Balanced-scaling principal component analysis as preprocessing step in multivariate curve resolution analysis of noisy data

In multivariate data, there are two types of error: Heteroscedastic and homoscedastic error. Heteroscedastic error refers to a group of measurements that have different measurement error variances, in contrast to homoscedastic errors, where all the measurements have an identical, independent and normally distributed (i.i.d) error structure. Some of the multivariate data analysis methods such as principal component analysis (PCA) assume homoscedastic measurement error structure. Thus PCA may be unable to extract useful information. The extension of PCA algorithm for heteroscedastic error structure is Maximum Likelihood Principal Component Analysis [1].In 1993, before MLPCA, a method has proposed for the analysis of data with heteroscedastic error structure, Balanced Scaling based on error estimates of measured data [2]. Balanced scaling as a pre-treatment method has been performed on data matrix, after that, customary PCA can be used for analysis of the scaled data. In this method, an estimate of the error matrix requires that its dimension is equal to the data matrix. Data have scaled using this estimate matrix.In this article, a comparison of the results obtained using three algorithms, Multivariate Curve Resolution Alternating Least Squares (MCR-ALS), MLPCA-MCR-ALS and Balanced Scaling-MCR-ALS is presented. The three approaches are applied to the analysis of a simulated environmental data set with error structures of different sizes. Special attention is paid to the case of highly heteroscedastic correlated noise. Balanced-Scaling can first be used as a preliminary step for MCR-ALS giving equivalent results to those obtained with MLPCA-MCR-ALS.In all cases, the results show that the solutions provided by Balanced Scaling-MCR-ALS are practically identical to those obtained by MLPCA-MCR-ALS.