Error local

The second method that may be used to measure local success is to monitor the accumulated error at each unit. This measures not how frequently the unit wins, but how closely the weights at the unit match the sample pattern 

In equation (4.1), xjk is the /-th data point in sample pattern k and w]a is the / -th weight at network unit a. After many sample patterns have been fed through a GCS that is being trained, some units will have accumulated a much larger local error than others this might be for either of the following reasons  

The unit wins the competition to be the BMU only rarely and, when it does win, the match between its weights and the sample pattern is poor. Although the unit rarely is the BMU, the poor match when it is chosen means that its local error grows to a large value. If the match at this unit is the best for a particular pattern, even though the match is poor, it is evident that no unit in the network is capable of representing the pattern adequately. 

The database cannot contain many samples that are similar to this particular pattern, for if it did the unit would match many patterns and, in due course, the weights at the BMU would match them well therefore, the poor match indicates that those samples for which this is the winning unit must be very different from each other and also few in number. As no unit in the network represents the samples well, the error in this region of the map is high and at least one new unit is required to represent a part of the variability in the samples. 

If alternative 2 applies, the same unit might be selected by the local error measure for insertion of a new unit as would be picked by the signal counter because, in both cases, the unit is frequently chosen as BMU. Alternative 1, however, picks out units that have a low signal counter rather than a high one. It follows that the course of evolution of a GCS will depend on the type of local measure of success that is used. 

---

The local error in the step from time t to t + t, i.e., the error, which is produced by calculating a discrete solution in this step instead of exactly solving the QCMD equations, is given as follows ... 

Unfortunately, this local error Cr cannot be calculated, since we do not know the exact solution to the QCMD equations. The clue to this problem is given by the introduction of an approximation to Let us consider another discrete evolution with an order q > p and define an error estimation via er t + z i) - z t). 

A growing neural gas has an irregular structure. A running total is maintained of the local error at each unit, which is calculated as the absolute difference between the sample pattern and the unit weights when the unit wins the competition to match a sample pattern. Periodically, a new unit is added close to the one that has accumulated the greatest error, and the error at the neighbors to this node share their error with it. The aim is to generate a network in which the errors at all units are approximately equal. 

Expansion of the network is guided by the principle that, when a new unit is added, it should be inserted at the position in the network where it can be of the greatest value this is determined by the local measure of success. There are several ways to define this location. We will consider the two most popular signal counters and local error. The difference between signal counters... 

If the local error is used to measure success, it too is decreased by a small amount after each sample pattern has been processed or each cycle is complete. 

If the local error is used instead of the signal counter, it is common to set the initial local error at the new unit to the average of the errors at all units to which it is connected ... 

The local error at each neighbor is then reduced by the amount of error that it has relinquished. 

In a completed map, every unit should have a similar probability of being the winning emit for a sample picked at random from the dataset. However, as the map evolves and the weights vectors adjust, the utility of an individual unit may change. Because the signal counter or the local errors are reduced every epoch by a small fraction, the value of this measure for units that are very rarely selected as BMUs will diminish to a value close to zero, indicating that these units contribute little to the network and, therefore, can be pruned out. 

An advantage to a formulation using Ri is that it also permits direct quantitative local error estimates. Thus, one can relax the stringent requirement of equidistribution, and directly enforce the error restrictions as follows ... 

Validation Once a structure has been determined, it is validated using a custom structure validation system (Badger and Hendle, 2002) to detect local errors. The system is based on PROCHECK (Morris efaJ., 1992), WHATCHECK(Vriend, 1990), SFCHECK (Vaguine ef ah, 1999), and PHISTATS and OVFRLAPMAP (from CCP4). 

The most sophisticated differential equation solver considered in this book and discussed in the next section includes such step size control. In contrast to most integrators, however, it takes a full back step when facing a sudden increase of the local error. If the back step is not feasible, for example at start, then only the current step is repeated with the new step size. 

Twopnt, solves systems of nonlinear, stiff, boundary-value problems [158]. It implements a hybrid, damped, modified Newton algorithm [159]. Local error is controlled using adaptive placement of mesh intervals. 

Solvation potentials can detect local errors as well as complete misfblds [64]. 

Rigorous and stiff batch distillation models considering mass and energy balances, column holdup and physical properties result in a coupled system of DAEs. Solution of such model equations without any reformulation was developed by Gear (1971) and Hindmarsh (1980) based on Backward Differentiation Formula (BDF). BDF methods are basically predictor-corrector methods. At each step a prediction is made of the differential variable at the next point in time. A correction procedure corrects the prediction. If the difference between the predicted and corrected states is less than the required local error, the step is accepted. Otherwise the step length is reduced and another attempt is made. The step length may also be increased if possible and the order of prediction is changed when this seems useful. 

A note is in order here on errors in the numerical solution of an ode. There are (regarding errors in a certain light) two kinds of errors. One is the local error, being the error added by a single step. The solution is always carried forward to a final point in t, using a number N of steps, and at that point we have a final, or global error. Unfortunately, this is always of a lower order than the local error. 

---