Data analyses correlation coefficients

With the description of a proper averaging model for data analysis, correlation coefficients of -0.99 were found between measured and computed 0 NMR tensors in salicylic acid and aspirin (Fig. 12.5c and d). 

An application of correlation analysis is the detection of related chemical de.scriptors when analyzing chemical data, correlation analysis should be used as a first step to identify those descriptors which are interrelated. 1 f two descriptors are strongly correlated, i.e, the correlation coefficient of two descriptors exceeds a certain value, e.g., r > 0.90, one of the descriptors can be excluded from the data set. 

Figure 2.15(a) shows the relationship between and Cp for the component characteristics analysed. Note, there are six points at q = 9, Cp = 0. The correlation coefficient, r, between two sets of variables is a measure of the degree of (linear) association. A correlation coefficient of 1 indicates that the association is deterministic. A negative value indicates an inverse relationship. The data points have a correlation coefficient, r = —0.984. It is evident that the component manufacturing variability risks analysis is satisfactorily modelling the occurrence of manufacturing variability for the components tested. 

The analysis of these data uses Eq. (9-108). The denominator is nearly unity, since values of r for related compounds are nearly unity. If there are two protons involved, then kD/kH in methanol is given by (1 - Jtp + - d )2- Indeed, the data fit this model with a correlation coefficient of 0.998, being 0.608 0.004. On the other hand, the model applied to water gives an imaginary expression for , necessitating the more complex picture. The data in water fit the equation Ad/Ah = (1 - d + 0.48 X xD) X (1 - xD +0.69 X. rp)2, which is indicative of the more complex proton involvement depicted. 

Linear regression analysis was performed on the relation of G"(s) versus PICO abrasion index. Figure 16.10 plots the correlation coefficient as a function of strain employed in the measurement of loss modulus. The regression results show poor correlation at low strain with increasing correlations at higher strains. These correlations were performed on 189 data points. 

Data Analysis Because of the danger of false conclusions if only one or two parameters were evaluated, it was deemed better to correlate every parameter with all the others, and to assemble the results in a triangular matrix, so that trends would become more apparent. The program CORREL described in Section 5.2 retains the sign of the correlation coefficient (positive or negative slope) and combines this with a confidence level (probability p of obtaining such a correlation by chance alone). 

Note that in data analysis we divide by n in the definition of standard deviation rather than by the factor n - 1 which is customary in statistical inference. Likewise we can relate the product-moment (or Pearson) coefficient of correlation r (Section 8.3.1) to the scalar product of the vectors (x - x) and (y - y) ... 

Scaling is a very important operation in multivariate data analysis and we will treat the issues of scaling and normalisation in much more detail in Chapter 31. It should be noted that scaling has no impact (except when the log transform is used) on the correlation coefficient and that the Mahalanobis distance is also scale-invariant because the C matrix contains covariance (related to correlation) and variances (related to standard deviation). 

Matrix formed by a set of correlation coefficients related to m variables in multivariate data sets, R = (rXi,Xj). It is relevant in multicomponent analysis. 

Alternatively, NIR spectroscopy has been applied to relate NIR data to mechanical properties [4], A multivariate data analysis was performed on a series of commercial ethene copolymers with 1-butene and 1-octene. For the density correlation, a coefficient of determination better than 99% was obtained, whereas this was 97.7% for the flexural modulus, and only 85% for the tensile strength. 

Fig. 2 Plot of P-Cl distances (in A) vs average P-N distances (in A) for P-chloro-NHPs (diamonds) and for all compounds (R2N)2PC1 (except P-chloro-NHPs) listed in the CSD data base (open squares). The solid and dashed lines represent the result of linear regression analyses. R2 is the square of the correlation coefficient in the regression analysis. (Reproduction with permission from [55])...

Because of the large difference in the behavior of the thin plywood and the gypsum board, the type of interior finish was the dominant factor in the statistical analysis of the total heat release data (Table III). Linear regression of the data sets for 5, 10, and 15 min resulted in squares of the correlation coefficients R = 0.88 to 0.91 with the type of interior finish as the sole variable. For the plywood, the average total heat release was 172, 292, and 425 MJ at 5, 10, and 15 min, respectively. For the gypsum board, the average total heat release was 25, 27, and 29 MJ at 5, 10, and 15 min, respectively. 

Linearity is evaluated by appropriate statistical methods such as the calculation of a regression line by the method of least squares. The linearity results should include the correlation coefficient, y-intercept, slope of the regression line, and residual sum of squares as well as a plot of the data. Also, it is helpful to include an analysis of the deviation of the actual data points for the regression line to evaluate the degree of linearity. 

A second use of this type of analysis has been presented by Stewart and Benkovic (1995). They showed that the observed rate accelerations for some 60 antibody-catalysed processes can be predicted from the ratio of equilibrium binding constants to the catalytic antibodies for the reaction substrate, Km, and for the TSA used to raise the antibody, Kt. In particular, this approach supports a rationalization of product selectivity shown by many antibody catalysts for disfavoured reactions (Section 6) and predictions of the extent of rate accelerations that may be ultimately achieved by abzymes. They also used the analysis to highlight some differences between mechanism of catalysis by enzymes and abzymes (Stewart and Benkovic, 1995). It is interesting to note that the data plotted (Fig. 17) show a high degree of scatter with a correlation coefficient for the linear fit of only 0.6 and with a slope of 0.46, very different from the theoretical slope of unity. Perhaps of greatest significance are the... 

The prenylflavonoid contents are listed in Table 2.71. The coefficient of variation of the within-day precision of the analysis varied between 3.8 and 11.4 per cent. The correlation coefficient of linearity was in each case over 0.998. The data presented indicate that the method can be applied for the separation, identification and quantitation of prenylflavonoids in hops and beer [191],... 

The distance between object points is considered as an inverse similarity of the objects. This similarity depends on the variables used and on the distance measure applied. The distances between the objects can be collected in a distance matrk. Most used is the euclidean distance, which is the commonly used distance, extended to more than two or three dimensions. Other distance measures (city block distance, correlation coefficient) can be applied of special importance is the mahalanobis distance which considers the spatial distribution of the object points (the correlation between the variables). Based on the Mahalanobis distance, multivariate outliers can be identified. The Mahalanobis distance is based on the covariance matrix of X this matrix plays a central role in multivariate data analysis and should be estimated by appropriate methods—mostly robust methods are adequate. 

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