Sum of Deviations

It has been known for a long time (Edgeworth [1887]) that minimizing of the linear sums of deviations... 

The objective to be minimized is a weighted sum of deviations of the produced amounts from the demanded amounts d at the due dates im. Overproduction and underproduction, i.e., positive differences — d% and d — p respectively, are weighted by the nonnegative factors am and fim. If the value of the objective function is represented by z e R the objective can be stated as follows ... 

However, if equation 5 is used (weights wj rather than wj2), then equation 6 does equal zero. When a zero sum of deviations is desirable, function 5 may be minimized, often without increasing the root-mean-square-error by an undue amount. 

The denominator - that is, the df - in this calculation may be puzzling at first, but, again, the principle is the same as we have seen before. Recall that, when estimating the sample variance, the df value is w - 1. This is because the sum of deviations has to equal 0. Given knowledge of K - 1 observations in the sample, we can determine the last observation It is the value that will ensure that the sum of all deviations adds to 0. In this case, the "minus 1" is applied for all k groups. This leads to ... 

Figure 1.1 Examples of (a) Levey-Jennings and (b) cumulative sum (cusum) control charts using inter-assay quality control data from an alprazolam GC-MS assay. Since the cusum chart persents the cumulative sum of deviations from the mean, it is more sensitive to small biases that develop over time, whereas Levey-Jennings charts are most useful for detecting changes in the precision of the assay...

Kinetic and electrochemical data, respectively, were fitted to eqs 28 and 29. Non-linear least-square fits of the observed rate constant and the formal redox potential versus [Ml were carried out using the Solver Function in Microsoft Excel-98. Sums of deviation-squared values were minimized by varying k, kivir, Ko, Kred, Eo, a and b in eqs 28-30. The ratios Yox/y ox and Yred/y rcd were assigned a value of 1. Additional limitations and constraints imposed on the adjustable parameters to improve fitting are discussed above. 

The total deviation can also be quantified as the sum of deviations (SUD). For a perfect fit this number should be zero. Other measures of deviation may also be appropriate, as we will see. However, minimization of the SSR is normally used as a criterion in the curve fitting routines available for this purpose. This may be a good way to find the region where the optimum solution will lie but not necessarily the final criterion for this purpose. 

ABO = average pyrrole ring bond order X = some of the ring bond orders or sum of deviation from average ring bond order. 

For short times after a pulse, prior to the exit of the material from the pulse, the concentration change of the pulse is conserved the sum of deviations of concentrations, weighted by stoichiometric coefficients, must be constant and equal to the change in concentration due to the initial pulse. This property is useful in confirming that all species produced from the pulse due to reactions have been detected, and in determining the correct stoichiometric coefficients of reactants and products. 

Fig. 5.13 Plots of time series for sums of deviations in concentrations for the species of the mechanism in fig. 5.11. In (a), the concentration of species Xj is perturbed in (b), the concentration of species X3 is perturbed. Plot (a) shows that the sum of the deviations, AXi + X2 + (AX3 + AX4)/2, is slowly decaying in time. From this conservation, we infer that species 2 produces two molecules of species 3, which produces species 4. Plot (b) shows a similar transient conservation of mass in species 2, 3, and 4 following a perturbation of 3 the initial flow is from 3 to (1/2)2 and over a longer time from 3 to 4. (From [1].)...

The validity of the models described can be tested by comparing experimentally measured reduced mobilities of several ions in the linear IMS with the predicted coefficients calculated according to the three models. The main features of interest were the correlations of mass with mobility and temperature with mobility another interesting feature is the effect of the drift gas on mobility coefficients (the last two are discussed in Chapter 11). Six parameters are needed in the modeling a, r, z, polarizability, reduced mass, and temperature. The last three arise from direct physical measurements, while the other parameters (fl, r, z) are optimized by a fitting procedure to minimize the deviation between calculated and measured mobility constants. The values of T and were calculated from a, r, and z, and the dimensionless collision cross section (1 was taken from Table 1 in Reference 9. In practice, a discrete value of a was chosen, and initial values for and z were estimated. The parameters Tq and z were then optimized to obtain a good fit with experimental data points by minimizing the squared sum of deviations between theory and experiment. Special attention... 

It is calculated as the sum of deviations from equilibrium bond lengths for all ring bonds. 

The objective function F is usually presented as a weighted sum of deviation values, in the form ... 

Empirical fitting is achieved in practice by calculation of observables for a given system using an initial guess of the potential parameters. Then the weighted sum of deviations between calculation and observation is computed and the potential parameters are adjusted to minimize this sum. The observable parameters employed are generally drawn from the following set ... 

Another simple device of use in determining consistency of judges is the mean score and sum of deviations from the mean. (Table X.) The differences among the four judges are readily apparent from the total deviations over all fats. 

Table 17.2 gives important time-series models that are commonly encountered in industrial process control, including statistical process control applications (see Chapter 21). Stationary disturbance models (a) and (b) have a fixed mean that is, the sums of deviations above and below the line are equal to zero, but case (a) rarely occurs in industrial processes. Nonstationary disturbance models (c) and (d) do not have a fixed mean but are drifting in nature. Case (c), so-called random walk behavior, is often used to describe stock market index patterns. Case (b) is called an autoregressive... 

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