Regression analysis error models

An experimental approach for the prediction of thrust force produced by a step drill using linear regression analysis and RBFN has been proposed. In the confirmation tests, RBFN (errors within 0.3 per cent) has been shown to be a better predictive model than multi-variable linear regression analysis (errors within 28 per cent) (Tsao, 2008). 

A linear regression analysis should not be accepted without evaluating the validity of the model on which the calculations were based. Perhaps the simplest way to evaluate a regression analysis is to calculate and plot the residual error for each value of x. The residual error for a single calibration standard, r , is given as... 

The first two examples show that the interaction of the model parameters and database parameters can lead to inaccurate estimates of the model parameters. Any use of the model outside the operating conditions (temperature, pressures, compositions, etc.) upon which the estimates are based will lead to errors in the extrapolation. These model parameters are effec tively no more than adjustable parameters such as those obtained in linear regression analysis. More comphcated models mav have more subtle interactions. Despite the parameter ties to theoiy, tliey embody not only the uncertainties in the plant data but also the uncertainties in the database. 

In a well-behaved calibration model, residuals will have a Normal (i.e., Gaussian) distribution. In fact, as we have previously discussed, least-squares regression analysis is also a Maximum Likelihood method, but only when the errors are Normally distributed. If the data does not follow the straight line model, then there will be an excessive number of residuals with too-large values, and the residuals will then not follow the Normal distribution. It follows, then, that a test for Normality of residuals will also detect nonlinearity. 

The model that utilized regression analysis was one that built upon previous work by the same authors [36,39]. In this case, the dataset was expanded to 125-129 drugs and the number of assessed descriptors increased to 210. Models for acidic and basic compounds were developed separately as well as a model using all compounds, and the advantages of analyzing acids and bases separately were minimal. Mean-fold errors were generally around 1.8. Descriptors that dominated the models included lipophilicity, fraction anionic or cationic, surface electrostatic potential, and parameters specific to aliphatic carbons and fluorine. 

The choice of electrical effect parameterization depends on the number of data points in the data set to be modeled. When using linear regression analysis the number of degrees of freedom, Ndf, is equal to the number of data points, Ndp, minus the number of independent variables, Ai,v, minus one. When modeling physicochemical data Ndf/Nj, should be at least 2 and preferably 3 or more. As the experimental error in the data increases, tVof/iViv should also increase. 

Intuitively, we assume that individual data points are generated by a normal distribution. A common variance is assumed for model error as in a conventional regression analysis. 

Regression analysis assumes that all error components are independent, have a mean of zero and have the same variance throughout the range of POM values. Through an examination of residuals, serious violations in these assumptions can usually be detected. The standardized residuals for each of the fitted models were plotted against the sequence of cases in the file and this scatterplot was examined visually for any abnormalities (10, 14). 

To evaluate each rate expression appearing in Table 3, the values of the a, c, and n were varied to obtain a minimum in the square of the relative error between the theoretically predicted and experimentally measured deposition rates. The results of this nonlinear regression analysis are presented in Table 4. It can be seen that only for Models 3 and 4 is d smaller than the errors in the measurements, +9.5% . 

Figure 3. Relationship between leaf area (A), epidermal cell density (B), stomatal density (C) and stomatal index (D) versus altitude for Nothofagus solandri leaves growing on the slope of Mt. Ruapehu, New Zealand (collected in 1999). Black diamonds indicate the mean of ten counting fields on each leaf, white squares are the averages of five to eight leaves per elevation, with error bars of 1 S.E.M. Nested mixed-model ANOVA with a general linear model indicates significant differences for all factors (p = 0.000). Averages per elevation were used for regression analysis A. y = -0.0212 + 73.1 R2 = 0.276 p = 0.147. B. y = 1.70 + 3122 R2 = 0.505 p = 0.048. C. y = 0.164 + 360 R2 = 0.709 p = 0.009. D. linear (dashed) y = 0.004 + 9.33 R2 = 0.540 p = 0.038 non-linear (solid) y = 0.00001 2 - 0.0206 + 21.132 R2 = 0.770.

It must be emphasized that the potential of multiple regression analysis to resolve sources of pharmacokinetic variations is much greater than has been realized by the particular canned1 model used previously. The technique itself is both sensitive and powerful. However, for multiple regression analysis to be used appropriately, a model must be developed that encompasses non-linear as well as linear relationships (JJ4). Error terms especially need to be appropriately modelled, rather than treated in a simply additive manner as in previous applications of this method. 

In this section we consider the statistical techniques, correlation and regression analysis, to study the interrelationship between two continuous random variables (Xi,X2), from the information supplied by a sample of n pairs of observations (xi.i, Xi,2), (X2,i, X2,2), , (x ,i, x ,2), from a population W. In the correlation analysis we accept that the sample has been obtained of random form, and in the regression analysis (linear or not linear) we accept that the values of one of the variables are not subject to error (independent variable X = Xi), and the dependent variable (X = X2) is related to the independent variable by means of a mathematical model (X = f(X) + s). 

Both the Precision and Tolerance options apply only to problems with constraints. The Precision parameter determines the amount by which a constraint can be violated. The Tolerance parameter is similar to the Precision parameter, but applies only to problems with integer solutions. Since adding constraints to a model that involves minimization of the error-square sum is not recommended, neither the Precision nor the Tolerance parameter is of use in non-linear regression analysis. 

For all mathematical models that are not naturally straight lines, non-linear regression analysis is often the best approach. The observed data and the corresponding dependent variable can be analyzed without transformation. Thus, the data and the error or variance are not distorted during the analysis. If necessary, clearly defined weighting schemes can be applied. Furthermore, multiple observation sets can be readily accommodated. 

In addition, the variance of impedance measurements depends strongly on frequency, and this variation needs to be addressed by the regression strategies employed. An assumed dependence of the variance of the impedance measurement on impedance values was employed in early stages of regression analysis, and this gave rise to some controversy over what assumed error structure was most appropriate. An experimental approach using measurement models, described in Chapter 21, was later developed, which eliminated the need for assumed error structures. 

A distinction is drawn in equation (21.1) between stochastic errors that are randomly distributed about a mean value of zero, errors caused by the lack of fit of a model, and experimental bias errors that are propagated through the model. The problem of interpretation of impedance data is therefore defined to consist of two parts one of identification of experimental errors, which includes assessment of consistency with the Kramers-Kronig relations (see Chapter 22), and one of fitting (see Chapter 19), which entails model identification, selection of weighting strategies, and examination of residual errors. The error analysis provides information that can be incorporated into regression of process models. The experimental bias errors, as referred to here, may be caused by nonstationary processes or by instrumental artifacts. 

Figure 23.1 Schematic flowchart showing the relationship between impedance measurements, error analysis, supporting observations, model development, and weighted regression analysis. (Taken from Orazem and Tribollet and reproduced with permission of Elsevier, Inc.)...

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