Recent Advances in Principal Component Analysis for Directional Data

The high dimensionality of the input data can pose multiple problems when implementing statistical techniques. The presence of many dimensions in the data can lead to challenges in visualizing the data, higher computational demands, and a higher probability of over-fitting or under-fitting in modeling. Furthermore, the curse of dimensionality contributes to these issues by stating that the necessary number of observations for accurate modeling increases exponentially as the number of dimensions increases. Dimension reduction tools help overcome this challenge. Principal Component Analysis (PCA) is the most widely used technique, intensively studied in classical linear spaces. However, in applied sciences such as biology, bioinformatics, astronomy and geology, there are many instances in which the data's support are non-Euclidean spaces. In fact, the available data often include elements of Riemannian manifolds such as the unit circle, torus, sphere, and their extensions. Therefore, the terms "manifold-valued" or "directional" data are used in the literature for these situations. When dealing with directional data, the linear nature of PCA might pose a challenge to achieve accurate data reduction. This paper therefore reviews and investigates the methodological aspects of PCA on directional data and their practical applications.