Recent research has pointed to a number of inherent disadvantages of unrotated principal components and empirical orthogonal functions when these techniques are used to depict individual modes of variation of data matrices in exploratory analyses. The various pitfalls are outlined and illustrated with an alternative method introduced to minimize these problems via available linear transformations known as simple structure rotations. The rationale and theory behind simple structure rotation and Procrustes target rotation is examined in the context of meteorological/climatological applications. This includes a discussion of the six unique ways to decompose a rotated data set in order to maximize the physical interpretability of the rotated results.



The various analytic simple structure rotations available are compared by a Monte Carlo simulation, which is a modification of a similar technique developed by Tucker (1983), revealing that the DAPPFR and Promax k = 2 rotations are the most accurate in recovering the input structure of the modes of variation over a wide range of conditions. Additionally, these results allow the investigator the opportunity to check the accuracy of the unrotated or rotated solution for specific types of data. This is important because, in the past, the decision of whether or not to apply a specific rotation has been a ‘blind decision’. In response to this, a methodology is presented herein by which the researcher can assess the degree of simple structure embedded within any meteorological data set and then apply known information about the data to the Monte Carlo results to optimize the likelihood of achieving physically meaningful results from a principal component analysis.

Rotation of principal components

Almost all modern rotation of factor loadings is based on optimizing a criterion, for example, the quartimax criterion for quartimax rotation. Recent advancements in numerical methods have led to general orthogonal and oblique algorithms for optimizing essentially any rotation criterion. All that is required for a specific application is a definition of the criterion and its gradient. The authors present the implementations of gradient projection algorithms, both orthogonal and oblique, as well as a catalogue of rotation criteria and corresponding gradients. Software for these is downloadable and free; a specific version is given for each of the computing environments used most by statisticians. Examples of rotation methods are presented by applying them to a loading matrix from Wehmeyer and Palmer.

Gradient Projection Algorithms and Software for Arbitrary Rotation Criteria in Factor Analysis

The Procrustes Program: Producing direct rotation to test a hypothesized factor structure.

Suggestions for proper procedures to use and pitfalls to avoid are offered to prospective authors of manuscripts on studies using factor analysis methods.

A Guide to the Proper Use of Factor Analysis in the Conduct and Reporting of Research: Pitfalls to Avoid

Recent research has pointed to a number of inherent disadvantages of unrotated principal components and empirical orthogonal functions when these techniques are used to depict individual modes of variation of data matrices in exploratory analyses. The various pitfalls are outlined and illustrated with an alternative method introduced to minimize these problems via available linear transformations known as simple structure rotations. The rationale and theory behind simple structure rotation and Procrustes target rotation is examined in the context of meteorological/climatological applications. This includes a discussion of the six unique ways to decompose a rotated data set in order to maximize the physical interpretability of the rotated results. The various analytic simple structure rotations available are compared by a Monte Carlo simulation, which is a modification of a similar technique developed by Tucker (1983), revealing that the DAPPFR and Promax k = 2 rotations are the most accurate in recovering the input structure of the modes of variation over a wide range of conditions. Additionally, these results allow the investigator the opportunity to check the accuracy of the unrotated or rotated solution for specific types of data. This is important because, in the past, the decision of whether or not to apply a specific rotation has been a ‘blind decision’. In response to this, a methodology is presented herein by which the researcher can assess the degree of simple structure embedded within any meteorological data set and then apply known information about the data to the Monte Carlo results to optimize the likelihood of achieving physically meaningful results from a principal component analysis.

/pdf/review-article-rotation-of-principal-components-5gpjtwvuyo.pdf

Review article rotation of principal components

Factorial transformation is viewed as an estimation problem in which the usual assumption of homogeneously distributed error cannot be applied, but may be replaced by a principle of maximum kurtosis. This leads to quartimax in the orthogonal case, and to “oblimax” in the oblique case. Oblimax is readily programmable, and typically provides results similar to those of subjective rotation. However, oblimax may encounter special difficulty in data which do not determine a simple structure, or which have been imprecisely factored.

The rationale for an “oblimax” method of transformation in factor analysis

Factor analysis is applied to a matrix of item intercorrelations for items of the WAIS Picture Completion subtest, using a sample of 228 college-oriented males. Three factors are found, and identified in terms of the three principal causes of failure on PC described by Rapaport. The factors are examined in relation to the overall framework of the WAIS reported by Saunders, and found not to require any new dimension. The multidimensional character of PC explains why the items of this subtest have been particularly resistant to scaling for difficulty.



Factor I is most appropriately labelled “Maintenance of Contact” versus “Loss of Contact” or withdrawal, is best measured by items 16 and 18, and is not linearly related to any other subtest. This factor, which apparently embodies the unique contribution of PC to the Wechsler battery, might also be called “Recognition of Incongruity.”



Factor II is most appropriately labelled “Maintenance of Perspective” versus “Loss of Distance,” is best measured by items 8 and 9, and is related to certain items in the Comprehension subtest.



Factor III is related to the effect of “uncertainty” on the subject, is best measured by items 2 and 14, and is related to total scores on the Object Assembly subtest.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/j.2333-8504.1959.tb00938.x

A factor analysis of the picture completion items of the wais12

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The rationale for an “oblimax” method of transformation in factor analysis

A factor analysis of the picture completion items of the wais12