Algebraic Correlation Factor

Let s explore in next whether the present Spectral regression gives the opportunity in defining another correlation index, beyond the standard statistical one given by Eq. (2.35) (Chicu Putz, 2009 Putz, 2012a, b). 

One starts with the simple connection between the observed, predicted and error vectors of Eq. (3.3), however specialized on their instantaneous entries the present discussion follows (Putz Putz, 2011)  

Finally, the Cauchy-Schwarz form (2.66) is employed on the right side term of (3.52), noting that the observed and predicted activities are of the same nature for a given molecule - i.e., either both positive or both negative - thus providing their scalar product as positively defined with these, the relation (3.52) immediately reads as the inequaUty  

Nevertheless, there remains to compare this new correlation factor, written in algebraically manner as the ration of predicted - to - observed norms of investigated molecular activity or of their effects, with the fashioned statistical counterpart given by Eq. (2.63) this issue will be addressed in what follows. 

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Therefore, there was proofed both the (qualitative) simplicity and the (quantitative) superiority of algebraic correlation factor. Many applications proof these statements also on dedicated molecular-biological or molecular-ecotoxicological cases. For instance, the specialization on modem bi-component molecular system, as concerned for the ionic liquids (IL) toxicological actions - is in next explicated by the general projective QSARform. 

TABLE 3.14 The Predicted Spectral Norm, the Statistic and the Algebraic Correlation Factors of the SPECTRAL-SAR Models of Table 3.13, Computed upon the General Eqs. (3.62), (3.222), and (3.86) Since the Entry Data of Table 3.11 Are Employed, Respectively ... 

The findings in Table 3.14 are twice relevant first, because it is clear that the spectral norm parallels the statistic correlation factor second, because, despite the introduced algebraic correlation factor does the same job in general, it poses slightly higher values on a systematic basis. In other words, one can say that in an algebraic sense the SPECTRAL-SAR... 

FIGURE 3.15 Norm correlation spectral space of the statistical and algebraic correlation factors against the spectral norm of the predicted SPECTRAL-SAR models of Table 3.14, respectively (Putz Lacrama, 2007). 

TABLE 3.17 Structure Activity Relationships for all Possible Correlation Models Considered from the Data in Table 3.16 Together with the Statistical (Simple Correlation Factor, Standard Error of Estimation SEE, Explained Variance EV, Student r-test, Fischer F-test) and Algebraic (Correlation Factor r ° and Norm-Length ) for each Considered Endpoint ... 

TABLE 3.18A Spectral Structure Activity Relationships (SPECTRAL-SAR) of the Ionic Liquids Toxicity of Table 3.17 Against the Daphnia magna Species, and the Associated Computed Spectral Norms, Computed Upon Eq. (3.62), With the Observed/ Recorded Result YEXP> =9.59481, Statistic and Algebraic Correlation Factors, Computed Upon Eqs. (3.87) (3.86) (Putz Lacrtoia, 2007 Putz, 2012b), Throughout the Possible Correlation Models Considered From the Anionic, Cationic, and Ionic Liquid 0+> and l+> States, see Eqs. (3.84) and (3.85), Respectively (Lacrama et al., 2007 Putz et al., 2007 Putz Putz, 2013a)... 

While the Algebraic Correlation Factor of Eq. (3.124) Uses the Measured Activity of II A = 26,9357 Computed Upon Eq. (3.123) With Data of Table 3.29 is the... 

FIGURE 3.25 Spectral representation of the endpoints employed in designing the bioactivity mechanism for the molecules of Table 3.29, according with the algebraic correlation factors of Eq. (3.124) in Table 3.31, across the shortest (three) paths identified fi om Table 3.32, while marking the fittest molecules orbital 3D-distribution for each considered model, i.e., molecule no. 12 (4 -5,7-Trimethoxyflavanone) for the models Ic and lib, molecule no. 13 (Flavone) for models Ib, la, and Ila, molecule no. 8 (6,2, 3 -7-Hydroxyflavanone) for model He, and molecule no. 3 (Naringenin) for model III, respectively (Putz et al., 2009b). 

Given a suitable algebraic correlation such as equation 2.19, the friction factor chart might be considered obsolete. Both / and fRe2 can be represented algebraically as functions of Re allowing both types of calculations to be done. In the case of the inverse problem, that is the calculation of the flow rate for a specified pressure drop, an alternative is to use an iterative calculation, a procedure that is particularly attractive with a pocket calculator or a spreadsheet. Using equation 2.19 for /, the procedure is as follows ... 

As such the so-called algebraic SPECTRAL-SAR correlation factor is defined as the ratio of the spectral norm of the predicted activity versus that of the measured one, see Eqs. (3.55) and (3.86). Forthe present case of the measured spectral norm of T pyriformis activity, = 6.83243,... 

Next, aiming to see whether the obtained models can provide us a mechanistic model of chemical-biological interaction of tested xenobiot-ics on T. pyriformis species, the introduced spectral norm is employed in conjunction with algebraic or statistic correlation factors to compute the spectral paths between these models. Such an endeavor may lead to an intra-species analysis of models and form the first step for designing of integrated test batteries (or an expert system) at the inter-species level of ecotoxicology. 

The statistical correlation factors always yield smaller values than the corresponding algebraically ones, see Table 3.18 ... 

Algebraic expressions for terms M and C were derived using Dewar s PMO method (for C in a version similar to the co-technique [57] in order to calculate carbocation stabilization energies). The size factor S is simply a cubic function of the number of carbon atoms [97], The three independent variables of the model were assumed to be linearly related to the experimental Iball indices (vide supra). By multilinear regression analysis (sample size = 26) an equation was derived for calculating Iball indices from the three theoretical parameters. The correlation coefficient for the linear relation between calculated and experimental Iball indices is r = 0.961. 

Spreading (constriction) resistance is an important thermal parameter that depends on several factors such as (1) geometry (singly or doubly connected areas, shape, aspect ratio), (2) domain (half-space, flux tube), (3) boundary condition (Dirchlet, Neumann, Robin), and (4) time (steady-state, transient). The results are presented in the form of infinite series and integrals that can be computed quickly and accurately by means of computer algebra systems. Accurate correlation equations are also provided. 

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