Trilinear Model Research Articles

Rapid estimation of rate constants using on-line SW-NIR and trilinear models

In this paper, two algorithms are presented to estimate reaction rate constants from on-line short-wavelength near-infrared (SW-NIR) measurements. These can be applied in cases where the contribution of the different species in the mixture spectra is of exponentially decaying character. From a single two-dimensional dataset two two-way datasets are formed by splitting the original dataset such that there is a constant time lag between the two two-way datasets. Next, a trilinear structure is formed by stacking these two two-way datasets into a three-way array. In the first algorithm, based on the generalized rank annihilation method (GRAM), the trilinear structure is decomposed by solving a generalized eigenvalue problem (GEP). Because GRAM is sensitive to noise it leads to rough estimations of reaction rate constants. The second algorithm (LM–PAR) is an iterative algorithm, which consists of a combination of the Levenberg–Marquardt algorithm and alternating least squares steps of the parallel factor analysis (PARAFAC) model using the GRAM results as initial values. Simulations and an application to a real dataset showed that both algorithms can be applied to estimate reaction rate constants in case of extreme spectral overlap of different species involved in the reacting system.

Demonstration of an efficient boundary layer parameterization for unbiased propagation estimation

Experimental results are presented indicating that only a few parameters describing capping inversion of the marine atmospheric boundary layer (MABL) need to be considered deterministically for accurate estimation of the MABL's effects on propagation loss, provided that (1) the right deterministic parameters are chosen and (2) other parameters are accounted for statistically. To perform this experiment, the refractivity structure associated with the capping inversion of the MABL is modeled using shifted empirical orthogonal functions (SEOFs) developed from the second‐order statistics of radiosondes. Propagation estimates calculated using the SEOFs as well as radiosondes and the heuristic trilinear model are compared with measured propagation. Using only two deterministic parameters, the results obtained using the SEOFs are essentially as good as can be obtained with a fading channel. Why this result is important in data assimilation for refractivity assessments is discussed.

Qualitative Assessment of Failure in Bolted Connections: Maximum Stress Criterion

Abstract This paper presents information regarding the distribution of critical stresses and regions where failure is likely to initiate in single bolt wood connections loaded in tension. Predicted stresses are the results of a three-dimensional numerical analysis of a connection consisting of a single steel pin and a wood member with a hole. Stresses of particular interest are: parallel-to-grain compression and shear and perpendicular-to-grain tension. Failure location is determined by considering the regions of the member where the material capacity is exceeded in at least one of the three aforementioned stresses. No stress interaction is assumed in this analysis. A recently developed and verified three-dimensional (3-D) finite element model was used in this study. Important features of this model include a trilinear constitutive model of the nonlinear behavior of wood and a mechanism to describe the contact between the pin and the hole with changing connection load. The connection geometries studied included a single ratio of end distance to pin diameter (e∕d=4) and three bolt aspect ratios (length of bolt in wood member/bolt diameter) (l∕d=2,5,and7). Stress contours are illustrated on the 3-D wood member surface, on the contact surface, and on planes of symmetry. Pin and wood deformations are also illustrated. The results show that for connections with an (l∕d)=2, stresses are nearly uniform through the wood member thickness with negligible pin yielding. For connections with the larger bolt aspect ratios, considerably more pin bending and yielding is predicted with nonuniform stress distributions through the member thickness. The maximum stress failure criterion is found to be useful for comparing numerically predicted stress distributions to failure modes observed experimentally. By making these comparisons, it is possible to make qualitative interpretations of the critical stress states to indicate the regions where failure is likely to occur.

Passive Structural Control Strategies Using Seismic Steel Energy Dissipators

The force versus deformation relationships of a steel frame constructed with the proposed hysteretic energy dissipators (HED) are characterized by a tri-linear model indicating the distinct yielding of the HED and the frame. The optimal stiffness and strength ratios between the HED and the frame are recognized from parametric analysis of the nonlinear response spectra. A design algorithm is presented to illustrate the strategies incorporated into the design of frame buildings for seismic response control, particularly on the reduction of the inelastic deformational demands imposed on the beam-to-column connections of a six-story example steel moment frame.

Partitioning of chromium and molybdenum in super duplex stainless steels with respect to nitrogen and nickel content

Experimental data of elemental partitioning of Cr, Mo, and Ni in a 5% Mn containing SAF 2507 super duplex stainless steel (SDSS) with varying nitrogen and nickel mass fractions are presented. Experimental results on phase equilibria and alloying element partitioning are compared to values calculated using Thermo-Calc software revealing good agreement. A trilinear model is proposed to describe the partitioning ratio of Cr, Mo and Ni with nitrogen and nickel mass percent in the austenite and the annealing temperature as variables. While nitrogen is found to reduce the partitioning of chromium and molybdenum, nickel enhances the partitioning ratio of these to alloying elements. Based on this semi-empirical model, a guideline for the future development of improved SDSS is formulated. With respect to corrosion resistance, a higher nitrogen level is not beneficial by itself but needs to be paralleled by increasing molybdenum and decreasing chromium contents.

Trilinear PARAFAC decomposition of synchronous fluorescence spectra of mixtures of the major metabolites of acetylsalicylic acid.

Mixtures of the three major metabolites of acetylsalicylic acid (salicylic, gentisic and salicyluric acid) were analyzed by synchronous molecular fluorescence spectroscopy. Three-way data matrices were generated by acquisition of spectra as a function of the pH (between 2 and 11) and of different relative concentrations of the three components. The PARAFAC trilinear model, without restrictions and using one factor per metabolite, was used in the data analysis. A full decomposition of the data matrices into the spectra, concentration and pH profiles was obtained. This result shows that molecular fluorescence spectroscopy can be used for the development of robust analytical methods for the simultaneous determination of the three major metabolites of acetylsalicylic acid in complex background samples.

Seismic behaviour of confined masonry walls

The results of tests of plain and confined masonry walls with h/l ratio equal to 1·5, made at 1:5 scale, have been used to develop a rational method for modelling the seismic behaviour of confined masonry walls. A trilinear model of lateral resistance–displacement envelope curve has been proposed, where the resistance is calculated as a combination of the shear resistance of the plain masonry wall panel and dowel effect of the tie-columns’ reinforcement. Lateral stiffness, however, is modelled as a function of the initial effective stiffness and damage, occurring to the panel at characteristic limit states. Good correlation between the predicted and experimental envelopes has been obtained in the particular case studied. The method has been also verified for the case of prototype confined masonry walls with h/l ratio equal to 1·0. Good correlation between the predicted and experimental values of lateral resistance indicates the general validity of the proposed method. © 1997 John Wiley & Sons, Ltd.

Comments on multilinear PLS

Recently, Bro published a paper on multilinear PLS (J. Chemometrics, 10, 47–61 (1996)) in which he proposed a generalization of PLS to multiway situations, called multilinear PLS, which is a mixture of a trilinear model (PARAFAC) and PLS. However, Bro does not give the equations for the prediction step. In this paper these prediction equations are given in both their full and closed forms. The least squares properties of the proposed multilinear PLS are established and a more comprehensive notation is given. Using this notation, it is clear that some other multiway analysis methods such as PARAFAC and Tucker1 models can be combined with PLS. Multiway methods such as Tucker2 and Tucker3 need a different approach. A framework is given for general two‐block multiway models.

Application of identification methodology to shaking table tests on reinforced concrete columns

Shaking table tests on twelve reinforced concrete model columns are described. The system identification method is used to process the test data and to find the restoring force for the scaled columns at different deformation stages. The analysis shows that a semidegrading trilinear model can be used for the cracking and yielding stages of the reinforced concrete columns, and a semidegrading trilinear model with a descending branch can be used for the failure stage.

Patterns of deterioration in senile dementia of the Alzheimer type.

To determine rates of decline in Alzheimer's disease. A longitudinal review of patients diagnosed as having dementia during life, tested serially with the Extended Scale for Dementia, and confirmed by autopsy as having Alzheimer's disease. Twenty-nine dead patients with Alzheimer's disease from the participants in the University of Western Ontario Dementia Study Project, confirmed at autopsy as having Alzheimer's disease. Analysis of the Extended Scale for Dementia data according to a trilinear model. In the middle phase of the trilinear model, there was a mean annual change of 13% (range, 2.5% to 51.7%). It is likely that the common method of averaging a group of different individual scores from the initial and middle phases of observation of Alzheimer's disease collapses together individuals at different stages of the disorder, some of whom are in the initial plateau phase and whose conditions are not declining rapidly. The trilinear model of decline avoids this difficulty and the present study provides postmortem confirmed figures on rate of change.

Identification of Fast and Slow Decliners in Alzheimer Disease

As the body of knowledge about Alzheimer disease (AD) expands, more evidence is appearing to suggest the existence of different subtypes. The delineation of subtypes is important for a variety of reasons, and there are probably several ways to identify subtypes. One crude but functionally promising distinction is between patients with early onset of AD and those with late onset. In theory, such a distinction appears to be a simple one, but critical methodologic issues are involved in this dichotomy, because a key aspect is estimating the rate of progression of the disease in different subtypes. This article discusses potential problems with various measures of deterioration and illustrates how the application of a novel technique--the trilinear model--can address many of these problems and provide additional valuable information. The trilinear model is especially well suited for use in pharmacologic studies because of the additional insight that it can provide.

PARAFAC: Parallel factor analysis

We review the method of Parallel Factor Analysis, which simultaneously fits multiple two-way arrays or ‘slices’ of a three-way array in terms of a common set of factors with differing relative weights in each ‘slice’. Mathematically, it is a straightforward generalization of the bilinear model of factor (or component) analysis ( x ij = Σ R r = 1 a irb jr ) to a trilinear model ( x ijk = Σ R r = 1 a irb jrc kr ). Despite this simplicity, it has an important property not possessed by the two-way model: if the latent factors show adequately distinct patterns of three-way variation, the model is fully identified; the orientation of factors is uniquely determined by minimizing residual error, eliminating the need for a separate ‘rotation’ phase of analysis. The model can be used several ways. It can be directly fit to a three-way array of observations with (possibly incomplete) factorial structure, or it can be indirectly fit to the original observations by fitting a set of covariance matrices computed from the observations, with each matrix corresponding to a two-way subset of the data. Even more generally, one can simultaneously analyze covariance matrices computed from different samples, perhaps corresponding to different treatment groups, different kinds of cases, data from different studies, etc. To demonstrate the method we analyze data from an experiment on right vs. left cerebral hemispheric control of the hands during various tasks. The factors found appear to correspond to the causal influences manipulated in the experiment, revealing their patterns of influence in all three ways of the data. Several generalizations of the parallel factor analysis model are currently under development, including ones that combine parallel factors with Tucker-like factor ‘interactions’. Of key importance is the need to increase the method's robustness against nonstationary factor structures and qualitative (nonproportional) factor change.

The Methodology of Studying Decline in Alzheimer's Disease

To present a new model for analyzing longitudinal data. The trilinear model is superior to the commonly used linear model that includes the flawed assumption that decline is uniform throughout the course of disease--an assumption that does not correspond to clinical observations. A longitudinal cohort sample was used to compare the linear and trilinear models. Simulated longitudinal data were generated to assess classification errors with the trilinear model. The subjects were 80 patients with Alzheimer's disease tested in a hospital out-patient clinic. The trilinear model describes Alzheimer's disease as proceeding through three periods: An initial period of stability before detectable decline, a period of decline, and a final period of stability during which there is no further detectable decline. A program for the Apple Macintosh computer is available at no charge to apply the model to data. The analyses indicated that the trilinear model provides a better reflection of decline in Alzheimer's disease than does the linear model. This advantage is present whether the periods of stability reflect a "true" lack of decline or insensitivity of a measurement instrument. The trilinear model provides not only a more accurate estimate of the average rate of change, but also (when possible) estimates of the point at which decline begins and ends. Also, more detailed comparisons of tests could be made by using the trilinear parameters. The trilinear model would benefit researchers engaged in longitudinal research of progressive disorders.

Eliminating complex eigenvectors and eigenvalues in multiway analyses using the direct trilinear decomposition method

AbstractThe direct trilinear decomposition method (DTDM) is an algorithm for performing quantitative curve resolution of three‐dimensional data that follow the so‐called trilinear model, e.g. chromatography–spectroscopy or emission‐excitation fluorescence. Under certain coditions complex eigenvalues and eigenvectors emerge when the generalized eigenproblem is solved in DTDM. Previous publications never treated those cases. In this paper we show how similarity transformations can be used to eliminate the imaginary part of the complex eigenvalues and eigenvectors, thereby increasing the usefulness of DTDM in practical applications. The similarity transformation technique was first used by our laboratory to solve the similar problem in the generalized rank annihilation method (GRAM). Because unique elution profiles and spectra can be derived by using data matrices from three or more samples simultaneously, DTDM with similarity transformations is more efficient than GRAM in the case where there are many samples to be investigated.

Resolution of the properties of hydrogen-bonded tyrosine using a trilinear model of fluorescence

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTResolution of the properties of hydrogen-bonded tyrosine using a trilinear model of fluorescenceJin Keun Lee, Robert T. Ross, Satheesh Thampi, and Sue LeurgansCite this: J. Phys. Chem. 1992, 96, 23, 9158–9162Publication Date (Print):November 1, 1992Publication History Published online1 May 2002Published inissue 1 November 1992https://doi.org/10.1021/j100202a016RIGHTS & PERMISSIONSArticle Views87Altmetric-Citations13LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (620 KB) Get e-Alerts

Resolution of the fluorescence spectra of plant pigment-complexes using trilinear models

The intensity of fluorescence from a pigment is separately linear in functions of excitation wavelength, emission wavelength, and any chemical treatment which alters overall fluorescence yield. This multiple linearity permits the use of an extension of principal components analysis to resolve overlapping spectra without the use of any additional information. The method is used to resolve the spectra of pigment complexes in pea thylakoids, using the concentration of Mg 2+ as the chemical treatment variable. Two components could be resolved accurately. One has little or no dependence on Mg 2+ ; the other, with an ecitation spectrum resembling LHC II, has a dependence on Mg 2+ which follows the Hill equation with a binding constant of 0.4–0.6 mM and a Hill coefficient of 2.4–3.1.

Differential memory in the trilinear model magnetotail

The previously proposed concept of “differential memory” is quantitatively demonstrated using an idealized analytical model of particle dynamics in the magnetotail geometry. In this model (the “trilinear” tail model), the magnetotail is divided into three regions. The particle orbits are solved exactly in each region, thus reducing the orbit integration to an analytical mapping. It is shown that the trilinear model reproduces the essential phase space features of the earlier model (Chen and Palmadesso, 1986), possessing well‐defined entry and exit regions, and stochastic, integrable (regular), and transient orbits, occupying disjoint phase space regions. Different regions have widely separated characteristic time scales corresponding to different types of particle motion. Using the analytical model, the evolution of single‐particle distribution functions is calculated.

STUDY ON OPTIMUM YIELD SEISMIC COEFFICIENT FOR DESIGN OF REINFORCED CONCRETE STRUCTURES BASED ON PROBABILITY THEORY

The purpose of this study is to determine an optimum yield seismic coefficient based on probability theory. RC structures are represented by a single-degree-of-freedom system with a degrading tri-linear model. Yield seismic coefficient and yield period are chosen as structural parameters, which are treated as deterministic. Frequency and magnitude of earthquake, attenuation equation and reflection of local site condition are considered to involve uncertainties.Ductility factor is taken as a measure for damage of structures. Dynamic analysis is performed on each set of structural parameters to obtain ductility factor. Then optimum yield seismic coefficient is determined so as to minimize expected total cost.

Tensorial resolution: A direct trilinear decomposition

AbstractModern instrumentation in chemistry routinely generates two‐dimensional (second‐order) arrays of data. Considering that most analyses need to compare several samples, the analyst ends up with a three‐dimensional (third‐order) array which is difficult to visualize or interpret with the conventional statistical tools.Some of these data arrays follow the so‐called trilinear model, These trilinear arrays of data are known to have unique factor analysis decompositions which correspond to the true physical factors that form the data, i.e. given the array ℝ, a unique solution can be found in many cases for each order X, Y and Z. This is in contrast to the well‐known second‐order bilinear data factor analysis, where the abstract solutions obtained are not unique and at best cannot be easily compared with the underlying physical factors owing to a rotational ambiguity.Trilinear decompositions have had the disadvantage, however, that a non‐linear optimization with many parameters is necessary to reach a least‐squares solution. This paper will introduce a method for reducing the problem to a rectangular generalized eigenvalue–eigenvector equation where the eigenvectors are the contravariant form (pseudo‐inverse) of the actual factors. It is shown that the method works well when the factors are linearly independent in at least two orders (e.g. Xir and Yjr are full rank matrices).Finally, it is shown how trilinear decompositions relate to multicomponent calibration, curve resolution and chemical analysis.

Trilinear Moment-Curvature Relationship for Reinforced Concrete Beams

A trilinear moment-curvature curve is constructed by depicting each of uncracked, cracked, and yielded behavior of reinforced concrete beams as linear segments. This curve and Branson's well-known formula for effective stiffness are qualitatively related to each other, and as a result the difference between predictions by the two models is inherently bounded. Moment-curvature and load-deflection curves derived by Branson's formula and a recent nonlinear model by Bazant and Oh have been compared with those produced by the trilinear model for several example beams. Good agreement is observed among these models. The main advantage of the trilinear model is in computer applications related to analysis and design of reinforced concrete beams and frames.