Quadratic loss function

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. 

When limiting ourselves to one quality characteristic (y), with preferred value T, a common way to define a measure of quality is the quadratic loss function (as used by Taguchi [13]), which is defined as L(y) = k(y-f) where A is a constant, coupling the loss (y-r) to, for example, an economic quantity. 

Figure 19.5 Quadratic Loss Function (according to Taguchi).

Basically, Taguchi works in terms of a quality loss function rather than quality and defined quality loss as the loss imparted by the product to society from the time it is shipped. By taking a target value as the best possible value for the quality characteristic under investigation, he used a simple quadratic loss function, with deviations from the target to show that a decrease in loss was associated with an increase in quality (Figure 19.5). It will be noted that a loss will occur even when the product is within the permitted specification, but this loss is minimal when the product is on target. 

The quadratic loss function corresponds to the classical least-squares criterion the Huber function was proposed as a robust loss function when the distribution of the data is unknown and the Laplacian one is less sensitive to outliers than the quadratic one. All three require all data points in order to make the calculations which, in turn, are time consuming and intensive. Cortes and Vapnik proposed the -insensitive loss function as an approximation to Huber s one that enables many fewer SVs to be used, which simplifies the calculation stages. In addition, its graphical distribution (Figure 6.11) reveals that samples within the -bands do not constitute SVs because the function there takes null values. 

As can be seen in Table 4.2, the conversion, X, was only about 40% at the maximum in production rate. Another optimization was performed using a quadratic loss function to penalize conversions of less than 85% (Figure 4.6). This optimization of production rate led to an optimum at higher residence time. 

MATLAB SVM Toolbox, http //www.igi.tugraz.at/aschwaig/ software.html. This is a MATLAB SVM classification implementation that can handle 1-norm and 2-norm SVM (linear or quadratic loss function) problems. 

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 ... 

The quadratic probability measure is related to the Brier quadratic score, which is a loss function for comparing two probability vectors, and is used for the elucidation of probabilities [3,4,5]. The QPM ranges from 0 to 1 with values closer to 1 being preferred, since this implies the classes can be differentiated with a higher degree of certainty. 

Figure 11 Effect of reactant consumption on the stability ofa first-order exothermic reaction, heat loss (a) against reciprocal adiabatic flame temperature (B ). Curve (a). Adler-Enig temperature-concentration inflection criterion curve (b) Bowes-Thomas temperature-time inflection criterion curve (c) Gray-Sherrington quadratic Uapunov function curve (d). Frank-Kamenetskii s empirical criterion (the curves show a )...

NPPC [22] is a binary classifier and it classifies a pattern by the proximity of a test pattern to one of the two planes as shown in Fig. 5. The two planes are obtained by solving two nonlinear programming problems (NPP) with a quadratic form of loss function. Each plane is clustered around a particular class of data by minimizing sum squared distances of patterns from it and considering the patterns of the others class at a distance of 1 with soft errors. Thus, the objective of NPPC is to find two hyperplanes ... 

The classical multiple regression has a well-known loss function that is quadratic in the prediction errors. However, the loss function employed in SVR is the e-insensitive loss function. Here, the Toss is interpreted as a penally or error measure. Usage of e-insensitive loss function has the following implications. If the absolute residual is off-target by e or less, then there is no loss, that is, no penalty should be imposed. However, if the opposite is fine, that is absolute residual is off-taiget by an amount greater than s, then a certain amount of loss should be associated with the estimate. This loss rises linearly with the absolute residual above e. 

Taguchi proposed that a measure of the loss to society, and to the customer, be a quadratic (square law) function of the difference between the target value and the actual value of the parameter of interest. This has become known as Taguchi s loss function. The goal posts idea just does not address this loss to society it does not penalize a product with a parameter that is not the target value. The loss function has been used as a measure of monetary loss associated with less than optimal performance as well as higher maintenance and repair costs. 

In this section we first introduce the definition of the f-insensitive loss function, then show that the same quadratic optimization technique that was used in Section 2.3 for constructing approximations to indicator functions provides an approximation to real-valued functions, involving the linear case and nonlinear case. 

Figure 2.9 shows the form of the linear and quadratic -insensitive loss function for zero and non-zero e. 

The quality loss is crucial in Taguchi s theory. It is based on the assumption that when a functional characteristic y deviates from the specified target value m, the customer and the society in general experiences an economical loss due to poorer product quality. This economic loss is expressed as the loss function L(y). Based on this, Taguchi defines the quality loss for not being on target by means of the quadratic quality loss function (Phadke (1989), Taguchi (1986)) ... 

Figure 44 Loss functions for support vector machines regression (a) quadratic (b) Laplace (c) Huber (d) e-insensitive.

The -insensitive loss function used in the SVM regression adds a new parameter s that significantly influences the model and its prediction capacity. Besides the -insensitive, other loss functions can be used with SVM regression, such as quadratic, Laplace, or Huber loss functions (Figure 44). 

Thus, the sensitivity of a thermistor is a quadratic function of operating temperature. The optimum sensitivity is typically 4% °C 1. Because of their nonlinear response, thermistors are sometimes linearized by placing a resistor of similar nominal value in parallel with the thermistor, with a resulting loss of sensitivity. This is not generally necessary for thermochemical sensors, particularly for the push-pull applications, because the temperature range involved is small. For direct temperature measurement, the detection limit of 10 4oC can be achieved with a conventional Wheatstone bridge. 

To this aim we will apply the function called Tauchi s Function Loss (1989) that suggests that the non-performance from target values produces quality losses financially due to customer dissatisfaction as a quadratic or parabolic curve. This estimation could be employed to include the losses of Failure as ... 

The quality is accepted by the customer when it is maintained inside a level of tolerance, between an upper and a lower value of the service (Fig. 3). In DNSP, the consiuner tolerance is defined by standards in the sector. So the cost of any deviation, in terms of quality, increases in a quadratic form from the target value as centre of the curve, see Fig. 3. In our case, focus only on intangible costs, the re-establishment time has been considered to reflect the maintenance impact in quality service. The Taguchi s Function Loss (TFL) is ... 

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