Nova formulação de ferramentas de estatística multivariada com incertezas experimentais

Abstract

When a researcher wants to analyze a set of data, assuming the randomness of the measurements, taking into account the statistical and instrumental errors involved in the process, experimental errors have a key role in the results of some statistical analysis. However, many statistical tools do not take them into account in their calculations and, therefore, this study proposes new formulations for the Principal Components, Fisher Linear Discriminant and Canonical Correlation analysis which take the experimental errors into account, and also proposes to evaluate the impact on the results of these new techniques. Since the three analysis have in common the fact that their results are tied to the data covariance matrix, the methodological procedure of this study consisted of using the weighted average of the variables by their experimental errors, in order to construct the covariance matrices. For purposes of propagating these errors to the results of the three analysis, it was chosen to use a numerical method similar to Monte Carlo, through algorithms developed to generate random results from the fluctuation of the data weighted average. In order to demonstrate the applicability of the new principal components model, it was reconstructed the principal components analysis of the variables for the diffuse interstellar band performed by Ensor et al. (2017) and the results were compared with the traditional approach that does not take into account the experimental errors. This new model of principal components provided an alternative way to choose the number of components to be used, through the values obtained for the relative errors concerning to the accumulated proportion of variance explained. For the other two analysis, simulations were performed to evaluate the applicability of the method in examples developed by the author. The discriminant analysis was the only technique that presented a change in its interpretation, providing as answer the probability of new observations belonging to each group and not a deterministic classification. The analysis of canonical correlations allowed for an evaluation of the data closer to the reality of the experiment, once both canonical variables and transformation vectors as well as canonical correlations have available now error bars. Therefore, it was possible to conclude that in the three analysis the insertion of the experimental errors enabled the researcher an interpretation of the results faithful to the real experiment, which may avoid a super or underestimation of parameters in the data analysis.