Vertex vectors sequential projection for self-modeling curve resolution of two-way data

A method for self-modeling curve resolution (SMCR) of two-way data is proposed. It is demonstrated that with an arbitrary p-normalization (p > 1), the two-way data points are located on a certain polyhedral hyper-“spherical” surface with the vertices constituted by the pure variables. This elucidates the geometry of an old discovery that two-way data points are bracketed by the pure variables. Thus a property of the polyhedral hyper-“sphere” is given, which states that the vertex vectors maximize a certain quadratic form over all points on the hyper-“spherical” surface. A procedure for determining pure variables in two-way data is then developed. Finally, an optimization algorithm to refine the resolution is suggested. With a good starting estimate that is as close as possible to the true pure profiles, the proposed method is expected to yield improved resolution compared to traditional resolution techniques. The proposed method is evaluated with two simulated data sets and two real chemical data sets from hyphenated chromatography–diode array detection (HPLC–DAD) of polyaromatic hydrocarbon in air particle samples. The results show that the proposed method gives satisfactory resolution for the four data sets.