The loss function

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 . 

Off-line quality control. A collection of methods to achieve the demanded quality. These methods enclose the following stages  

However, the shape of such a loss function depends on the process or product which is considered and is often difficult to establish. In general the goals of quality improvement (or the shapes of the belonging loss-functions) are often simplified to three types, which are  

The quality characteristic of type nominal the best which uses a specific target value combined with one particular loss function, the quadratic, is probably the most commonly used  

If the constant in the quadratic equation is chosen properly then the quadratic loss function can be used for direct calculation of the financial loss induced by an off-target quality characteristic. 

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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-... 

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 ). 

Because Taguchi has made no immediate coupling between the concept of loss-functions (the more philosophical part) and the off-line quality control methods (the more concrete part), the loss functions probably serve best as an important background concept rather than a practical working tool. 

If we find the structural factor from some independent experimental data, say, from those concerning the scattering of neutrons, relation (3.25) will enable us to find the loss function also. Since the structural factor is determined by the density-density electron correlation function, relation (3.25) implies that the excitation of plasmon oscillations is determined by the correlation in electron motion. 

A popular choice for the loss function L(-) is the log-score defined as... 

The tangent of 8 exhibits a prominent peak in this region, and the viscoelastic loss functions G"(co) and J" m) may also exhibit an absorption in the transition zone. The relative location of the maxima of the loss functions G"(co) and if they exist, can be obtained by taking into... 

Here, is again the surface work, S is the surface energy as previously defined and is the loss function dependent on crack speed, temperature and the strain, eo, applied to the specimen. The theory gives explicitly in terms of the energy density distribution in the specimen and the plastic or visco-elastic hysteresis of the material. 

The EELS results provide reliable information about the density of electrons in the valence band of solids, electron excitations in solids, and volume density of the material. The EELS spectrum was obtained from the spectrum of the Is carbon photoemission line corresponding to the electron energy of about 1 keV. We have followed the procedure described in ref. [15] and removed the elastic Is line to obtain the loss function in order to evaluate the energy losses as shown in Figures 11.9 and 11.10(a). 

An algorithm for fitting the Tucker2 model may be developed in line with the Tucker3 algorithms. The loss function for the Tucker2 model is... 

An algorithm for solving Equation (6.48) can be based on alternating least squares. Starting values are chosen and then iterations are performed to minimize the loss function between brackets in Equation (6.48). The full algorithm is described in Algorithm 6.8. Several parts of the algorithm are explained below. 

Assuming that all other parameters are given, W can be solved by rewriting the loss function of Equation (6.48) as... 

The vector py only appears in the second term of the loss function. Given the other parameters, py can be estimated as py = (XW)+y. This is a simple least squares step. The matrix Px consists of three Tucker-type matrices B, C and G. Given the other parameters, these matrices B, C and G can be updated by proper Tucker3 steps (see Algorithm 6.4) where the score matrix A = XW is kept constant. 

A natural way to handle missing data is to fit the model only to the nonmissing data. Thus, only optimizing the loss function over nonmissing elements. The loss function for... 

Consider a PARAFAC model of an / x J x K array X. The model is usually fitted by minimizing the loss function... 

As shown above the parameters can be estimated in two steps. Centering the data across the first mode will remove the offsets p, and the bilinear model is subsequently fitted to the centered data Z, thus minimizing the loss function... 

The loss function minimized when fitting a model with a weighted criterion is... 

The proof has been given by Kruskal [1977] and Gabriel [1978], Understanding that centering is a projection, it is simple to prove the above theorem. Let the loss function be... 

As subtracting the overall level does not remove the offset, another approach must apparently be adopted for handling situations with one common offset. There are at least two different ways of treating the problem. The best way is to optimize the loss function of the problem directly in a one-step procedure, rather than trying to use a two-step procedure... 

The problem with one common offset (Appendix 9.F) can be dealt with in the following way. The loss function for a bilinear model with constant overall offset is expressed as... 

We have also measured the intensity of the optically allowed transition relative to the other valence excitations in the loss function. As q increases, the intensity decreases rapidly along [100], while the intensity along [110] changes little. This indicates that even the oscillator strength of the dipole-allowed transition is anisotropic. 

Fig. 8.11. Measured peak position of the dipole-allowed transition in the loss function versus the orientation angle of q relative to [100], for a fixed q of 0.38 in the x — 7 plane. The solid line is a fit of the data to —cos(2/) for the purpose of illustration.

These examples show how description of the local drug concentration, C(t), can be obtained for substitution into the loss function, A (/, ). For leukemias the plasma values may be what is required. For solid tumors, however, the local drug concentration would be of primary concern and would require a more comprehensive pharmacokinetics model. 

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