Sum of squared errors

The number of neurons to be used in the input/output layer are based on the number of input/output variables to be considered in the model. However, no algorithms are available for selecting a network structure or the number of hidden nodes. Zurada [16] has discussed several heuristic based techniques for this purpose. One hidden layer is more than sufficient for most problems. The number of neurons in the hidden layer neuron was selected by a trial-and-error procedure by monitoring the sum-of-squared error progression of the validation data set used during training. Details about this proce-... 

A number of modifications to eliminate some less favorable aspects of the Levenberg-Marquardt method were considered by Fletcher. For instance, the arbitrary initial choice of the adjustable parameter A, if poor, can cause an excessive number of evaluations of squared error, before a realistic value is obtained. This is especially noticeable if v, i.e., J R x), is chosen to be small, i.e., v = 2. Another disadvantage of the method is that the reduction of A to v at the start of each iteration may also cause excessive evaluations, especially when V is chosen to be large, i.e., = 10. The... 

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

We still can define a sum-of-squares error criterion (to be minimized) by selecting the parameter set /3 so as to... 

Thus for each mode a factorization (decomposition into scores and loadings) is done, expressed by three matrices and a three-way core-array G. The matrices A, B, C, and G are computed by minimizing the sum of squared errors the optimum number of factors, P, Q, R, can be estimated by cross validation. In a similar manner the Tucker2 model can be defined which reduces only two of the three modes, as well as the Tuckerl model which reduces only one of the three modes. 

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

An important point is the evaluation of the models. While most methods select the best model at the basis of a criterion like adjusted R2, AIC, BIC, or Mallow s Cp (see Section 4.2.4), the resulting optimal model must not necessarily be optimal for prediction. These criteria take into consideration the residual sum of squared errors (RSS), and they penalize for a larger number of variables in the model. However, selection of the final best model has to be based on an appropriate evaluation scheme and on an appropriate performance measure for the prediction of new cases. A final model selection based on fit-criteria (as mostly used in variable selection) is not acceptable. 

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 calibration phase focuses on the determination of the planarization length itself. This is a crucial characterization phase since once the planarization length is determined, the effective density, and thus the thickness evolution, can be determined for any layout of interest polished under similar process conditions. The determination of planarization length is an iterative process. First, an initial approximate length is chosen. This is used to determine the effective density as detailed in the previous subsection. The calculated effective density is then used in the model to compute predicted oxide thicknesses, which are then compared to measured thickness data. A sum of square error minimization scheme is used to determine when an acceptably small error is achieved by gradient descent on the choice of planarization length. 

This is similar to a least-squares regression, where the mean error is zero, but the sum of square error is not. We will first deal with the x-component of our convective transport terms ... 

Thus, the sum-of-squares error based on the aggregated sums is computed as... 

The aggregated sum-of-squares error for each batch, which is also needed, is calculated as... 

The sum-of-squares errors for each batch is computed using Equation (24) ... 

The aggregate sum-of-squares error is obtained using Equation (23) ... 

When comparing the test and reference products, dissolution profiles should be compared using a similarity factor (fz). The similarity factor is a logarithmic reciprocal square root transformation of the sum of squared error and is... 

The fit of the theoretical isotherms calculated using the Si and Ei parameters in comparison with the experimental data is satisfactory, as shown in Figures 2 and 3. The sum of squares error calculated by the expression... 

The sum of squared errors between the observed value and predicted value is minimized by taking the partial derivative (SSE)/b with respect to each parameter and setting each result equal to zero ... 

The moving average is a process similar to exponential smoothing. The exponential smoothing method (see also Section 6.4.2) has exponentially decreasing coefficients of the recent values. In a MA model the single coefficients b1 b2,. .., b were calculated by minimization of the sum of squared errors. 

For both the subdistribution and the GEX fit methods a Marquardt algorithm for constrained non-linear regression was used to minimize the sum of squares error (.10). The FORTRAN program CONTIN was used for the constrained regularization method. All computations were performed on a Harris H-800 super mini computer. 

The similarity factor /2 [137-139] is a logarithmic reciprocal transformation of the sum of squared errors and is a measurement of the similarity in the percentage dissolution between the two curves ... 

The lack-of-fit sum of square error is simply the difference between these two numbers or 2.705, and may be defined by... 

The PRESS errors can then be compared widi the RSS (residual sum of square) errors for each object for straight PCA (sometimes called the autoprediction error), given by... 

Using the eigenvalues obtained in question 1, calculate die residual sum of squares error for 1-5 PCs and autoprediction. 

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. 

Terms were deleted from the regression equation until an equation containing only terms having a confidence level of greater than 80% to improve fit (in terms of sum of squares error) was obtained. Coefficients (di s) for the resulting simplified equations are summarized in Table II with an estimate of fit. 

Table 1 Summary of parameter estimates and their standard deviation, correlation coefficient (r), and sum of square errors (SSE) for the best fit of experimental retention data (some of them are shown in Figs. 1, 3, and 5) as a function of the mobile-phase concentration of the nR by Eq. 8...

Table 2 Summary of parameter estimates and their standard deviation, correlation coefficient (r), and sum of square errors (SSE) for g...

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