Weighted sum of squares

It is also possible to determine A and B that minimize the weighted sum of squares. The weighting funetion is the square of the independent variable, and the funetion to be minimized is... 

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 of double-closed data Z, with weighted means, weighted sums of squares and weight coefficients added in the margins, from Table 32.4... 

Weighted sums of squares c of the row-profiles in F around the origin of can be expressed as distances of chi-square ... 

The global weighted sum of squares c of the transformed data Z can be shown to be equal to the global interaction 52 between rows and columns ... 

The trace of A is equal to the traces of the weighted cross-product matrices, which in turn are equal to the global weighted sum of squares c or global interaction 5 (eq. (32.27)) ... 

In implicit estimation rather than minimizing a weighted sum of squares of the residuals in the response variables, we minimize a suitable implicit function of the measured variables dictated by the model equations. Namely, if we substitute the actual measured variables in Equation 2.8, an error term arises always even if the mathematical model is exact. 

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... 

We will now investigate the sampling properties of the statistic representing the weighted sum of squared residuals i1 given by equation (5.4.13). We first observe that the slightly different expression (y — l)rSy i(y —is zero since... 

With these definitions, the mathematical derivation of the FCV family of clustering algorithms depends upon minimizing the generalized weighted sum-of-squared-error objective functional... 

In the ordinary weighted least squares method, the most probable values of source contributions are achieved by minimizing the weighted sum of squares of the difference between the measured values of the ambient concentration and those calculated from Equation 1 weighted by the analytical uncertainty of those ambient measurements. This solution provides the added benefit of being able to propagate the measured uncertainty of the ambient concentrations through the calculations to come up with a confidence interval around the calculated source contributions. 

In this analysis, the weighted sum of squared differences / between the fit with n rate constants kj and data points at measured wavelengths z>, and n, time t, is minimized ... 

S 6) based on Bayes theorem. For single-response data, S 6) is a weighted sum of squares of deviations of the observations pu from the fitted functions /t, (0) of the current model in events w = 1.NEVT ... 

If all classes do not contain n members but do contain a variable number then = 2 i( ) which is the weighted sum of squares. In each class the... 

This discussion of mathematical modeling is limited to methods based on the assumption of error in the y (dependent) term only. The objective or minimization function will generally be restricted to the sum of the weighted residuals between the observed and calculated data, weighted sum of squared residuals (WSS). The objective of the mathematical modeling approaches is to adjust the parameter values so that a minimum value of the WSS is achieved. 

Fitting the model to the observed data is an important task. Each mathematical model studied consists of independent and dependent variables, constants (possibly), and parameters that have to be estimated. The objective is to reduce the overall difference between the observed data and the calculated points by adjusting the values of the parameters. As mentioned earlier, the methods to be discussed assume that there is no error in the independent variable. Also, in general, the criteria of best fit will be the weighted sum of squared residuals between the observed and calculated data, the WSS. The chosen model can be validated in part by accurately describing the observed data. 

Non-linear regression analyses involve relatively complex calculations and thus are well suited to computer assistance. However, the program must have a well developed sequence of steps or algorithm to follow. Some methods are better than others. The program is asked to find the minimum point on a weighted sum of squares (objective or minimized function) surface. For two parameters, this can be represented as a three-dimensional surface (Fig. 4). 

This method is more informative, but it can be quite slow. Its major objective is to produce a weighted sum of squares surface diagram. In addition the minimum calculated WSS is estimated. The three-dimensional plot in Fig. 4 was calculated by this method. The calculation is set up by inputting the upper and lower limits of each parameter of interest. This range is split into a number of... 

The standard way of devising a RTO scheme is the so-called two-step approach [1], also referred to as repeated identification and optimization in the literature. In the first step, the values of (a subset of) the adjustable model parameters 6 are estimated by using the available process measurements. This is typically done by minimizing the lack of closure in the steady-state model equations (2), such as the weighted sum of squared errors between measured outputs y and predicted outputs y [17]. 

In this work we consider a benehmark eontrol problem of the isothermal operation of a eontinuous stirred tank reactor (CSTR) where the Van de Vusse reaetions take place [12, 13] (i.e. A B -> C and 2A -> D). The performance index is defined as the weighted sum of squares of errors between the setpoint and the estimated model output predieted for the time step in the future with witk) = D.D for all w(t< ) = 10,000foj- k=Mp The... 

The regression involved minimization of the weighted sum of squared deviations between the predicted and actual y values as a function of the four model parameters and was consequently an exploration of a four dimensional response surface. The source code listing for the sum of squares calculation routine, wexpred.m, is given in Section 6.3. 

Data reconciliation can be formulated as an optimization problem. The adjusted values of the measurements are determined by minimizing the weighted sum of squared measurement adjustments (i.e., the difference between measured values of the process variables and the process variable values that would satisfy the requisite conservation laws) subject to the conservation laws ... 

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