Summed squared error function

If the performance index or cost function J takes the form of a summed squared error function, then... 

The objective function, is typically formulated as the summed squared error between experimental measurements, y, and model predictions, y. The relationship should properly describe the experimental error present and best utilize available experimental data. In the common least squares estimation, the measurement error is assumed to be normally distributed and takes the form... 

The objective function is a minimum sum squared error of the monitored deformation and the calculated deformation ... 

In order to train a neural controller, a multilayered network with linear activation functions was initially considered. During the training process, a large sum-squared error occurred due to the unbounded nature of the linear activation function that caused a floating point overflow. To avoid the floating point overflow we used the h3rperbolic tangent activation functions in the hidden layers of the network. The network was unable to identify the forward... 

Introduction of Equation (2.3) into a sum-of-squares error objective function gives... 

They apply another objective function as OLS (not simple minimization of sum of squared errors). 

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... 

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... 

The number of reflection intensities measured in a crystallographic experiment is large, and commonly exceeds the number of parameters to be determined. It was first realized by Hughes (1941) that such an overdetermination is ideally suited for the application of the least-squares methods of Gauss (see, e.g., Whittaker and Robinson 1967), in which an error function S, defined as the sum of the squares of discrepancies between observation and calculation, is minimized by adjustment of the parameters of the observational equations. As least-squares methods are computationally convenient, they have largely replaced Fourier techniques in crystal structure refinement. 

Given the n observations, our aim is to obtain the best estimates X of the m unknown parameters to be determined. Gauss proposed the minimization of the sum of the squares of the discrepancies, defining the error function S, which, after assignment of a weight w, to each of the observations, is given by... 

As mentioned earlier, the sum of the squared error is found to be the most convenient measure of the error because much of the calculation may be done analytically. In the following sections the sum of the squared error will be formulated for each of the constraints, and the form of the equations in the unknown Fourier coefficients for each constraint will be determined. Values of both artificial and experimental data will then be substituted in these equations to determine these unknown Fourier spectral components of the extended spectrum. From these, the completely restored function may be determined. 

In any case, the cross-validation process is repeated a number of times and the squared prediction errors are summed. This leads to a statistic [predicted residual sum of squares (PRESS), the sum of the squared errors] that varies as a function of model dimensionality. Typically a graph (PRESS plot) is used to draw conclusions. The best number of components is the one that minimises the overall prediction error (see Figure 4.16). Sometimes it is possible (depending on the software you can handle) to visualise in detail how the samples behaved in the LOOCV process and, thus, detect if some sample can be considered an outlier (see Figure 4.16a). Although Figure 4.16b is close to an ideal situation because the first minimum is very well defined, two different situations frequently occur ... 

The parameters in Eq. (2.59) are usually determined from the condition that some function mean-square deviations between the experimental and calculated curves (the error function). The search for the minimum of the function Nelder-Mead algorithm.103 As an example, Table 2.2 contains results of the calculation of the constants in a self-accelerating kinetic equation used to describe experimental data from anionic-activated e-caprolactam polymerization for different catalyst concentrations. There is good correlation between the results obtained by different methods,as can be seen from Table 2.2. In order to increase the value of the experimental results, measurements have been made at different non-isothermal regimes, in which both the initial temperature and the temperature changes with time were varied. 

Minimize the sum of the squared errors using the Mathcad Minimize function. 

Minimize the sum of the squared errors using the Mathcad Minimize function. Guesses ai2 = 0.5 a2i = 1.0... 

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. 

A commonly used error function is the root-mean-square error, which is the square root of the sum-of-square errors calculated from all patterns across the entire training file. Other error functions (cost functions) may also be defined (Van Ooyen Nienhuis, 1992 Rumelhart et al., 1995), depending on the particular application. 

The usual criterion of best fit is the sum of errors squared (the SSD discussed above) rather than the absolute magnitude of the errors. This procedure is mathematically justified when the errors in the data follow the Gaussian (or normal) distribution. Under these conditions the error distribution function is given by Eqn. 9.6 in which % is the measurement, p the mean, and a the standard deviation cf. Sect. 

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