Squared error function

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

The Universal Approximation Theorem states that a single layer net, with a suitably large number of hidden nodes, can well approximate any suitably smooth function. Hence for a given input, the network output may be compared with the required output. The total mean square error function is then used to measure how close the actual... 

J = least squares error function to be minimized K = absolute permeability Kr = relative permeability /Cr = relative permeability of non-wetting phase Kr = relative permeability of wetting phase... 

Most of the applications of fuzzy cluster analysis in chemistry apply the fuzzy-c-means algorithm. It relies on the general least-squares error functional... 

Plot the probability density obtained from E in Problem 9 as a function of r, that is, simply square the function above with an appropriate scale factor as determined by trial and error. Comment on the relationship between your plot and the shell structure of the atom. 

The unknown parameters of the model, such as film thicknesses, optical constants, or constituent material fractions, are varied until a best fit between the measured P and A and the calculated P/ and A/ is found, where m signifies a quantity that is measured. A mathematical function called the mean squared error (MSE) is used as a measure of the goodness of the fit ... 

Probabilistic techniques of estimation provide some Insights Into the potential error of estimation. In the case of krlglng, the variable pCic) spread over the site A is first elevated to the status of a random function PC c). An estimator P (2c) is then built to minimize the estimation variance E [P(2c)-P (2c) ], defined as the expected squared error ( ). The krlglng process not only provides the estimated values pCiyc) from which a kriged map can be produced, but also the corresponding minimum estimation variances 0 (39 ) ... 

The unknown parameter vector k is obtained by minimizing the corresponding least squares objective function where the weighting matrix Q, is chosen based on the statistical characteristics of the error term e, as already discussed in Chapter 2. 

Optimization of the PPR model is based on minimizing the mean-squares error approximation, as in back propagation networks and as shown in Table I. The projection directions a, basis functions 6, and regression coefficients /3 are optimized, one at a time for each node, while keeping all other parameters constant. New nodes are added to approximate the residual output error. The parameters of previously added nodes are optimized further by backfitting, and the previously fitted parameters are adjusted by cyclically minimizing the overall mean-squares error of the residuals, so that the overall error is further minimized. 

In the equations above, the mean square error, the sample variance, and the finite sampling bias are all explicitly written as functions of the sample size N. Both the variance and bias diminish as /V — oc (infinite sampling). However, the variance... 

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

The preceding results are applied to develop a strategy that allows us to isolate the source of gross errors from a set of constraints and measurements. Different least squares estimation problems are resolved by adding one equation at a time to the set of process constraints. After each incorporation, the least square objective function value is calculated and compared with the critical value. 

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 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... 

Table 3. Various properties evaluated using the hybrid functional of Eq. (36). We report root-mean-square errors in energies (in eV) as a function of the mixing parameter a, and the optimum value of a. The properties and data sets on which they are evaluated are described in the text. All results were extracted from Tables V and VII of Ref. [13]. (1 eV = 23.06 kcal/mole.)...

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... 

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