一种三维矩阵的奇异值分解算法

提出了一种三维矩阵的奇异值分解算法,该法适合处理具有三维矩阵数据的模式识别和分类模型等领域实际问题,该算法与二维矩阵奇异值分解算法类似,通过求解约束条件极值问题获得,该算法与已有的三线性分解算法比较,相对简单,计算速度快,适合处理数据量大的实际问题,该算法也很容易推广到更高维阵列的光谱数据。     

“Parallel factor analysis of multi-excitation ultraviolet resonance Raman spectra for protein secondary structure determination”

Protein secondary structural analysis is important for understanding the relationship between protein structure and function, or more importantly how changes in structure relate to loss of function. The structurally sensitive protein vibrational modes (amide I, II, III and S) in deep-ultraviolet resonance Raman (DUVRR) spectra resulting from the backbone C–O and N–H vibrations make DUVRR a potentially powerful tool for studying secondary structure changes. Experimental studies reveal that the position and intensity of the four amide modes in DUVRR spectra of proteins are largely correlated with the varying fractions of α-helix, β-sheet and disordered structural content of proteins. Employing multivariate calibration methods and DUVRR spectra of globular proteins with varying structural compositions, the secondary structure of a protein with unknown structure can be predicted. A disadvantage of multivariate calibration methods is the requirement of known concentration or spectral profiles. Second-order curve resolution methods, such as parallel factor analysis (PARAFAC), do not have such a requirement due to the “second-order advantage.” An exceptional feature of DUVRR spectroscopy is that DUVRR spectra are linearly dependent on both excitation wavelength and secondary structure composition. Thus, higher order data can be created by combining protein DUVRR spectra of several proteins collected at multiple excitation wavelengths to give multi-excitation ultraviolet resonance Raman data (ME-UVRR). PARAFAC has been used to analyze ME-UVRR data of nine proteins to resolve the pure spectral, excitation and compositional profiles. A three factor model with non-negativity constraints produced three unique factors that were correlated with the relative abundance of helical, β-sheet and poly-proline II dihedral angles. This is the first empirical evidence that the typically resolved “disordered” spectrum represents the better defined poly-proline II type structure.     

A flexible trilinear decomposition algorithm for three-way calibration based on the trilinear component model and a theoretical extension of the algorithm to the multilinear component model

There is a great deal of interest in decompositions of multilinear component models in the field of multi-way calibration, especially the three-way case. A flexible novel trilinear decomposition algorithm of the trilinear component model as a modification of an alternating least squares algorithm for three-way calibration is proposed. The proposed algorithm (constrained alternating trilinear decomposition, CATLD) is based on an alternating approximate least-squares scheme, in which two extra terms are added to each loss function, making it more efficient and flexible. The analysis of simulated three-way data arrays shows that it converges fast, is insensitive to initialization, and is insensitive to the overestimated number of components used in the decomposition. The analysis of real excitation–emission matrix (EEM) fluorescence and real high performance liquid chromatography–photodiode array detection (HPLC–DAD) data arrays confirms the results of the simulation studies, and shows that the proposed algorithm is favorable not only for EEMs but also for HPLC–DAD data. The three-way calibration method based on the CATLD algorithm is very efficient and flexible for direct quantitative analysis of multiple analytes of interest in complex systems, even in the presence of uncalibrated interferents and varying background interferents. Additionally, a theoretical extension of the proposed algorithm to the multilinear component model (constrained alternating multilinear decomposition, CAMLD) is developed.     

Trilinear equations, Bell polynomials, and resonant solutions

A class of trilinear differential operators is introduced through a technique of assigning signs to derivatives and used to create trilinear differential equations. The resulting trilinear differential operators and equations are characterized by the Bell polynomials, and the superposition principle is applied to the construction of resonant solutions of exponential waves. Two illustrative examples are made by an algorithm using weights of dependent variables.     

Trilinear Monomials with Mixed Sign Domains: Facets of the Convex and Concave Envelopes

Convex underestimators of nonconvex functions, frequently used in deterministic global optimization algorithms, strongly influence their rate of convergence and computational efficiency. A good convex underestimator is as tight as possible and introduces a minimal number of new variables and constraints. Multilinear monomials over a coordinate aligned hyper-rectangular domain are known to have polyhedral convex envelopes which may be represented by a finite number of facet inducing inequalities. This paper describes explicit expressions defining the facets of the convex and concave envelopes of trilinear monomials over a box domain with bounds of opposite signs for at least one variable. It is shown that the previously used approximations based on the recursive use of the bilinear construction rarely yield the convex envelope itself.     

Vertex vector sequential projection for the resolution of three-way data

Usually, the PARAFAC2 method is utilized for handling retention time shifts in resolving chromatographic three-way data. It requires all profiles shift the same amount, which, unfortunately, seems unlikely the case in the practice of chromatographic analysis of multi-component samples. The present authors deal with the problem by unfolding the three-way data array along a certain direction into one matrix and setting up a multi-bilinear model. Then, a new method called vertex vector sequential projection (VVSP) is proposed to select pure variables and then the alternating least squares (ALS) procedure is used to iteratively improve the fit of the data to the multi-bilinear model. With a good estimate that is as close as possible to the pure variables, a fast convergence can be expected. Moreover, no prerequisite on the shifting is required and the multi-bilinear model provides a plausible manner to make use of the multi-sample information. An additional advantage is that the present fitting procedure is easier to adjust when constraints such as non-negativity, unimodality, etc., are to be imposed on the loading matrix. The proposed method is evaluated with simulated and real chemical data sets. Satisfactory resolution results are obtained, which demonstrates the performance of the proposed method.     

Well-posedness and the small time large deviations of the stochastic integrable equation governing short-waves in a long-wave model

This paper is concerned with the stochastic integrable equation governing short-waves in a long-wave model. Firstly, the local well-posedness for this system is established by fixed point argument and (bilinear) trilinear estimates. Then the small time asymptotics of the equation is proved. The corresponding results for the stochastic Hunter–Saxton equation can be obtained by the same methods.     

Detection capability of tetracyclines analysed by a fluorescence technique: comparison between bilinear and trilinear partial least squares models

According to the committee decision of 12 August 2002 (2002/657/EC) the capability of detection, CCβ, must be set in all analytical methods not only at concentration levels close to zero but also at the maximum permitted limit (PL). In this work we describe a methodology which evaluates the capability of detection of a fluorescence technique with soft calibration models (bilinear and trilinear PLS) to determine tetracyclines (group B1 substances from annex 1 of Directive 96/23/EC). Its estimation is based on the generalisation of the procedure described in International Union of Pure and Applied Chemistry and in the ISO standard 11843 for univariate signals which evaluates the probabilities of false positive (α) and false negative (β). The capability of detection, CCβ, estimated from the second-order signal and the trilinear PLS model is 9.93 μg l⁻¹ of tetracycline, 17.75 μg l⁻¹ of oxytetracycline and 26.31 μg l⁻¹ of chlortetracycline, setting α and β at 0.05. The capability of detection, CCβ, determined around the PL (100 μg kg⁻¹ in milk and muscle) with the second-order signal is 109.4 μg l⁻¹ of tetracycline, 117.0 μg l⁻¹ of oxytetracycline and 124.9 μg l⁻¹ of chlortetracycline, setting α and β at 0.05. The results were compared with those obtained with zero and first-order signals. The effect of the interferences on the capability of detection was also analysed as well as the number of standards used to build the models and their calibration range.When a tetracycline is quantified in presence of uncalibrated ones by means of the trilinear PLS model the errors oscillate between 14.70% for TC and 9.57% for OTC.     

Classification of 9-dimensional trilinear alternating forms over GF(2)

Let V be a finite-dimensional vector space over a finite field and let f be a trilinear alternating form over V. For such forms, we introduce two new invariants. Together with a generalized radical polynomial used for classification of forms in dimension 8 over GF(2), they are sufficient to distinguish between all trilinear alternating forms in dimension 9 over GF(2). To prove the completeness of the list of forms, we computed their groups of automorphisms. There are 31 degenerate and 317 nondegenerate forms. We point out some forms with either small or large automorphism group.     

Autotopisms and Isotopisms of Trilinear Alternating Forms

For a trilinear alternating form f on a vector space V, a generalization of the group of automorphisms group of autotopisms Atp(f), is introduced. An autotopism of f is a triple (α, β, γ) of automorphisms of V satisfying f(u, v, w) = f(α(u), β(v), γ(w)) for all u, v, w ∈ V. Basic results concerning this group are presented, and it is shown that the subgroup of Atp(f) containing autotopisms with identity in one coordinate is Abelian and that a mapping in this group has no fixed points if and only if its order is not a power of two.

Moreover, the notion of equivalence of two trilinear alternating forms is generalized in a similar way, and a partial result is given.

Examples of forms with both trivial (Atp(f) = Aut(f)) and nontrivial groups of autotopisms are presented.