Mathematics linear regression

The following paper discusses some of the problems that maybe encountered when using linear regression to model data that have been mathematically transformed into a linear form. 

Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then linear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treatment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for... 

Data given in Tables 1-6 clearly show a significant dependence of P2 and p4 on amine concentration, that is, at least one of the apparent rate constants kj contains a concentration factor. Thus, according to the mathematical considerations outlined in the Analysis of Data Paragraph, both p2, P4 exponents and the derived variables -(P2 + p)4> P2 P4 ind Z (see Eqns. 8-12) are the combinations of the apparent rate constants (kj). To characterize these dependences, derived variables -(p2+p)4, P2 P4 and Z (Eqns. 8,11 and 12) were correlated with the amine concentration using a non-linear regression program to find the best fit. Computation resulted in a linear dependence for -(p2 + p)4 and Z, that is... 

Thus, Tis a linear function of the new independent variables, X, X2,. Linear regression analysis is used to ht linear models to experimental data. The case of three independent variables will be used for illustrative purposes, although there can be any number of independent variables provided the model remains linear. The dependent variable Y can be directly measured or it can be a mathematical transformation of a directly measured variable. If transformed variables are used, the htting procedure minimizes the sum-of-squares for the differences... 

The formulation of the parameter estimation problem is equally important to the actual solution of the problem (i.e., the determination of the unknown parameters). In the formulation of the parameter estimation problem we must answer two questions (a) what type of mathematical model do we have and (b) what type of objective function should we minimize In this chapter we address both these questions. Although the primary focus of this book is the treatment of mathematical models that are nonlinear with respect to the parameters nonlinear regression) consideration to linear models linear regression) will also be given. 

Multiple linear regression (MLR) is a classic mathematical multivariate regression analysis technique [39] that has been applied to quantitative structure-property relationship (QSPR) modeling. However, when using MLR there are some aspects, with respect to statistical issues, that the researcher must be aware of ... 

The linearity of a method is defined as its ability to provide measurement results that are directly proportional to the concentration of the analyte, or are directly proportional after some type of mathematical transformation. Linearity is usually documented as the ordinary least squares (OLS) curve, or simply as the linear regression curve, of the measured instrumental responses (either peak area or height) as a function of increasing analyte concentration [22, 23], The use of peak areas is preferred as compared to the use of peak heights for making the calibration curve [24],... 

Predominantly, Freundlich s fitted adsorption isotherms computed by means of simple linear regression were proposed for the mathematical description of the process studied. Unlike the Langmuir equation, the Freundlich model did not reduce to a linear adsorption expression at very low nor very high solute concentrations, as above resulted. 

Since most quantitative applications are on mixtures of materials, complex mathematical treatments have been developed. The most common programs are Multiple Linear Regression (MLR), Partial Least Squares (PLS), and Principal Component Analyses (PCA). While these are described in detail in another chapter, they will be described briefly here. 

Using multivariable linear regression, a set of equations can be derived from the parameterized data. Statistical analysis yields the "best equations to fit the en irical data. This mathematical model forms a basis to correlate the biologicsd activity to the chemical structures. 

Response or effect surface analysis (RSA) uses multiple linear regressions to produce a statistically based mathematical relationship between the doses of each of the chemicals in a mixmre and the effect parameter. The equation for a mixmre containing three compounds would be... 

Regression generally means the fitting of mathematical equations to experimental data ( 3). Nonlinear regression, unlike linear regression, encompasses methods vdiich are not limited to fitting equations linear in the coefficients (e.g. simple polynomial forms). 

Figure 6. Experimentally observed and mathematically simulated regression lines of foam capacity of different percentages of glandless cottonseed flour in suspensions at various pH values. Experimental 4%, 10%, and 16% suspensions were run at pH 3.5, 6.5, and 9.5 to test the reliability of the multiple linear regression analysis. Quantitative data used in this analysis are in Figures 2 and 4.

The linearity of an analytical method is its ability to elicit test results that are directly, or by means of well-defined mathematical transformation, proportional to the concentration of analytes in samples within a given range. Linearity is determined by a series of three to six injections of five or more standards whose concentrations span 80-120% of the expected concentration range. The response should be—directly or by means of a well-defined mathematical calculation—proportional to the concentrations of the analytes. A linear regression equation applied to the results should have an intercept not significantly differ-... 

Spath, 1991] Spath, H. (1991). Mathematical Algorithms for Linear Regression. Academic Press. 

Movement to optimum by an inadequate linear model is also possible in cases when doing the mentioned eight trials is not acceptable. The values of linear regression coefficients are considerably above the values of those for interactions, the more so since linear effects are not aliased/confounded with interaction effects. Although the movement to optimum by an inadequate linear model is mathematically incorrect, it may be accepted in practice with an adequate risk. Note that when trying to optimize a process one should aspire towards both the smallest possible interaction effects and approximate or symmetrical linear coefficients. In problems of interpolation models, the situation is exactly the opposite since it insists on interaction effects, which may be significant. 

The property of the method of steepest ascent lies in the fact that movement along the gradient of a function must be preceded by a local description of the response surface by means of full or fractional factorial experiments [49]. It has been demonstrated that by processing FUFE or FRFE experimental outcomes we may obtain a mathematical model of a research subject in the form of a linear regression ... 

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