Output errors

In Figure 8.8, sinee the observer dynamies will never exaetly equal the system dynamies, this open-loop arrangement means that x and x will gradually diverge. If however, an output veetor y is estimated and subtraeted from the aetual output veetor y, the differenee ean be used, in a elosed-loop sense, to modify the dynamies of the observer so that the output error (y — y) is minimized. This arrangement, sometimes ealled a Luenberger observer (1964), is shown in Figure 8.9. 

The general principle behind most commonly used back-propagation learning methods is the delta rule, by which an objective function involving squares of the output errors from the network is minimized. The delta rule requires that the sigmoidal function used at each neuron be continuously differentiable. This methods identifies an error associated with each neuron for each iteration involving a cause-effect pattern. Therefore, the error for each neuron in the output layer can be represented as ... 

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. 

For an aquatic model of chemical fate and transport, the input loadings associated with both point and nonpoint sources must be considered. Point loads from industrial or municipal discharges can show significant daily, weekly, or seasonal fluctuations. Nonpoint loads determined either from data or nonpoint loading models are so highly variable that significant errors are likely. In all these cases, errors in input to a model (in conjunction with output errors, discussed below) must be considered in order to provide a valid assessment of model capabilities through the validation process. 

Output Errors. Output errors are analogous to input errors they can lead to biased parameter values or erroneous conclusions on the ability of the model to represent the natural system. As noted earlier, whenever a measurement is made, the possibility of an error is introduced. For example, published U.S.G.S. stream-flow data often used in hydrologic models can be 5 to 15% or more in error this, in effect, provides a tolerance range within which simulated values can be judged to be representative of the observed data. It can also provide a guide for terminating calibration efforts. 

Output errors can be especially insidious since the natural tendency of most model users is to accept the observed data values as the "truth" upon which the adequacy and ability of the model will be judged. Model users should develop a healthy, informed scepticism of the observed data, especially when major, unexplained differences between observed and simulated values exist. The FAT workshop described earlier concluded that rt is clearly inappropriate to allocate all differences between predicted and observed values as model errors measurement errors in field data collection programs can be substantial and must be considered. 

Fig. 7. Tracking output error for the robust and the nonrobust controllers where error and y are dimensionless...

The second approach we consider here consists of updating accordingly the bounds of the outputs in order to ensure feasibility as follows. Given bounds on the output, y "" control input as follows [5] ... 

Output Error (OE) model. When the properties of disturbances are not modeled and the noise model H q) is chosen to be identity tie = 0 and rid = 0), the noise source w k) is equal to e k), the difference (error) between the actual output and the noise-free output. 

Kozub and Garcia [149] point out that in many practical cases, rating of output error characteristics relative to MVC is not practical or achievable. They propose autocorrelation patterns for first-order exponential output error decay trend ... 

For the closed-loop performance bound given in Eq. 9.9, the variance of the output error is... 

Time series models of the output error such as Eq. 9.5 can be used to identify the dynamic response characteristics of e k) [148]. Dynamic response characteristics such as overshoot, settling time and cycling can be extracted from the pulse response of the fitted time series model. The pulse response of the estimated e k) can be compared to the pulse response of the desired response specification to determine if the output error characteristics are acceptable [148]. 

M Verhaegen and P Dewilde. Subspace model identification. Part I The output error state space model identification class of algorithms. Int. J. Control 56 1187-1210, 1992. 

NN training consists of finding a parameter vector W that minimizes the mean square output error ... 

The control cards OUTPUT, ERRORS and PRINTPIE print values computed by the program for the equilibrium composition of a problem. Other control cards cause the printing out or punching out of the data as it currently is in the program. The remaining output control cards control the amount or spacing of the output. 

OUTPUT OUTPUT ERRORS ARITH BAR DIVIDE LEAVE EXIT MATINV PHCALC RCALC FIND PAGE... 

The third set, the external set is used to check performance of the trained network, and to compare with other configuration or topologies. At the end of training process a well-fit model is the desired result. If the network has properly trained then it will be able to generalize to unknown input data with the same relationship it has learned. This capability is evaluated with the test data set by feeding in these values and computing the output error. 

These parameters must change in the direction determined by the negative of the output error function s gradient ... 

The term 1/N averages the output error depending on the number of WNN s outputs. 

Figures 12.2 and 12.3 show the error of the evaluated self-replication rate constant as a function of the experimental error and of the variation of the input flux for emulated experiments of type (a) and (b), respectively. In the case of emulated experiments of type (a) for which the evaluation of the rate constant is based on linearized kinetic equations, the error of the evaluated rate constant depends strongly on the variation of the input perturbation. The range of the final output error (—40%,+10%) is distorted in comparison with the range of the experimental error (—20%,+20%). For small values of the input perturbation, between 20% and 40%, the output error is surprisingly small— between 10% and 0%. As the input perturbation increases, the accuracy of the method deteriorates rapidly and for large perturbations the output error is almost twice as big as the experimental error. For the emulated experiments of type (b), where the rate coefficient is evaluated from our exact response law (12.105) without linearization, the situation is different. For input perturbations between 20% and 70% the error of the evaluated rate coefficient has about the same range of variation as the experimental error (-20%,+20%) and does not depend much on the size of the perturbation. For very large input perturbations, between 70% and 80%, the output error increases abruptly. In fig. 12.4 we show the difference of errors of the evaluated self-replication...

Fig. 1 2.2 The error of the evaluated selfreplication rate constant (output error) versus the experimental error and the inpnt perturhation for an emulated response experiment of type (a). The rate constant is evalnated hy using a linearized evoln-tion eqnation. The range of the outpnt error (—40%, +10%) is strongly distorted in comparison withy the range of the experimental error (—20%, +20%). The outpnt error varies greatly with the perturbation size for large pertnrbations it is about twice the experimental error, whereas in other regions there is error compensation. (From [11].)...

Fig. 1 2.4 The difference of errors of the evaluated self-replication rate constant (output error) evaluated from emulated experiments of type (a), with linearization, and type (b), without linearization, respectively. The figure shows that the variations of the input perturbation and of the experimental error have a different effect on the two types of response methods. The biggest difference occurs for large perturbations, because for large perturbations the linear approach is very inaccurate. (From [11].)...

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