Bilinear Decomposition Research Articles

Early plant disease diagnosis through handheld UV-Vis transmittance spectrometer with DD-SIMCA one-class classification and MCR-ALS bilinear decomposition

A novel non-destructive analytical method for early diagnosis of two bacterial diseases, Pseudomonas syringae and Xanthomonas euvesicatoria, in tomato plants, using ultraviolet-visible (UV-Vis) transmittance spectroscopy and chemometric tools, is developed. Plant-pathogen interactions caused tissue damage that generated non-linear data patterns compared to the control set (healthy samples), which complicates traditional discrimination models, even when employing non-linear discriminant approaches. Alternatively, an authentication task to conduct one-class classification relying on a data-driven version of soft independent modeling of class analogy (DD-SIMCA) is a wise choice due to its quadratic approach, proper to deal with non-linear data. DD-SIMCA detached the target class (control healthy plant leaflet tissues) from all other samples (target class and non-target class of plant leaflet tissues inoculated with two bacteria, even before the manifestation of macroscopic lesions associated with the diseases) by capturing the main similarities within the samples of the target class through the full distance that acts as a classification analytical signal, reaching 100% sensitivity in the training and validation sets. Multivariate curve resolution constrained analysis allowed the description of the bacterial inoculation process on diseased tissues through pure spectral signatures. DD-SIMCA results indicate that non-target class of samples with higher proximity to the acceptance boundary suggested that they were at earlier stages of infection compared to more distant ones, presenting lower full distance values. These findings reveal that a handheld UV-Vis transmittance spectrometer is sufficiently sensitive to be used in acquiring biological data with suitable chemometric modeling for early disease diagnosis and prompt intervention.

Modeling Soil Organic Carbon Content Using Mid-Infrared Absorbance Spectra and a Nonnegative MCR-ALS Analysis

A new approach based on mid-IR absorbance spectra is proposed for modeling total organic carbon (TOC) content of soils. This approach involves a first-time bilinear decomposition of soil mid-IR absorbance spectra using nonnegative multivariate curve resolution (MCR) with an alternating least squares (ALS) algorithm. An MCR-ALS-derived component signifies a chemically meaningful combination of soil constituents. A new mechanistic model has been developed to link the soil composition, expressed in terms of ratios of MCR-ALS-based concentration scores of the identified components, to soil TOC value. Nonnegative MCR-ALS decomposition, performed for 213 mid-IR absorbance spectra of soil samples collected in the north and south of Israel, yielded four components. Fitting the mechanistic model-derived TOC to the experimental TOC values exhibited a TOC content threshold that affected model performance. TOC content <1.0 % w w-1 was represented by the root mean square deviation of 0.18% and 62% of the variance explained, whereas for larger TOC values, a sharp decline in model performance was observed. The existence of this TOC threshold in determining model performance suggested that successful TOC modeling (below 1%) could be indirect and related to IR spectral fingerprints of minerals binding soil organic matter (SOM) and forming organo-mineral complexes. Thus, a SOM fraction having weak interactions with soil minerals was poorly accounted for in a particular set of soil samples. This dependency of the model performance on soil TOC range proposes that it might be possible to differentiate between soil samples based on their different dominating SOM pools, mineral-associated ones and those having weak interactions with minerals. Further studies, especially in soils with high SOM content, are needed to validate our findings.

Multivariate curve resolution-alternating least-squares and second-order advantage in first-order calibration. A systematic characterisation for three-component analytical systems

BackgroundRecent interest has been focused on the application of multivariate curve resolution-alternating least-squares (MCR-ALS) to systems involving the measurement of first-order and non-bilinear second-order data. The latter pose important challenges to bilinear decomposition models, due to the phenomenon of rotational ambiguity in the solutions, even under the application of the full set of chemical constraints that is usually employed in MCR-ALS calibration. ResultsAfter the analysis of several simulated and experimental datasets, important conclusions regarding the role of the selectivity patterns in the constituent spectra have been drawn concerning the achievement of the second-order advantage. Theoretical considerations based on the calculation of the areas of feasible solutions helped to support the observations regarding the predictive ability of MCR- ALS in the various datasets. SignificanceThe understanding of the impact of rotational ambiguity in obtaining the second-order advantage with both first-order and non-bilinear second-order data is of paramount importance in the future development of analytical protocols of complex samples.

Multiplication between elements in martingale Hardy spaces and their dual spaces

In the first part of the paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces Hp(0<p⩽1) and functions in their dual spaces. Our decompositions are based on martingale paraproducts. The second part of the paper concerns applications of the method developed for the classical martingales in the first part. In particular, we build a connection between Hardy spaces on spaces of homogenous type equipped with a doubling measure and Hardy spaces with respect to the corresponding dyadic martingales. Using the method introduced in the first part, we obtain analogous results for dyadic martingales on spaces of homogenous type, which, thanks to the aforementioned connection, yield conclusions for products between elements in Hardy spaces and those in their duals on spaces of homogenous type. The key property of martingale Hardy spaces in the study is that they admit the atomic decomposition, for which we provide an interpretation via duality.

Two new methods for the estimation and interpretation of the range of feasible profiles in multivariate curve resolution and their implications to analytical chemistry

AbstractTwo new models have been recently introduced for studying the remaining rotational ambiguity in the bilinear decomposition of matrix data. One of the models is N‐BANDS, which yields two extreme profiles per sample component, corresponding to maximum or minimum signal contribution function or relative component area under its concentration profile. It is highly useful for computing the relative root mean square error due to rotational ambiguity in estimated analyte concentrations (RMSERA), which numerically quantifies the impact of the phenomenon in terms of prediction uncertainty. Since N‐BANDS successfully consider the presence of instrumental noise in the data, it is extremely useful for the analysis of real data sets. The other model is SW‐N‐BANDS, which is similar to N‐BANDS, but is applied in a sensor wise manner, that is, computing the maximum and minimum intensity value at each sensor. It provides the boundaries of the full set of feasible profiles, and helps to better understand the behavior of a given component under the application of several constraints. Both models are described in light of both simulations and experimental data, illustrating their main characteristics of importance to analytical chemistry studies.

A Fast Minimization Algorithm for the Euler Elastica Model Based on a Bilinear Decomposition

A Fast Minimization Algorithm for the Euler Elastica Model Based on a Bilinear Decomposition

Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators

Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators

Products and commutators of martingales in H1 and BMO

Let f:=(fn)n∈Z+ and g:=(gn)n∈Z+ be two martingales related to the probability space (Ω,F,P) equipped with the filtration (Fn)n∈Z+. Assume that f is in the martingale Hardy space H1 and g is in its dual space, namely the martingale BMO. Then the semi-martingale f⋅g:=(fngn)n∈Z+ may be written as the sumf⋅g=G(f,g)+L(f,g). Here L(f,g):=(L(f,g)n)n∈Z+ with L(f,g)n:=∑k=0n(fk−fk−1)(gk−gk−1) for any n∈Z+, where f−1:=0=:g−1. The authors prove that L(f,g) is a process with bounded variation and limit in L1, while G(f,g) belongs to the martingale Hardy-Orlicz space Hlog associated with the Orlicz functionΦ(t):=tlog⁡(e+t),∀t∈[0,∞). The above bilinear decomposition L1+Hlog is sharp in the sense that, for particular martingales, the space L1+Hlog cannot be replaced by a smaller space having a larger dual. As an application, the authors characterize the largest subspace of H1, denoted by H1b with b∈BMO, such that the commutators [T,b] with classical sublinear operators T are bounded from H1b to L1. This endpoint boundedness of commutators allows the authors to give more applications. On the one hand, in the martingale setting, the authors obtain the endpoint estimates of commutators for both martingale transforms and martingale fractional integrals. On the other hand, in harmonic analysis, the authors establish the endpoint estimates of commutators both for the dyadic Hilbert transform beyond doubling measures and for the maximal operator of Cesàro means of Walsh–Fourier series.

The structure of flavor mixing and reconstruction of the mass matrix

The fermion flavor structure is investigated by bilinear decomposition of the mass matrix after EW symmetry breaking, and the roles of factorized matrices in flavor mixing and mass generation are explored. On a new Yukawa basis, the minimal parameterization of flavor mixing is realized containing two relative phases and two free [Formula: see text] rotation angles. It is shown that flavor mixing can be addressed from four independent parameters. The validity of the flavor mixing structure is checked in both the lepton and quark sectors. Under the decomposition of flavor mixing, fermion mass matrices are reconstructed in the hierarchy limit. A flat mass matrix with all elements equal to 1 arises naturally from the requirement that homology exists between up-type and down-type fermion mass matrices. Some hints of a flat matrix and flavor breaking are also discussed.

Low-rank matrix factorization with nonconvex regularization and bilinear decomposition

Low-rank matrix factorization (LRMF) is a powerful approach that can recover useful information with low-rank features from corrupted data and has been broadly applied in various scenarios such as computer vision and statistics. Existing convex relaxation-based LRMF models' performance and solutions are often limited by unsatisfied recovery accuracy and prolonged execution time due to excessive relaxation bias and heavy computational burden. In this paper, we first explore one class of parameterized non-convex penalty function and generalize it as a non-convex relaxation substitution term for the original rank function in the LRMF, which can effectively enhance the recovery capability for low-rank data. In order to reduce the computational burden of the nuclear-norm-based optimization scheme and make LRMF applicable to large-scale data. We propose to perform bilinear decomposition on the low-rank matrix and derive the bilinear factorization formula under the parameterized non-convex alternative. With the above considering, a low-rank matrix recovery model via bilinear decomposition and non-convex regularization, denoted as NCBF, is formulated. Further, the optimal solution algorithm for the NCBF model is developed in the framework of the block coordinate descent, combining simultaneously three strategies of decoupling, proximal gradient descent, and block prox-linear. Also, the convergence condition for the proposed algorithm has been given. We perform extensive experiments using synthetic data and real-world image datasets. The synthetic data test results demonstrate the dual superiority of the bilinear decomposition scheme in terms of operation time and estimation accuracy. Application of denoising to both generic and CT image datasets, experimental results show that NCBF outperforms several existing methods under both subjective and objective metrics evaluations.

Predicting the uniqueness of single non-negative profiles estimated by multivariate curve resolution methods

In many kinds of chemical data, one or more species are unknown and the only efficient way to identify and/or quantify them is by mathematical resolution of the mixture spectra. The major problem with such mathematical decompositions is the possibility of obtaining a range of feasible solutions instead of a unique solution due to insufficient prior information about the system under study. However, even with the minimal non-negativity assumptions, there may be some levels of uniqueness, i.e., full/partial/fractional, in the results of the bilinear decomposition of chemical data which is very important to detect. In this study, a procedure is proposed to predict the uniqueness of the resolved non-negative profiles obtained by MCR-ALS (or analogous methods like NMF, EFA, SIMPLISMA, ITTFA, HELP, etc.). This uniqueness prediction is based on the data-based uniqueness (DBU) theorem and the general rule of uniqueness (GRU) presented in previous studies. The proposed procedure is easy to implement, has no additional computational cost, and is general for different systems with any number of components. Several simulated and experimental datasets containing different numbers of components were used to examine and evaluate the proposed procedure.

Least-squares bilinear clustering of three-way data

A least-squares bilinear clustering framework for modelling three-way data, where each observation consists of an ordinary two-way matrix, is introduced. The method combines bilinear decompositions of the two-way matrices with clustering over observations. Different clusterings are defined for each part of the bilinear decomposition, which decomposes the matrix-valued observations into overall means, row margins, column margins and row–column interactions. Therefore up to four different classifications are defined jointly, one for each type of effect. The computational burden is greatly reduced by the orthogonality of the bilinear model, such that the joint clustering problem reduces to separate problems which can be handled independently. Three of these sub-problems are specific cases of k-means clustering; a special algorithm is formulated for the row–column interactions, which are displayed in clusterwise biplots. The method is illustrated via an empirical example and interpreting the interaction biplots are discussed. Supplemental materials for this paper are available online, which includes the dedicated R package, lsbclust.

Modular Ternary Additive Problems with Irregular or Prime Numbers

Our initial problem is to represent classes $$m$$ modulo $$q$$ by a sum of three terms, two being taken from rather small sets $$\mathcal A$$ and $$\mathcal B$$ and the third one having an odd number of prime factors (the so-called irregular numbers in S. Ramanujan’s terminology) and lying in an interval $$[q^{20r},q^{20r}+q^{16r}]$$ for some given $$r\ge1$$ . We show that it is always possible to do so provided that $$|\mathcal A|\cdot|\mathcal B|\ge q(\log q)^2$$ . The proof leads us to study the trigonometric polynomials over irregular numbers in a short interval and to seek very sharp bounds for them. We prove in particular that $$\sum_{q^{20r}\le s \le q^{20r}+q^{16r}}e(sa/q)\ll q^{16r}(\log q)/\sqrt{\varphi(q)}$$ uniformly in $$r$$ , where $$s$$ ranges over the irregular numbers. We develop a technique initiated by Selberg and Motohashi to do so. In short, we express the characteristic function of the irregular numbers via a family of bilinear decompositions akin to Iwaniec’s amplification process, and that uses pseudo-characters or local models. The technique applies to the Liouville function, to the Mobius function and also to the von Mangoldt function, in which case it is slightly more difficult. It is however simple enough to warrant explicit estimates, and we prove, for instance, that $$\bigl|\sum_{X<\ell\le 2X}\Lambda(\ell)\, e(\ell a/q)\bigr|\le 1300 \sqrt{q}\,X/\varphi(q)$$ for $$250\le q\le X^{1/24}$$ and any $$a$$ prime to $$q$$ . Several other results are also proved.

Estimating the boundaries of the feasible profiles in the bilinear decomposition of multi-component data matrices

The boundaries of the range of feasible profiles in the bilinear decomposition of a data matrix are estimated by suitably modifying the already published N-BANDS algorithm. Instead of maximizing and minimizing the profile norms, the new sensor-wise N-BANDS approach calculates the profiles showing extreme values of the elements at each sensor in both data modes. This produces two sets of profiles for each sensor, whose envelops define the boundaries of the range of feasible solutions: the upper envelop for maximum element values represents the upper boundary, and the lower envelop the lower boundary. A three-component data set was studied in this manner under non-negativity constraints in both data modes. The sensor-wise N-BANDS results agree with those provided by the FACPACK program, which computes the area of feasible solutions and the associated set of feasible profiles. Computer times required for truly multi-component systems appear to be feasible using sensor-wise N-BANDS (up to eight sample components were checked). As an example, a six-component system was analyzed with the presently described approach, with results which are consistent with the expectations based on the consideration of the degree of profile overlapping among the participating species.

Simultaneous Determination of the Acid Dissociation Constants of Phenolics by Multivariate Analysis of pH and Ultraviolet-Visible Spectrophotometric Measurements

In the development of new commercial products in the pharmaceutical and food industries, acid dissociation constant (Ka) values of active and inactive ingredients are important factors for the estimation of the biological activity. In this regard, new spectrophotometric approaches based on parallel factor analysis (PARAFAC) and multivariate curve resolution-alternating least squares (MCR-ALS) to pH-absorbance dataset are described for the simultaneous determination of the acid dissociation constants of the phenolics ferulic acid, 4-hydroxybenzoic acid, syringic acid, protocatechuic acid, vanillic acid, and isovanillic acid that have more than one dissociation equilibrium without using classical titration procedures. The acid dissociation constants were determined by the pH profiles obtained by bilinear MCR-ALS and trilinear PARAFAC decomposition of pH-ultraviolet and pH-ultraviolet-visible spectroscopic datasets of the phenolics. For the quantification of the acid dissociation constants with the MCR-ALS and PARAFAC approaches, the estimated spectral profiles provided an opportunity for the direct observation of the individual spectral bands of each phenolic without the use of additional software. The results showed that MCR-ALS and PARAFAC applications to spectrophotometric measurements at various pH values were alternatives for solving acid dissociation constant problems that cannot be determined by classical methods.

Real-variable characterizations of local Orlicz-slice Hardy spaces with application to bilinear decompositions

Recently, both the bilinear decompositions [Formula: see text] and [Formula: see text] were established. In this paper, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains [Formula: see text], a variant of the local Orlicz Hardy space, introduced by Bonami and Feuto as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms, and various maximal functions, which are new even for [Formula: see text]. The relationship [Formula: see text] is also clarified.

N‐BANDS: A new algorithm for estimating the extension of feasible bands in multivariate curve resolution of multicomponent systems in the presence of noise and rotational ambiguity

AbstractA new algorithm named N‐BANDS has been developed for the estimation of the combined effect of noise and rotational ambiguity in the bilinear decomposition of data matrices using the popular multivariate curve resolution–alternating least‐squares model. It is based on a nonlinear maximization and minimization of a component‐wise signal contribution function (SCF), with a single‐objective function and a separate module for applying a variety of constraints. The algorithm can be applied to multicomponent systems and efficiently estimates the extreme component profiles corresponding to maximum and minimum SCF in the presence of varying amounts of instrumental noise. Simulated systems mimicking multicomponent and multisample analytical calibration protocols have been studied, having uncalibrated interferents in the test samples. Different noise structures (independent and identically distributed, proportional and correlated) were added to the data matrices, with results that indicate an increase in the extension of feasible bands as the noise level increases.

Bilinear decomposition and divergence-curl estimates on products related to local Hardy spaces and their dual spaces

Let p∈(0,1), α:=1/p−1, and, for any τ∈[0,∞), Φp(τ):=τ/(1+τ1−p). Let Hp(Rn), hp(Rn), and Λnα(Rn) be, respectively, the Hardy space, the local Hardy space, and the inhomogeneous Lipschitz space on Rn. In this article, applying the inhomogeneous renormalization of wavelets, the authors establish a bilinear decomposition for multiplications of elements in hp(Rn) [or Hp(Rn)] and Λnα(Rn), and prove that these bilinear decompositions are sharp in some sense. As applications, the authors also obtain some estimates of the product of elements in the local Hardy space hp(Rn) with p∈(0,1] and its dual space, respectively, with zero ⌊nα⌋-inhomogeneous curl and zero divergence, where ⌊nα⌋ denotes the largest integer not greater than nα. Moreover, the authors find new structures of hΦp(Rn) and HΦp(Rn) by showing that hΦp(Rn)=h1(Rn)+hp(Rn) and HΦp(Rn)=H1(Rn)+Hp(Rn) with equivalent quasi-norms, and also prove that the dual spaces of both hΦp(Rn) and hp(Rn) coincide. These results give a complete picture on the multiplication between the local Hardy space and its dual space.

A chemometric comparison of different models in fluorescence analysis of dabigatran etexilate and dabigatran.

In this paper, a simple, rapid, low-cost and potential method was established for the simultaneous quantitative analysis of dabigatran etexilate (DABE) and dabigatran (DAB) in spiked biological fluids. It combined excitation-emission matrix fluorescence (EEMF) with different second-order calibration methods, including the self-weighted alternating normalized residue fitting (SWANRF) algorithm based on trilinear decomposition model, the multivariate curve resolution - alternating least-squares (MCR-ALS) based on bilinear decomposition model and the unfolded partial least-square coupled with residual bilinearization (U-PLS/RBL) based on latent variables model. The proposed method showed "second-order advantage", that is, satisfactory quantitative results were successfully obtained even in the presence of unknown interferences and serious spectral overlap. The recoveries of DABE and DAB in spiked biological fluids were 91.7%-101.7% for SWANRF, 95.9%-117.8% for MCR-ALS, 83.0%-109.6% for U-PLS/RBL, respectively. Figures of merit and other statistical parameters were also calculated to assess the performance of the proposed method. Moreover, the modeling procedures and characteristics of three different models in EEMF analysis were discussed and compared.

Pointwise Multipliers of Zygmund Classes on"Equation missing"

It is well known that Lipschitz spaces on the torus are an algebra. It is no more the case in the non compact situation because of the behavior at infinity. This is a companion article to Bonami et al. (J Math Pures Appl (9) 131:130–170, 2019), where pointwise multipliers on Lipschitz spaces on $${\mathbb {R}}^n$$ are characterized for non-integer values of the parameter. In this article, the authors first establish two equivalent characterizations of a modified Zygmund space, and then characterize the pointwise multipliers on Lipschitz spaces on $${\mathbb {R}}^n$$ for the integer values of the parameter, in particular, for the Zygmund class, via the intersection of the Lebesgue space $$L^\infty ({\mathbb {R}}^n)$$ and the modified Zygmund space. This result can be used to show that the bilinear decomposition of the pointwise product of the Hardy space and its dual, in the integer values of the parameter, obtained in the aforementioned reference is sharp in the dual space sense.