The Impact of Measurement Error on Principal Component Analysis

Abstract

We investigate the effect of measurement error on principal component analysis in the high‐dimensional setting. The effects of random, additive errors are characterized by the expectation and variance of the changes in the eigenvalues and eigenvectors. The results show that the impact of uncorrelated measurement error on the principal component scores is mainly in terms of increased variability and not bias. In practice, the error‐induced increase in variability is small compared with the original variability for the components corresponding to the largest eigenvalues. This suggests that the impact will be negligible when these component scores are used in classification and regression or for visualizing data. However, the measurement error will contribute to a large variability in component loadings, relative to the loading values, such that interpretation based on the loadings can be difficult. The results are illustrated by simulating additive Gaussian measurement error in microarray expression data from cancer tumours and control tissues.

Robust Segmentation of Planar and Linear Features of Terrestrial Laser Scanner Point Clouds Acquired from Construction Sites

“…The directional vector components ( ; Figure 2 ) of the simulated linear points are affected less by the point measurement errors compared to the normal vector of the simulated planar points. In the work of [ 61 ], it was shown that, in cases where the largest eigenvalue is larger than the error variance (such as the case of a linear feature), the relative impact of the measurement error on the eigenvector corresponding to the largest eigenvalue is small, which agrees with the results presented here. The best fit line for each of the curves was used to determine the value of the initial threshold.…”

“…This impact can theoretically be derived by variance propagation from the functional TLS positioning model and the stochastic properties of the instrumental random errors. Some research has aimed to determine the impact of point measurement errors on the principal components [ 60 , 61 ]. However, the exact analytical relationship between the principal components of error-free points and the principal components of the points including measurement errors when their covariance matrix contains non-zero off-diagonal elements has not been developed [ 60 ].…”

Deep XMM-Newton observations of the northern disc of M 31

“…High values of the first two admixture coefficients (α 1 and α 2 ) have been found to generally indicate variable sources (Sokolovsky et al 2017;Moretti et al 2018). For a general discussion of the impact of measurement errors on the PCA analysis see Hellton & Thoresen (2014).…”

Principal component analysis within nuclear structure

“…Apparently, analysis of this kind of error has not been studied so far. It is, therefore, hoped that findings of [21] should also apply to our study. First, we start by applying two tests [22] on the correlation matrices of each R4/2 ratio:…”