Residuals sum of squares

Vcaic,i is obtained by feeding the appropriate r,- value into the regression equation. Anothe common squared term is the residual sum of squares (RSS), which is the sum of square of the differences between the observed and calculated y values. TSS is equal to the sur of RSS and ESS. The is then given by ... 

Equations 8-109, 8-110, 8-111, and 8-112 are redueed to an ordinary tanks-in-series model when N = i and h = 0. For the equivalent number of ideal CSTRs, N is obtained by minimizing the residual sum of squares of the deviation between the experimental F-eurve and that predieted by Equation 8-109. The objeetive funetion is minimized from the expression... 

A non-linear regression analysis is employed using die Solver in Microsoft Excel spreadsheet to determine die values of and in die following examples. Example 1-5 (Chapter 1) involves the enzymatic reaction in the conversion of urea to ammonia and carbon dioxide and Example 11-1 deals with the interconversion of D-glyceraldehyde 3-Phosphate and dihydroxyacetone phosphate. The Solver (EXAMPLEll-l.xls and EXAMPLEll-3.xls) uses the Michaehs-Menten (MM) formula to compute v i- The residual sums of squares between Vg(,j, and v j is then calculated. Using guessed values of and the Solver uses a search optimization technique to determine MM parameters. The values of and in Example 11-1 are ... 

Equation (6-19) was said to provide a fit as good as or better than those with other equations. The parameters were evaluated by fixing C and carrying out a linear least-squares regression of In k on T C was then altered and the procedure was repeated. The residual sum of squares was taken as a criterion of best fit. 

To ht the parameters of a model, there must be at least as many data as there are parameters. There should be many more data. The case where the number of data equals the number of points can lead to exact but spurious fits. Even a perfect model cannot be expected to fit all the data because of experimental error. The residual sum-of-squares is the value of after the model... 

The slopes bj are connected with activation energies of individual reactions, computed with the constraint of a common point of intersection. We called them the isokinetic activation energies (163) (see Sec. VI). The residual sum of squares So has (m - 1)X— 2 degrees of freedom and can thus serve to estimate the standard deviation a. Furthermore, So can be compared to the sum of squares Sqo computed from the free regression lines without the constraint of a common point of intersection... 

If the explicit solution cannot be used or appears impractical, we have to return to the general formulation of the problem, given at the beginning of the last section, and search for a solution without any simplifying assumptions. The system of normal equations (34) can be solved numerically in the following simple way (164). Let us choose an arbitrary value x(= T ) and search for the optimum ordinate of the point of intersection y(= log k) and optimum values of slopes bj to give the least residual sum of squares Sx (i.e., the least possible with a fixed value of x). From the first and third equations of the set eq. (34), we get... 

Parameter estimation to fit the data is carried out with VARY YM Y1 Y2, FIT M, and OPTIMIZE. The result is optimized values for Ym (0.7835), Y1 (0.6346), and Y2 (1.1770). The statistical summary shows that the residual sum of squares decreases from 0.494 to 0.294 with the parameter optimization compared to that with starting values (Ym=Yl=Y2=l. 0. ) The values of after optimization of Ym, Yl, and Y2 are shown in Figure 2, which illustrates the anchor-pivot method and forced linearization with optimization of the initiator parameters through Yl and Y2. 

If replicates are not available a test based on the ratio of the regression sum of squares to the residual sum of squares can be applied ... 

In a more recent variant of cross-validation one replaces PRESS(r- ) in the denominator by the residual sum of squares RSS(r ) ... 

The total residual sum of squares, taken over all elements of E, achieves its minimum when each column Cj separately has minimum sum of squares. The latter occurs if each (univariate) column of Y is fitted by X in the least-squares way. Consequently, the least-squares minimization of E is obtained if each separate dependent variable is fitted by multiple regression on X. In other words the multivariate regression analysis is essentially identical to a set of univariate regressions. Thus, from a methodological point of view nothing new is added and we may refer to Chapter 10 for a more thorough discussion of theory and application of multiple regression. 

Subsequently, Watts performed a parameter estimation by using the data from all temperatures simultaneously and by employing the formulation of the rate constants as in Equation 16.19. The parameter values that they found as well as their standard errors are reported in Table 16.18. It is noted that they found that the residuals from the fit were well behaved except for two at 375°C. These residuals were found to account for 40% of the residual sum of squares of deviations between experimental data and calculated values. 

Another measure for the precision of multivariate calibration is the so-called PRESS-value (predictive residual sum of squares, see Frank and Todeschini [1994]), defined as... 

We will not repeat Anscombe s presentation, but we will describe what he did, and strongly recommend that the original paper be obtained and perused (or alternatively, the paper by Fearn [15]). In his classic paper, Anscombe provides four sets of (synthetic, to be sure) univariate data, with obviously different characteristics. The data are arranged so as to permit univariate regression to be applied to each set. The defining characteristic of one of the sets is severe nonlinearity. But when you do the regression calculations, all four sets of data are found to have identical calibration statistics the slope, y-intercept, SEE, R2, F-test and residual sum of squares are the same for all four sets of data. Since the numeric values that are calculated are the same for all data sets, it is clearly impossible to use these numeric values to identify any of the characteristics that make each set unique. In the case that is of interest to us, those statistics provide no clue as to the presence or absence of nonlinearity. 

FDA/ICH recommendation Linear regression with report of slope, intercept, correlation coefficient, and residual sum of squares Objective Can be computerized Uses standard statistics Doesn t work as a test of linearity... 

Linearity is evaluated by appropriate statistical methods such as the calculation of a regression line by the method of least squares. The linearity results should include the correlation coefficient, y-intercept, slope of the regression line, and residual sum of squares as well as a plot of the data. Also, it is helpful to include an analysis of the deviation of the actual data points for the regression line to evaluate the degree of linearity. 

Associated with each data point is a certain degree of freedom, which will be used to attribute more information to, say, 500 data points than to 5 data points. In particular, if N data points are used, the total sum of squares is said to possess N degrees of freedom. The predicted rates estimated from a model containing p parameters have p degrees of freedom, and the remaining N — p degrees of freedom are possessed by the residual sum of squares. 

Here, y is the average of all of the replicated data points. If the residual sum of squares is the amount of variation in the data as seen by the model, and the pure-error of squares is the true measure of error in the data, then the inability of the model to fit the data is given by the difference between these two quantities. That is, the lack-of-fit sum of squares is given by... 

If there are n replications at q different settings of the independent variables, then the pure-error sum of squares is said to possess (n — 1) degrees of freedom (1 degree of freedom being used to estimate y) while the lack-of-fit sum of squares is said to possess N — p — q(n — 1) degrees of freedom, i.e., the difference between the degrees of freedom of the residual sum of squares and the pure-error sum of squares. 

This is, then, the regression sum of squares due to the first-order terms of Eq. (69). Then, we calculate the regression sum of squares using the complete second-order model of Eq. (69). The difference between these two sums of squares is the extra regression sum of squares due to the second-order terms. The residual sum of squares is calculated as before using the second-order model of Eq. (69) the lack-of-fit and pure-error sums of squares are thus the same as in Table IV. The ratio contained in Eq. (68) still tests the adequacy of Eq. (69). Since the ratio of lack-of-fit to pure-error mean squares in Table VII is smaller than the F statistic, there is no evidence of lack of fit hence, the residual mean square can be considered to be an estimate of the experimental error variance. The ratio... 

An important point is the evaluation of the models. While most methods select the best model at the basis of a criterion like adjusted R2, AIC, BIC, or Mallow s Cp (see Section 4.2.4), the resulting optimal model must not necessarily be optimal for prediction. These criteria take into consideration the residual sum of squared errors (RSS), and they penalize for a larger number of variables in the model. However, selection of the final best model has to be based on an appropriate evaluation scheme and on an appropriate performance measure for the prediction of new cases. A final model selection based on fit-criteria (as mostly used in variable selection) is not acceptable. 

While in the regression case the optimization criterion was based on residual sum of squares, this would not be meaningful in the classification case. A usual error function in the context of neural networks is the cross entropy or deviance, defined as... 

Name ethylparaben Calibration Id 4902 Fit Type Linear R 0.999998 R 2 0.999996 A -7.166783e+003 B 5.661624e+003 Slope Residual Sum of Squares 3.112287e+008 Standard Error 1,247455e+004 Units ng... 

Formal tests are also available. The ANOVA lack-of-fit test ° capitalizes on the decomposition of the residual sum of squares (RSS) into the sum of squares due to pure error SSs and the sum of squares due to lack of fit SSiof. Replicate measurements at the design points must be available to calculate the statistic. First, the means of the replicates (4=1,. .., m = number of different design points) at all design points are calculated. Next, the squared deviations of all replicates U — j number of replicates] from their respective mean... 

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