Redundant variables

Whitley DC, Ford MG, Livingstone DJ. Unsupervised forward selection a method for eliminating redundant variables. J Chem Inf Comput Sci 2000 40 1160-8. 

You can see that the model for a realistic process can become extremely complex what is important to remember is that steps 1 and 3 in Table 1.1 provide an organized framework for identifying all of the variables and formulating the objective function, equality constraints, and inequality constraints. After this is done, you need not eliminate redundant variables or equations. The computer software can usually handle redundant relations (but not inconsistent ones). 

The process has four separate subsystems for the degree-of-freedom analysis. Redundant variables and redundant constraints are removed to obtain the net degrees of freedom for the overall process. The 2 added to Nsp refers to the conditions of temperature and pressure in a stream +1 represents the heat transfer Q. 

Redundant variables in interconnecting streams being eliminated ... 

As was discussed in Chapters 3 and 4, variable classification allows us to obtain a reduced subsystem of redundant equations that contain only measured and redundant variables. These are used in the reconciliation procedure. 

Let us consider the system of g overmeasured (redundant) variables in m balance equations. Assuming that all of the errors are normally distributed with zero mean and variance I>, it has been shown that the least squares estimate of the measurement errors is given by the solution of the following problem ... 

Because many redundant variables are used, it is possible to quantitate analjTies withoverlapping features. 

Thus the Hessian will become singular if we include rotations between the active orbitals. Redundant parameters must not be included in the Newton-Raphson procedure.They are trivial to exclude for the examples given above, but in more general cases a redundant variable may occur as a linear combination of S and T and it might be difficult to exclude them. One of the advantages of the CASSCF method is that all parameters except those given above are non-redundant. 

Information can be defined as the scatter of points in a measurement space. Correlations between measurement variables decrease the scatter and subsequently the information content of the space [39] because the data points are restricted to a small region of the measurement space because of correlations among the measurement variables. If the measurement variables are highly correlated, the data points could even reside in a subspace. This is shown in Figure 9.2. Each row of the data matrix is an object, and each column is a measurement variable. Here x3 is perfectly correlated with x, and x2, since x3 (third column) equals x3 (first column) plus x2 (second column). Hence, the seven data points lie in a plane (or two-dimensional subspace), even though each point has three measurements associated with it. Because x3 is a redundant variable, it does not contribute any additional information, which is why the data points lie in two dimensions, not three dimensions. 

We must deduct redundant variables and add redundant restrictions as follows ... 

Figure E5.5a shows a simplified flowsheet. All the units except the separator and lines are adiabatic. The liquid ammonia product is essentially free of Nz, Hz, and A, and assume that the purge gas is free of NH3. Treat the process as four separate units for a degree-of-freedom analysis, and then remove redundant variables and add redundant constraints to obtain the degrees of freedom for the overall process. The fraction conversion in the reactor is 25%.

If we have a set of k experimental variables at least (k + 1) individual experiments must be run to make an evaluation of each individual variable possible. This means that to keep the number of experiments acceptably low, redundant variables should be removed. 

A. Redundant Variables and the Configuration State Function Expansion... 

Section V consists of a detailed discussion of redundant variables. The special case of MCSCF wavefunction optimization for two-electron systems is discussed in some detail. The relation between the configuration expansion space and the orbital variation space is quite straightforward for this case and this simplicity may be used to advantage in understanding the generalization to arbitrary numbers of electrons. There are two aspects of redundant variables that are important in the MCSCF method. First, if redundant variables are allowed to remain in the wavefunction variation space, then the optimization procedure becomes undefined or at least numerically ill-conditioned. Secondly, if the redundant variables are known for a given wavefunction then this flexibility may be used to transform the wavefunction to a form that is qualitatively easier to understand. The qualitative interpretation of MCSCF wavefunctions is one of the assets of the MCSCF method. 

If the orbitals within each electron pair of the RCI expansion are restricted to be singlet coupled, the expansion may be written as n /2f (2 S = 0) (2 S = 0) or as (nV2)" (n" RCI, S=0)" and, for singlet states, consists of 3" CSFs where n" is the number of active electrons. In this expansion, each open-shell orbital occupation corresponds to a single CSF. For a single correlated electron pair, the open-shell CSF may be eliminated without restricting the wavefunction with an appropriate orbital choice. This is considered in more detail in the discussion of redundant variables in Section V. For more than one... 

The set of variables in the MCSCF optimization process determine the changes in the CSF mixing coefficients and the orbital expansion coefficients during each iteration. When particular choices of CSFs are employed in the wavefunction expansion or when particular relations between the CSF expansion coefficients are satisfied, redundant variables will occur within this... 

As discussed earlier, many CSF expansion spaces are purposely chosen to be full with respect to some subset of the orbitals. It was mentioned that this resulted in the occurrence of redundant variables in the optimization procedure. In this section, the details of this relation between these redundant variables and the CSF expansion space are examined, first for the two-electron case and then for the general iV-electron case. 

The general discussion of redundant variables will be preceded by a detailed discussion of the special case of two-electron wavefunctions. The formal manipulations are easier to understand for this case than in the general N-electron case. Additionally, the exact full-CI wavefunctions may be included in these formal manipulations. Finally, several features of the two-electron case may then be used in understanding the more general cases. For the discussion of this special case, the approach of Mclver is used to define a matrix of determinants of the form... 

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