Row-closure

Comparison between rows of a contingency table X is made easier after dividing each element of the table by its corresponding row-sum. This operation is called row-closure as it forces all rows of the table to possess the same unit sum. After closure, the rows of the table are called row-profiles. These can be represented in the form of stacked histograms such as shown in Fig. 32.1. 

The average or expected row-profile is obtained by dividing the marginal row in the original table by the global sum. The matrix F of deviations of row-closed profiles from their expected values is defined by  

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It is important to realize that closure may reduce the rank of the data matrix by one. This is the case with row-closure when n>p, and with colunm-closure when n < p. It is always the case with double-closure. This reduction of the rank by one is the result of a linear dependence between the rows or columns of the table that results from closure of the data matrix. 

Note that double-closure yields the same results as those produced by row-closure in 5 and by column-closure in 5. From an algorithmic point of view, double-closure is the more attractive transformation, although row- and column-closure possess a strong didactic appeal. 

A special type of homogeneous measurements is found in a compositiorml table which describes chemical samples by means of the relative concentrations of their components. By definition, relative concentrations in each row of a compositional table add up to unity or to 100%. Such a table is said to be closed with respect to the rows. In general, closure of a table results when their rows or columns add up to a constant value. This operation is only applicable to homogeneous tables. Yet another type of homogeneous table arises when the rows or columns can be ordered according to a physical parameter, such as in a table of spectroscopic absorptions by chemical samples obtained at different wavelengths. 

Atmospheric data from Table 31.1, after double-closure. The weights w are proportional to the row- and column-sums of the original data table. They are normalized to unit sum. 

In the literature we encounter three common transformations of the contingency table. These can be classified according to the type of closure that is involved. By closure we mean the operation of dividing each element in a row or column of a table by its corresponding marginal sum. We reserve the word closure for the specific operation where the elements in a row or column of the table are reduced to unit sum. This way, we distinguish between closure and normalization, as the latter implies an operation which reduces the elements of a table to unit sums of squares. In a strict sense, closure applies only to tables with non-negative elements. 

Double-closure is the joint operation of dividing each element of the contingency table X by the product of its corresponding row- and column-sums. The result is multiplied by the grand sum in order to obtain a dimensionless quantity. In this context the term dimensionless indicates a certain synunetry in the notation. If x were to have a physical dimension, then the expressions involving x would appear as dimensionless. In our case, x represents counts and, strictly speaking, is dimensionless itself. Subsequently, the result is transformed into a matrix Z of deviations of double-closed data from their expected values ... 

Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

We assume that Z is a transformed nxp contingency table (e.g. by means of row-, column- or double-closure) with associated metrics defined by W and W. Generalized SVD of Z is defined by means of ... 

In CFA we can derive biplots for each of the three types of transformed contingency tables which we have discussed in Section 32.3 (i.e., by means of row-, column- and double-closure). These three transformations produce, respectively, the deviations (from expected values) of the row-closed profiles F, of the column-closed profiles G and of the double-closed data Z. It should be reminded that each of these transformations is associated with a different metric as defined by W and W. Because of this, the generalized singular vectors A and B will be different also. The usual latent vectors U, V and the matrix of singular values A, however, are identical in all three cases, as will be shown below. Note that the usual singular vectors U and V are extracted from the matrix. ... 

The log-linear model (LLM) is closely related to correspondence factor analysis (CFA). Both methods pursue the same objective, i.e. the analysis of the association (or correspondence) between the rows and columns of a contingency table. In CFA this can be obtained by means of double-closure of the data in LLM this is achieved by means of double-centring of the logarithmic data. 

For the same reason as for double-closure, double-centring always reduces the rank of the data matrix by one, as a result of the introduction of a linear dependence among the rows and columns of the data table. 

A // = 40kJmol-1 AS = —204JK 1mol 1). The activation parameters indicate associative activation for both steps (212). Associative chelate ring closure here is consistent with the mechanistic pattern established in organometallic systems (213), where this process is associative for [M(CO)5(diimine)] where M = Mo or W (though dissociative for M = first-row Cr (214)) and the diimine does not carry bulky substituents. 

Rank(Cg) = 1. In this case, all components of Cg are either 1 or —1, and it has only one independent row (column). If the allowable region at t = 0 is one-dimensional, then it will remain one-dimensional for all time (assuming that the rank does not change). This limiting case will occur when all scalars can be written as a function of the mixture fraction (e.g., the conditional-moment closure). 

As expected, the trace in Figure 5-19 is less ordered than the equivalent in Figure 5-18. Concentration profiles are governed by the law of mass action and closure and thus the trace, following the rows of US, is structured accordingly. No such law governs the relative shape of the absorption spectra and the trace following the columns of SV. 

We have to be careful. The symmetry between columns and rows of the matrix Y is not complete. Closure is a property of the concentration profiles only and thus applies only in one dimension. The command mean (Y, 1) computes the mean of each column of Y and the resulting mean spectrum is subtracted from each individual spectrum. 

The Diels-Alder reaction is the best known and most widely used pericyclic reaction. Two limiting mechanisms are possible (see Fig. 10.11) and have been vigorously debated. In the first, the addition takes place in concerted fashion with two equivalent new bonds forming in the transition state (bottom center, Fig. 10.11), while for the second reaction path the addition occurs stepwise (top row, Fig. 10.11). The stepwise path involves the formation of a single bond between the diene (butadiene in our example) and the dienophile (ethylene) and (most likely) a diradical intermediate, although zwitterion structures have also been proposed. In the last step, ring closure results with the formation of a second new carbon carbon bond. Either step may be rate determining. 

This table is derived as follows. We have I A =AI = A, and 11 = I. The only remaining product is AA, which by closure must be either A or /. If A A were equal to A, then A would occur twice in column 2 (and twice in row 2), thereby violating the theorem proved above. 

Weinberger and Baoh lint demonstrated that the Gabriel ring closure occurs with inversion at the substituted oarbon atom, since an optically active /row -2,3-chphenylaziridine was formed by cyclizaticn... 

The closure constraint is applied to closed reaction systems, where the principle of mass balance is fulfilled. With this constraint, the sum of the concentrations of all of the species involved in the reaction (the suitable elements in each row of the C matrix) is forced to be equal to a constant value (the total concentration) at each stage in the reaction [27, 41, 42], The closure constraint is an example of an equality constraint. 

Scaling die rows to a constant total is useful if the absolute concentrations of samples cannot easily be controlled. An example might be biological extracts the precise amount of material might vary unpredictably, but the relative proportions of each chemical can be measured. This method of scaling introduces a constraint which is often called closure. The numbers in the multivariate data matrix are proportions and... 

Baldwin has formulated rules for ring closure in a systematic manner . For digonal systems these are (/) 3- and 4-exo-dig, disfavoured ( ) 5- to 1-exo-dig, favoured (i/7) 3 to 1-endo-dig, favoured. In his terminology the choice in equation (58) is between S-exo-dig di di 6-endo-dig. On the basis of his survey Baldwin concludes that enrfo-ring closures at digonal carbon predominate. For the possibilities we encountered most often, namely, 5- to 7-rlngs, we find that first-row nucleophilic sites, e.g. O, N, C, favour 5-exo-dig and 6-exo-dig closures. These will be illustrated here and in later sections. 

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