Principal Components Analysis: Centring and Rotations

It is rare to perform PCA on raw data, and usually some transformation or pre-processing is performed prior to PCA. One of the commonest methods is column centring, that is, the mean of each column is subtracted from the values of the data, so x cent ij = x ij − x ¯ j $$ {{}^{\mathit{\mathsf{cent}}}\mathit{\mathsf{x}}}_{\mathit{\mathsf{ij}}}={\mathit{\mathsf{x}}}_{\mathit{\mathsf{ij}}}-{\overline{\mathit{\mathsf{x}}}}_{\mathit{\mathsf{j}}} $$ , where there are I rows and J columns. The principle is that PCA looks at variability around the mean, or more strictly, at the variance. In previous articles, we have not centred the columns [1-3], which is a legitimate approach to PCA and can be used to understand the underlying principles of the method but is not very widespread and most applications centre the columns. The consequences of centring are not as straightforward as might initially be imagined. We will illustrate the principles of centring using four 6 × 2 datasets numbered 1 to 4 as presented in Table 1, consisting of objects A to F, and will use primarily graphical approaches. As many readers will choose to centre data prior to PCA, having a good understanding is useful. Readers should be able to reproduce corresponding numerical results using your favourite package but some of the graphics might not be available in most software. By convention, a clockwise rotation is denoted using a negative rotation angle and a vector is a row vector, so a rotation would conventionally be represented by T′ = P X′ where in our case T′ and X′ are 2 × 6 transposes of the scores and original data respectively and P a 2 × 2 loadings or rotation matrix pre-multiplying the data matrix. Similarly, we can obtain different results for the relative variability or variance of the two components. For uncentred data, the sum of squares of PC1 is 2315.6 (93.1%) and PC2 is 170.4 (6.9%) For the centred data, the sum of squares of PC1 is 362.9 (71.9%) and PC2 is 141.5 (28.1%). Similarly, we can obtain different results for the relative variability or variance of the two components. For uncentred data, the sum of squares of PC 1 is 2742.3 (86.3%) and PC 2 is 434.7 (13.7%) For the centred data, the sum of squares of PC 1 is 570.3 (62.8%) and PC 2 is 337.2 (37.2%). So far, this article has primarily been about geometry, but what are the practical consequences? The first and most obvious is that centring may usually change the patterns of the scores. When there are more than two components, rotations can be very complicated algebraically, but we will not have the room in this introductory article to discuss this in detail, but a 3D rotation, for example, is often expressed as three separate 2D rotations around each of the axes [5, 6]. Changes in signs can be viewed as reflections in hyperplanes. The directions of PC axes can vary considerably according to whether the data are centred and often show much greater deviation than in these simple two dimensional examples. As such, centring is not straightforward to visualise and understand when the number of variables is large. Whether it is appropriate to centre does depend on the questions being asked. For example, in spectroscopy of mixtures, we may be primarily interested in the properties of individual chemical components above a baseline, so centring might not be appropriate. In many areas, we are interested in variability around a mean and so it is best to centre the data. This article discusses the influence of column centring on PC scores.