Analysis of latent structures in linear models

2014

The most popular models for the analysis of multi-way data structures, i.e., the Tucker and CANDECOMP-PARAFAC (CP) models, were developed in the domain of numerical psychology. However, data collected by many modern analytical chemical instruments are multi-way data structures per definition, and many applications have shown that data analysis gains in robustness and information outcome by deriving meaningful parameters from appropriately chosen multi-linear models of such measurements. The current presentation covers the two basic models of widest general interest and two applications from the sugar industry. Both applications are based on fluorescence measurements of aqueous samples. In the first application, we motivate that the parameters estimated by the three-way CP model are estimates of the pure underlying spectral profiles of the observed chemical species. Hence, the CP model is applied in a curve-resolution approach and the estimated parameters allow us to uniquely identif...

2020

The purpose of data analysis in chemistry is to obtain information from the experimental data. Modern data acquisition systems provide a lot of data. In the preceding volume we have seen that data reduction and modeling may proceed through determination of the means or applications of linear, nonlinear, or multiple regressions. However, these methods often use a limited amount of acquired data.

Chemometrics and Intelligent Laboratory Systems, 1992

A new approach to nonlinear empirical modelling is proposed. Let X and Y be two data matrices, centered and optionally scaled. The aim is to find a functional relationship Y = F(X, parameters). In this approach the function F is determined by minv,e,b[ II u -i II /II u II12uur2(a-1) with constraints II w )I = II c II = 1, where t = Xw, u = Yc, and b are the parameters of the nonlinear function between t and u, i.e. u =f(t, b). Here a,* is the variance of t. The parameter a defines a continuum of models and gives a weight between maximizing the fit in the inner relationship and the variance of the latent variable score t. Y is now calculated as in linear partial least squares (PLS). This procedure is repeated up to a chosen number of dimensions with residual matrices X + X -tpT and Y + Y -;iT. The vector li is the fitted value of u and 4 is constructed from li as in linear PLS. The function f will be called the link function and it can be any parametric function between the two latent variables. This will be referred to as nonlinear (NL) PLS and its performance is illustrated by four examples using quadratic and logistic link functions: one synthetic, one from product development, one from near infrared spectroscopy and one from molecular pharmacology.

Journal of Chemometrics, 2008

In spectroscopy the measured spectra are typically plotted as a function of the wavelength (or wavenumber), but analysed with multivariate data analysis techniques (multiple linear regression (MLR), principal components regression (PCR), partial least squares (PLS)) which consider the spectrum as a set of m different variables. From a physical point of view it could be more informative to describe the spectrum as a function rather than as a set of points, hereby taking into account the physical background of the spectrum, being a sum of absorption peaks for the different chemical components, where the absorbance at two wavelengths close to each other is highly correlated. In a first part of this contribution, a motivating example for this functional approach is given. In a second part, the potential of functional data analysis is discussed in the field of chemometrics and compared to the ubiquitous PLS regression technique using two practical data sets. It is shown that for spectral data, the use of B-splines proves to be an appealing basis to accurately describe the data. By applying both functional data analysis and PLS on the data sets the predictive ability of functional data analysis is found to be comparable to that of PLS. Moreover, many chemometric datasets have some specific structure (e.g. replicate measurements, on the same object or objects that are grouped), but the structure is often removed before analysis (e.g. by averaging the replicates). In the second part of this contribution, we suggest a method to adapt traditional analysis of variance (ANOVA) methods to datasets with spectroscopic data. In particular, the possibilities to explore and interpret sources of variation, such as variations in sample and ambient temperature, are examined.

2016

The current study looks at several ways to investigate latent variables in longitudinal surveys and their use in regression models. Three different analyses for latent variable discovery will be briefly reviewed and explored. The latent analysis procedures explored in this paper are PROC LCA, PROC LTA, PROC TRAJ, and PROC CALIS. The latent variables will then be included in separate regression models. The effect of the latent variables on the fit and use of the regression model compared to a similar model using observed data will be briefly reviewed. The data used for this study was obtained via the National Longitudinal Study of Adolescent Health, a study distributed and collected by Add Health. Data was analyzed using SAS® 9.4. This paper is intended for any level of SAS user. This paper is also written to an audience with a background in behavioral science and/or statistics. Introduction to Latent Variables When being introduced to statistics, students are usually presented with ...

Computational and Mathematical Methods in Medicine, 2009

Linear latent structure analysis is a new approach for investigation of population heterogeneity using high-dimensional categorical data. In this approach, the population is represented by a distribution of latent vectors, which play the role of heterogeneity variables, and individual characteristics are represented by the expectation of this vector conditional on individual response patterns. Results of the computer experiments demonstrating a good quality of reconstruction of model parameters are described. The heterogeneity distribution estimated from 1999 National Long Term Care Survey (NLTCS) is discussed. A predictive power of the heterogeneity scores on mortality is analysed using vital statistics data linked to NLTCS.

Chemometrics and intelligent laboratory …, 2001

Two examples are used as illustrations: First, a Quantitative Structure–Activity Relationship (QSAR)/Quantitative Structure–Property Relationship (QSPR) data set of peptides is used to outline how to develop, interpret and refine a PLSR model. Second, a data set from the ...

A practical consultation problem is used to explain and illustrate the model of latent structure analysis. After an introduction, the model, its identification, estimation and testing are discussed; in a final section the hypothesis that visually handicapped people can be classified as "plucky" or "not plucky" is investigated.

Journal of Chemometrics, 2012

In this work, we present a new approach to path modeling based on an extended multiple covariance criterion: system extended multiple covariance (SEMC). SEMC is suitable to measure the quality of any structural equations system. We show why SEMC may be preferred to criteria based on usual covariance of components and also to criteria based on residual sums of squares. We give a pursuit algorithm ensuring that SEMC increases and converges. When one wishes to extract more than one component per variable group, a problem arises of component hierarchy. To solve it, we define a local nesting principle of component models that makes the role of each component statistically clear. We then embed the pursuit algorithm in a more general algorithm that extracts sequences of locally nested models. We finally provide a component backward selection strategy. The technique is applied to cigarette data to model the generation of chemical compounds in smoke through tobacco combustion. Copyright

Frontiers in analytical science, 2024

This perspective article reviews how the chemometrics community approached non-linear methods in the early years. In addition to the basic chemometric methods, some methods that fall under the term "machine learning" are also mentioned. Thereafter, types of non-linearity are briefly presented, followed by discussions on important aspects of modeling related to non-linear data. Lastly, a simulated data set with non-linear properties is analyzed for quantitative prediction and batch monitoring. The conclusion is that the latent variable methods to a large extent handle non-linearities by adding more linear combinations of the original variables. Nevertheless, with strong nonlinearities between the X and Y space, non-linear methods such as Support Vector Machines might improve prediction performance at the cost of interpretability into both the sample and variable space. Applying multiple local models can improve performance compared to a single global model, of both linear and non-linear nature. When non-linear methods are applied, the need for conservative model validation is even more important. Another approach is pre-processing of the data which can make the data more linear before the actual modeling and prediction phase.