Measured variables selection

Thus we encounter the cascade control system, where the final element is manipulated through an intermediate or secondary controlled variable whose value is dependent on the primary. In ratio control systems, a specification is set on a designated mathematical combination of two or more measured variables. Selective control embodies the logical assignment of the final element to whichever controlled variable (of several) is in danger of violating its specified limits. Finally, adaption is the act of automatically modifying a controller to satisfy a combination of func-... 

Process Measurements. The most commonly measured process variables are pressures, flows, levels, and temperatures (see Flow LffiASURELffiNT Liquid-levell asurel nt PressureLffiASURELffiNT Temperaturel asurel nt). When appropriate, other physical properties, chemical properties, and chemical compositions are also measured. The selection of the proper instmmentation for a particular appHcation is dependent on factors such as the type and nature of the fluid or soHd involved relevant process conditions rangeabiHty, accuracy, and repeatabiHty requited response time installed cost and maintainabiHty and reHabiHty. Various handbooks are available that can assist in selecting sensors (qv) for particular appHcations (14—16). 

The type and extent of automatic processing to be carried out immediately after a chromatogram has been measured is controlled by a "processing" variable selected from a table displayed by the SETUP program. In our implementation, the choices are... 

Replicate samples are collected to evaluate the measurement variability of held and laboratory procedures. When sampling a water source, replicate samples (two or more) should be collected sequentially. Select wells for replicate sampling that are known to have a measurable concentration of the compound of interest. 

We should not have to estimate variables that we can measure. It is logical to design a reduced-order estimator which estimates only the states that cannot be measured or are too noisy to be measured accurately. Following our introductory practice, we will consider only one measured output. The following development assumes that we have selected x, to be the measured variable. Hence, the output is... 

FIGURE 6.16 Bar graph of (3-carotene and lycopene skin levels measured with selective RRS for seven subjects. White bars represent (3-carotene levels, black bars the lycopene levels. Note strong intersubject variability of (3-carotene to lycopene concentration ratios, indicated above the bar graphs. 

Using a classification algorithm we can determine the measured variables that are overmeasured, that is, the measurements that may also be obtained from mathematical relationships using other measured variables. In certain cases we are not interested in all of them, but rather in some that for some reason (control, optimization, reliability) are required to be known with good accuracy. On the other hand, there are unmeasured variables that are also required and whose intervals are composed of over measured parameters. Then we can state the following problem Select the set of measured variables that are to be corrected in order to improve the accuracy of the required measured and unmeasured process variables. 

In some cases, we do not want all the variables to be determinable only those that are required. Consequently, we must identify which of the measurable variables have to be measured. Let p be the set of variables that for various reasons should be known correctly p may be composed of measured and unmeasured variables. Sometimes we are not interested in the whole system being determinable, so we want to select which of the process variables have to be measured to have complete determinability of the variables in set p. This problem can be stated as follows Select the necessary measurements for the subset of required variables to be determinable. 

A number of performance criteria are not primarily dedicated to the users of a model but are applied in model generation and optimization. For instance, the mean squared error (MSE) or similar measures are considered for optimization of the number of components in PLS or PC A. For variable selection, the models to be compared have different numbers of variables in this case—and especially if a fit criterion is used—the performance measure must consider the number of variables appropriate measures are the adjusted squared correlation coefficient, adjR, or the Akaike S information criterion (AIC) see Section 4.2.3. 

The following criteria are usually directly applied to the calibration set to enable a fast comparison of many models as it is necessary in variable selection. The criteria characterize the fit and therefore the (usually only few) resulting models have to be tested carefully for their prediction performance for new cases. The measures are reliable only if the model assumptions are fulfilled (independent normally distributed errors). They can be used to select an appropriate model by comparing the measures for models with various values of in. 

The following three performance measures are commonly used for variable selection by stepwise regression or by best-subset regression. An example in Section 4.5.8 describes use and comparison of these measures. 

The most reliable approach would be an exhaustive search among all possible variable subsets. Since each variable could enter the model or be omitted, this would be 2m - 1 possible models for a total number of m available regressor variables. For 10 variables, there are about 1000 possible models, for 20 about one million, and for 30 variables one ends up with more than one billion possibilities—and we are still not in the range for m that is standard in chemometrics. Since the goal is best possible prediction performance, one would also have to evaluate each model in an appropriate way (see Section 4.2). This makes clear that an expensive evaluation scheme like repeated double CV is not feasible within variable selection, and thus mostly only fit-criteria (AIC, BIC, adjusted R2, etc.) or fast evaluation schemes (leave-one-out CV) are used for this purpose. It is essential to use performance criteria that consider the number of used variables for instance simply R2 is not appropriate because this measure usually increases with increasing number of variables. 

An important point is the evaluation of the models. While most methods select the best model at the basis of a criterion like adjusted R2, AIC, BIC, or Mallow s Cp (see Section 4.2.4), the resulting optimal model must not necessarily be optimal for prediction. These criteria take into consideration the residual sum of squared errors (RSS), and they penalize for a larger number of variables in the model. However, selection of the final best model has to be based on an appropriate evaluation scheme and on an appropriate performance measure for the prediction of new cases. A final model selection based on fit-criteria (as mostly used in variable selection) is not acceptable. 

For each chromosome (variable subset), a so-called fitness (response, objective function) has to be determined, which in the case of variable selection is a performance measure of the model created from this variable subset. In most GA applications, only fit-criteria that consider the number of variables are used (AIC, BIC, adjusted R2, etc.) together with fast OLS regression and fast leave-one-out CV (see Section 4.3.2). Rarely, more powerful evaluation schemes are applied (Leardi 1994). 

FIGURE 4.20 Scheme of a GA applied to variable selection. The first chromosome defines a variable subset with four variables selected from m = 10 variables. Fitness is a measure for the performance of a model built from the corresponding variable subset. The population of chromosomes is modified by genetically inspired actions with the aim to increase the... 

Stepwise Perform a stepwise variable selection in both directions start once from the empty model, and once from the full model the AIC is used for measuring the performance. 

Revisit variable selection A profound expansion or shift in the relevant analyzer response space, or in the measurement objectives, could result in the need to reconsider the optimal set of variables to use for the calibration model. This action is usually considered when new or additional calibration data are obtained. 

Unlike MUR, PCR and PLS are methods that can be used without explicitly selecting variables. This is accomplished by transforming the measured variables (e.g., absorbance values at many wavelengths) into new variables (often referred to as factors) that are used in the matrix calculations. The difference between PCR and PLS is in how this variable transformation is performed. Both PCR and PLS have good diagnostic tools and in general the results are similar. These methods are often preferred over MLR unless the number of variables is small or circumstances dictate the explicit reduction in the number of varialj es. 

As discussed in the introduction, the solution of the inverse model equation for the regression vector involves the inversion of R R (see Equation 5 23). In many anal al chemistry experiments, a large number of variables are measured and R R cannot be inverted (i.e., it is singular). One approach to solving this problem is called stepwise MLR where a subset of variables is selected such that R R is not singular. There must be at least as many variables selected as there are chemical components in the system and these variables must represent different sources of variation. Additional variables are required if there are other soairces of variation (chemical or physical) that need to be modeled. It may also be the case that a sufficiently small number of variables are measured so that MIR can be used without variable selection. 

The only completely rigorous method to determine an optimum subset of variables is to test all possible combinations. The problem with this approach is that even with as few as 100 variables the number of combinations becomes too large to consider. For example, to select an optimum subset of three variables from amongst 100 measurement variables, 161,700 different combinations need to be tested. Because of the large number of possibilities, various strategies have been developed to select variables with a reasonable amount of work. The most common approaches include step forward, step backward,... 

Selected Variable Plot (Model and Variable Diagnostic) The variables that have been induded in the model should be examined to see if they are reasonable given knowledge about the chemistry of the samples and measurements. Figure 5.68 displays the calibration spectra with vertical lines indicating the variables selected for the models for components A and B. 

Candidate Tariables were chosen using a mixed-variable selection method and validated based on prediction ability. Separate models (with different measurement variables) were estimated for each of the components. The final models and measures of performance are as follows (see Table 5.11 for a description of these figures of merit) ... 

Calibration Design 9 samples, selected using a mixture design Preprocessing baseline correction using the average of the first 10 measurement variables. 

In Section 5.3 the inverse methods of MLR and PLS/PCR are discussed. The one challenge in using the inverse approach is in the inversion of a matrix. The two approaches discussed for solving the inversion problem are to select variables (MLR) or to estimate factors to use in place of the original measurement variables (PLS/PCR). 

One particular challenge in the effective use of MLR is the selection of appropriate X-variables to use in the model. The stepwise and APC methods are some of the most common empirical methods for variable selection. Prior knowledge of process chemistry and dynamics, as well as the process analytical measurement technology itself, can be used to enable a priori selection of variables or to provide some degree of added confidence in variables that are selected empirically. If a priori selection is done, one must be careful to select variables that are not highly correlated with one other, or else the matrix inversion that is done to calculate the MLR regression coefficients (Equation 8.24) can become unstable, and introduce noise into the model. 

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