Squares for residuals

The F statistic can also be useful in recognizing suspected outliers within a calibration sample set if the F value decreases when a sample is deleted, the sample was not an outlier. This situation is the result of the sample not affecting the overall fit of the calibration line to the data while at the same time decreasing the number of samples (N). Conversely, if deleting a single sample increases the overall F for regression, the sample is considered a suspected outlier. F is defined as the mean square for regression divided by the mean square for residual (see statistical terms in this table). Statistic Coefficient of Multiple Determination Abbreviations R or r ... 

The SEP is calculated as the root mean square differences (RMSD), also known as a mean square for residuals for IV - 1 degrees of freedom. It allows for comparison between NIR-observed predicted values and wet laboratory values. The SEP is generally greater than the SEC but could be smaller in some cases due to chance alone. When calculating the SEP, it is critical that the constituent distribution be uniform and the wet chemistry be very accurate for the validation sample set. If these criteria are not met for validation sample sets, the calculated SEP may not have validity as an accurate indicator of overall calibration performance. To summarize, the SEP is the square root of the mean square for residuals for - 1 degrees of freedom, where the residual equals actual minus predicted for samples outside the calibration set. 

Comments The cross-validation algorithm is performed identically to PRESS with the exception that rather than sum of squares for residual as the reported results, the cross-validation uses the square root of the mean squares for residuals, using N — I degrees of freedom for each model as ... 

The quadr atic curve fit leads to a number of residuals equal to the number of points in the data set. The sum of squares of residuals gives SSE by Eqs. (3-23) and MSE by Eq. (3-30), except that now the number of degrees of freedom for n points is... 

In order to examine whether this sequence gave a fold similar to the template, the corresponding peptide was synthesized and its structure experimentally determined by NMR methods. The result is shown in Figure 17.15 and compared to the design target whose main chain conformation is identical to that of the Zif 268 template. The folds are remarkably similar even though there are some differences in the loop region between the two p strands. The core of the molecule, which comprises seven hydrophobic side chains, is well-ordered whereas the termini are disordered. The root mean square deviation of the main chain atoms are 2.0 A for residues 3 to 26 and 1.0 A for residues 8 to 26. 

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

Table 11. Sum of the square of residuals (SSR) for A1 and A3 through A5, using the S basis vectors for Al.

Van der Voet [21] advocates the use of a randomization test (cf. Section 12.3) to choose among different models. Under the hypothesis of equivalent prediction performance of two models, A and B, the errors obtained with these two models come from one and the same distribution. It is then allowed to exchange the observed errors, and c,b, for the ith sample that are associated with the two models. In the randomization test this is actually done in half of the cases. For each object i the two residuals are swapped or not, each with a probability 0.5. Thus, for all objects in the calibration set about half will retain the original residuals, for the other half they are exchanged. One now computes the error sum of squares for each of the two sets of residuals, and from that the ratio F = SSE/JSSE. Repeating the process some 100-2(K) times yields a distribution of such F-ratios, which serves as a reference distribution for the actually observed F-ratio. When for instance the observed ratio lies in the extreme higher tail of the simulated distribution one may... 

In this case we minimize a weighted sum of squares of residuals with constant weights, i.e., the user-supplied weighting matrix is kept the same for all experiments, Q,=Q for all i=l,...,N and Equation 3.7 reduces to... 

Comparisons between optimized and X-ray structures were once again made by calculating root-mean-square (RMS) deviations. When comparing all heavy atoms in the protein, the total RMS deviation is approximately 1.7 A, irrespective of method for the model system or the ONIOM implementation (mechanical, ONIOM-ME, or electronic embedding, ONIOM-EE). The largest deviations occur for residues in the vicinity of the second monomer. Therefore, adding the second monomer to the model should improve the calculated geometries. 

The other popular technique in use today is the drag-sled technique, which consists of attaching a cloth dosimeter onto the bottom of a square block of some material (such as aluminum) of a predetermined size which has a weight added to the top side. The weighted sled is then pulled one time over a prescribed area of treated turf. The cloth dosimeter is taken off the sled, extracted, and analyzed for residues. 

Fig. 7. Contours of sums of squares of residual rates for isooctene hydrogenation, Eq. (46).

This is the general matrix solution for the set of parameter estimates that gives the minimum sum of squares of residuals. Again, the solution is valid for all models that are linear in the parameters. 

Figure 5.3 plots the squares of the individual residuals (rj, and nd the sum of squares of residuals (SS,), for this data set as a function of different values of feo demonstrating that = 4 is the estimate of Pq that does provide the best fit in the least squares sense. 

In general, the sum of residuals (not to be confused with the sum of squares of residuals) will equal zero for models containing a Pq term for models not containing a Po term, the sum of residuals usually will not equal zero. 

We begin by examining more closely the sum of squares of residuals between the measured response, yi and the predicted response, (y, = 0 for all i of this model), which is given by... 

Assume the model = 0 + r, is used to describe the nine data points in Section 3.1. Calculate directly the sum of squares of residuals, the sum of squares due to purely experimental uncertainty, and the sum of squares due to lack of fit. How many degrees of freedom are associated with each sum of squares Do and SS add up to give SS l Calculate and What is the value of the Fisher F-ratio for lack of fit (Equation 6.27)7 Is the lack of fit significant at or above the 95% level of confidence ... 

In Section 6.4, it was shown for replicate experiments at one factor level that the sum of squares of residuals, SS can be partitioned into a sum of squares due to purely experimental uncertainty, SS, and a sum of squares due to lack of fit, SSi f. Each sum of squares divided by its associated degrees of freedom gives an estimated variance. Two of these variances, and were used to calculate a Fisher F-ratio from which the significance of the lack of fit could be estimated. 

The sum of squares corrected for the mean, SS, is equal to the sum of squares due to the factors, plus the sum of squares of residuals, SS,. This result can be obtained from the partitioning... 

Figure 9.4 emphasizes the relationship among three sums of squares in the ANOVA tree - the sum of squares due to the factors as they appear in the model, SSf (sometimes called the sum of squares due to regression, SS ) the sum of squares of residuals, SS, and the sum of squares corrected for the mean, (or the total sum of squares, SSj, if there is no Pq term in the model). 

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