Dealing with three-way data containing missing values by new weighted method for second-order calibration

Multi-way data arrays contain missing values for several reasons, such as various malfunctions of instruments, responses being outside instrument ranges, irregular measurement intervals between samples and data postprocessing. In the present study, one new method, weighted alternating penalty trilinear decomposition (W-APTLD), based on the weighted trilinear model and the idea of alternative trilinear decomposition was given to analyze three-way data arrays containing missing values. In addition, one improved core consistency diagnostic method (W-CORCONDIA) was proposed to estimate the chemical ranks of three-way data arrays containing missing values. The results of one simulation and two real data sets demonstrate that the new method W-APTLD could be used to deal with missing values and reserves the second-order advantage. When meeting excessive factors, W-APTLD could give more accurate results than weighted PARAFAC (W-PARAFAC), PARAFAC with single imputation (PARAFAC-SI) and incomplete data PARAFAC (INDAFAC). The convergence rate of W-APTLD was much faster than W-PARAFAC and PARAFAC-SI but slower than INDAFAC. Better than W-PARAFAC and PARAFAC-SI, W-APTLD could overcome the problem due to severe collinearity. In addition, this new method could be extended to analyze higher-way data arrays containing missing values.