Orthogonal rotation

Orthogonal rotation produces a new orthogonal frame of reference axes which are defined by the column-vectors of U and V. The structural properties of the pattern of points, such as distances and angles, are conserved by an orthogonal rotation as can be shown by working out the matrices of cross-products ... 

After an orthogonal rotation one can also perform a backrotation toward the original frame of reference axes ... 

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. 

The columns of V are the abstract factors of X which should be rotated into real factors. The matrix V is rotated by means of an orthogonal rotation matrix R, so that the resulting matrix F = V R fulfils a given criterion. The criterion in Varimax rotation is that the rows of F obtain maximal simplicity, which is usually denoted as the requirement that F has a maximum row simplicity. The idea behind this criterion is that real factors should be easily interpretable which is the case when the loadings of the factor are grouped over only a few variables. For instance the vector f, =[000 0.5 0.8 0.33] may be easier to interpret than the vector = [0.1 0.3 0.1 0.4 0.4 0.75]. It is more likely that the simple vector is a pure factor than the less simple one. Returning to the air pollution example, the simple vector fi may represent the concentration profile of one of the pollution sources which mainly contains the three last constituents. 

Figure 13.4 Orthogonal rotatable central composite design. Square points 2, star points 2 2,...

A section has been added to Chapter 1 on the distinction between analytic vs. enumerative studies. A section on mixture designs has been added to Chapter 9. A new chapter on the application of linear models and matrix least squares to observational data has been added (Chapter 10). Chapter 13 attempts to give a geometric feel to concepts such as uncertainty, information, orthogonality, rotatability, extrapolation, and rigidity of the design. Finally, Chapter 14 expands on some aspects of factorial-based designs. 

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). 

An alternative approach for selecting the number of retained factors may be found by examing the partition of variance after the orthogonal rotation. It can be argued that a factor with a variance of less than one contains less Information than did one of the original variables. However, since the objective of the rotation is to redistribute the variance from the artificially compressed state that results from the matrix diagonalization, it appears to be useful to examine a number of solutions with differing numbers of retained factors. The rotated solutions that contain factors with total variance less than one can then be rejected. For this example, the fine fraction results yield... 

The trace C matrix is, therefore, equal to a non-orthogonal rotation of QT... 

Thus the problem of deducing E and C is the determination of the proper non-orthogonal rotation matrix, R. 

An orthogonal rotation is thus described by the "replacement" operators Ey -... 

These criteria lead to different numeric transformation algorithms. The main distinction between them is orthogonal and oblique rotation. Orthogonal rotations save the structure of independent factors. Typical examples are the varimax, quartimax, and equi-max methods. Oblique rotations can lead to more useful information than orthogonal rotations but the interpretation of the results is not so straightforward. The rules about the factor loadings matrix explained above are not observed. Examples are oblimax and oblimin methods. 

Step 1. PCA is usually performed as the Varimax orthogonal rotation of PCs. This rotation gives a more straightforward interpretation of extracted PCs by increasing higher factor loadings and decreasing lower ones. 

Where X is an antisymmetric matrix containing the independent (orthogonal) rotation parameters. Expanding the energy in X about the origin... 

Four by Four. A convenient way of describing a symmetry operation is by using, not the 3x3 matrix that could represent three orthogonal rotations or three translations (but not both), but rather a 4 x 4 augmented matrix Q. For instance, we can represent the symmetry operator number 3, namely (—x + y, —x, z + 2/3) in space group P 3i 2 1 ( 152) as the matrix Q3 ... 

Figure 1 Three line-position and line intensity roadmaps as a function of rotation angle (deg), for rotations about three orthogonal rotation axes, obtained from solid-state 31P cross-polarization (CP) NMR spectra at fixed magnetic fields of 4.7 T, for single-crystals (space group C2/m) of TMPS (tetramethyldiphosphine sulfide). One sees the spectra from two site types occurring in abundance ratio 1 2, each with a spin-spin doublet structure. Taken from Ref. 31.

The orthogonal rotation matrix R(co,) which will perform the rotation using an angle set co is defined as ... 

R(6) is an orthogonal rotation matrix. The symbol tilde ( ) means here transposed and M(0) is the Mueller matrix of the given optical element corresponding to 0°. If e.g. a linear polarizer is rotated at an angle 6 the above transformation should look like as follows ... 

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