Factor rotation methods

To transform the abstract factors determined in the first step into interpretable factors, rotation methods are applied. If definite target vectors can be assumed to be contained in the data, for example, a spectrum under a spectrochromatogram, the rotation of data is performed by using a target. This technique is known as target-transform factor analysis TTFA, c Example 5.6). 

The aim of factor analysis is to calculate a rotation matrix R which rotates the abstract factors (V) (principal components) into interpretable factors. The various algorithms for factor analysis differ in the criterion to calculate the rotation matrix R. Two classes of rotation methods can be distinguished (i) rotation procedures based on general criteria which are not specific for the domain of the data and (ii) rotation procedures which use specific properties of the factors (e.g. non-negativity). 

Prior Applications. The first application of this traditional factor analysis method was an attempt by Blifford and Meeker (6) to interpret the elemental composition data obtained by the National Air Sampling Network(NASN) during 1957-61 in 30 U.S. cities. They employed a principal components analysis and Varimax rotation as well as a non-orthogonal rotation. In both cases, they were not able to extract much interpretable information from the data. Since there is a very wide variety of sources of particles in 30 cities and only 13 elements measured, it is not surprising that they were unable to provide much specificity to their factors. One interesting factor that they did identify was a copper factor. They were unable to provide a convincing interpretation. It is likely that this factor represents the copper contamination from the brushes of the high volume air samples that was subsequently found to be a common problem ( 2). 

Varimax rotation is a commonly used and widely available factor rotation technique, but other methods have been proposed for interpreting factors from analytical chemistry data. We could rotate the axes in order that they align directly with factors from expected components. These axes, referred to as test vectors, would be physically significant in terms of interpretation and the rotation procedure is referred to as target transformation. Target transformation factor analysis has proved to be a valuable technique in chemo-metrics. The number of components in mixture spectra can be identified and the rotated factor loadings in terms of test data relating to standard, known spectra, can be interpreted. 

On the right-hand side of relation (6.3) are the L, P and A factors. The Lorentz factor, L, takes account of the relative time each reflection spends in the reflecting position. It depends on the precise diffraction geometry used. For example, in the rotation method... 

If the condition 0 < ( 7 — jS) < tt is fulfilled, the second exponential factor in the last form of exp [iknR] goes to zero as p = R — oo. The channel function then behaves as that of a bound state. It is also important to note that this complex transformation of the coordinate does not affect the decreasing asymptotic behavior and the square integrability of a bound sfafe wavefunc-tion. This means that any method available for bound sfafe calculafions can be used for resonance calculations. A variant of the complex rotation method consists in transforming the reaction coordinate only after some value, say Rq. The form given to the coordinate is then Rq + [R — Ro)exp(k). This procedure is called exterior scaling [42,43]. 

Multivariate calibrations must also be documented for robustness. There are several ways that this can be done. For example, minor changes in sample positioning can be used for robustness testing, for example the effect of rotation of an oval tablet on the predictions of PLS method. Another important issue relevant to method robustness is number of factors in the PLS model. If the PLS model has too many factors, the method will not be robust because the PLS method is fitting noise in the data [14]. 

The limitation of PCA to detect common factors and the application of rotation methods will be illustrated using the pipelines artificial examples (14). Data for each pipeline are simulated according to the following equalities ... 

Rotation Methods An optimal loading matrix is obtained by rotation of factors. One distinguishes orthogonal and oblique (correlated) rotations. In the case of an orthogonal rotation, the coordinate system is rotated. The aim is that the new coordinate axis cut the swarm of points in an optimal way. This can be often better achieved by an oblique rotation. If the data can be described by an orthogonal rotation in an optimal way, then an oblique method will also lead to coordinate axes that are perpendicular to each other. 

The various factor analysis methods which became widely available during the 1970s are ideally suited to the examination of conservative mixing (Klovan Imbrie, 1971). They can be used to simultaneously classify sites and identify independently varying compositional components. Dean et al. (1988 1993) apply Q-mode factor analysis to total elemental analyses of lake surface sediments, with a view to regional classification and sediment source characterization. The approach uses a varimax rotation. This is a numerical procedure that rigidly rotates the selected axes to maximize or minimize the variable scores on each axis. This helps in the interpretation of the factors as real end-members. 

The first study on curve resolution, carried out by Kaiser [1] in 1958, proposed the varimax method, wherein factor rotation was used in factor analysis. Studies by Lawton and Sylvestre of Kodak clearly picked up on curve resolution technology as a means of reaction analysis in chemistry (1971, 1974) [2]. The idea of employing rotating matrices was first used in iterative target transformation factor analysis... 

When factor analysis has been done to determine the factors that affect the survival of SMEs in the province, the factors were extracted using principle component to see how many of the 23 variables could be factors. It was considered by eigenvalue that exceeds 1.0 the eigenvalue is indicative of the ability of the emerging factors to explain the variability of the original variables. Besides, in this research, we also applied the Varimax rotation method and the KMO statistics, which are used to measure the suitability of the information available, and KMO > 0.6 would be considered suitable data to use for factor analysis techniques. The results showed that the KMO = 0.8123, which was over 0.6, so the information was appropriate to use technical analysis. The results showed there were live factors that had eigenvalue over 1.0, so the analysis grouped the factors into live factors as in Table 3. 

For given operating conditions and submergence, the dry cake production rate increases with the speed of rotation (eq. 10) and the limiting factor is usually the minimum cake thickness which can stiU be successfiiUy discharged by the method used in the filter. Equation 11 shows the dependence of the sohds yield on cake thickness ... 

The isotope effects of reactions of HD + ions with He, Ne, Ar, and Kr over an energy range from 3 to 20 e.v. are discussed. The results are interpreted in terms of a stripping model for ion-molecule reactions. The technique of wave vector analysis, which has been successful in nuclear stripping reactions, is used. The method is primarily classical, but it incorporates the vibrational and rotational properties of molecule-ions which may be important. Preliminary calculations indicate that this model is relatively insensitive to the vibrational factors of the molecule-ion but depends strongly on rotational parameters. 

The NMR spectra can be used to obtain kinetic information in a completely different manner from that mentioned on page 294. This method, which involves the study of NMR line shapes, depends on the fact that NMR spectra have an inherent time factor If a proton changes its environment less rapidly than 10 times per second, an NMR spectrum shows a separate peak for each position the proton assumes. For example, if the rate of rotation around... 

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