Functionality multiple regression modeling

A table of correlations between the variables from the instrumental set and variables from the sensory set may reveal some strong one-to-one relations. However, with a battery of sensory attributes on the one hand and a set of instrumental variables on the other hand it is better to adopt a multivariate approach, i.e. to look at many variables at the same time taking their intercorrelations into account. An intermediate approach is to develop separate multiple regression models for each sensory attribute as a linear function of the physical/chemical predictor variables. 

McWatters and Holmes ( ) developed multiple regression models of the effects of pH and salt concentration on functional properties of soy flour. Design of the experiment and selection of the factors to be Included were based. In part, on earlier findings that emulsion capacity of defatted peanut meal was Inhibited around the Isoelectric point (ca pH 4.0)... 

The higher-level parameters in the multiple regression model enable quantification of how chemicals influence each other in relation to the measured response. Suppose that fil 2 (the estimated function parameter for the first-level interaction term between chemical 1 and chemical 2) in a regression model... 

Empirical multiple linear regression models were developed to describe the foam capacity and stability data of Figures 2 and 4 as a function of pH and suspension concentration (Tables III and IV). These statistical analyses and foaming procedures were modeled after data published earlier (23, 24, 29, 30, 31). The multiple values of 0.9601 and 0.9563 for foam capacity and stability, respectively, were very high, indicating that approximately 96% of the variability contributing to both of these functional properties of foam was accounted for by the seven variables used in the equation. 

Table III. Empirical multiple linear regression model describing foaming capacity as a function of pH and suspension concentration.

Use of multiple regression techniques in the study of functional properties of food proteins is not new I76) Most food scientists have some familiarity with basic statistical concepts and some access to competent statistical advice. At least one good basic text on statistical modelling for biological scientists exists (7 ). A number of more advanced texts covering use of regression in modelling are available (, ). ... 

Table 8.3 lists several examples of MCI-Koc regression models that have appeared in the literature. A more comprehensive compilation of MCI-Koc regression models appear in Gaw-lik et al. (1997). As with the property-Koc relationships previously described, if an appropriate class-specific relationship can be found, it is likely to provide the best estimate. However, if an appropriate class-specific relationship is not available or if the compound of interest has multiple functional groups that make it difficult to classify, one of the more general expressions should be used. 

The true model parameters (ft, ftj(...) are partial derivatives of the response function / and cannot be measured directly. It is, however, possible to otain estimates, ft, bV], bVl, of these parameters by multiple regression methods in which the polynomial model is fitted to known experimental results obtained by varying the settings of xr. These variations will then define an experimental design and are conveniently displayed as a design matrix, D, in which the rows describe the settings in the individual experiments and the columns describe the variations of the experimental variables over the series of experiments. 

Occasionally, in a multiple linear regression model using an empirical equation, two or more independent variables are correlated among themselves, that is, one independent variable is a linear function of another independent variable. Intercorrelated variables are said to exhibit multicollinearity. A nonchemical example of multicollinearity might exist in the relationship between... 

A linear mathematical function that relates descriptor values to a biological activity can be created using multiple linear regression. For n observations and p independent variables the general linear regression model is represented as... 

A DuPont Model 990 Thermal Analysis Console with Model 910 DSC accessory was interfaced to a minicomputer system by means of a microcomputer for automated data collection. A program to provide the analysis of reaction kinetics data by the single dynamic scan method for DSC kinetics was developed. Features of this program include a fit of the data to a single equation by multiple regression techniques to yield the reaction order, the energy of activation and the Arrhenius frequency factor. The rate constant k(T) is then calculated and conversion data as a function of time and temperature can be generated at the operator s option. 

It must be remembered that RSM uses multiple regression techniques to determine the coefficients for the Taylor expansion equation which best fits the data. The RSM does not determine the function which describes the data. The Taylor equation only approximates the true function. The RSM process fits one of a series of curves to the data. Most RSM programs use only the first and second order terms of the Taylor equation to the data, which limits the number of curves available to fit the data. The first order T aylor equation is a linear model. Therefore, the only curves available are a series of straight lines. The second order Taylor equation is a nonlinear model where two types of curves are available a peak or a saddle surface. Over a narrow range, these curves will approximate the true function that exists in nature but they are not necessarily the function that describes the response. 

Comillon et al. 2007) The nonlinear multiple regression consists in modeUing the N observational data by a function / which is a nonlinear combination of the model variables x, to predict an indicator. The polynomial regression can be written as ... 

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