Classification methods measured variables

An elegant classification strategy using projection matrices was proposed by Crowe et al. (1983) for linear systems and extended later (Crowe, 1986, 1989) to bilinear ones. Crowe suggested a useful method for decoupling the measured variables from the constraint equations, using a projection matrix to eliminate the unmeasured process variables. 

Points with a constant Euclidean distance from a reference point (like the center) are located on a hypersphere (in two dimensions on a circle) points with a constant Mahalanobis distance to the center are located on a hyperellipsoid (in two dimensions on an ellipse) that envelops the cluster of object points (Figure 2.11). That means the Mahalanobis distance depends on the direction. Mahalanobis distances are used in classification methods, by measuring the distances of an unknown object to prototypes (centers, centroids) of object classes (Chapter 5). Problematic with the Mahalanobis distance is the need of the inverse of the covariance matrix which cannot be calculated with highly correlating variables. A similar approach without this drawback is the classification method SIMCA based on PC A (Section 5.3.1, Brereton 2006 Eriksson et al. 2006). 

HCA is a common tool that is used to determine the natural grouping of objects, based on their multivariate responses [75]. In PAT, this method can be used to determine natural groupings of samples or variables in a data set. Like the classification methods discussed above, HCA requires the specification of a space and a distance measure. However, unlike those methods, HCA does not involve the development of a classification rule, but rather a linkage rule, as discussed below. For a given problem, the selection of the space (e.g., original x variable space, PC score space) and distance measure (e.g.. Euclidean, Mahalanobis) depends on the specific information that the user wants to extract. For example, for a spectral data set, one can choose PC score space with Mahalanobis distance measure to better reflect separation that originates from both strong and weak spectral effects. 

VARIABLE (Process). The quantity or characteristic that is the object of measurement in an instrumentation or automatic control system. Odier terms used include measurement variable, instrumentation variable, and process variable, The latter term is commonly used in the manufacturing industries, Numerous ways to classify variables have been proposed—by methods of measurement, by end-measurement objectives, and so on. One of the most convenient and meaningful classifications is the physical and/or chemical nature of the variable, as follows ... 

Figure 13 shows the membership function foe each state variable and the method of control in each phase. All the rules and classifications are those described in Table 3. We used the total CO2 evolved, pH, and pH slope as measurement variables for control. S,... 

A second distinction among the different classification methods proposed in the literature concentrates on the mathematical form of the functional relationship representing the classification rules in terms of the measured variables (or, alternatively, on the geometrical shape of the decision boundaries in the multidimensional space). In this framework, the main differentiation is made between linear and non-linear methods, even if the latter can be sometimes further subdivided according to the kind of non-linearity they implement (e.g. quadratic, polynomial, etc.). In linear methods, the classification mles result in decision boundaries, which are linear functions of the original variables (i.e. which correspond to linear surfaces in the hyperspace spanned by the variables a line in two dimensions, a plane in three... 

A mathematically very simple classification procedure is the nearest neighbour method. In this method one computes the distance between an unknown object u and each of the objects of the training set. Usually one employs the Euclidean distance D (see Section 30.2.2.1) but for strongly correlated variables, one should prefer correlation based measures (Section 30.2.2.2). If the training set consists of n objects, then n distances are calculated and the lowest of these is selected. If this is where u represents the unknown and I an object from learning class L, then one classifies u in group L. A three-dimensional example is given in Fig. 33.11. Object u is closest to an object of the class L and is therefore considered to be a member of that class. 

When the balance equations are formulated around individual units only, it is possible that the classification by output set assignment may not be satisfactory. Some variables classified as indeterminable may actually be determinable if we consider additional balances around groups of units. An erroneous measurement classification is also possible. The problem is in the system of equations used in the classification rather than in the assignment method. The most common problem arises because of the presence of parallel streams between two units. 

TTie classification of kinetic methods proposed by Pardue [18] is adopted in the software philosophy. TTie defined objective of measurement in the system is to obtain the best regression fit to a minimum of 10 data points, taken over either a fixed time (i.e. the maximum time for slow reactions) or variable time (for reactions complete in less than 34 min, which is the maximum practical observation time). In an analytical system generating information at the rate of SO datum points per second, with reactions being monitored for up to 2040 s, effective data-reduction is of prime importance. To reduce this large quantity of analytical data to more manageable proportions, an algorithm was devised to optimize the time-base of the measurements for each individual specimen. 

In addition to the somewhat empirical and difficult development of NIR applications, thorough documentation must be produced. NIR methods have to comply with the current good manufacturing practice (cGMP) requirements used in the pharmaceutical industry. Various regulatory aspects have to be carefully considered. For example, NIR applications in classification, identification, or quantification require extensive model development and validation, a study of the risk impact of possible errors, a definition of model variables and measurement parameters, and... 

Figure 8.1 Schematic classification of complexation measurement methods as a function of the perturbations that they can create at the discriminator (sensitive part of the analytical system that enables differentiation of the chemical species of interest from the other components present) and in solution. The compound reacting with the discriminator and the nature of the discriminator are shown in parentheses, a Constant cell volume methods are less perturbing than variable volumes, b Possibility of ligand release by organisms, c Possibility of interactions with the indicator (ligand with suitable absorbance or fluorescence properties added into the test solution in spectro-metric methods), d Possibility of contamination of very dilute media by ISE membranes (redrawn from Buffle, 1988).

Real-time optimization (RTO) schemes improve process performance by adjusting selected optimization variables using available measurements. The goal of this closed-loop adaptation is to drive the operating point towards the true plant optimum in spite of inevitable structural and parameter model errors. RTO methods can be classified in different ways. This section presents one such classification based on the parameters that can be adapted, as illushated in Fig. 1 note that repeated numerical optimization is used in the methods of columns 1 and 2, but not in those of column 3. 

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