Regression uncertainty, prediction

Uncertainty in the linear regression is estimated by determining the standard errors of adjustable parameters and predictions from the linear model. With a reliable estimate of the variance of the response variable, o, the known value is used for uncertainty predictions. Otherwise, the variance is estimated in terms of the sum of square residuals. The residual at each measurement is defined as... 

The expression x (J)P(j - l)x(j) in eq. (41.4) represents the variance of the predictions, y(j), at the value x(j) of the independent variable, given the uncertainty in the regression parameters P(/). This expression is equivalent to eq. (10.9) for ordinary least squares regression. The term r(j) is the variance of the experimental error in the response y(J). How to select the value of r(j) and its influence on the final result are discussed later. The expression between parentheses is a scalar. Therefore, the recursive least squares method does not require the inversion of a matrix. When inspecting eqs. (41.3) and (41.4), we can see that the variance-covariance matrix only depends on the design of the experiments given by x and on the variance of the experimental error given by r, which is in accordance with the ordinary least-squares procedure. 

A valuable inference that can be made to infer the quality of the model predictions is the (l-a)I00% confidence interval of the predicted mean response at x0. It should be noted that the predicted mean response of the linear regression model at x0 is y0 = F(x0)k or simply y0 = X0k. Although the error term e0 is not included, there is some uncertainty in the predicted mean response due to the uncertainty in k. Under the usual assumptions of normality and independence, the covariance matrix of the predicted mean response is given by... 

Mathematical model that describes the relationship between random variables (usually x and y) by means of regression coefficients and their uncertainties as well as uncertainties of model and the prediction. 

Discriminant Analysis (DA) is a multivariate statistical method that generates a set of classification functions that can be used to predict into which of two or more categories an observation is most likely to fall, based on a certain combination of input variables. DA may be more effective than regression for relating groundwater age to major ion hydrochemistry and well construction because it can account for complex, non-continuous relationships between age and each individual variable used in the algorithm while inherently coping with uncertainty in the age values used for... 

In Hitchell s work unequal variance of the response data was compensated for by weighting the data by the variance at each level. The regression parameters and the confidence band around the regression line were estimated by least squares ( ) The overall level of uncertainty, OL, was divided between the variation in response values and the variance in the regression estimation. His overall a was 0.05. The prediction interval was estimated around a single response determination. 

Statistical Prediction Errors (Model and Sample Diag Jostic) Uncertainties in the concentrations can be estimated because the predicted concentrations are regression coefficients from a linear regression (see Equations 5.7-5.10). These are referred to as statistical prediction errors to distinguish them from simple concentration residuals (c — c). Tlie statistical prediction errors are calculated for one prediction sample as... 

Multiple linear regression (MLR), although a popular technique, does not meet the requirements of the experimental design describe above. MLR can only deal with one dependent variable at a time and assumes that all variables are orthogonal (uncorrelated), and tiiat they are all completely relevant to the experiment, en dealing with an experimental system for the first time, it is not always possible to predict which variables will be relevant to the experiment, and which will not. So a technique is needed that can reconcile such uncertainties. 

J. A. Fernandez Pierna, L. Jin, F. Wahl, N. M. Faber and D. L. Massart, Estimation of partial least squares regression prediction uncertainty when the reference values carry a sizeable measurement error, Chemom. Intell. Lab. Syst., 65, 2003, 281-291. 

The ANOVA results for the regression are compiled in Table 4. The / -test value is smaller than the associated critical F- value (there is no significant lack of fit for the OLS-regression). Figure 3 shows the absolute and relative measurement uncertainties for the calibration range. The uncertainties are attributed to the predicted... 

In the assessment of the uptake of a chemical after dermal exposure, for instance, the dermal permeability of the skin is often estimated using the Potts-Guy quantitative structure-activity relationship (Guy Potts, 1992), which was derived from an experimental data set of in vitro measured steady-state skin permeations (Wilschut et al., 1995). Uncertainty in the use of a value for the skin permeation obtained this way comes from questions of how well a regression model based on Kow and molecular weight predicts the skin permeability of a chemical that was not in the original data set, and how representative the steady-state permeability measured in vitro is for a (possibly) non-steady-state permeability in vivo (see also IPCS, 2006b). 

We have seen how consideration of theoretical deposition velocities has identified potential biases in economic assessments. An additional consideration is the relative uncertainties in the determination of theoretical vs. experimental deposition velocities. The heat transfer data on which the theoretical deposition velocities are based are generally very precise, within a few percent. In contrast, the damage functions developed by Lipfert et al. (3) for metals from extant corrosion test data are only capable of predicting corrosion losses at a given time and place within a factor of two, although the individual regression coefficients are much better than that. Most of the uncertainty in the experimental approach is felt to be in test site characterization rather than... 

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