Error cost function

IAE = Integral of the Absolute Error, cost functional for determining quality of control (the higher the value, the worse the control)... 

The first MDS method is the metric MDS, characterized by minimizing the squared error cost function ... 

Lonally, the templates were chosen by trial and error or exhaustive enumeration. A itafional method named ZEBEDDE (ZEolites By Evolutionary De novo DEsign) en developed to try to introduce some rationale into the selection of templates et al. 1996 Willock et al. 1997]. The templates are grown within the zeolite by an iterative inside-out approach, starting from a seed molecule. At each jn an action is randomly selected from a list that includes the addition of new (from a library of fragments), random translation or rotation, random bond rota-ing formation or energy minimisation of the template. A cost function based on erlap of van der Waals spheres is used to control the growth of the template ale ... 

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

After fitting the parameters of the model to the data, the the best tuning constants were found. The cost functional to minimize was the integral of the absolute value of the error (lAE) ... 

The idea of selecting waveforms adaptively based on tracking considerations was introduced in the papers of Kershaw and Evans [3, 4], There they used a cost function based on the predicted track error covariance matrix. 

We have, on the other hand done simple simulations for the case of one-step ahead and two-step ahead scheduling. In the latter case, the revisit times and waveforms are calculated while the target states are propagated forward over two measurements, with the cost function being the absolute value of the determinant of the track error covariance after the second measurement. Only the first of these measurements is done before the revisit calculation is done again for that target, so that the second may never be implemented. 

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. 

An additional complication in formulating the objective function is the quantification of uncertainty. Economic objective functions are generally very sensitive to the prices used for feeds, raw materials, and energy, and also to estimates of project capital cost. These costs and prices are forecasts or estimates and are usually subject to substantial error. Cost estimation and price forecasting are discussed in Sections 6.3 and 6.4. There may also be uncertainty in the decision variables, either from variation in the plant inputs, variations introduced by unsteady plant operation, or imprecision in the design data and the constraint equations. Optimization under uncertainty is a specialized subject in its own right and is beyond the scope of this book. See Chapter 5 of Diwekar (2003) for a good introduction to the subject. 

Since GSA requires an estimate of the magnitude of cost function response at the global optimum, an estimate of the residual sum of squares for the best fit was needed. Based on work with similar data and some trial and error, a value of le-4 was found to give good results. 

Jurs cost function. A function based on the mean square error s of both the training and test set [Sutter, Peterson et al., 1997] ... 

To proceed some measure of error of estimation is needed. The error is quantified with the help of the cost function. The choice of the cost function depends on the given problem. Two most often used cost functions are as follows ... 

The main idea of MDL approach can be summarized as follows the MDL cost function represents two conflicting requirements, we would like to compress signal to the highest possible degree, but simultaneously we would like to have as small reconstruction error as possible. These requirements are represented by two terms the first term describes the reconstruction error depending on the number of retained coefficients, and the second one is a penalty function, increasing with the number of the retained coefficients ... 

The optimal number of retained wavelet coefficients, and the reconstruction error strongly depend on the applied filter. It means that the MDL cost function can be used for the filter optimization. Filter for which the MDL achieves minimal value or the minimal number of retained wavelet coefficients can be considered the optimal one for data compression. 

The values of the parameter w,j of the network are computed by an iterative algorithm that minimizes a cost function Jbased on the squared modeling error ... 

Measurements y t,0) obtained by n sensors from the real system carry some uncertainty. Therefore, the residuals e t, 0) = y t, 0) — y(t, 0) between measured and computed vectors, the cost function and the estimated parameters 0 are uncertain. Each residual ej 0) = e tk-q- +j, 0) has an error Sj. It is assumed that all these errors ej, j = 1,..., q + 1, are independent and that each of its n components is normally distributed with mean zero and a known variance i = 1,..., n. If the known variances are quite different, a weighted least squares problem may be considered. 

A cost function to be minimised in an iterative parameter estimation procedure may be formulated by using either differences between outputs from a real system and computed outputs from a model or by means of ARR residuals. As output errors, as well as ARR residuals are generally nonlinear functions of the component parameters, multiple fault parameter isolation becomes a well-known nonlinear least squares problem. For real-time FDI, ARR residuals obtained from a DBG have the advantage that they make the parameter estimation independent of any initial conditions of the process that are hardly known and will have to be estimated along with component parameters. In off-line simulation, the real system may be replaced by a behavioural model. Measured data is then generated by assuming realistic consistent initial conditions and by solving the equations of the behavioural model. 

Clearly, if the density N t) is exactly evaluated, L takes the value of the cost function C, since the added term vanishes. What is required is for L to be insensitive to variations or errors in the density N. We write such a variation as 5N about N. The corresponding variation of the Lagrangian... 

If the adjoint function satisfies these equations and boundary conditions, Lis a stationary expression, insensitive to small errors in the density, whose numerical value will yield C. Inspection shows that the Lagrangian has a certain symmetry such that, if N satisfies its equation and boundary conditions, then the Lagrangian is stationary to errors in iV (stationary, in fact, to large errors, since /o and M are not functions of the costate variable). In practice, both equations are perturbed by a change in the control variable and simultaneous errors are made in both functions. For small control perturbations, we anticipate small perturbations in the state and costate variables and that the resulting expression is in error in the cost function only through terms involving the product of small errors. We write... 

In the b form, limiting ourselves to a time optimal problem for simplicity, the control period is fixed, and the Hamilton density does not now vanish. Since the end of the trajectory is not bound to any particular target curve, we must take both adjoint functions to vanish at the end time if the Lagrangian is to be stationary for arbitrary errors in the density. On the other hand, the cost functional is now the post-shutdown xenon peak, which is determined only by the end state Nf(tf). Thus, the integrand of the cost function has a delta function form ... 

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