Orthogonally rotated factor

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. 

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... 

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. 

A rotation can be described by an orthogonal rotation matrix R. Orthogonality implies that R = r , i.e. R R = RR = I (unit matrix). Insertion of RR in the factor model gives... 

Suppose for example a trial wavefunction has the form of Eq. (226) and is written explicitly in terms of the n pair-expansion variables and the n(n — l)/2 variables in the matrix K. The theory of the orthogonal rotation group (or of the more general unitary group) may be invoked to factor the orthogonal matrix exp(K) according to... 

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 varimax criterion serves the purpose of an orthogonal rotation, where the variance of the squared loadings within a common factor is maximized. As a result, as many common factors should be retained that are described by as few features (variables) as possible. Large eigenvalues and loadings are increased, but small ones... 

Prior to ANOVA, ratings for the 16 mood scales underwent factor analysis (principal components) of the correlation matrix and subsequent orthogonal rotation... 

In principle, in the absence of noise, the PLS factor should completely reject the nonlinear data by rotating the first factor into orthogonality with the dimensions of the x-data space which are spawned by the nonlinearity. The PLS algorithm is supposed to find the (first) factor which maximizes the linear relationship between the x-block scores and the y-block scores. So clearly, in the absence of noise, a good implementation of PLS should completely reject all of the nonlinearity and return a factor which is exactly linearly related to the y-block variances. (Richard Kramer)... 

The design in Figure 13.4 is similar to the design in Figure 13.3, but the center point has been replicated a total of eight times, not four. This makes the design not only rotatable but also orthogonal in the coded factor space that is, the estimate of one factor effect (i.e pj, Pj, P , P22, or P j) is independent of the estimates of all other factor effects (see Section 12.10). 

It seems unlikely that reaction following this pathway can proceed with simultaneous formation of both bonds. If the n,n triplet is involved, a spin inversion must occur before bonding. Another factor which might be important even with the n,it singlet is that the unshared (j) electron and the n orbited are orthogonal. This means that some bond rotation may be required before complete bonding. 

We have used the Born Oppenheimer approximation to factor 4 0/3, I,ma into electronic and nuclear parts and have further assumed that the former are orthogonal to enable us to reduce V. Both wave functions may be approximated by products of electronic, nuclear rotation and vibrational wave functions. The last of these may be factored out at once, and... 

This empirical statistical function, based on the residual standard deviation (RSD), reaches a minimum when the correct number of factors are chosen. It allows one to reduce the number of columns of R from L to K eigenvectors or pure components. These K independent and orthogonal eigenvectors are sufficient to reproduce the original data matrix. As they are the result of a mathematical treatment of matrices, they have no physical meaning. A transformation (i.e. a rotation of the eigenvectors space) is required to find other equivalent eigenvectors which correspond to pure components. 

In this example, orthogonality of all factor effects has been achieved by including additional center points in the coded rotatable design of Equation 11.81. Orthogonality of some experimental designs may be achieved simply by appropriate coding (compare Equation 11.26 with Equation 11.20, for example). Because orthogonality is almost always achieved only in coded factor spaces, transformation of... 

For tensors of higher rank we must ensure that the bases are properly normalized and remain so under the unitary transformations that correspond to proper or improper rotations. For a symmetric T(2) the six independent components transform like binary products. There is only one way of writing xx xx, but since xx x2 = x2 xx the factors xx and x2 may be combined in two equivalent ways. For the bases to remain normalized under unitary transformations the square of the normalization factor N for each tensor component is the number of combinations of the suffices in that particular product. F or binary products of two unlike factors this number is two (namely ij and ji) and so N2 = 2 and x, x appears as /2x,- Xj. The properly normalized orthogonal basis transforming like... 

This example refers to response dependence on two factors (k=2). Orthogonal second-order design in this case, according to Table 2.164, has nine design points (N=9). The design matrix with outcomes of design points is shown in Table 2.165. The same case has been elaborated in the previous section, in Example 2.43, by application of rotatable second-order design. However, the connection between coded and real values of factors for the same null point is now different ... 

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