Multilinear regressions

Another problem is to determine the optimal number of descriptors for the objects (patterns), such as for the structure of the molecule. A widespread observation is that one has to keep the number of descriptors as low as 20 % of the number of the objects in the dataset. However, this is correct only in case of ordinary Multilinear Regression Analysis. Some more advanced methods, such as Projection of Latent Structures (or. Partial Least Squares, PLS), use so-called latent variables to achieve both modeling and predictions. 

Furthermore, QSPR models for the prediction of free-energy based properties that are based on multilinear regression analysis are often referred to as LFER models, especially, in the wide field of quantitative structure-activity relationships (QSAR). 

Multiple linear regression analysis is a widely used method, in this case assuming that a linear relationship exists between solubility and the 18 input variables. The multilinear regression analy.si.s was performed by the SPSS program [30]. The training set was used to build a model, and the test set was used for the prediction of solubility. The MLRA model provided, for the training set, a correlation coefficient r = 0.92 and a standard deviation of, s = 0,78, and for the test set, r = 0.94 and s = 0.68. 

The models are applicable to large data sets with a rapid calculation speed, a wide range of compounds can be processed. Neural networks provided better models than multilinear regression analysis. 

The variables that are combined hnearly are In / 17T, and In C, Multilinear regression software can be used to find the constants, or only three sets of the data smtably spaced can be used and the constants found by simultaneous solution of three linear equations. For a linearized Eq. (7-26) the variables are logarithms of / C, and Ci,. The logarithmic form of Eq. (7-24) has only two constants, so the data can be plotted and the constants read off the slope and intercept of the best straight line. 

More than just a few parameters have to be considered when modelling chemical reactivity in a broader perspective than for the well-defined but restricted reaction sets of the preceding section. Here, however, not enough statistically well-balanced, quantitative, experimental data are available to allow multilinear regression analysis (MLRA). An additional complicating factor derives from comparison of various reactions, where data of quite different types are encountered. For example, how can product distributions for electrophilic aromatic substitutions be compared with acidity constants of aliphatic carboxylic acids And on the side of the parameters how can the influence on chemical reactivity of both bond dissociation energies and bond polarities be simultaneously handled when only limited data are available ... 

Szentpaly, L.V. 1984. Carcinogenesis by polycyclic aromatic hydrocarbons a multilinear regression on new type PMO indices. Jour. Amer. Chem. Soc. 106 6021-6028. 

These topics are explained in specialist books and briefly by CHAPRA CANALE. Polynomial and multilinear regression programs are in POLYMATH and AXUM, nonlinear in CONSTANTINIDES and AXUM. No periodic data are regressed in the present collection. 

Examples Polynomial regression is applied in problem Pi.03.02. Several examples of POLYMATH multilinear regression are in sections P3.06, P3.08 and P3.10. A non-linear regression is worked out in PI.02.07. 

POLYMATH multilinear regression could be used to find the constants w, p and q but the problem will be solved directly. 

The constants can be found by multilinear regression or by solving sets of three suitably spaced data. When only one exponent is involved, a linear plot is made, as in Figure 3.1. 

The constants are found by multilinear regression or by solving an appropriate number of linear eqautions. The constants of the simpler equation,... 

After linearization by taking logarithms, the three constants can be found by multilinear regression. 

The three constants k, a and 3 are found by multilinear regression with data of (Ca, t) of this arrangement of the material balance,... 

When values of the derivative can be obtained from the experimental data, the constants are found by multilinear regression as usual, or by plotting when only one exponent is to be found. 

The exponents a and b could be found independently by POLYMATH multilinear regression if data were available with unequal starting... 

The numerical values are found from the data with POLYMATH multilinear regression. Hence,... 

POLYMATH multilinear regression handles the rate equation in linearized form, In r = a - b/T + q ln(C/C0)... 

Use POLYMATH multilinear regression with the equation in the form,... 

The numbers are obtained by POLYMATH multilinear regression. The plot indicates only poor fit of the data. The negative constant in the denominator also casts doubt on the correlation. 

The constants of the seven equations equations are found by POLYMATH Multilinear Regression. They are recorded in the second table. POLYMATH graphs of the regressions are shown. 

The constants are found by POLYMATH Multilinear Regression. The effect of temperature on the constants is represented by the Arrhenius equation, k = exp(A - B/T°K). 

Values of y, xt = 1/CS and x2 = Cs are tabulated. POLYMATH multilinear regression is used to find the constants. The regression equation of the points with Cso = 30 is... 

---