Oblique rotation

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. 

Kiers HAL, SIMPLIMAX oblique rotation to an optimal target with simple structure, Psychometrika, 1994, 59, 579. 

Kiers HAL, Three-way SIMPLIMAX for oblique rotation of the three-mode factor analysis core to simple structure, Computational Statistics and Data Analysis, 1998b, 28, 307-324. 

Extraction Method Principal Component Analysis, Rotation Method Oblimin Oblique Rotation with Kaiser Normalization. Correlation of Components Asian to Caucasian 0.307, Asian to African -0.615, Caucasian to African -0.119... 

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. 

For an oblique rotation by multiplication with a transformation matrix, a matrix of the so-called factor structure, is obtained. This matrix contains correlation of common factors and features ... 

Example of an oblique vector rotation for a hypothetical three variable system with Pr, Ph and Hopane (Hop). The dotted lines indicate and represent a set of principal component (PC) vectors Vj, V2, and V. Vj has a negative value for Ph, a chemically unreasonable composition. The oblique rotation rotates each of the PC vectors to the sample vector (solid lines with data points) nearest it, resulting in positive values for all variables... 

Al-Shammari AA, Rahman SU, Chin D-T (2004) An oblique rotating barrel electrochemical reactor for removal of copper ions from waste water. J Appl Electrochem 34 447-453... 

Dysfunctions of motion created by the sacrum moving on the ilium are commonly unilateral anterior or posterior sacral dysfunctions, a unilateral shear of the sacrum along the articulation or oblique rotational sacral dysfunctions. If the fifth lumbar vertebra is not involved in the oblique rotation, the dysfunction is described as a sacral rotation dysfunction. if the fifth lumbar vertebra rotates in a direction opposite that of the sacrum, it is described as a sacral torsion. 

The collective study of the various atmospheres and surfaces in the Solar System constitutes the field of comparative planetology. Wide ranges in surface gravity, solar flux, internal heat, obliquity, rotation rate, mass, and composition provide a broad spectrum of boundary conditions for atmospheric systems. Analyses of data within this context lead to an understanding of physical processes applicable to all planets. Once the general physical principles are identified, the evolution of planetary systems can be explored. 

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