Row-profile

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 ... 

In 5 the row-profiles are centred about the origin, as can be shown by working out the expression for the weighted column-means m ... 

Weighted sums of squares c of the row-profiles in F around the origin of can be expressed as distances of chi-square ... 

The distance 5, is a measure of the difference of the I th row-closed profile with respect to the average or expected row-profile. 

The distance 5 - between two row-profiles / and i can also be expressed formally as a distance of chi-square ... 

The second kind of transformations are done for chemical reasons and comprise the called "constant-row sum" and "normalization variable" (Johnson Ehrlich, 2002). Dealing with compositional data, concentrations can vary widely due to dilution away from a source. In the case of contaminated sediment investigations, for example, concentrations may decrease exponentially away from the effluent pip>e. However, if the relative propwrtions of individual analytes remain relatively constant, then we would infer a single source scenario coupled with dilution far away from the source. Thus, a transformation is needed to normalize concentration/ dilution effects. Commonly this is done using a transformation to a fractional ratio or percent, where each concentration value is divided by the total concentration of the sample Row profile or constant row-sum transformation because the sum of analyte concentrations in each sample (across rows) sums unity or 100% ... 

When n > p, the rank of data matrix is p (if all features are independent). Thus, autoscaling and minimax transformations do not change data dimensionaUty because these treatments do not induce any boimd between features. Row profiles instead build a relationship... 

The following chart shows a typical net effective gas radiating temperature profile, along with the individual burner row by burner row profiles. 

The wall thickness estimation in tangential projection technique is based on the evaluation of profile plots along the pipe diameter as shown in fig. 1 (lowest row). 

State-of-the-art for data evaluation of complex depth profile is the use of factor analysis. The acquired data can be compiled in a two-dimensional data matrix in a manner that the n intensity values N(E) or, in the derivative mode dN( )/d , respectively, of a spectrum recorded in the ith of a total of m sputter cycles are written in the ith column of the data matrix D. For the purpose of factor analysis, it now becomes necessary that the (n X m)-dimensional data matrix D can be expressed as a product of two matrices, i. e. the (n x k)-dimensional spectrum matrix R and the (k x m)-dimensional concentration matrix C, in which R in k columns contains the spectra of k components, and C in k rows contains the concentrations of the respective m sputter cycles, i. e. ... 

Water molecules are placed in the lower half of the grid, leaving the upper half empty. A temperature is selected using the Fb and J parameters and the CA is allowed to run for a specified time. The number of water molecules in each row of the upper half of the grid is counted. The grid is defined as a cylinder with the upper and lower boundaries stationary. This prevents water movement past the bottom boundary. A profile of evaporation versus temperature can be obtained by varying the simulated temperature. Use Example 3.5 in the Program CASim. 

Some analytical instruments produce a table of raw data which need to be processed into the analytical result. Hyphenated measurement devices, such as HPLC linked to a diode array detector (DAD), form an important class of such instruments. In the particular case of HPLC-DAD, data tables are obtained consisting of spectra measured at several elution times. The rows represent the spectra and the columns are chromatograms detected at a particular wavelength. Consequently, rows and columns of the data table have a physical meaning. Because the data table X can be considered to be a product of a matrix C containing the concentration profiles and a matrix S containing the pure (but often unknown) spectra, we call such a table bilinear. The order of the rows in this data table corresponds to the order of the elution of the compounds from the analytical column. Each row corresponds to a particular elution time. Such bilinear data tables are therefore called ordered data tables. Trilinear data tables are obtained from LC-detectors which produce a matrix of data at any instance during the... 

As indicated before, the columns and the rows of a bilinear or trilinear dataset have a particular meaning, e.g., a spectrum and a chromatogram or the concentration profiles of reactants and the reaction products in an equilibrium or kinetic study. The resulting data table is made up by the product of the tables of these pure factors, e.g., the table of the elution profiles of the pure compounds and the table of the spectra of these compounds. One of the aims of a study of such a table is the decomposition of the table into its pure spectra and pure elution profiles. This is done by factor analysis (Chapter 34). 

One finds an illustration of the measures of location and spread in the margins of Table 32.6. The geometric properties of row-closed and column-closed profiles are summarized in Fig. 32.5. 

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. ... 

All other cases can be readily derived by analogy. In the case of row-closed profiles we have to perform the following substitutions ... 

The rows of X are mixture spectra and the columns are chromatograms at the p = 20 wavelengths. Here, columns as well as rows are linear combinations of pure factors, in this example pure row factors, being the pure spectra, and pure column factors, being the pure elution profiles. 

The columns of V are the abstract factors of X which should be rotated into real factors. The matrix V is rotated by means of an orthogonal rotation matrix R, so that the resulting matrix F = V R fulfils a given criterion. The criterion in Varimax rotation is that the rows of F obtain maximal simplicity, which is usually denoted as the requirement that F has a maximum row simplicity. The idea behind this criterion is that real factors should be easily interpretable which is the case when the loadings of the factor are grouped over only a few variables. For instance the vector f, =[000 0.5 0.8 0.33] may be easier to interpret than the vector = [0.1 0.3 0.1 0.4 0.4 0.75]. It is more likely that the simple vector is a pure factor than the less simple one. Returning to the air pollution example, the simple vector fi may represent the concentration profile of one of the pollution sources which mainly contains the three last constituents. 

When the rows of a data matrix follow a certain pattern, e.g. the appearance and disappearance of compounds as a function of time, a fixed-size window EFA is applicable. This is the case, for instance, for data sets generated by hyphenated measurement techniques such as HPLC with DAD. Fixed-size window EFA [22] can be applied for detecting the presence of minor compounds (< 1 %) and for the resolution of a data set into its components (pure spectra and elution profiles). 

The pure variable technique can be applied in the column space (wavelength) as well as in the row space (time). When applied in the column space, a pure column is one of the column factors. In LC-DAD this is the elution profile of the compound which contains that selective wavelength in its spectrum. When applied in the row space, a pure row is a pure spectrum measured in a zone where only one compound elutes. 

The aim of all the foregoing methods of factor analysis is to decompose a data-set into physically meaningful factors, for instance pure spectra from a HPLC-DAD data-set. After those factors have been obtained, quantitation should be possible by calculating the contribution of each factor in the rows of the data matrix. By ITTFA (see Section 34.2.6) for example, one estimates the elution profiles of each individual compound. However, for quantitation the peak areas have to be correlated to the concentration by a calibration step. This is particularly important when using a diode array detector because the response factors (absorptivity) may considerably vary with the compound considered. Some methods of factor analysis require the presence of a pure variable for each factor. In that case quantitation becomes straightforward and does not need a multivariate approach because full selectivity is available. 

L oxygen exposure with a (2 x 1) structure present image (b) is after 42 L oxygen exposure with both (2 x 1) and (3 x 1) states present line profiles of the rows running in the < 100 > direction also shown, inter-row spacings are twice and three times the Cu-Cu distance in the < 110 > direction (c). Also shown is the image of a c(6 x 2) structure present as a minor component (b, d). (Reproduced from Ref. 16). 

K well-ordered chains running in the < 110 > direction separated by the atom resolved structure of the Cu(l 10) surface with a spacing between the rows of 0.36 nm (see line profile), (b) The spacing within the chains is 0.51 nm (see line profile), i.e. close to twice the Cu-Cu distance within the copper rows running in the < 110 > direction. (Reproduced from Refs. 16, 18). 

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