Input Variable Selection

Iman, R.L., Helton, J.C. and Campbell, J.E. An approach to sensitivity analysis of computer models Part I—Introduction, input variable selection and preliminary variable assessment. Journal of Quality Technology 1981 13 174— 183. 

There are two parts in this chapter. In part 1 (section 8.3), the above-mentioned models, i.e. the ANN model and statistical model are used to predict the fiber diameter of melt blown nonwoven fabrics from the processing parameters. The results are expected to give an indication of the relative roles of these models in predicting the fiber diameter of melt blown nonwoven fabrics. In part 2 (section 8.4), to meet the demand of establishing small-scaled ANN models, an input variable selection method was developed to help model the structure-property relations of nonwoven fabrics for filtration application. The structural parameters were selected by utilizing this method. The ANN models of structure-property relations of nonwovens were established. This section will establish a reasonably good ANN model that can generalize well and consider more structural parameters as the model inputs. 

In this section, an input variable selection method to rank the structural parameters was developed. This two-part method can deal with nonlinear relationships between input variables and output and no large number of data was required for running it. The first part takes the human knowledge on nonwoven products into account (VA/t). The second part was a data sensitivity criterion based on a distance method (S/t). 

Input variables specified for each error iUialy/Lil and how they were selected,... 

Our selection of the initial state, x0, and the value of the manipulated variables vector, u(t) determine a particular experiment. Here we shall assume that the input variables u(t) are kept constant throughout an experimental run. Therefore, the operability region is defined as a closed region in the [xoj.xo, , Xo,n, U u2,...,u,]T -space. Due to physical constraints these independent variables are limited to a very narrow range, and hence, the operability region can usually be described with a small number of grid points. 

Murray and Reiff(1984) showed that the use of an optimally selected square wave for the input variables can offer considerable improvement in parameter estimation. Of course, it should be noted that the use of constant inputs is often... 

Partition-based methods address dimensionality by selecting input variables that are most relevant to efficient empirical modeling. The input space is partitioned by hyperplanes that are perpendicular to at least one of the input axes, as depicted in Fig. 6d. 

Partition-based modeling methods are also called subset selection methods because they select a smaller subset of the most relevant inputs. The resulting model is often physically interpretable because the model is developed by explicitly selecting the input variable that is most relevant to approximating the output. This approach works best when the variables are independent (De Veaux et al., 1993). The variables selected by these methods can be used as the analyzed inputs for the interpretation step. 

Pm is the number of partitions or splits spm = 1 and indicates the right or left of the associated step function v(p, m) indicates the selected input variables in each partition and tpm represents the location of the split in the corresponding input space. The indices p and m are used for the split and node or basis function, respectively. The basis functions given by Eq. (34) are of a fixed, piecewise constant shape. 

The variables selected as design variables (fixed by the designer) cannot therefore be assigned as output variables from an f node. They are inputs to the system and their edges must be oriented into the system of equations. 

If, for instance, variables r3 and u4 are selected as design variables, then Figure 1.11 shows one possible order of solution of the set of equations. Different types of arrows are used to distinguish between input and output variables, and the variables selected as design variables are enclosed in a double circle. 

The third step is to select the number of iterations or calculations of dose that are to be performed as a part of each simulation. For the analysis here, a total of 10,000 iterations based on the selection of input variables from each defined distribution were performed as part of each simulation. The large number of iterations performed, as well as the Latin hypercube sampling (non-random sampling) technique employed by the Crystal Ball simulation program, ensured that the input distributions were well characterized, that all portions of the distribution (such as the tails) were included in the analysis, and that the resulting exposure distributions were stable. 

This construction works because G is free and we are dealing with free interpretations. When we come to a test we can nondeterministically select either path - seme free interpretation will take either path. The final output under any free interpretation must be either f. -.f x for some input variable x or else for some constant o. In the first case, G guesses this will... 

Let us review what we did with the depression example so far. First, we conjectured a taxon and three indicators. Next, we selected one of these indicators (anhedonia) as the input variable and two other indicators (sadness and suicidality) as the output variables. Input and output are labels that refer to a role of the indicator in a given subanalysis. We cut the input indicator into intervals, hence the word Cut in the name of the method (Coherent Cut Kinetics), and we looked at the relationship between the output indicators. Specifically, we calculated covariances of the output indicators in each interval, hence the word Kinetics —we moved calculations from interval to interval. Suppose that after all that was completed, we find a clear peak in the covariance of sadness and suicidality, which allows us to estimate the position of the hitmax and the taxon base rate. What next Now we need to get multiple estimates of these parameters. To achieve this, we change the... 

Indicator and sample selection are not the only choices a researcher has to make when using MAXCOV. A decision also has to be made about interval size, that is, how finely the input variable will be cut. Sometimes it is possible to use raw scores as intervals that is, each interval corresponds to one unit of raw score (e.g., the first interval includes cases that score one on anhedonia, the second interval includes cases that score two). This is what we used in the depression example. This approach usually works when indicators are fairly short and the sample size is very large, since it would allow for a sufficient number of cases with each raw score. In our opinion, this is the most defensible method of interval selection and should be used whenever possible. However, research data usually do not fit the requirements of this approach (e.g., the sample size is too small). Instead, the investigator can standardize indicators and make cuts at a fixed distance from each other (e.g.,. 25 SD), thereby producing intervals that encompass a few raw scores. 

A total of 185 emission lines for both major and trace elements were attributed from each LIBS broadband spectrum. Then background-corrected, summed, and normalized intensities were calculated for 18 selected emission lines and 153 emission line ratios were generated. Finally, the summed intensities and ratios were used as input variables to multivariate statistical chemometric models. A total of 3100 spectra were used to generate Partial Least Squares Discriminant Analysis (PLS-DA) models and test sets. 

McKay MD, Conover WJ, Beckman RJ. 1979. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21 239-245. 

Validation of models is desired but can be difficult to achieve. Models are empirically validated by examining how output data (predictions) compare with observed data (such comparisons, of course, must be conducted on data sets that have not been used to create or specify the model). However, model validations conducted in this manner are difficult given limitations on data sources. As an alternative approach, model credibility can be assessed by a careful examination of the subcomponents of the model and inputs. One should ask the question Does the selection of input variables and the way they are processed make sense Also, confidence in the model may be augmented by peer reviews and the opinion of the scientific community. Common faults and shortcomings are... 

In order to calcnlate the individual and societal risks that are relevant to solvent selection, mathematical models must be used. In a fuzzy logic approach, the characterization of the effects of solvent use is strictly connected with the fuzzy representation of the input variables, that are based on individual risk and societal risk (Bonvicini et al., 1998). 

Support Vector Machine (SVM) is a classification and regression method developed by Vapnik.30 In support vector regression (SVR), the input variables are first mapped into a higher dimensional feature space by the use of a kernel function, and then a linear model is constructed in this feature space. The kernel functions often used in SVM include linear, polynomial, radial basis function (RBF), and sigmoid function. The generalization performance of SVM depends on the selection of several internal parameters of the algorithm (C and e), the type of kernel, and the parameters of the kernel.31... 

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