Sum of residuals

A disadvantage of the conventional control charts is that a small or gradual shift in the observed process parameter is only confirmed long after it has occurred, because the shift is swamped in statistical (analytical) noise. A simple way out is the Cusum chart (cumulated sum of residuals, see program CUSUM.exe), because changes in a parameter s average quickly show up, see Fig. 1.32. The... 

Purpose A technique to detect deviations from random scatter in the residuals (symmetrical about 0, frequent change of sign) Cumulative sum of residuals detects changes in trend or average. Here, an average is subtracted to yield residuals these residuals are then summed over points 1. .. k. .. N, with the sum being plotted at every point x k). Two uses are possible ... 

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. 

In step 3, a multiline-fitting program was run to optimize the pK a values to minimize the sum of residual squares between calculated and observed mobilities from Eq. (17). Figure 2 shows an example of the MS Excel spreadsheet for pK a calculation. The solver function of MS Excel could be used to perform the multiline-fitting analysis. 

Pore distribution curves can be obtained plotting the change in the sum of residuals, which can be calculated by experimental elution volumes of all standards against the mean diameter of the PS standards [119]. 

Feeding low levels of olaquindox (2.0 and 6.0 ppm in feed) to laying hens for 21 days, resulted in residues in eggs that reached a plateau after some 10 days of medication (19). The sum of residues of the parent compound and the N" -monooxy metabolite, which was the only metabolite observed, was higher in egg white than in yolk, the amount of the metabolite accounting for 15-20% of the total residues. After cessation of the medication, the residues in both yolk and white declined below 2 ppb in about 5 days. 

Sum of residues which may be hydrolyzed to N-Methyl-1,3-propanediamine and expressed as morantel equivalents... 

Sum of residues on shoulder pads (pg/cm2) Sum of residues on forearm pads (pg/cm2) Sum of residues from hand rinses (pg)... 

Since Eq. (16) is nonlinear, one must use a nonlinear least-squares fitting program or, as described here, make use of the Solver option available as an add-in tool in Excel. An example of the use of Solver is given in Chapter HI. In the present apphcation, initial estimated values for the four fitting parameters (Q, Cj, P, and j8) are entered into four worksheet cells. For each of the Ndata points, these cells are used to calculate r and then to obtain a theoretical < >obs value from Eq. (16). The difference between the experimental and theoretical < >obs value (residual) is squared and the sum of these squares (essentially proportional to is placed in a test location. Solver is then run iteratively to adjust the fitting parameters so as to minimize this sum of residuals squared. 

The constants In eqn. 4 have been previously defined except for ai, ay, etc., which are arbitrary, adjustable parameters. The °Fey data of Fig. 1 were fit to eqn. 4 using a Fletcher-Powell program to minimize the sums of residuals. For the ground state of the Fe atom the proper choice of n Is 5 (11). C5 was calculated to be -3.1775 x 10 cm using the formulas and... 

The isotherm parameters were determined using Nelder Mead simplex method by minimizing the sum of residual, namely, the differences between experimental and estimated adsorption amount. Figure 2 showed the adsorption isotherms of TCE on MCM-48 at 303, 308, 313, 323 K. As one can be expected, the adsorption capacity was decreased with increasing temperature. The hybrid isotherm model for a pure adsorbate was found to fit the individual isotherm data very well. The parameters of the hybrid equations are listed in Table 1. 

The LOI is defined as the weight loss of the heat-treated sample on its subsequent calcination at 1000°C for 1 hr. Thus, it gives the sum of residual water and OH groups which can be dehydroxylated at 1000 °C. 

The objective of any modeling exercise is to place a calculated line (based on some relevant mathematical model) as close to the data collected as possible. The difference between individual data points and the calculated line (in a vertical direction— no error in the x terms) is called the residual. The sum of residuals could be zero even with very large residuals for individual points if the negative residuals canceled the positive values. An absolute residual might solve this problem, but more usefully, the squared residual will also achieve the desired result. This is the least-squares criterion. An extension of this is to weight each data point by the inverse of the estimated variance. This term is the objective function, WSS, shown in Eq. (1), calculated for n data points ... 

Molten crystals (5) cooled to -15 C, leading to crystals (7) and residual liquor (8). 9 - Sum of residual liquors at room temperature. 

Note also that the x-ray method measures the existing stress, whether it be solely residual or the sum of residual and applied. It therefore has the capability of measuring the actual service stress in a machine or structure under a service load. 

The system of equations (7.189) and (7.190) cannot be solved analytically (except for z=l). The estimation of reaction rates and comparison with experimental data should be done by minimization of the sum of residual squares, while the value of surface coverage from balance equations should be solved numerically using, for instance, the Newton-Raphson procedure. 

Small variances guarantee that the parameter is accurately estimated and small correlation coefficients indicate that the parameters camiot be mutually compensated. The variances are very much dependent on the precision of the experiments since they are directly proportional to the weighted sum of residual squares (Q), while the correlation coefficient depends heavily on the model structure as such. Special tricks to suppress the correlation between parameters exist, and should always be used. 

The second term in the numerator is zero, because it contains a sum of residuals about the average. Remembering that by definition the... 

In the second approach, reaction curves are calculated with sets of preset parameters by iterative numerical integration from a preset staring point. Such calculated reaction curves are fit to a reaction curve of interest the least sum of residual squares indicates the best fitting (Duggleby, 1983, 1994 Moruno-Davila, et al., 2001 Varon, et al., 1998 Yang, et al., 2010). In this approach, calculated reaction curves still utilize reaction time as the paedictor variable and become discrete at the same intervals as the reaction curve of interest. Clearly, there is no transformation of data from a reaction curve in this approach. 

If the intercept is included in the model, the sum of residuals equals zero. 

There are really many different variants of this method based on the selection of the support points and the elements to discretize the interval. Some of them have special names that highlight their approach. For instance, if the points are selected as the roots of an orthogonal polynomial and if the elements have only one point in common, the method is said to be a finite-element orthogonal collocation. On the other hand, if each element consists of three points and the adjacent elements share two points, the method is said a finite-difference method. In some cases, when the elements have common points, the single residual is not zeroed, but the sum of residuals is calculated in the same point using all the elements that are sharing it. The aim of these variants is to find a well-conditioned system of equations with a structure that makes its solution particularly efficient when the number of variables is rather large. 

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