Anal. Chem. 1989, 61, 1168-1169
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Numerical and Statistical Properties of Target Factor Analysis Methods Sir: In a recent paper Brayden et al. ( I ) presented a new version of the iterative target factor analysis (TFA) method and proved convergence for their method. As the authors indicate, TFA and its iterative form have been proven to be useful tools in many applications for analytical chemistry (2). Unfortunately, the mathematical context of the method with respect to both its numerical and statistical aspects are not well understood and represented in those chemical applications, which has led to unnecessary complication in formulation and argumentation. Chart I of ref 1, that occupies two journal pages, exemplifies this problem. This note will attempt to represent TFA and the iterative TFA in the light of current numerical analysis and statistical literature. It is our hope that this will help to enhance communication with mathematicians and statisticians and convince chemists to express their data analysis methods using the concise formulation of modern mathematics. THEORY Throughout this paper, lower-case boldface letters denote vectors and upper-case boldface letters denote matrices. All vectors are column vectors and the transpose of vectors or matrices is denoted by the superscript T, e.g. aT or AT. The inverse and the pseudoinverse are denoted as, A-' and At, respectively. Given an I X J data matrix R, without loss of generality, it is assumed that the rows correspond to samples and the columns to variables (wavelengths, time, sensors, etc). The first step in most multivariate data analyses, after an optional translation of the origin and in some cases scaling, is to obtain a bilinear representation of R. Here, the singular value decomposition (SVD) (3) representation is used R = USVT
+E
(1)
where U and V are the two sets of eigenvector matrices of RRT and R?R, respectively. S is a diagonal matrix of the singular values (square roots of the eigenvalues) ordered such that s1 L s2 ... L sK Z 0. Only K singular vectors are used to reconstruct R, and therefore E is the matrix of residuals. Determining K is beyond the scope of this communication and argumentation on several approaches can be found ( 4 ) . Consider the matrices
H = RR+ = ULTT
(2)
P = R+R = VVT
(3)
and
where Rt = VS-'UT is the pseudoinverse (3) and is of rank K because it is constructed of the first K singular vectors. The matrices H and P (whose rank is also K ) are important in statistics and of special importance to understand TFA. those matrices are called projection matrices (3). Projection matrices have mainly two properties, symmetry (PT= P) and idempotency (P2= P). The matrix H is called the h a t matrix in the statistical literature ( 5 , 6 ) . The diagonal elements of H or P also have special significance and are called the leverage values. They explained the influence each sample or variable has in the multivariate model. Other properties of the projection matrices can be found in the literature ( 5 , 6 ) . Target Factor Analysis. TFA was introduced by Malinowski (2) as a means to acquire physical or chemical insight on the real sources for variation in the data. The idea is to
take a test vector, t, that has physical significance (e.g. pure spectrum) and test whether it is present in the data matrix R. Lorber (7) showed that the testing procedure can be represented as
i
= Pt
(4)
where P is defined in eq 3. Equation 4 is for testing on the column space; tests on the row space are done by multiplying the test vector with H. Malinowski (2) developed several empirical measures for inference about whether the test vector actually may be represented by R. However, multivariate statistics already developed tools based on residuals calculated by 6 = t - Pt = (I - P)t (5) Assuming that the residuals are independently and equally distributed with common variance, 2,the residuals covariance matrix is Var(6) = a2(I - P)
(6)
This result can be used directly to estimate 2. Powerful methodology developed in the statistical literature (e.g. ref 5 and 6) for outlier detection are directly applicable in TFA. Outlier detection methodology may help to determine which individual elements of the test vector, t,, are incorrect. Brayden et al. ( I ) devoted two pages in their paper to prove that either h,, or p,, equal the sum of squares of the elements in the corresponding row or column. By presentation of the TFA method with projection matrices this result is an obvious property of projection matrices (idempotency). They also prove that the range of the leverage value is limited between zero and one. This is also a basic property of projection matrices ( 6 ) . Iterative Target Factor Analysis. There are several variations on how to use TFA in an iteration mode. The basic idea is to seek improvement over the basic testing procedure to get an improved estimate of the test vector. Of course once that t is multiplied by the projection matrix, further multiplication off will result in f again. Therefore, in any iterative TFA procedure some of the elements of f are changed before further multiplication. In the procedure presented in ref 1 all elements of f except one obtain their original values. Another procedure used for chromatographic data with a multiple wavelength detection system ( 4 9 ) has negative values of f (which contradict the knowledge that chromatographic elution profiles should be positive) set to zero. All these methods, including ref 1, can be described in the framework of the method called the EM algorithm in the statistical literature (IO). The EM algorithm is a general algorithm for statistical analysis with missing data as well as other nontrivial statistical models and consists of two steps. In the E step, where E stands for estimation, the missing values are estimated; and in the M step, where M stands for maximization, maximum likelihood principles are used to estimate the model parmeters. The procedure is repeated until convergence is achieved. Dempster et al. (11) proved convergence of the EM algorithm. Other topics like estimating rate of convergence, bias, and failure also were investigated (10).
The method presented by Brayden et al. ( I ) may be described as the EM algorithm without the M step. It is differeqt from earlier works (8,9)which are applied to completely unknown systems. The proof they present is only for the case
0003-2700/89/0361-1168$01.50/0'C 1989 American Chemical Society
Anal. Chem. 1989, 67, 1169-1171
when the starting estimate for the single missing value is zero. It is easily proven that for any starting point there will be convergence and the rate of convergence is equal to the leverage value. However, sinice convergence for the whole class of EM methods has already been proved, there is no reason to prove it for each individual case. The procedure used for chromatographic data with a multiple wavelength detection system (8, 9) may also be viewed as a subset of the EM algorithm with the M step. But rather than using the maximum likelihood principle, the knowledge of nonnegativity is inserted in the M step to zero all negative values.
LITERATURE CITED (1) Brayden, T. H.; Poropatic, P. A,; Watanabe, J. L. Anal. Chem. 1988, 6 0 , 1154. (2) Mallnowski, E. R.; Howery, D. G. Factor Analysis in Chemistry; Wiley: New York, 1980. (3) Golub, G. H.; Van Loan, F. Matrix Computations; John Hopkins University Press: Baltimore, MD, 1983. (4) Wold, S.; Slostrom, M. J. Chemom. 1987, 1 , 243. (5) Hoaglin, D. C.; Welsch, R. E. Am. Statist. 1978, 3 2 , 17-22. (6) Weisberg, S. Applied Linear Regression: Second Edition; Wiley: New York, 1985. (7) Lorber. A. Anal. Chem. 1984, 5 6 , 1004.
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(8) Gemperline, P. J. J. Chem. Inf. Comput. Sci. 1984, 2 4 , 206. (9) Vandeginste, B. G. M.; Leyten, F.; Gerritsen, M.; Noor, J. W.; Kateman, G.;Frank, J. J. Chemom. 1987, 1 , 57. (IO) Little, R. J. A.; Rubin. D.B. Statistical Analysis with Missing Data; Wiley: New York, 1987. (11) Dempster, A. P.; Laird, N. M.; Rubin, D. B. J. R. Statist. SOC.B 1977, 39. 1. 'On leave from Nuclear Research Centre-Negev, PO Box 9001, BeerSheva, Israel.
Avraham Lorber' Bruce R. Kowalski* Center for Process Analytical Chemistry, and Laboratory for Chemometrics Department of Chemistry BG-10 University of Washington Seattle, Washington 98195 RECEIVED for review June 15,1988. Resubmitted December 9,1988. Accepted January 1, 1989. This work was supported by the Center for Process Analytical Chemistry (CPAC), a National Science Foundation Industry/University Cooperative Research Center a t the University of Washington.
Contribution of Membrane Components to the Overall Response of Anion Carrier Based Solvent Polymeric Membrane Ion-Selective Electrodes Sir: Anti-Hofmeister behavior of anion carrier based ionselective electrodes originates from strong complexation of some anions with anion carriers. In positively charged anion carriers, especially in the case of carriers having a monovalent positive charge such as metalloporphyrins and metallocorines (1-3), this complexation reduces significantly the net charge densities in membranes. On the other hand, recent studies on the nature of negative sites in neutral carrier based solvent polymeric membrane electrodes have revealed that poly(viny1 chloride) (PVC) contains ionic or ionizable impurities that function as negative sites (4-6). Thus, plasticized PVC membranes without additions of sensing materials (blank membranes) are not potentiometrically inert and respond to certain ions (blank response). Indeed, several researchers have found ideal responses of blank membranes to proton and ionic surfactants (7, 8). By considering the two facts mentioned above, one can predict a possibility that a normal anionic function originating from the positively charged anion carrier will be strongly affected by the blank response of a membrane matrix when a positively charged anion carrier complexes too strongly with a specific anion and when the membrane matrix is not free from ionic impurities as in the case of plasticized PVC membranes (4-6). During examinations of performances of solvent polymeric nitrite-selective electrode membranes based on nitrite salts of cobalt(II1) complexes of two different porphyrins, remarkable pH dependence different from that reported in similar works (1,3) was observed. In order to clarify whether this anomalous pH-dependent response can be ascribed to the complexes themselves or rather to membrane matrices, the present work was conducted.
EXPERIMENTAL SECTION Reagents. Ligands a,/3,y,b-tetrakis(4-n-octyloxyphenyl)porphyrin (TOOPP) and a,P,y,6-tetraphenylprphyrin(TPP)and a plasticizer, 2-nitrophenyl octyl ether (0-NPOE),were obtained from Dojindo Lab. PVC was purchased from Katayama Chemical, and thin-layer chromatography (TLC) plates were Silicagel 70 Plate (Wako Chemical). All other chemicals and solvents were of reagent grade. Distilled and subsequently deionized water was used throughout. To assure high purity of o-NPOE for measurements in liquid membranes where a rather large amount was required, it was prepared according to the reported procedure (9). After 10 washings of the crude product with aqueous NaOH, it was purified by distilling twice under reduced pressure. Preparation of Anion Carriers. First, Co(I1) complexes of both porphyrins were prepared according to the method of Adler et al. (IO) and oxidized into Co(II1) form as described (11). Because only a very small amount was oxidized in the case of Co(I1)-TOOPP, the above mentioned procedure was modified as follows: First the Co(I1)-TOOPP complex was dissolved in a small amount of tetrahydrofuran, and then methanol was added, resulting in a precipitation of the complex in the form of very fine particles. With this modification, oxidation proceeds in a manner similar to that for the nonalkylated complex. Both Co(II1) complexes were obtained as chloride salts. Oxidation of both complexes was checked by TLC and CHN analyses. Nitrite forms of complexes were prepared from the chloride salts by using a solvent extraction technique (12). emf Measurements of Plasticized PVC Membrane Electrodes. Plasticized PVC membranes containing 1, 2, and 3% nitrite forms of complexes of both ligands were prepared according to the reported method (13). Blank membranes without additions of sensing materials were also prepared. All membranes had almost the same matrix, which consisted of PVC and o-NPOE in weight ratios from 10:25 to 1023. A small piece of membrane was glued to a tip of an electrode body (Denki Kagaku Keiki Co., Ltd., Tokyo) having an internal silver-silver chloride electrode.
0003-2700/89/0361-1169$01.50/00 1989 American Chemical Society
