Closure, double

If the original data represent counts or frequencies one defines  

In CFA the means m , mp, m and norms d , dp, d are computed by weighted sums and weighted sums of squares  

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. 

When all weight coefficients w and are constant then we have the special case 1. 1 

The effect of double-closure is shown in Table 31.8. For convenience, we have subtracted a constant value of one from all the elements of Z in order to emphasize the analogy of the results with those obtained by log double-centering in Table 31.7. The marginal means in the table are average values for the relative deviations from expectations and thus must be zero. 

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There is a close analogy between double-closure (eq. (31.51)) and log doublecentering (eq. (31.49)) which can be rewritten as ... 

Fig. 31.11. Biplot of chromatographic retention times in Table 31.2, resulting from correspondence factor analysis, i.e. after double-closure of the data. The line segments have been added to emphasize contrasts in the same way as in Fig. 31.10.

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. 

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. 

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. 

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