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3 (BPLS) may be found by use of SVD. After decomposition of ^ Martens, the Martens factorization () can be transformed according to X = ^ Martens ^W + E PLS = U SVD S SVD V SVD ^W + E PLS = U U S V SVD ^W + E PLS = (U S ) ^WV SVD + EPLS = ^ B ^V B + E PLS ; () resulting in ^ B = U S and ^V B = ^WV SVD. he Wold factorization () can rst be transformed to a Martens factorization according to X = ^ Wold ^P Wold ^W ^W + E PLS = ^ Martens ^W + E PLS ; () which may then be transformed to a bi orthogonal factorization according to (). Alternatively we may obtain ^ B and ^V B directly by an SVD of ^X = ^ Wold ^P Wold ^W ^W = ^ Martens ^W taken to the speci ed number of components. Note that after the unifying transformations above the loading weight matrix ^W is replaced by the loading (weight) matrix ^V B, i.e. there is no longer a need to distinguish between loadings and loading weights. Permutations As a result of the SVD decomposition used in () the ordering of components according to explaining power may get lost. In ^X = ^t B;^v B; + ^t B;^v B; + + ^t B;A^v B;A (7) the third component may for example explain more of the response variable y than the second component etc. his does not, however, a ect the total explaining power of all A components, where A is determined through validation using an ordinary PLS procedure. he ordering according to explaining power may be restored by augmenting () with a square and orthonormal permutation matrix, i.e. X = ^ B QQ ^V B + E PLS = ^ B Q ^V B Q + EPLS = ~ B ~V B + E PLS : (8) For the common case of a very low number A of total components the permutation to use is easily found by a systematic search (see example in Section ). Other cases are of little interest in a correspondence context. Final predictor It can be shown 7 that the PLS predictor based on observations collected in an X matrix and a y vector (assuming a scalar response) can be written as ^b = ^W ^W X X ^W ^W X y; (9) where ^W is found by either the Wold or the Martens algorithm. In the transformations above ^W is replaced by ^V B = ^WV SVD. Since V SVD R AA is invertible we thus nd ^b = ^V B V SVD V ^V SVD B X X ^V B V SVD V ^V SVD B X y = ^V B ^V B X X ^V B ^V B X y: () he predictor is thus unaltered after replacement of ^W by ^V B, and for the same reason it is also unaltered by the permutation matrix Q in (8).

4 Discussion on BPLS properties In the same way as in PCA, the BPLS factorization results in a score matrix with orthogonal columns and a loading matrix with orthonormal columns. his makes a comparison with PCA natural. he PCA factorization () may be found from solutions of the eigenvalue problem associated with the spectral decomposition 8 X X^p i = ^p i^i ; () X X = ^p ^^p + ^p ^^p + : : : + ^p p^p^p p = ^P^^P ; () where ^ ^ : : : ^ p, ^P ^P = I and ^P^P = I, and where ^ is diagonal. Using A components this results in X X = ^p ^^p + ^p ^^p + : : : + ^p A^A^p A + E PCAE PCA = ^P PCA ^ PCA ^P PCA + E PCAE PCA = ^P PCA ^ PCA ^ PCA ^P PCA + E PCAE PCA ; () which is also found from (). he BPLS factorization (), on the other hand, uses a loading matrix ^V B that is a linear combination ^V B = ^PL B = ^Pl ^Pl ^Pl A () such that and a score matrix such that ^V B ^V B = L B ^P ^PL B = L B L B = I; () ^ B = X ^V B = X^PL B () ^ B ^ B = L B ^P X X^PL B (7) is diagonal, just as ^ PCA ^ PCA = ^ PCA. However, this does not imply that ^V B can be found as a solution of an eigenvalue problem, except for A = p, in which case L B = I and thus ^V B = ^P. Note that ^W in the ordinary PLS factorizations also is a linear combination of ^P with L Martens L Martens = I, but that ^ Martens ^ Martens is non-diagonal 9. Also ^V Wold = ^W ^W ^P Wold is a linear combination of ^P, but then with L Wold L Wold = I. Although in itself interesting, further relations between the BPLS and other factorizations are beyond the scope of the present correspondence context. Score and loading correspondence General discussion As indicated in the introduction, correspondence between PLS scores and loadings is related to correspondence in several other multivariate display techniques used in PCA, correspondence factor analysis, spectral map analysis, factor analysis in the strict statistical sense etc.. he common step in these methods is the factorization of the data matrix X, but the methods di er with respect to the processing of the data prior to the factorization, and to the factorization method used. Comparison of factorization methods We will here use PCA as a reference. From the general factorization () and the relation ^V Wold = ^W ^W ^P Wold used in () follow the least squares solutions 8> ^ = X ^V ^V ^V = <>: X^P PCA = X ^V PCA PCA X ^W ^P Wold ^W = X ^V Wold Wold PLS X ^W = X ^V Martens X ^V B Martens PLS BPLS, (8)

5 where the orthonormality of ^P PCA, ^W and ^V B is used. Using the notation X = x x x p = N and ^ = ^t ^t ^t A = ^ ^ ^ N it follows that a given observation i results in scores 8 >< ^ i = >: ^P i PCA PCA i ^W ^P ^W Wold Wold PLS i ^W Martens PLS ^V i B BPLS, (9) where ^P PCA, ^W, and ^V B are orthonormal, while ^W ^P Wold ^W is not. Introducing the notation ^P PCA = ^p ^p ^p A = ^ ^ ^ p, ^W = ^w ^w ^w A = ^! ^! ^! p, ^V B = ^v B; ^v B; ^v B;A = ^#B; ^#B; ^#B;p and ^V Wold = ^v Wold; ^v Wold; ^v Wold;A = ^#Wold; ^#Wold; ^#Wold;p, and assuming centered data, a speci c observation i = x ij results in 8 < ^ i = : x ij ^ j x ij ^! j x ij ^#B;j PCA Martens PLS BPLS, () while ^ i = x ij ^# Wold;j Wold PLS. () Assuming orthogonal coordinate systems, the vector ^ i in the score plots thus has the same direction as the vector ^ j, ^! j or ^# j in the corresponding loading or loading weight plots for PCA, Martens PLS and BPLS. For x ij = the vectors will coincide (see example in Section ). For the Wold PLS solution, on the other hand, the vector ^ i and the corresponding vector in any of the possible loading or loading weight plots ( ^V Wold, ^W or ^P Wold ) will not have the same directions. he reason for this is that the ^V Wold matrix used in the factorization is not orthogonal, and plotting projections of ^W or ^P Wold instead of ^V Wold does not remedy the situation (see example in Section ). Relation to predictive power he correspondence discussion and results above are limited to the di erent factorization methods, and are thus not related to the predictive power of the di erent regression methods. his means that the good interpretational properties of PCA and BPLS to a certain extent may be undermined by prediction errors. Industrial data example he example uses multivariate regression data from a paper production plant ;. he problem considered here is to monitor a given paper quality y i (the second column in the rst data set) from six known process variables i = i i i i i i (columns to 9 in the rst data set), and for the purpose of nding PLS factorizations all N = 9 samples of i and y i are used. he rst three process variables i, i and i were varied systematically through an experiment, taking the values, and -. he next three variables were constructed as i = i, i = i and i = i. he three constructed variables i i, i i and i i are also included in the data set, but for the paper quality chosen they have little predictive power, and for clarity of presentation they are not used in the present example.

7 PC PC (Wold) and W(PLS) (Wold) and V(Wold)=P(Wold) (Martens) and W(PLS) (BPLS) and V(BPLS)..... PC... PC Figure. Loadings/loading weights ^V Wold, ^W and ~V B (o) for the modeling data, and scores ^ test Wold, ^ test Martens and ~ test B (x) for the X test data () with the Wold PLS, Martens PLS and BPLS factorizations. Note the total correspondence for the Martens PLS and BPLS factorizations only. Since the X-variables are correlated, the test data () are not realistic in the present case. However, a realistic test observation is test = : () he result of this is shown in Fig., where the de ciency of the ^ test Wold and ^W plot is clearly demonstrated. Use of ^ test Wold and ^V Wold = ^P Wold gives in fact a somewhat more correct picture of the in uences of variables and, although total correspondence is found only by use of ^ test Martens and ^W or ~ test B and ~V B. 7

9 [] hielemans A, Lewi PJ, Massart DL. Similarities and Di erences among Multivariate Display echniques Illustrated by Belgian Cancer Mortality Distribution Data. Chemometrics Intell. Lab. Syst. 988;:77-. [] Kvalheim OM, Karstang V. Interpretation of Latent-Variable Regression Models. Chemometrics and Intelligent Laboratory Systems 989;7:9-. [] Martens H, Næs. Multivariate Calibration, Wiley: New York, 989;-. [] Manne R. Analysis of two partial-least-squares algorithms for multivariate calibration. Chemometrics Intell. Lab. Syst. 987;: [] Esbensen KH. Multivariate Data Analysis - in practice, Camo ASA: rondheim, Norway, ;- 8. [7] Helland IS. On the structure of partial least squares regression. Communications in statistics 988;7:8-7. [8] Johnson AJ, Wichern DW. Applied Multivariate Statistical Analysis, Prentice-Hall: Englewood Cli s, NJ, 99;8. [9] Kalivas JH. Interrelationships of multivariate regression methods using eigenvector basis sets. J. Chemometrics 999; :-. [] Aldrin M. Moderate projection pursuit regression for multivariate response data. Computational Statistics and Data Analysis 99; :-. [] StatLib-Datasets Archive Website. [ June 999]. 9

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