Regression-type models

A more common use of informatics for data analysis is the development of (quantitative) structure-property relationships (QSPR) for the prediction of materials properties and thus ultimately the design of polymers. Quantitative structure-property relationships are multivariate statistical correlations between the property of a polymer and a number of variables, which are either physical properties themselves or descriptors, which hold information about a polymer in a more abstract way. The simplest QSPR models are usually linear regression-type models but complex neural networks and numerous other machine-learning techniques have also been used. 

Quantitative models for predicting quality can be classified into two categories (1) fundamental process models, which are based on physical and chemical events that occur in the autoclave, and (2) regression-type models, which are based on a statistical fit of the observed product quality to the input raw material properties and the process conditions used. 

Traditional regression-type models have been linear and quadratic regression models. Linear and quadratic regression models unfortunately impose further constraints upon the nature of the process nonlinearity as such, these models are limited in the range of their applicability. A relatively new nonlinear regression-type model—the Artificial Neural Network (ANN)—is not as limited, and is worthy of additional discussion. 

Continuous toxicity data can be generally described as using a regression-type model as depicted in Figure 5.14. This is a simple linear regression model using only one parameter to describe the toxicity. The resulting expression used to describe the relationship between toxicity and the parameter is a typical linear equation ... 

An alternative method for the construction of regression type models for multivariate response sets is a technique known as canonical correlation analysis (CCA). CCA operates by the construction of a linear combination of q responses... 

Note we are not advocating always using flat priors. We only want to illustrate that when we do, the posterior will be the same shape as the likelihood. Hence, the likelihood can be thought of as an unsealed posterior when we have used flat priors. When the integral of the flat prior over its whole range is infinite, the flat prior will be improper. Despite this, the resulting posterior which is the same shape as the likelihood will usually be proper. For many models, such as the regression-type models that we will discuss in Chapters 8 and 9, it is ok to use improper flat priors. 

FIGURE 5.2 Diagram of three different types of linear models with n standards. Left the simplest model has a slope and no intercept. The center model adds a nonzero intercept. The right model is typically noted in the literature as the multiple linear regression (MLR) model because it uses more than one response variable, and n>(m+ 1) with an intercept term and n> m without an intercept term. This model is shown with a nonzero intercept. 

The analysis of a supersaturated design is usually conducted by using some type of sequential model-fitting procedure, such as stepwise regression. Abraham et al. (1999) and Holcomb et al. (2003) have studied the performance of analysis methods for supersaturated designs. Techniques such as stepwise model fitting and all-possible-regressions type methods may not always produce consistent and reliable results. Holcomb et al. (2003) showed that the performance of an analysis technique in terms of its type I and type II error rate can depend on several factors,... 

In the study of Teuschler et al. (2000), the models for analyzing the data were selected beforehand, and it was also decided to only focus on environmentally relevant mixtures. The authors indicated that these 2 factors were decisive for choosing the concentration levels to test. The concentration levels were not selected in relation to a specific endpoint, using the toxic unit approach. This may have been avoided because several different hepatotoxic endpoints have been measured simultaneously. The concentrations tested enabled the use of 3 types of models a multiple regression CA model, the interaction-based HI, and the proportional-response addition method. A major problem with mixture toxicity research in general is the... 

The quality of a regression can also be assessed by visual inspection of plots. Of course, some plots are more sensitive than others to the level of agreement between model and experiment. As will be demonstrated in this chapter, the plot types can be categorized as given in Table 20.1. The comparison of plot types is presented in the subsequent sections for regression of models to a specific impedance data set. 

An important first step in any model-based calculation procedure is the analysis and type of data used. Here, the accuracy and reliability of the measured data sets to be used in regression of model parameters is a very important issue. It is clear that reliable parameters for any model cannot be obtained from low-quality or inconsistent data. However, for many published experimentally measured solid solubility data, information on measurement uncertainties or quality estimates are unavailable. Also, pure component temperature limits and the excess GE models typically used for nonideality in vapor-liquid equilibrium (VLE) may not be rehable for SEE (or solid solubility). To address this situation, an alternative set of consistency tests [3] have been developed, including a new approach for modehng dilute solution SEE, which combines solute infinite dilution activity coefficients in the hquid phase with a theoretically based term to account for the nonideality for dilute solutions relative to infinite dilution. This model has been found to give noticeably better descriptions of experimental data than traditional thermodynamic models (nonrandom two liquid (NRTE) [4], UNIQUAC [5], and original UNIversal Eunctional group Activity Coefficient (UNIEAC) [6]) for the studied systems. 

Deterministic, statistically regressive, stochastic models, and physical representations in water tanks and wind tunnels have been developed. Solutions to the deterministic models have been analytical and numerical, but the complexities of analytical solution are so great that only a few relatively simple cases have been solved. Numerical solutions of the more complex situations have been carried out but require a great amount of computer time. Progress appears to be most likely for the deterministic models. However, for the present the stochastically based Gaussian type model is the most useful in modeling for regulatory control of pollutants. 

Taking these findings into consideration for neat PP, a fit was attempted using nonlinear regression with model (4), where the nth-order (Fn) reaction type was used for all steps of the reaction (Fig. 7, Table 2). 

Taking these fittings for PP-MAPP-Cloisite 20A, a best approximation was attempted using nonlinear regression with model (5), based on the best fit quality (correlation coefficient) (Fig. 8), where the one-dimensional diffusion (DO reaction type was used for the second step of the reaction (Table 3), whereas the n -order reaction models were chosen for the first and third steps respectively. 

There are several major limits of the simple regression-type statistical analysis model. One is that PbB outcomes for a specific location depend on the site-specific conditions governing the eventual statistical relationship. A second limit arises from typical absence of any means for evaluating Pb exposure pathways from source(s) to exposed individuals, their relative importance to the outcome measure, and evaluation of any role for host factors affecting the resulting PbB values, e.g., child age or family socioeconomic status. 

Simple regression analysis tends to produce inferentials that are arithmetically simple and may therefore be readily built into the DCS using standard features. More complex regressed types, such as neural networks, will require a separate platform and probably some proprietary software. They can therefore be more costly. First-principle models may be provided in pseudo-code that the engineer can convert to code appropriate to the DCS. This with its testing and documentation can be time-consuming. Inferentials delivered as black boxes may require less implementation effort but can only be maintained by the supplier. 

Models of this type allow solving the simplest predictive tasks or making the basic description of simple linear systems. Usually, however, they are developed further in the direction of extending their application possibilities. In practice, the ARMA type model, due to the emergence of the measurement noise, inaccuracies, etc., is usually extended to a simple stochastic model of the ARX type (Auto Regressive with auXUiary input) it is extended by an additional signal, i.e. white noise. In this model, as input, we can also use the values of other (known) time series, data in the current time moments or historical data. 

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