Correlation multiple

Multiple correlations for ash composition and ash fusibdity are discussed in the Coal Conversion Systems Technical Data Book (part lA, U.S. Dept, of Energy, 1984). 

Then vkt is calculated from the vX values as (-ln(l-vX)). The independent function Temperature vx is expressed as 1000 K/vT for the Arrhenius function. Finally the independent variable vy is calculated as In(vkt). Next a linear regression is executed and results are presented as y plotted against Xi The results of regression are printed next. The slope and intercept values are given as a, and b. The multiple correlation coefficient is given as c. 

The a s are dimension constants, with a value of 1. is the multiple correlation coefficient, the fraction of total variance in the data accounted for by the model. 

This work describes one approach for optimizing recovering systems using a simulation package in conjunction with standard statistical techniques such as designed experiments, multiple correlation analyses and optimization algorithms. The approach is illustrated with an actual industrial process. 

The "data" obtained from the model can be regressed using any multiple correlation program, e.g., SAS (7) or a BFGoodrich... 

R r Multiple correlation coefficient. R indicates the percentage of the variability of the relative biological response that can be accounted for by the selected independent variables. 

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. 

Here, the notation (, I C, X2) stands for the squared multiple correlation coefficient (or coefficient of determination) of the multiple regression of y, on Xj and X2. The improvement is quite modest, suggesting once more that there is only a weak (linear) relation between the two sets of data. 

We can go one step further, however. Each of the above multiple regression relations is between a single variable (response) of one data set and a linear combination of the variables (predictors) from the other set. Instead, one may consider the multiple-multiple correlation, i.e. the correlation of a linear combination from one set with a linear combination of the other set. Such linear combinations of the original variables are variously called factors, components, latent variables, canonical variables or canonical variates (also see Chapters 9,17, 29, and 31). 

Recently the data concerning to interaction of propanthiole with chlorine dioxide in 8 solvents have been published [1], In this work it was shown, that the dependence of process rate from solvents properties is satisfactory described for seven solvents, after the exclusion of data for ethyl acetate, by the Koppel-Palm four parameters equation (coefficient of multiple correlation R 0,96) at determining role of medium polarity (coefficient of pair correlation between lg(k) and (s - l)/(2e + 1) - r 0.90). Chemical mechanism of the reaction including the formation of ion-radical RS H and radical RS has been proposed by authors [ ] ... 

Fig. 11 Multiple correlations of bond lengths and angles in various four- and six-coordinate tin systems. Average C-Sn-X and C-Sn-Y angles are plotted against the changes AY and AY in the lengths of the Sn-X and Sn-Y bond lengths for individual structures [52]. (Where Sn-X and Xn-Y are of different lengths, the data refer to opposite pairs.) Reprinted with permission from Britton and Dunitz (1981).

R = multiple correlation coefficient the other symbols have the same meanings as in connection with equations 12 and 13 in Section III.D.)... 

The coefficient of multiple determination ranges from 0 (indicating that the factors, as they appear in the model, have no effect on the response) to 1 (indicating that the factors, as they appear in the model, explain the data perfectly ). The square root of the coefficient of multiple determination is the coefficient of multiple correlation, R. 

Table II. Multiple correlation analysis using Infestation...

Table III. Multiple correlation analysis using adult female dry weight as the dependent variable. (r2 = 0.35 ...

Multiple correlation coefficient an index which measures the joint effect of several variables on some response. 

Multiple regression programs also calculate auxiliary statistics, designed to help decide how well the calibration fits the data, and how well it can be expected to predict future samples. For example, two of these statistics are the standard error of calibration (SEC) and the multiple correlation coefficient (R). The SEC (also called standard error of estimate, or residual standard deviation) and the multiple correlation coefficient indicate how well the calibration equation fits the data. Their formulas are given in Table 3. 

The multiple correlation coefficient R is a dimensionless measure of how well the calibration fits the data. R can have values between -1 and +1, but in a calibration situation only positive values exist. A value close to zero indicates that the calibration fails to relate the spectra to the reference values. As the correlation coefficient increases, the spectra become better and better predictors of the reference values. Because the multiple correlation coefficient is dimensionless, it is a useful way of comparing data or results with different units, and that are difficult to compare in other ways. However, its value gives no indication of how well the calibration equation can be expected to perform on future samples. 

GLC, atomic absorption and mass spectrophotometry, enzymatic, and specific colorimetric procedures seem to be the likely candidates for routine use in the future. Automation will certainly be common. GLC is now used to detect imitation muscat wines (127). Characteristic flavor byproducts of yeasts may be detected and measured. Multiple correlation of the amounts of the more influential major and minor constituents with wine quality is the goal of such research. A simple apparatus for the simultaneous determination of the redox potential (two platinum electrodes), pH, specific conductivity, oxygen, and carbon dioxide (ion-specific electrode) has been devised (128). Molecular oxygen in wines has been determined by several procedures—polarography (129) and GLC being the latest. 

Such a high correlation coefficient indicates that the regression model describes the experimental data extremely well. Apart from the mentioned multiple correlation coefficient the following partial coefficient of determination ... 

---