Application of TuckFMIN for Robust Estimating Tucker3 Solutions

Despite MCR-FMIN [1] with trilinearity constraint which can be considered as a two-way analysis of three-way data sets, TuckFMIN is pave the way for the multi-way analysis of three-way data sets. TuckFMIN is an extended objective function minimization to preserve the Tucker3 features. Starting from higher-order singular value decomposition (HO-SVD)[2] loadings, TuckFMIN represents a new approach for obtaining three proper rotation matrices for transforming the HO-SVD loadings to physically and chemically meaningful solutions. In TuckFMIN, there are three rotation matrices S, T and U that are responsible for converting physically meaningless HO-SVD loadings (A, B, C) and non-sparse core of G to physically meaningful ones ( )and sparse core . For each rotation matrix a special objective function (f(S), f(T) and f(U))is regarded that is defined based on constraints non-fulfillment. FMIN is proposed for minimization of f(total), where f(total)=f(S)+f(T)+f(U). Every objective function is defined basis on suitable constraints such as normalization, non-negativity, trilinearity, sparsity, and etc [3]. For example f(S) is defined as: . Trilinearity scalar function is defined because of the trilinear structure of . This trilinear scalar function can be efficient for obtaining robust convergence for full rank or rank-deficient data sets and decrease the risk of converging to the local minimum. TuckFMIN can sparse HO-SVD core by rotation of core elements using sparsity constraint. Simulated fluorescence data was exemplified to evaluate the feasibility of proposed method. For sake of comparison between different methods PARAFAC-ALS, HO-SVD and TuckFMIN are used. LOF (lack of fit) from TuckFMIN modeling is always as same as LOF of HO-SVD and less than LOF of the PARAFAC-ALS for noise free data sets. Despite the PARAFAC-ALS decomposition, TuckFMIN has the best fitting regardless to rank-deficiency or levels of noise . By means of simulation study, it is demonstrated that TuckFMIN with trilinearity and sparsity constraints can be helpful for faster convergence and obtaining the reproducible results.