refer to matrices that hold the parameters (the parameter matrix P should not be confused with the projection matrix defined in eq 1). Application of the classical model implies estimating all parameters in K whereas application of the inverse model implies estimating only the elements in P that are associated with the analyte of interest, i.e., the regression vector b (a row in P). It seems natural that in the classical model the NAS vector is calculated with a projection matrix that contains explicit information about all substituents whereas in the inverse model it should, in principle, be possible to calculate the NAS vector using only the parameter estimates for the analyte of interest. It follows that the mathematical form of eq 1 more closely reflects the philosophy of classical calibration, although it has been shown2,3 that a suitable projection matrix can be calculated in the inverse model too. In contrast, eq 5 fully harmonizes with the rationale behind inverse calibration. The transparent relationship between NAS vector and model parameters can be seen as an added advantage of the alternative computation method.
where r⊥ is the vector orthogonal to the NAS vector (see Figure 1). Inserting eq 3 in eq 2 gives
GENERALIZATION TO MULTIWAY CALIBRATION MODELS It is worth noting that interest in multiway calibration methods is rapidly increasing among chemometricians. (An example of multiway data results from the coupling of a chromatographic separation and excitation emission matrix (EEM) fluorescence measurement.) For multiway calibration methods such as multilinear partial least squares (PLS), the model parameters can be collected in a regression vector similar to standard PLS.7-10 The
(4) Wentzell, P. D.; Andrews, D. T.; Kowalski, B. R. Anal. Chem. 1997, 69, 2299-2311. (5) Xu, L.; Schechter, I. Anal. Chem. 1996, 68, 2392-2400. (6) Sa´nchez, E.; Kowalski, B. R. J. Chemom. 1988, 2, 247-264.
(7) Faber, N. M.; Lorber, A.; Kowalski, B. R. J. Chemometrics 1997, 11, 419461. (8) Bro, R. J. Chemometrics 1996, 10, 47-61. (9) Smilde, A. K. J. Chemometrics 1997, 11, 367-377.
r ) r* + r⊥
(3)
Analytical Chemistry, Vol. 70, No. 23, December 1, 1998
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calculation procedure symbolized by eq 5 applies to those methods as well where it is understood that the multiway mixture responses are suitably unfolded (“strung out”) to a vector so that the generalization of eq 2 applies. Especially for multiway calibration
problems, the conventional methods for calculating the NAS vector could easily become unpractical.
(10) de Jong, S. J. Chemom. 1998, 12, 77-81.
AC980319Q
5110 Analytical Chemistry, Vol. 70, No. 23, December 1, 1998
Received for review March 18, 1998. Accepted September 2, 1998.
