Noise variables

FIGURE 4.2 Linear combinations of the x-variables (data set in Table 4.2) are useful for the prediction of property y. For the left plot, xh x2, and x3 have been used to create an OLS-model for y, Equation 4.2 for the right plot x and x2 have been used for the model, Equation 4.4. R2 is the squared Pearson correlation coefficient. Both models are very similar the noise variable x3 does not deteriorate the model. 

In recent years much attention has been focused on the impact of the use of statistics, and in particular experimental design, to improve the quality of products and processes. An important component of the quality of a product is its robustness or stability in the presence of what Taguchi has called noise variables. 

These noise variables can be from a variety of sources, such as environmental conditions, deterioration of components, or variation in product components and manufacturing processes. It is possible that variation due to these sources will cause variation in the key characteristics of a product or process, resulting in a product of inferior quality. 

A.E. Freeny and V.N. Nair, Robust parameter design with uncontrolled noise variables, Statistica Sinica, 2 (1992). 

There are three eomponents to the empirieal approach of developing a predietive elas-sifier. The first component is determining whieh genes to inelude in the predietor. This is generally called feature seleetion. Including too many noise variables in the predictor usually reduces the aeeuracy of predietion. The second component is speeifieation of the mathematical function that will provide a prediction for any given expression veetor. 

As a general approach in RSM, one uses the black box principle (see Figure 8.1a). According to this principle, any technological process can be characterized by its input variables, xh i= I,. .., the output or response variables, yb i - 1,. .., s and the noise variables, w i= I,. .., Z. One then considers two ways of performing an experiment, active or passive. There are several important presumptions for active experiments ... 

The set of the noise variables, in comparison with the input variables, is assumed to have insignificant influence on the process. 

It is assumed that the summation of the noise variables over each of the. y responses can be represented by one variable i = 1,...,. y. Without any loss of generality, we assume that values of the error variables are distributed normally having variance c2 and mean zero, e, A/"(0, a1). Based on these assumptions, one can represent the black box principle in a slightly different way (see Figure 8.1b). Now the measurable response variable, yb can be represented as ... 

To apply the Taguchi approach, we must define two sets of variables controlled variables and noise variables. The controlled variables are process variables that can be adjusted or controlled to influence the quality of the products produced by the process. Noise variables represent those factors that are uncontrollable in normal process operating conditions. A study must be conducted to define the range of variability encountered during normal plant conditions for each of the controlled variables and the noise variables. These values of the variables are coded in the range [-1, 1],... 

Noise variables are factors that negatively affect the performance of a product, service, or process, but which are difficult to control. Noise comes from three primary sources ... 

As you move to a more detailed design, make a list of control factors and noise variables that may cause variation in performance (Exhibit 38.1). Control factors include anything that you, the provider, can reasonably manage—product specifications, equipment settings, storage, and so on. Noise variables are factors that are beyond your control—material variations (within tolerances), customer use and abuse, environmental conditions, and the like. 

EXHIBIT 38.1 In this example, we see the factors that the skin patch manufacturer can control (top row), as well as noise variables (bottom row) that are beyond the manufacturer s control but have the potential to negatively affect product performance. 

By this point, you ve developed a conceptual design for your innovative product or service and have made a list of noise variables that could negatively impact performance. Now, entering the parameter design phase of robust design, you can use physical experiments and/or simulations to refine the design so that it is less affected by noise. For example ... 

Use Design of Experiments (DOE) (Technique 50) to test how varying control factors affect your design. You should also test control factors against noise variables to stress the design and ultimately reduce sensitivity to customer use and abuse, as well as deterioration. 

Parameter design helps you determine optimal functional requirements for your innovation based on inputs, outputs, and the effect of noise variables. 

The use of several variables in the original data set increases the complexity of the data and therefore the - model complexity noise, variable correlation, redundancy of... 

The use of several variables in describing objects increases the complexity of the data and therefore the —> model complexity, noise, variable correlation, redundancy of information provided by the variables, and unbalanced information and not useful information give the data an intrinsic complexity that must be resolved. This happens in the case of spectra, each constituted, for example, by 800-1000 digitalized signals, which are highly correlated variables. Usually, —> variable reduction and variable selection improve the quality of models (in particular, their predictive power) and information extracted from models. Chemometrics provides several useful tools able to check the different kinds of information contained in the data [Frank and Todeschini, 1994]. 

To simplify the computation we assume that the noise variable Z is not correlated with S and with W, although the derivation can be extended to the case where E ZS) and E ZW) are non-zero. As a first step we determine the optimal value of (3 for fixed a. Direct differentiation yields (3°p = and by substituting this expression in (14), one finds... 

Some of these, like the granulation time, may also be control factors. When the noise and control factors have been chosen, a design can be set up in the control factors. This should be based on the results already obtained in a process study, if available. It is then multiplied by a design in the noise variables, generally factorial... 

The designs are similar to those involving control and noise variables. A design of the appropriate order for the process variables (c.f., control variables), is multiplied by a full factorial design for the scale or apparatus variables. It is thus a product design and may be considered as consisting of an inner and an outer array if the "quality/variability" method of analysis is to be used. 

We saw in chapter 7 that models for minimizing variability include interaction terms between noise variables and control variables, but omit interactions between two noise variables. They also are incomplete models. The experimenter may occasionally wish to use a "mechanistic", non-polynomial model, one resulting from, or inspired by a thermodynamic, or kinetic equation. 

According to the results of the study carried out by Derksen and Keselman [13], and concerning several automatic variable selection methods, in a typical case 20 to 74 percent of the selected variables are noise variables. The number of the noise variables selected varies with the number of the candidate predictors and with the degree of collinearity among the true predictors (due to the well-known problem of variance inflation when variables are correlated any model containing correlated variables are unstable). Sereening out noise variables while retaining true predietors seems to be a possible solution to the chance correlation problem in stepwise MLR. 

S. Derksen, H.J. Keselma, Backward, Forward and Step Wise Automated Subset Selection Algorithms Frequency of Obtaining Authentic and Noise Variables, British Journal of Mathematical and Statistical Psychology. 45 (1992), 265-282. 

Although the PLS method is claimed to be a robust modeling technique, experience shows that too many noise variables, i.e. variables that do not contribute to prediction, obscure the result. For prediction such additional variables are most often useless or even detrimental. This effect can clearly be demonstrated by different PLS analyses of the data set given in Table 22, using different combinations of the independent variables X-1, X-2, X-3, and X-4 (Table 23) three vectors were chosen as the maximum number of PLS components. If all X variables are included, the results are only slightly better than those obtained from regression analysis using X-4 as the only independent variable (Tables 22 and 23). 

Redundancy can be assumed to be adaptive. The precise reasons for signal redundancy may vary and include environmental (here chemical) noise, variable biosynthesis of compounds (as in diet-dependent cues), or grave consequences of misreading, as in predator alarm signals. 

A two-step approach was used for the determination of method detection and quantification limits for the sulfur analytes, as described in Lee Aizawa (2003). The two step approach takes into consideration several factors that affect the analyte signal, including instrumental noise, variability in instrumental sensitivity, and variability in method efficiency, matrix effects and interference, and is simple to follow. Other methods, such as the Hubaux-Vos approach for the calculation of the detection limit can also be used, as reported in Fedrizzi et al. (2007). However, this later approach is complicated, time consuming and does not take either the variabiUty in method efficiency or the matrix effects into consideration (Lee Aizawa, 2003). A brief discussion on how to conduct the method vahdation using the two steps approach is mentioned in this section. 

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