Varimax rotation orthogonality

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). 

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. 

Finally, after all of the above have been taken into account, it is possible to "rotate" a set of factors with different types of rotation yielding somewhat different species loading (9 ). If the speciation patterns for the points are not distinct however, rotation of the factors cannot achieve a separation. We have found Varimax rotation, which maximizes the squared loadings of variables in each factor while retaining the orthogonality of the factors to be most useful. 

Statistical manipulations on the USDA database (cluster analysis, principal component analysis with varimax rotation e.g., Everitt, 1980) revealed subsets of represent ve species, as idealized in Fig 2b, but with dif ent variables (orthogonal principal components) than traditional fractions as measured by USDA. A set cf species from each orthogonal subset appears in Table 1. The Latin names, and where available, the common names of the biomass species are given. The extractives ranges are ash content, 4 to 17% protein content, 5 to 14% polyphenol, 3 to 11% and oil content, 1 to 4%. However no species contains extremes of all 4 variables. Nor can species be found, retaining native compositions, at extremes of just one extractive composition, while the other fractions are present at constant levels. Thus we use orthogonal but non-intuitive compositions in this work, then rank pyrolysis effects in terms of traditional extractives content to get an understanding of their impact on biomass pyrolysis. 

Fig. 2. Factor analysis of the Keasling/Moffett [48] data plot of the tests in the space spanned by the three orthogonally varimax-rotated common factors. Groups of variables are toxicity ( ) simple reflexes ( ) — traction (2), chimney (3), dish (4) anticonvulsant activity (o) — thiosemi carbazide (5) and nicotine antagonism (6), electroshock (7) anticholinergic actions (A) — central (8) and peripheral tremorine antagonism (9), mydriatic activity (10) fighting (A) (11).

Table 3. Factor analysis of the Keasling/Moffett data. Factor loadings aik o/the tests with respect to the three orthogonally varimax-rotated common factors, communality and uniqueness of the variables (++, high loadings +, moderate loadings)...

Varimax rotation A numerical technique applied to factor analysis, yielding an orthogonal solution in which variable loadings on each factor are maximized or minimized. This has the effect of making factors more easily interpretable. 

The 55-item Likert scale was analysed using a principal components analysis and inspection of the scree plot indicated that six factors be rotated. A varimax rotation was performed on the six factors and the orthogonal factors generated are shown in Table 2. To create factor scores, item scores were aggregated for each factor and Cronbach s alphas were calculated to establish their reliability five of the factors had high enough alphas to allow the aggregated factors to be used in further analyses, and these factors are those shown in table 2. Factor 5 was difficult to interpret and had an alpha of only. 435, so it is omitted. 

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. 

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

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... 

---