Intercorrelation coefficient

Intercorrelation coefficients are then computed. These tell when one descriptor is redundant with another. Using redundant descriptors increases the amount of fitting work to be done, does not improve the results, and results in unstable fitting calculations that can fail completely (due to dividing by zero or some other mathematical error). Usually, the descriptor with the lowest correlation coefficient is discarded from a pair of redundant descriptors. 

The models characterized by small intercorrelation coefficients are preferred to the models of large intercorrelation coefficients arbitrarily it can be assumed that intercorrelation coefficients of an absolute value over 0.8 should not be accepted. 

Since Eqs. 48 and 49 were found to be identical within the standard errors on coefficients, it indicated that the coefficients in Eq. 48 were not greatly affected by the intercorrelation of descriptors. On this basis they conclude that Eq. 48 is as good as, or better than, the models based on subsets of the data used in the study. 

In real applications, where there is noise in the data, it is rare to have two x variables exactly correlated to one another. However, a high degree of correlation between any two x variables leads to an unstable matrix inversion, which resnlts in a large amonnt of noise being introduced to the regression coefficients. Therefore, one mnst be very wary of intercorrelation between x variables when nsing the MLR method. 

The elements of fhe vector y are the reference values of the response variable, used for building the model. The uncertainty on the coefficient estimation varies inversely with the determinant of the information matrix (X X) which, in the case of a unique predictor, corresponds to its variance. In multivariate cases, the determinant value depends on the variance of fhe predictors and on their intercorrelation a high correlation gives a small determinant of the information matrix, which means a big uncertainty on the coefficients, that is, unreliable regression results. 

Unfortunately, electronic tongue variables are very often considerably intercorrelated in voltammetric profiles, for instance, currents evaluated at two consecutive potential values frequently carry almost the same information, so that their correlation coefficient is nearly 1. In such cases, standard OLS is absolutely not recommendable. Furthermore, the number of objects required for OLS regression must be at least equal to the number of predictors plus 1, and it is difficult to satisfy such a condition in many practical cases. 

Fig. 2a-c. The strongest intercorrelations among investigated topological indices, a average value for correlation coefficients obtained in the three series of structures b correlation coefficients for the polyalkylcyclohexanes with 6-10 carbon atoms c correlation coefficients for the polyalkylcyclo- hexanes with 10 carbon atoms... 

Although there is a strong negative correlation between partition coefficient and aqueous solubility (Hansch et al., 1968 Chiou et al., 1977), and a strong positive correlation between % and molecular volume (Dearden et al., 1988), the use of the partial least squares (PLS) method in this study allows the simultaneous use of intercorrelated descriptors. Nevertheless, the use of four descriptors to model the bioconcentration factor of only 11 compounds contravenes the Topliss and Costello (1972) rule, and renders the QSAR of dubious validity. 

If the relationship between dependent (Y) and independent (X) variables were perfect, the Y values for any m + 1) compounds could be used resulting in a square matrix. If the m parameters, X, are independent (orthogonal, i.e., have no intercorrelation), the matrix may be inverted and the coefficients, Uj, calculated. However, the relationships are seldom perfect so using another set of compounds would lead to another set of coefficients with values different from the previous set. This process could be repeated until all combinations had been tried giving, ultimately, a range of values (a distribution) for each coefficient. For even a medium sized data set, this is a daunting task ... 

Our basic assumption of the intercorrelation of the migration coefficients and surface absorption coefficients to the migration through fissures is verified. However, a great deal of effort must be spent studying the effects of other solute species, the chemical nature of the plutonium itself, and the kinetics of the absorption process before any understanding of the macroscopic characteristics of the transport of plutonium can be reached. 

Selection of independent variables. A wide range of different parameters, such as log P or n, a, MR, and steric parameters, should be tried molecular orbital (MO) parameters and indicator variables should not be overlooked. The parameters selected for the best equation should be essentially independent [i.e., the intercorrelation coefficients r should not be larger than 0.6-0.7 exceptions are combinations of linear and squared terms, such as (log P)2 and log P, which are usually highly interrelated, with rvalues > 0.9],... 

Sometimes variable filters are applied before the real variable selection is performed (e.g., variables that have no or nearly no variance or variables that are highly intercorrelated with another variable both procedures are fine). On the other hand, the elimination of variables that, taken alone, show no correlation with the biological activity values is a procedure that should not be applied. There is a certain chance that this variable might be able to explain the data set in combination with another variable. A better preselection procedure is the selection from the best of all possible models with three different X variables thousands of such models can be calculated within seconds, using Eq. (25) (rY Yi ... vm)—multiple correlation coefficient rYx vector of rYxt correlation coefficients Rvv matrix of rxi,xj correlation coefficients) [49]. If necessary, highly intercorrelated variables can be eliminated afterward ... 

The simplest means to obtain such a quantitative relationship is to use multiple linear regression (MLR) available in any statistical software package. In order to avoid statistically insignificant relationships or chance correlations, one should always apply the following rules of thumb (1) the ratio of compounds to descriptors should be >5 (2) the descriptors should not be intercorrelated (inter-descriptor correlation coefficient should be less than r2<0.5). 

The VIE for each estimated coefficient bj can be computed as VIE, = 1/(1 - Rf), where Rf is the coefficient of determination obtained from regressing Xj on the other predictor variables. As Rj approaches 1 (i.e., nearly linear dependent) the VIE for the estimated coefficient will tend to infinity. VIFs larger than 10 suggest problems with intercorrelation. 

Table 1.5 Intercorrelations between descriptors of chemical structures for a set of diverse organic compounds, characterized by the correlation coefficient the second figure (in parentheses) gives the number of compounds available for each pair of parameters (modified from Nendza and Russom, 1991). 

As it is the case here, n descriptors with the highest absolute correlation coefficients to the target variable often form a significantly worse descriptor set in MLR than the set obtained by BSS. A reason for that maybe strong intercorrelation of descriptors, as we saw in Section 7.5. In order to account for this, we calculated the following reference value for a descriptor subset H ... 

A potentially more appropriate statistical tool, the Durbin-Watson statistic [29], can be used to assess the linearity of the NIR quantitative method. This statistic allows the analyst to establish the lack of intercorrelation between data points in the regression. The correlation coefficient R only describes the tendency of the line, not the trueness cffit to a linear model. If there is no intercorrelation of the residuals described by the Durbin-Watson statistic, then a linear model is appropriate and may be used. 

From what we have seen before, if two descriptors are highly intercorrelated, this does not mean that they are trivially related. Trivially related are two descriptors whose coefficient of regression is 0.9999999... or 1 exactly ... 

The low correlation coefficient of 0.747 for the pH is caused by the small pH range of about 0.6 pH unit. On the other hand, determination of pH in cheese by NIRS is the result of many overlapping overtones that are also very weak. Consequently, it seems impossible to make a high precision calibration for the measurement of pH by NIRS. Mathematical manipulation of the raw NIR data of the cheese samples shows a high correlation or intercorrelation between the parameters — water sol. N/tot. N, TCA sol. N/tot. N, and water-soluble primary amines — and the total protein content. This is to be expected because during cheese ripening proteins are broken down to peptides and amino acids, mainly by enzymatic processes. Because the measurement of protein by NIRS is based on the absorption of casein molecules as well as a variety of peptides and amino acids, it is very difficult to resolve the protein absorption bands in a lot of smaller bands and to correlate these bands to the constituents from which they arise. The only possibility probably is to use higher order mathematical data transformations. 

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