Varimax rotation

Two different types of ANN models were developed. In the first type, prediction was performed based on the original PCs. In the second type of ANNs developed, scores of rotated (varimax rotation) PCs (ANN-RPCs) with eigenvalues greater than 1 were selected as input. For this model, prediction of WQI was performed using two to six rotated principal components separately. 

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. 

Several applications of varimax rotation in analytical chemistry have been reported. As an example the varimax rotation is applied on the HPLC data table of... 

PAHs introduced in Section 34.1. A PCA applied on the transpose of this data matrix yields abstract chromatograms which are not the pure elution profiles. These PCs are not simple as they show several minima and/or maxima coinciding with the positions of the pure elution profiles (see Fig. 34.6). By a varimax rotation it is possible to transform these PCs into vectors with a larger simplicity (grouped variables and other variables near to zero). When the chromatographic resolution is fairly good, these simple vectors coincide with the pure factors, here the elution profiles of the species in the mixture (see Fig. 34.9). Several variants of the varimax rotation, which differ in the way the rotated vectors are normalized, have been reviewed by Forina et al. [2]. 

H.F. Kaiser, The varimax criterion for analytic rotation in factor analysis. Psychometrika, 23 (1958) 187-200. 

M. Forina, C. Armanino, S. Lanteri and R. Leardi, Methods of Varimax rotation in factor analysis with applications in clinical and food chemistry. J. Chemom., 3 (1988) 115-125. 

Factor analysis with the extraction of two factors and varimax rotation can be carried out in R as described below. The factor scores are estimated with a regression method. The resulting score and loading plots can be used as in PCA. 

A Varimax rotation (Vl ) is often used to achieve it. However, simple structure may not be the most useful criterion for environmental source resolution since an element may be present in an aerosol sample because of its emission by several sources. The variance should, therefore, be spread over several factors rather than concentrated in one. 

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). 

Kaiser, H. F. Computer Program for Varimax Rotation in Factor Analysis, Educational and Psychological Measurement, 1959, 19, 413. 

Computer Programs O) Initial factors were extracted using a principal components solution. The number of factors to be kept for rotation to a final solution was selected from a plot of the variance explained by each factor (its eigenvalue) versus its ordinal number. Usually, factors with eigenvalues larger than about 1.0 were kept. Final solutions were obtained using Varimax rotations. 

Iterative Target Testing is another approach. The preliminary approximations of the real factors are chosen, based on the first (VARIMAX) rotation of the abstract PCA solution. With iterative target testing the factors are transformed to the best approximations. It can be considered as LSO, where PCA and VARIMAX are supplying the model. Clusters with an arbitrary number of peaks can be deconvoluted. Six component systems are tested (Vandeginste et al. 

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. 

Let us now compare the loadings obtained after utilizing a standard rotational procedure (Varimax). In Tab. 5-7 we find the first factor loaded only by Al and Fe. (Using insider knowledge , should we relate this to the fact that these are the only elements in the interlaboratory comparison determined by a gravimetric method. .. ) All other factors express the importance of single elements. The sole exception is Ti If we keep our limit of 0.7 for a coefficient Ti does not contribute to any of the factors significantly. 

Tab. 5-7. Varimax rotated loadings of five factors, F, and communalities hf...

Tab. 9-5. Factor loading matrix of the Flettstedt soil profile after varimax rotation (loadings less than 0.5 in absolute value are set to zero)...

Ondrick, C. W. Srivastava, G. S. "C0RFAN - Fortran IV Computer Program for Correlation, Factor Analysis and Varimax Rotation University of Kansas, 1970 Computer Contribution 42 92 pp. 

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. 

The next step is to choose approximate concentration profiles as targets. There are many ways to select these initial vectors, and any method used to provide initial CR estimates can be useful for this purpose. Historically, the vectors obtained after performing varimax rotation onto the scores were used [47] and also the needle targets (i.e., vectors with only one non-null element equal to 1), which are the simplest representation of a peak-shaped profile [48, 75],... 

Figure 7.2 Authentication of monovarietal virgin olive oils results of applying factor analysis to the volatile compounds, (a) Maximum likelihood and Varimax rotation, (b) Principal components and Varimax rotation. Note A, cv. Arbequina C, cv. Coratina K, cv. Koroneiki P, cv. Picual (source SEXIA Group-Instituto de la Grasa, Seville, Spain).

The program used in this study was a modified version of Dixon s BMD08M factor analysis with varimax rotation (12), The principal components analysis was conducted using covariance matrices. Five factors were created from the data set. Examination of the individual proportion of the total variance contributed by each of the factors demonstrated that 96.3% of the total variance could be accounted for by the first three factors. These three factors were used in the following cluster analysis. 

Statistical analyses. Three-way analyses of variance treating judges as a random effect were performed on each descriptive term using SAS Institute Inc. IMP 3.1 (Cary, North Carolina). Principal component analysis of the correlation matrix of the mean intensity ratings was performed with Varimax rotation. Over 200 GC peaks... 

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