The informative converse paradox: Windows into the unknown

Model-based interpretation of empirical data is useful. But unanticipated phenomena (interferences) can give erroneous model parameter estimates, leading to wrong interpretation. However, for multi-channel data, interference phenomena may be discovered, described and corrected for, by analysis of the lack-of-fit residual table — although with a strange limitation, which is here termed the Informative Converse paradox: When a data table (rows × columns) is approximated by a linear model, and the model-fitting is done by row-wise regression, it means that only the column-wise interference information can be correctly obtained, and vice versa. These “windows into the unknown” are here explained mathematically. They are then applied to multi-channel mixture data — artificial simulations as well as spectral NIR powder measurements — to demonstrate discovery after incomplete row-wise curve fitting and column-wise multivariate regression. The analysis shows how the Informative Converse paradox is the basis for selectivity enhancement in multivariate calibration. Data-driven model expansion for statistical multi-response analyses (ANOVA, N-way models etc.) is proposed.