Some techniques give simplified versions of PCs, in order to aid interpretation. Indeed, many adaptations of the basic methodology for different data types and structures have been developed, two of which will be described in § 3a,d. The main uses of PCA are descriptive, rather than inferential an example will illustrate this.Īlthough for inferential purposes a multivariate normal (Gaussian) distribution of the dataset is usually assumed, PCA as a descriptive tool needs no distributional assumptions and, as such, is very much an adaptive exploratory method which can be used on numerical data of various types. In either case, the new variables (the PCs) depend on the dataset, rather than being pre-defined basis functions, and so are adaptive in the broad sense. The choice between these analyses will be discussed. PCA can be based on either the covariance matrix or the correlation matrix. In § 2, the formal definition of PCA will be given, in a standard context, together with a derivation showing that it can be obtained as the solution to an eigenproblem or, alternatively, from the singular value decomposition (SVD) of the (centred) data matrix. Substantial books have been written on the subject and there are even whole books on variants of PCA for special types of data. Since then its use has burgeoned and a large number of variants have been developed in many different disciplines. The earliest literature on PCA dates from Pearson and Hotelling, but it was not until electronic computers became widely available decades later that it was computationally feasible to use it on datasets that were not trivially small.
Finding such new variables, the principal components (PCs), reduces to solving an eigenvalue/eigenvector problem. This means that ‘preserving as much variability as possible’ translates into finding new variables that are linear functions of those in the original dataset, that successively maximize variance and that are uncorrelated with each other. statistical information) as possible.Īlthough it is used, and has sometimes been reinvented, in many different disciplines it is, at heart, a statistical technique and hence much of its development has been by statisticians. Its idea is simple-reduce the dimensionality of a dataset, while preserving as much ‘variability’ (i.e. Many techniques have been developed for this purpose, but principal component analysis (PCA) is one of the oldest and most widely used.
In order to interpret such datasets, methods are required to drastically reduce their dimensionality in an interpretable way, such that most of the information in the data is preserved.
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