Overdetermination

The analogous procedure for a multivariate problem is to obtain many experimental equations like Eqs. (3-55) and to extract the best slopes from them by regression. Optimal solution for n unknowns requires that the slope vector be obtained from p equations, where p is larger than n, preferably much larger. When there are more than the minimum number of equations from which the slope vector is to be extracted, we say that the equation set is an overdetermined set. Clearly, n equations can be selected from among the p available equations, but this is precisely what we do not wish to do because we must subjectively discard some of the experimental data that may have been gained at considerable expense in time and money. 

If the source fingerprints, for each of n sources are known and the number of sources is less than or equal to the number of measured species (n < m), an estimate for the solution to the system of equations (3) can be obtained. If m > n, then the set of equations is overdetermined, and least-squares or linear programming techniques are used to solve for L. This is the basis of the chemical mass balance (CMB) method (20,21). If each source emits a particular species unique to it, then a very simple tracer technique can be used (5). Examples of commonly used tracers are lead and bromine from mobile sources, nickel from fuel oil, and sodium from sea salt. The condition that each source have a unique tracer species is not often met in practice. 

A stmcture, dehned by a set of nodes and a set of stmts that interconnect the nodes, may be interconnected in a way that under-constrains the nodes, may determinately constrain the nodes or it may be overdetermined or indeterminate. 

Overdetermination of the system of equations is at the heart of regression analysis, that is one determines more than the absolute minimum of two coordinate pairs (xj/yi) and xzjyz) necessary to calculate a and b by classical algebra. The unknown coefficients are then estimated by invoking a further model. Just as with the univariate data treated in Chapter 1, the least-squares model is chosen, which yields an unbiased best-fit line subject to the restriction ... 

As a result of compelling three-dimensional models and remarkably high levels of precision, it is often assumed that structural elucidation by single crystal X-ray diffraction is the ultimate structural proof. Spatial information in the form of several thousands of X-ray reflection intensities are used to solve the position of a few dozen atoms so that the solution of a structure by X-ray diffraction methods is highly overdetermined, with a statistically significant precision up to a few picometers. With precise atomic positions, structural parameters in the form of bond distances, bond... 

Equation (6.79) is valid for exactly determined systems (m = n). In case of overdetermined systems, m > n, the condition number is given by... 

In case of serious overlappings, multivariate techniques (see Sect. 6.4) are used and p ) > n sensors (measuring points zjt) are measured for n components. From this an overdetermined systems of equations results and, therefore, non-squared sensitivity matrixes. Then the total multicomponent sensitivity is given by... 

In the previous chapter we presented the problem of fitting data when there is more information (in the form of equations relating the several variables involved) available than the minimum amount that will allow for the solution of the equations. We then presented the matrix equations for calculating the least squares solution to this case of overdetermined variables. How did we get from one to the other ... 

Thus, when the attention of the mathematicians of the time turned to the description of overdetermined systems, such as we are dealing with here, it was natural for them to seek the desired solution in terms of probabilistic descriptions. They then defined the best fitting equation for an overdetermined set of data as being the most probable equation, or, in more formal terminology, the maximum likelihood equation. 

We will begin our discussion by demonstrating that, for a non-overdetermined system of equations, the algebraic approach and the least-square approach provide the same solution. We will then extend the discussion to the case of an overdetermined system of equations. 

Starting from this expression, one can execute the derivation just as in the case of the full equation (i.e., the equation including the constant term), and arrive at a set of equations that result in the least square expression for an equation that passes through the origin. We will not dwell on this point since it is not common in practice. However, we will use this concept to fit the data presented, just to illustrate its use, and for the sake of comparison, ignoring the fact that without the constant term these data are overdetermined, while they are not overdetermined if the constant term is included - if we had more data (even only one more relationship), they would be overdetermined in both cases. 

Unfortunately, the required number of constraints is not negotiable. Regardless of the difficulty of determining these values in sufficient number or the apparent desirability of including more than the allowable number, the system is mathematically underdetermined if the modeler uses fewer constraints than components, or overdetermined if he sets more. 

The use of this extended planning model will only be problematic if extra reference points, e.g., initial tank storage levels, have to be considered. This may lead to overdetermination of the model (i.e., conflicting level values for a given point in time) and it may be necessary to solve a data reconciliation problem. ... 

Due to this overdetermination problem there exists a strong prejudice in the chemical industry that production planning for batch production is completely different from production planning for continuous production. This leads to the ineffective separation between process engineering and business data processing in continuous and semicontinuous production. 

Nf < 0 The problem is overdetermined. If NF < 0, fewer process variables exist in the problem than independent equations, and consequently the set of equations has no solutions. The process model is said to be overdetermined, and least squares optimization or some similar criterion can be used to obtain values of the unknown variables as described in Section 2.5. 

Typically, process data are improved using spatial, or functional, redundancies in the process model. Measurements are spatially redundant if more than enough data exist to completely define the process model at any instant, that is, the system is overdetermined and requires a solution by least squares fitting. Similarly, data improvement can be performed using temporal redundancies. Measurements are temporally redundant if past measurement values are available and can be used for estimation purposes. Dynamic models composed of algebraic and differential equations provide both spatial and temporal redundancy. 

A measured process variable, belonging to subset x, is called redundant (overdetermined) if it can also be computed from the balance equations and the rest of the measured variables. 

In Chapters 3 and 4 we have shown that the vector of process variables can be partitioned into four different subsets (1) overmeasured, (2) just-measured, (3) determinable, and (4) indeterminable. It is clear from the previous developments that only the overmeasured (or overdetermined) process variables provide a spatial redundancy that can be exploited for the correction of their values. It was also shown that the general data reconciliation problem for the whole plant can be replaced by an equivalent two-problem formulation. This partitioning allows a significant reduction in the size of the constrained least squares problem. Accordingly, in order to identify the presence of gross (bias) errors in the measurements and to locate their sources, we need only to concentrate on the largely reduced set of balances... 

Note that the coefficient vector found from (5.65) is overdetermined. Indeed, by summing the component governing equations and the initial and boundary conditions, it is... 

In most cases of interest, however, the system represented by equation (5.3.3) is overdetermined and we must enforce the closure condition with a different method. Let us return to a standard mass-balance least-square problem, such as, for instance, calculating the mineral abundances from the whole-rock and mineral chemical compositions. If xu x2,.. -,x are the mineral fractions, which may be lumped together in a vector x, the closure condition... 

Arranging the whole-rock mineral concentrations for each element i (i=l,...,m) in a vector y, and putting the concentration of element i in the phase j at the ith row and jth column of the matrix Amxn (mineral matrix), the usual overdetermined system is obtained... 

This equation can be combined with either Equation (4.27) or (4.28), and in each procedure alternative values can be computed for AH ,(j y and for A//uipala-An additional method makes use of all three equations simultaneously to take advantage of all data available. In such an overdetermined set of data, one can use the method of least squares (see Appendix A) for more than one independent variable and matrix methods to solve the resulting equations [9]. 

Having obtained a set of mineral components that satisfactorily reproduces Che data, we have defined Che matrix of Equation 5. Given jC and C, Equation 5 Is Chen used Co solve for For Che alblte cluster with 107 samples and 4 mineral components, F Is 107 X 4 matrix containing the mass fractions of each mineral component In each sample. Because these mass.fractions sum to 1.0 for each sample, assuming we are accounting for all the mineral matter, we solve an overdetermined set of simultaneous equations of Che form... 

No single spectrometer is the perfect input to Eq. (2). Each experimental plan has limitations and many of these are associated with accessible compositions, temperatures and pressures. In addition, there is the issue of having an overdetermined system as remarked previously. Taken together, we arrive at the conclusion ... 

The phase problem of X-ray crystallography may be defined as the problem of determining the phases ( ) of the normalized structure factors E when only the magnitudes E are given. Since there are many more reflections in a diffraction pattern than there are independent atoms in the corresponding crystal, the phase problem is overdetermined, and the existence of relationships among the measured magnitudes is implied. Direct methods (Hauptman and Karle, 1953) are ab initio probabilistic methods that seek to exploit these relationships, and the techniques of probability theory have identified the linear combinations of three phases whose Miller indices sum to... 

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