Relations between Measurement Tables

As an example consider the data presented in Tables 35.1-35.4. These tables are extracted from a much larger data base obtained in an international cooperative study on the sensory aspects of olive oils [1]. Table 35.1 gives the mean scores for 16 samples of olive oil with respect to six appearance attributes given by a Dutch sensory panel. Table 35.2 gives similar scores for the same samples as judged by a 

Olive oils mean scores for appearance attributes from Dutch sensory panel 

Sample ID Yellow Green Brown Glossy Transp Syrup 

British panel. Note that the sensory attributes are to some extent different. Table 35.3 gives some information on the country of origin and the state of ripeness of the olives. Finally, Table 35.4 gives some physico-chemical data on the same samples that are related to the quality indices of olive oils acid and peroxide level, UV absorbance at 232 nm and 270 nm, and the difference in absorbance at wavelength 270 nm and the average absorbance at 266 nm and 274 nm. 

Given these tables of multivariate data one might be interested in various relationships. For example, do the two panels have a similar perception of the different olive oils (Tables 35.1 and 35.2) Are the oils more or less similarly scattered in the two multidimensional spaces formed by the Dutch and by the British attributes How are the two sets of sensory attributes related Does the 

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A comparison between the original measurements, and the new estimates and the relation between measurement adjustment and the standard deviation (relative error found by data adjustment) is given in Table 2. 

To reduce an angle to the first quadrant of the unit circle, that is, to a degree measure between 0° and 90°, see Table 1-1. For function values at major angle values, see Tables 1-2 and 1-3. Relations between functions and the sum/ difference of two functions are given in Table 1-4. Generally, there will be two angles between 0° and 360° that correspond to the value of a function. 

Canonical Correlation Analysis (CCA) is perhaps the oldest truly multivariate method for studying the relation between two measurement tables X and Y [5]. It generalizes the concept of squared multiple correlation or coefficient of determination, R. In Chapter 10 on multiple linear regression we found that is a measure for the linear association between a univeiriate y and a multivariate X. This R tells how much of the variance of y is explained by X = y y/yV = IlylP/llylP. Now, we extend this notion to a set of response variables collected in the multivariate data set Y. 

As an example we try to model the relation between the sensory data of Table 35.1 and the instmmental measurements of Table 35.4. The PLS analysis results are shown in Table 35.8. The first PLS dimension loads about equally high on... 

The Monin-Obukhov length L is not a parameter that is routinely measured. Colder (1972), however, established a relation between the stability classes of Pasquill, the roughness height zo (see Section VII,B), and L. The results of his investigation are shown in Fig. 4. Alternatively, the local wind speed and cloud cover measurements are used to estimate the Pasquill stability class (Table IV). In addition, Colder developed a nonogram for relating the gradient Richardson number R-, to the more easily determined bulk Richardson number / (, ... 

To circumvent the problems of ring cleavage in cyclobutabenzene (38) and thereby extend the series, the related reaction of protodesilylation has been used. The rate constants and partial rate factors for protodesilylation of benzocycloalke-nes in perchloric acid-methanol at 50 °C were measured (Table 3). In this system the enhancement of rate between o-xylene (5) and tetralin (2) is much less dramatic, but the loss of a-reactivity as a function of decreasing ring size stands out clearly a factor of 15 separates the rates of a exchange for tetralin versus cyclobutaben-... 

Table 11. Relation between electrocatalytic activity (given in the potential 20 measured at a current density of 20 mA/cm2) and conductivity in Fe-polyphthalocyanine samples (from Ref. e5>)...

Nearly all the studies listed in Table 10.1 use intrinsic viscosity as one of the quantities measured. Most of the estimates of A thus depend on the assumed relation between g and g0 there is no agreement about the correct relationship, and as Section 4 shows, there is not really an adequate basis for this relation. It can be seen from the Table that g = g%, used in the more recent papers, tends to give higher values of A than the relation (f = g% used earlier, or g = h3 used by two authors. Though different authors studied different polymers (and no paper states the synthesis conditions used for the polymers it is reporting on) it seems likely that the big differences in A are to a large extent unreal and due to the different assumptions used in treating the data. 

The boiling point, refractive index, and density of the olefin derivative of any paraffin were shown, by use of Table III, to stand in the onier of their olefin type. Table X contains the engine data of the olefin derivatives of 2-methylpentane and 3-methylpentane, recorded in the order of their olefin type. No consistent relations between octane numbers or critical compression ratios are obvious—but the blending octane numbers of these branched olefins, as measured by both the research and Motor methods, do generally stand in the order of their type. Two olefins of type III form exceptions, the exceptions being in one case too high and in the other case too low. 

For the Fc+/Fc and BCr+/BCr couples, the electrode reactions are reversible or nearly reversible in most non-aqueous solvents and the half-wave potentials are not much influenced by solvent impurities such as water. Thus, the half-wave potentials will not vary widely even when they are measured by different persons or at different laboratories. Moreover, the potentials of the Fc+/Fc and BCr+/BCr couples are considered almost solvent-independent. This justifies the comparison of potential data in different solvents, as far as they are based on this proposal. The data in Table 6.4 are useful in determining the mutual relations between the potentials of the Ag+/Ag and Hg2+/Hg electrodes and the Fc+/Fc and BCr+/BCr cou-... 

In 1960, the eleventh General Conference on Weights and Measures recommended the International System of Units (Systeme International d Unites), abbreviated as SI units, for use in science SI units are essentially the rationalized mks system of units. Relations between SI units and Gaussian units are given in Table A.4 of the Appendix. Table A.5 allows one to convert equations from SI to Gaussian units. 

E. P. Perman, and H. Mollier measured the relation between the vap. press., temp., and percentage cone, of aq. soln. of ammonia. The latter s results for the percentage cone, of the soln. at different temp, and press, are shown in Table XXIV,... 

Table 2.1 Tabulated relationship of n(E-EV2) and Xrev12 for the relation between the measured current, /, and the applied potential, E, in cyclic voltammetry for a reversible system...

A second source of type II errors appears when the difference between two groups is not to be found in any single variable as such but in the relation between the variables. Consider the small set of simulation data summarized in Table 6.2, in which the effect of a treatment of one group is compared with that of a control group on two measured responses. 

As can be seen from Figures 6.5 and 6.6, there are several similarities between PLS and PCA. For example, both methods make a linear model of the data table X by means of a score vector, t (one score for each object), and a loading vector, p, which measures the importance of the variables. However, in PCA, neither t nor p is influenced (computationally) by anything but the variation in the measurements. Hence, if it is attempted to relate the measurements X to some external event (for example, drug treatment) via the PC t-scores, it must be realised that, unless this external event is a sufficiently large... 

Making some assumptions on the chemical filiation between some organo-sulfur compounds, it was possible to establish the mathematical variation law for the concentration ratio of the various detected species and consequently to deduce the depletion rate constant of these compounds. From the measurements at the "Pointe de Penmarc h" in September 1983, the DMS lifetime estimations obtained are reported in Table I. This method for determining chemical lifetimes can only be applied for local and intensive sources. The most critical point concerns the chemical relation between the various sulfur compounds which should be verified in order to validate these estimations. However, the other assumptions do not seem to have a significant influence on the lifetime estimation within an order of magnitude. 

See Section 2.1 ofEisenberg (1976) for a full discussion of the relation between molal and molar units. Table 21.1 lists partial molar volumes of different salts. The difference between molar and molal units is about 1% for a 0.3 MKC1 solution, and rises to only 3% for 1 MKC1. These percentages are smaller than the error in most measurements of Kohs. 

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