Effect of noise in Principal Component Analysis

This paper demonstrates the effect of independent noise in principal components of k normally distributed random variables defined by a population covariance matrix. We prove that the principal components determined by a joint distribution of the original sample affected by noise can be essentially different in comparison with those determined from the original sample. However when the differences between the eigenvalues of the population covariance matrix are sufficiently large compared to the level of the noise, the effect of noise in principal components proved to be negligible. We support the theoretical results by using simulation study and examples. We also compare the results about the eigenvalues and eigenvectors in the two dimensional case with other models examined before. This theory can be applied in any field for the decomposition of the time series in multivariate analysis