Closure constraint in multivariate curve resolution

Multivariate curve resolution techniques try to estimate physically and/or chemically meaningful profiles underlying a set of chemical or related measurements. However, the estimation of profiles is not generally unique and it is often complicated by intensity and rotational ambiguities. Constraints as further information of chemical entities can be imposed to reduce the extent of ambiguities. Not only a long list of constraints has been introduced but also some of them can be applied in different ways. Either investigating constraint effects on the extent of rotational ambiguity or how they can be applied during curve resolution can shed light on curve resolution studies. The motivation behind this contribution is to pave the way to a clarification about the closure constraint. Considering simulated equilibrium and kinetic spectrophotometric data sets, different approaches to closure implementation were applied to demonstrate the geometrical explanation of closure constraint and its effect on multivariate curve resolution-alternating least squares results. Besides, the closure constraint is compared with normalization and it is proved that the closure constraint is a Borgen norm and has the same effect as other Borgen norms in multivariate curve resolution. Finally, to further examine the closure constraint, a real data set was investigated.