On a black hole effect in bilinear curve resolution based on least squares

27 July 2022, Version 3
This content is a preprint and has not undergone peer review at the time of posting.


Least squares-based estimations lay behind most chemometric methodologies. Their properties, though, have been extensively studied mainly in the domain of regression, in relation to which the effect of well-known deleterious factors (like object leverage or data distributions deviating from ideal conditions) on the accuracy of the prediction of an external response variable have been thoroughly assessed. Conversely, much less attention has been paid to what these factors might yield in alternative scenarios, where least squares approaches are still utilised, yet the objectives of data modelling may be very different. As an example, one can think of multivariate curve resolution (MCR) problems which are usually addressed by means of Multivariate Curve Resolution-Alternating Least Squares (MCR-ALS). In this respect, this article wants to offer a perspective on the basic principles of MCR-ALS from the regression point of view. In particular, the following critical aspects will be highlighted: i) in the presence of minor components, if the number of analysed data points is too large, the leverage of those that may be essential for a MCR-ALS resolution might become too low for guaranteeing its correctness and ii) in order to overcome this black hole effect and improve the accuracy of the MCR-ALS output, data pruning can be exploited. More in detail, this communication will provide a practical illustration of such aspects in the field of hyperspectral imaging where even single experimental runs may lead to the generation of massive amounts of spectral recordings.


least squares
curve resolution

Supplementary materials

On a black hole effect in bilinear curve resolution based on least squares - Supporting Information
The original paper describes the effect that the number of analysed data points can have on the quality and the reliability of the solutions that least squares-based unmixing approaches (namely, Multivariate Curve Resolution-Alternating Least Squares - MCR-ALS) may provide. All the illustration examples initially reported were conceived to characterise such an effect in scenarios where selective information was encoded in the collected measurements. Here, additional tests were conducted in the absence of selectivity.


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