Regression predicted response

A check of lack of fit of the regression model (2.116), is done in accordance with the formulas from Sect. 2.4.3, where all design points are replicated the same number of times (r 25). The obtained predicted response values of the regression model are also given in Table 2.167. Variance of lack of fit is calculated thus ... 

The most important among the known criteria of design optimality is the requirement of D- and G-optimality. A design is said to be D-optimal when it minimizes the volume of the scatter ellipsoid for estimates of regression equation coefficients. The property of G-optimality provides the least maximum variance of predicted response values in a region under investigation. 

Although calibration and regression are closely related, they do not coincide with each other. With regression, the response Y, is estimated from the independent variable Xt by means of the mathematical model Yt = f(X,) +On the other hand, in analytical chemistry the independent variable (that is, the concentration) is predicted from the response variable (i.e., the instmment response). One should be aware that, from a statistical viewpoint, this is not correct. 

U we put the values of the variables in experiment i in the model we can calculate a predicted value of the response, For the series of experiments it is thus possible to compute the sum of squares of the predicted responses. This is called the sum of squares due to regression, SSR, is the vector of predicted responses it is... 

A response surface model which has been determined Ity regression to experimental data can be used to predict the response for any given settings of the experimental variables. The presence of a random experimental error is, however, transmitted into the model and gives a probability distribution of the model parameters. Hence, the precision of the predictions by the model will depend on the precision of the parameters of the response surface model. The error variance of a predicted response, V(yi), for a given setting, Xj = [Xjj,. .. x ] of the experimental variables is determined by the variance function, d, introduced in Chapter 1. 

For SS EsiD we take the deviations from the regression line, or residuals, shown in figure 4.1. The residual sum of squares SSk sid has already been defined as the sum of squares of the differences between experimental data iy and predicted response values (y,). This sum of squares therefore represents deviations of the experimental data from the model. 

Using common regression techniques these ) -coefficients are estimated (as )-coefficients) leading to eqns (3.13) and (3.14), respectively where f is the predicted response from the model and e the residual. 

Figure 7.15 compares the four model responses with the experimental data. Excel was used to fit Models 3 and 4, while linear regression was used for Models 1 and 2. A step response model with 50 coefficients (and Ar = 2 min) provides a predicted response that is indistinguishable from the experimental data (solid line) it is shown as Model 1. A step response model with 120 coefficients and Af = 1 min would provide an exact fit of the 120 data points. 

If possible one should evaluate the significance of the regression coefficients as explained in Section 2.1.2 and eliminate from the model those considered nonsignificant. A new multiple regression is then performed with the simplified model. It is preferable also to validate the fit of the model and its prediction accuracy. The former is usually done by considering the residuals between the experimental and predicted responses, because this does not require replicate determinations as is needed for the ANOVA procedure. The validation of the prediction accuracy requires that additional experiments are carried out, which are then predicted with the model. The selection of the optimal conditions is often, but not necessarily, done with the aid of visual representation of the response surface, describing y as a function of pairs of variables. The final decision is often a multicriterion problem where multicriterion decision making techniques are applied (Section 5). 

Figure 7 The training set (/ objects is used to find the regression coefficients vector b. With a test set ( J objects), the predicted responses can be caicuiated using the same b. Usuaiiy the responses for the test set are kept in the background for diagnostic checking.

L. Breiman and J.H. Friedman, Predicting multivariate responses in multiple linear regression. J. Roy. Stat. Soc. B59 (1997) 3-37. 

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. 

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