Rotation of principal components

TLDR

Unrotated principal components and empirical orthogonal functions have inherent disadvantages for depicting individual modes of variation, prompting the need for rotation techniques such as simple structure and Procrustes target rotation, especially in meteorological/climatological analyses where rotation decisions were previously made blindly. The study outlines the pitfalls of unrotated PCs and introduces simple structure rotations as an alternative method to mitigate these problems. The authors present a methodology that decomposes rotated data using six approaches to maximize interpretability and employs a Monte Carlo simulation to assess simple structure and guide optimal rotation choices for meteorological datasets. Monte Carlo simulations show that DAPPFR and Promax k = 2 rotations most accurately recover input structure across varied conditions, enabling investigators to evaluate the accuracy of unrotated or rotated solutions for specific data types.

Abstract

Abstract 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.