Loss Functions

Fig. 7. Optical density of solid Coo on Suprasil based on two different optical techniques (+, ). For comparison, the solution spectrum for Coo dissolved in decalin (small dots) is shown. The inset is a plot of the electron loss function -7m[(l + e)] vs E shown for comparison (HREELS) [78].

CR 3nd tp are the contributions from chain recoiling and interfacial dynamics (i.e. drag forces and disentanglement), respectively, and / ve is the viscoelastic loss function which has interfacial and bulk parts. / is a characteristic length of the viscoelastic medium, t is the contact time and n is the chain architecture factor. Fig. 21 illustrates the proposed rate dependency of adhesion energy. 

The energy release rate (G) represents adherence and is attributed to a multiplicative combination of interfacial and bulk effects. The interface contributions to the overall adherence are captured by the adhesion energy (Go), which is assumed to be rate-independent and equal to the thermodynamic work of adhesion (IVa)-Additional dissipation occurring within the elastomer is contained in the bulk viscoelastic loss function 0, which is dependent on the crack growth velocity (v) and on temperature (T). The function 0 is therefore substrate surface independent, but test geometry dependent. 

Figure 9-8. Top Loss functiun of highly textural liexuphenyl films for different values of niomcmut transfer parallel to the molecular axis. The tines connect plasinons related to (he same excitation. Bo tom Loss function of textured hexaphenyt films for different values of momentum transfer pcrpcndii. ular to the molecular axis. In both graphs tire spectra were uorinali/al lo obtain equal areas under the curves between 6 and 8 cV - taken front Kef. 1138].

A second approach considers that the regions of equivalent parameter values must enclose parameters for which the loss function is nearly the same or at any rate less different than some threshold. In other words, the equivalence regions should take the form 015(0) < c 5(6) for some appropriate constant of. Note that in this case the shape of the regions would not necessarily be ellipsoidal, or even convex In fact, we might postulate in general the existence of multiple minima surrounded by disjoint equivalence neigh-... 

C. Taguchi Loss Functions as Continuous Quality Cost Models.401... 

This Section addresses cases with a continuous performance metric, y. We identify the corresponding problem statements and results, which are compared with conventional formulations and solutions. Then Taguchi loss functions are introduced as quality cost models that allow one to express a quality-related y on a continuous basis. Next we present the learning methodology used to solve the alternative problem statements and uncover a set of final solutions. The section ends with an application case study. 

It is also important to realize that Taguchi loss functions not only bring into consideration both issues of location and dispersion of z but also provide a consistent format for combining them. By taking expectations on both sides of Eq. (19), and after a few algebraic rearrangements, we can show that the expected loss, [L(z)] is... 

If, besides the quality-related measure, z, one also wishes to include operating costs, in the analysis, because quality loss functions express quality costs on a monetary basis, commensurate with operating costs, the final global performance metric, y, which reflects total manufacturing cost, is simply the sum of both quality and operating costs (Clausing, 1993),... 

The total loss function, y, given by Eq. (23), is not directly measured and has to be computed from information that is available and collected from the process, consisting of (x, z) pairs. After defining an adequate loss function, L z), and considering operating costs, f, one can identify the (x, y) pairs that correspond to each of the initial (x, z) data records. 

Both situations with categorical and continuous, real-valued performance metrics will be considered and analyzed. Since Taguchi loss functions provide quality cost models that allow the different objectives to be expressed on a commensurate basis, for continuous performance variables only minor modifications in the problem definition of the approach presented in Section V are needed. On the other hand, if categorical variables are chosen to characterize the system s multiple performance metrics, important modifications and additional components have to be incorporated into the basic learning methodology described in Section IV. 

Since these loss functions express quality costs on a common and commensurate basis, extending the learning methodology of Section V to a situation with P objectives is straightforward. All one has to do is replace the original definition of the y performance metric [Eq. (23)] by the following more general version ... 

Similarly, for the case where a continuous performance metric, y, is employed, the following loss functions were defined (Saraiva and Stephanopoulos, 1992c) ... 

In the bottom-up approach the initiative to start the learning process is taken by one of the infimal decision units. Since solutions found at this unit may include connection variables, the request for given values of these variables is propagated backward, to unit A + 1, through temporary loss functions. After successive backpropagation steps, the participation of several other fhe operators associated with them, a final decision... 

Firstly, various criteria for estimation, different from the least square E [P(x)-P (x)], may now be considered. Consider a general loss function L(e), function of the error of estimation e p(x) -p (x). The objective Is to build an estimator that would minimize the expected value of that loss function, and more precisely. Its conditional expectation given the N data values and configuration. 

For any predetermined loss function L, this conditional expectation appears as a function of both the estimated value p (x) and the N data values, p, - 1 1,...,N. The optimization process consists of determining the particular value p (2 ) that would minimize expression (4). The solution Is straightforward for some particular functions L ... 

More generally, the loss function need not be symmetric L(e) L(-e). Indeed, underestimation of a pollutant concentration may lead to not cleaning a hazardous area with the resulting health hazards. These health hazards may be weighted more than the costs of cleaning unduly due to an overestlmatlon of the pollutant concentration. The optimal estimators linked to asymmetric linear loss functions are given In Journel (3 ). 

An answer to the previous problems is provided by the conditional distribution approach, whereby at each node x of a grid the whole likelihood function of the unknown value p(x) is produced instead of a single estimated value p (x). This likelihood function allows derivation of different estimates corresponding to different estimation criteria (loss functions), and provides data values-dependent confidence intervals. Also this likelihood function can be used to assess the risks a and p associated with the decisions to clean or not. 

Figure 1. Electronic and nuclear energy loss function of (a) Au implanted in Si02 and (b) He implanted in Si02. The fraction of the electronic or nuclear 5 stopping power with respect to the total (S tot = S + for Au (c) and He (d). (Reprinted from Ref [1], 2005, with permission from Italian Physical Society.)...

In non-metric MDS the analysis takes into account the measurement level of the raw data (nominal, ordinal, interval or ratio scale see Section 2.1.2). This is most relevant for sensory testing where often the scale of scores is not well-defined and the differences derived may not represent Euclidean distances. For this reason one may rank-order the distances and analyze the rank numbers with, for example, the popular method and algorithm for non-metric MDS that is due to Kruskal [7]. Here one defines a non-linear loss function, called STRESS, which is to be minimized ... 

Considering an initial electron energy much larger than kBT, Rips and Silbey show that the distribution of thermalization time is given by the first two moments of the energy loss function (e) per unit time,... 

Nonlinear mapping (NLM) as described by Sammon (1969) and others (Sharaf et al. 1986) has been popular in chemometrics. Aim of NLM is a two-(eventually a one- or three-) dimensional scatter plot with a point for each of the n objects preserving optimally the relative distances in the high-dimensional variable space. Starting point is a distance matrix for the m-dimensional space applying the Euclidean distance or any other monotonic distance measure this matrix contains the distances of all pairs of objects, due. A two-dimensional representation requires two map coordinates for each object in total 2n numbers have to be determined. The starting map coordinates can be chosen randomly or can be, for instance, PC A scores. The distances in the map are denoted by d t. A mapping error ( stress, loss function) NLm can be defined as... 

Sammon s NLM is one form of multidimensional scaling (MDS). There exist a number of other MDS methods with the common aim of mapping the similarities or dissimilarities of the data. The different methods use different distance measures and loss functions (see Cox and Cox 2001). 

Instead of the misclassification error, a loss function can be used that allows to consider different risks for the different types of wrong classifications for instance assigning healthy people to be sick can be given another risk than assigning sick people to be healthy. 

Ross, P.J. (1988), Taguchi Techniques for Quality Engineering Loss Function, Orthogonal Experiments, Parameter and Tolerance Design, McGraw-Hill Book Company, New York, NY. 

The loss function, which is the concrete form of Taguchi s definition of quality The quality of a product is the loss caused by the product to society from the time the product is shipped . 

---