Nonredundant measurement

For measurement adjustment, a constrained optimization problem with model equations as constraints is resolved at a fixed interval. In this context, variable classification is applied to reduce the set of constraints, by eliminating the unmeasured variables and the nonredundant measurements. The dimensional reduction of the set of constraints allows an easier and quicker mathematical resolution of the problem. 

Classification of the measured variables included in NA1 and NA2 as redundant. The other measurements are categorised as nonredundant. Measured variable classification results for this example are in Table 7. 

From the classification it was found that, for this specific problem, there are 10 redundant and 6 nonredundant measured variables, and all the unmeasured process variables are determinable. Symbolic manipulation of the equations allowed us to obtain the three redundant equations used in the reconciliation problem ... 

Let us further consider a nonredundant measured variable Xj. Let again 2 e iW (8.5.31). We have a curve on parametrized as... 

A nonredundant measured variable is not affected by the solvability condition it can be freely varied in some interval leaving the model equations solvable, if the remaining fixed measured variables obey the condition of solvability. The necessary condition reads, according to (8.5.39)... 

The authors used these measures to construct three indicator sets. First, eight nonredundant PSWQ items were selected to have high correlations with the PSWQ total score. Second, a set of four item pairs was created by... 

Steady-state process variables are related by mass and energy conservation laws. Although, for reasons of cost, convenience, or technical feasibility, not every variable is measured, some of them can be estimated using other measurements through balance calculations. Unmeasured variable estimation depends on the structure of the process flowsheet and on the instrument placement. Typically, there is an incomplete set of instruments thus, unmeasured variables are divided into determinable or estimable and indeterminable or inestimable. An unmeasured variable is determinable, or estimable, if its value can be calculated using measurements. Measurements are classified into redundant and nonredundant. A measurement is redundant if it remains determinable when the observation is deleted. 

A measured process variable, belonging to subset x, is called nonredundant (just-measured) if it cannot be computed from the balance equations and the rest of the measured variables. 

Remark 4. As indicated by Crowe et al. (1983), measured variable classification is performed by examining the matrix associated with the reconciliation equations. The zero columns of G or Gx correspond to variables that do not participate in the reconciliation, so they are nonredundant. The remaining columns correspond to redundant measurements. 

Unmeasured temperatures or concentrations that correspond to enthalpy or component flows in Vd are determinable if the total flow rate of the stream is measured. Otherwise, they are indeterminable. Measured total flow rates are nonredundant and unmeasured total flow rates are indeterminable. The analysis of intensive constraints between variables may change previous classification. 

In hardware sizing and capacity planning, two sources of data are used. The first is the server capacity, often given as a SPEC mark, or the time in which a standard set of procedures is executed. A less-accurate measure of computing capacity is processor speed in cycles per second. Memory size may play a role in some calculations. BLAST queries, for example, are limited by the fetches from the disk the rule is, the more memory the better. Optimally, the entire nonredundant GenBank database can be stored in memory rather than on disk. The second component of the capacity analysis is the computation load, modeled as a typical workload factors. 

Now that all the superfluous information has been removed as shown in Table 3.4, what remains can be translated in association rules that represent the simplest possible rules that conserve the nonredundant information in the data. These rules are characterized by their strength as measured by the support for a given rule, which is based on the number of molecules in tlie dataset that support the rule. Since the example involves a very small (hypothetical) dataset, support for the rules will only be illustrative of the methodology. As will be seen in Section 3.3, a more meaningful analysis can be made on a real drug-induced phospholipi-dosis dataset that is focus of this work (vide infra). Although that dataset is also not overly large, it nonetheless represents data associated with an actual problem of interest. 

Chapters 7 and 8 are devoted to the problems of solvability. We call a set of equations solvable when there exists some vector of solutions, not necessarily unique. In Chapter 3, we have shown that the set of mass balance equations is always solvable if no variable has been fixed a priori. With redundant measured variables, the equations need not be (and usually are not) solvable, unless the fixed variables have been adjusted. Then certain unknown (unmeasured) variables are uniquely determined (observable), other still not (unobservable variables). Certain measured variables can be nonredundant they... 

Graph 2-7a represents one balancing equation around one node. There is also one unknown flow which can be calculated from the equation. The system is just solvable and the unmeasured flow is observable. The two measured flows are nonredundant. 

In practice we can meet with even more complicated situations - see Fig. 2-7d. Streams 1, 2, 4 and 5 are measured and redundant (one stream can be calculated from the others). Stream 6 is measured, but nonredundant. Streams 3 and 7 are unmeasured and observable. Streams 8 and 9 are unmeasured and unobservable. The general classification of balancing variables is presented in Fig. 2-8. Anyway, we can see that even in a relatively simple flowsheet with the... 

The j-th measured variable is nonredundant if and only if arc j is deleted by the graph reduction (merging the nodes connected by unmeasured streams). 

As an example, recall Fig. 3-6 where the j -th variable is nonredundant. As drawn, all the unmeasured variables are observable. Note that if we deleted the j -th measurement, the three unmeasured streams j , i, k forming a circuit would become unobservable. 

The structure of the graph G along with the partition (3.6.1) allows one to classify all the variables (mass flowrates) with respect to the solvability see Section 3.3. According to the standard terminology, the variables mj, i J are called measured, the remaining (j J°) unmeasured. This classification will be used henceforth any a priori fixed variable of a model will be called measured , else unmeasured . The classification of the measured variables follows immediately from the partition (3.6.3) if i e J then m is redundant, else (i G J ) nonredundant. The nonredundant variables are also characterized by the property that i g J if and only if arc i closes a circuit in G with certain unmeasured streams (/ g J°). The nonredundant variables are unaffected by the solvability condition. 

The definition of a redundant variable thus depends only on the matrix (B, A), not on the value x. A measured variable that is not redundant is called nonredundant. If it is the h-ih. variable then the condition reads a = 0 where a is the h-th column vector of A . If in particular L = M, we have no additional condition, thus no redundant variable, and all the measured variables are nonredundant. Summarizing... 

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