Conditionality of the principal component method in the problem of calculating the composite index of a system from a series of observations

The construction of the integral characteristic of the system based on a number of observations containing errors can be considered as the problem of extracting useful signal against a background of noise. Principal component analysis (PCA) is widely used to find the most characteristic features in the data and to reveal their structure. However, in practice, PCA is rarely used to determine the weights of composite indices. This is partly because the weights determined using PCA often contradict intuitive ideas about the properties of the analyzed system. One of the possible reasons for the occurrence of such situations may be the poor conditioning of the PCA - the influence of irremovable data errors on the determination of the eigenvalues and eigenvectors in PCA. The covariance matrices for different observations can be interpreted as a perturbation of the true unknown covariance matrix that determines the structure of the system. The error in calculating the eigenvalues of an unknown covariance matrix can be estimated a posteriori from the available values of the observed covariance matrices. The obtained estimates should be used when choosing computationally significant principal components that satisfy the criterion of information content, which works correctly in the case of noisy data.