Column-closure

Comparison between columns of a contingency table X is also made easier after dividing each element of the table by its corresponding column-sum. This operation is referred to as column-closure as it makes all columns of the table possess the same unit sum. After closure, the columns of the table are called column-profiles. These can also be visualized in the form of stacked histograms as is shown in Fig. 32.2. 

The average or expected column-profile results from dividing the marginal column by the global sum. The matrix G of deviations of column-profiles from their expected values is given by  

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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. 

Brenguier J. L., Chuang P. Y., Fouquart Y., Johnson D. W., Parol F., Pawlowska H., Pelon J., Schuller L., Schroder F., and Snider J. (2000) An overview of the ACE-2 Cloudy-Column closure experiment. Tellus Ser. B-Chem. Phys. Meteorol. 52, 815-827. 

All the y-sultines were obtained as diastereomeric mixtures (ca 1 1, by NMR), and each one of y-sultines ( + )-49 and ( + )-51 (R = t-Bu) was separated into two diastereomers A and B by column chromatography. The oxidation of y-sultines (— )-49A and (+ )-49B to the corresponding optically active sultones (+ )-52A,B, which lack a chiral sulfur, may be taken as proof that the observed optical activity in the sultines is also due to the y-carbon. This result seems to exclude the intermediacy of vinylsulfene in the reaction mechanism, since its disrotatory closure would lead to racemic y-carbon in the product. 

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 ... 

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. 

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. 

Fig. 5.5(b) Single unit responses in neocortex of dog effect of N-P duct closure [= columns] on response to conspecific odours. Ou = own urine, Of = own faeces, Su = strange urine, Sf = strange faeces, and C. = dry food for dog (from Onoda et al., 1981). 

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. 

Figure 9. Shellfish toxin data from Newagen, Maine. Upper panel is times of State closure to the taking of shellfish (>80pg/100g) for years 1972 to 1982. Bottom panel is "start dates" indicating when initial levels over quarantine are witnessed. These dates fall in categories 1) transition from mixed to stratified water column (May) and vice versa (Sept.) and 2) during summer meteorological events.

Paraffin conversion to naphthenes is very unfavorable (last column of Table IV). For paraffins to be converted to naphthenes by ring closure, naphthenes must be at very low concentrations. If appreciable naphthenes exist, such as at short catalyst contact times, naphthene ring opening to paraffins can occur. Again, equilibria improve with carbon number. Eight-and nine-carbon paraffins behave quite similarly. 

Fixed-Roof Tanks. The effect of internal pressure on plate structures, including tanks and pressure vessels, is important to tank design. If a flat plate is subjected to pressure on one side, it must be made quite thick to resist bending or deformation. A shallow cone-roof deck on a tank approximates a flat surface and is typically built of 3/ 16-in. (4.76-mm) thick steel (Fig. 4a). This is unable to withstand more than a few inches of water column pressure. The larger the tank, the more severe the effect of pressure on the structure. As pressure increases, the practicality of fabrication practice and costs force the tank builder to use shapes more suitable for internal pressure. The cylinder is an economic and easily fabricated shape for pressure containment. Indeed, almost all large tanks are cylindrical. The problem, however, is that the ends must be closed. The relatively flat roofs and bottoms or closures of tanks do not lend themselves to much internal pressure. As internal pressure increases, tank builders use roof domes or spheres. The spherical tank is the most economic shape for internal pressure storage in terms of required thickness, but it is generally more difficult to fabricate than a dome- or umbrella-roof tank because of its compound curvature. 

In Fig. 2.7, Rt and R2 are either simple alkyl or aryl chains (methyl or phenyl) or incorporate functional groups (e.g. cyanopropyl, trifluoropropyl). Combined in different proportions, Ri and R2 modify the polarity and the characteristics of the columns. One of the processes used to obtain a bonded polydimethylsiloxane phase is to allow a solution of tetradimethylsiloxane to flow through the column, then heat to 400 °C after evaporation of solvent and closure of the the extremities (Fig. 2.7). 

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. 

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