Am3+-F- Interaction in Mixed System of Dimethyl Sulfoxide and Water

Variance of experimental observations is illustrated in Fig. 1, where a representative selection of eleven of totally 26 experimental spectra is given. Spectral curves given in dashed lines have previously been recognized as two-component spectra [20], while solid lines represent spectra due to more than two species. Two data matrices will be discussed in the following : Data matrix A composed of all 26 spectra and data matrix Β omitting those spectra recognized as two-factor spectra in [20], Data matrix Β therefore does not contain experimental evidence used to identify the single component spectrum of (U02)2(0H)2 in Ref. [20]. In the following, data matrices A and B, resp. are symbolized by X if it is not necessary to distinguish among A and B. From the 26 UV-Vis spectra collected in 0.2 nm steps in the range 340 to 580 nm, an absorption matrix A is constructed with 181 rows containing absorption data in the range 340 nm to 520 nm in 1 nm steps. Absorption matrix Β is likewise constructed from 16 out of 26 UV-Vis spectra. Therefore a total of 4706 experimental observations is analyzed simultaneously in data matrix A, and 2896 observations in data matrix B. Data matrix X is decomposed by AFA into the eigenvector matrices E% and C% as well as the respective eigenvalues λ, = s of i-th column and row eigenTable 2. Results of AFA for the first 6 of 16 eigenvalues (matrix B) λ RE *2 χ expectation values 1.3468468 0.0103233 215256 196 0.0192484 0.0008624 1385.4 169 0.0001105 0.0003030 157.14 144 0.0000981 0.0001761 48.183 121 0.0000033 0.0000922 11.900 100 0.0000008 0.0000503 3.1443 81 Table 3. Root mean square error RE, imbedded error IE and extracted error XE as a function of the number η of relevant factors for the first eight factors of matrix A Number η RE IE XE of factors 1 9.173 · 10"' 2.452 · 10~ 8.839 · IO" 2 8.132· 10" 3.074· 10" 7.529 · 10" 3 3.105 · 10" 1.437 · 10" 2.752 · 10 4 4 2.013 · 10" 1.076· IO" 1.702 · 10" 5 1.136 · 10 4 6.789 · 10" 9.109 · 10" 6 8.083 · 10" 5.292 · 1 0 5 6.110· 10" 7 7.171 · IO" 5.071 · 10" 5.071 · 10" 8 6.279 · 10" 4.746 · 10" 4.110· 10" vector, resp. Table 2 summarizes the results of AFA for the first six eigenvalues, obtained for matrix B. The root mean square error in the data matrix RE [30] and Bartlett's χ value [31] together with its expectation values are given for the first six eigenvalues. For two factors, the remaining root mean square error in the data matrix RE is about 8 · 10" absorption units, above the detection limit of the given spectrometer. For three factors, RE is reduced to 3 · 10" absorption units, corresponding to the detection limit assessed for the given spectrometer from an analysis of background spectra. Hence, three factors are suggested by the RE criteria. The same conclusion has to be drawn from the χ criterion in Table 2. A factor is considered to be significant if its associated χ value is larger than its expectation value. Comparison of both columns in Table 2 indicates the borderline between the third and fourth eigenvalue. Compared to the previously reported analysis [20], data matrix Β contains less and different spectra. These differences do not result in different conclusions concerning the dimensionality of the data space. Other criteria, e.g. Scree test [32] or uncertainty in the eigenvalue [33], do not lead to differing conclusion. Hence, these criteria are not repeated here. The same conclusions hold for matrix A, as can be seen by comparing the RE data in Tables 2 and 3. Abstract data matrices E% and Ct are reduced to the first three eigenvectors each, thus forming the rX3 matrix E* and the 3 X c matrix C*. If the original data matrices are recalculated from matrices E* and C* according to X ' = E* C*, the AFA-reproduced data matrix X ' will differ from X, because the summationsdata matrices E% and Ct are reduced to the first three eigenvectors each, thus forming the rX3 matrix E* and the 3 X c matrix C*. If the original data matrices are recalculated from matrices E* and C* according to X ' = E* C*, the AFA-reproduced data matrix X ' will differ from X, because the summations 224 Günther Meinrath are extended only over the first three eigenvectors containing significant information. The difference