Factor rotations

For instance, the first row of the matrix X defines a point with the coordinates (x, y,) in the space defined by the two orthogonal axes = (I 0) and = (01). Factor rotation means that one rotates the original axes = (1 0) and = (0 1) over a certain angle 9. With orthogonal rotation in two-dimensional space both axes are rotated over the same angle. The distance between the points remains unchanged. 

Factor rotation involves the calculation of [/ , /,] from [x,- y,], given a rotation angle with respect to the original axes. Suppose that after rotation the matrix X is transformed into a matrix F ... 

Factor rotation by target transformation factor analysis (TTFA)... 

The factor rotation by which the factors are transformed into more interpretable variables and can be tested concerning hypothetical data structures, respectively. There are various techniques of factor rotation with specific advantages in several fields of application (Frank and... 

Table 3.7. Levels for a three-factor, rotatable, central composite design.

In the case of very long molecules rotation in the liquid may be hindered by steric factors. Rotation would also be reduced by powerful intennolecular forces and by association. 

Another process utilized for better interpretation of the factor analysis results is factor rotation. A much easier interpretation can be done if a new rotation is done such that only a few variables have weights different from zero in matrix Z or Z . Considering for example the matrix Z, this process can be achieved formally by replacing matrix F with matrix G such that ... 

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. 

With many sampling stations and sampling periods it is possible to do a curve resolution giving the number of sources, the pure source profiles and the contribution of each profile to each sample [Hopke 1991], This curve resolution problem was earlier solved by factor analysis, but lately three-way solutions have been applied. The establishment of source profiles relies a lot on factor rotation. Bilinearity is very much dependent on the dispersion process. 

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). 

As examples for orthogonal and oblique factor rotations, the varimax, quartimax, and oblimax criteria will be considered. 

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... 

Laboratory tests have been performed according to four experimental programs due to a three-factor rotatable second order design with two central points [2](Table 1). 

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