Linear statistical methods

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

Rao S.P., Linear Statistical Methods and Their Using, Moscow Nauka, 1968, 548 p. 

Non-linear reaction rate expressions that are adequate for reactor design frequently contain more unknown parameters than can be evaluated by either classical kinetics or non-linear statistical methods. Sucl expressions are encountered in both heterogeneous and homogeneous catalysis, and in biochemistry. Fifteen o more non-linear parameters are not uncommon in rate models for complex industrial reactions. 

The heuristic approach described in this paper utilizes linear statistical methods to formulate the basic hyperbolic non-linear model in a particularly useful dimensionless form. Essential terms are identified and others rejected at this stage. Reaction stoichiometry is combined with the inherent mathematical characteristics of the dimensionless rate expression t< reduce the number of unknown parameters to the critical few that must be evaluated by non-linear estimation. Typically, only four or five parameters remain at this point, and initial estimates are available for these. The approach is equally applicable to cases where the rate-limiting mechanism is known and where it is not. 

Most physicochemical properties and biological activities are multidimensional in essence. Their study in terms of structure-property or structure-activity relationships requires the examination of multidimensional spaces which are hardly perceivable by humans. Under these conditions, numerous linear and nonlinear methods are routinely used in environmental QSAR for data reduction and graphical display (Figure 2). In the same way, even if in most cases linear statistical methods allow the derivation of powerful QSAR models, it has been shown that numerous environmental phenomena are better simulated by means of nonlinear statistical tools such as artificial neural networks. ... 

The previously mentioned data set with a total of 115 compounds has already been studied by other statistical methods such as Principal Component Analysis (PCA), Linear Discriminant Analysis, and the Partial Least Squares (PLS) method [39]. Thus, the choice and selection of descriptors has already been accomplished. 

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

There are instances where it is important to know if a given regression line is linear. For example, simple competitive antagonism should yield a linear Schild regression (see Chapter 6). A statistical method used to assess whether or not a regression is linear utilizes analysis of covariance. A prerequisite to this approach is that there... 

An extensive introduction into robust statistical methods is given in Ref. 134 a discussion of non-linear robust regression is found in Ref. 135. An example is worked in Section 3.4. 

Oncogenic Risk Calculations. On the basis of the expos ire analysis and potential oncogenic risk (oncogenic potency might be more descriptive), a risk analysis will be performed according to statistical methods like linear extrapolation (one-hit model) or multistage estimation (9.). 

With regard to linear projection based methods, the latent variables or scores determined by linear multivariate statistical methods such as... 

Bakshi, B. R., and Utojo, U., Unification of neural and statistical methods that combine inputs by linear projection, Comput Chem. Eng. 22(12), 1859-1878 (1998). 

Before collecting data, at least two lean/rich cycles of 15-min lean and 5-min rich were completed for the given reaction condition. These cycle times were chosen so as the effluent from all reactors reached steady state. After the initial lean/rich cycles were completed, IR spectra were collected continuously during the switch from fuel rich to fuel lean and then back again to fuel rich. The collection time in the fuel lean and fuel rich phases was maintained at 15 and 5 min, respectively. The catalyst was tested for SNS at all the different reaction conditions and the qualitative discussion of the results can be found in [75], Quantitative analysis of the data required the application of statistical methods to separate the effects of the six factors and their interactions from the inherent noise in the data. Table 11.5 presents the coefficient for all the normalized parameters which were statistically significant. It includes the estimated coefficients for the linear model, similar to Eqn (2), of how SNS is affected by the reaction conditions. 

Less strict descriptions of linearity are also provided. One recommendation is visual examination of a plot (unspecified, but presumably also of the method response versus the analyte concentration). Another recommendation is to use statistical methods , calculation of a regression line is advised. If regression is performed, the correlation... 

In general, parameter estimation by statistical methods from experimental data in which the number of measurements exceeds the number of parameters falls into one of two categories, depending on whether the function to be fitted to the data is linear or nonlinear with respect to the parameters. A function is linear with respect to the parameters, if for, say, a doubling of the values of all the parameters, the value of the function doubles otherwise, it is nonlinear. The right side of equation 3.4-17 is nonlinear. We can put it into linear form by taking logarithms of both sides, as in equation 3.4-4 ... 

Statistical methods can be applied to obtain values of parameters in both linear and nonlinear forms (i.e., by linear and nonlinear regression, respectively). Linearity with respect to the parameters should be distinguished from, and need not necessarily be associated with, linearity with respect to the variables ... 

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. 

The first QSPR models for skin tried to establish linear relationships between the descriptors and the permeability coefficient. In many cases validation of these models using, for example, external data sets was not performed. Authors of more recent models took advantage of the progress in statistical methods and used nonlinear relationships between descriptors and predicted permeability and often tried to assess their predictive quality using some validation method. 

We will describe an accurate statistical method that includes a full assessment of error in the overall calibration process, that is, (I) the confidence interval around the graph, (2) an error band around unknown responses, and finally (3) the estimated amount intervals. To properly use the method, data will be adjusted by using general data transformations to achieve constant variance and linearity. It utilizes a six-step process to calculate amounts or concentration values of unknown samples and their estimated intervals from chromatographic response values using calibration graphs that are constructed by regression. 

Another QSAR study utilizing 14 flavonoid derivatives in the training set and 5 flavonoid derivatives in the test set was performed by Moon et al. (211) using both multiple linear regression analysis and neural networks. Both statistical methods identified that the Hammett constant a, the HOMO energy, the non-overlap steric volume, the partial charge of C3 carbon atom, and the HOMO -coefficient of C3, C3, and C4 carbon atoms of flavonoids play an important role in inhibitory activity (Eqs. 3-5, Table 5). 

In a separate study ( ) aerosol species mass distributions were successfully used to calculate the contribution of each species to the extinction coefficient. Unfortunately, such detailed data is not usually available. At most air monitoring stations, only the total aerosol species mass concentrations, M -, are determined from filter samples. Statistical methods have been used to infer chemical species contributions to the particle light extinction coefficient ( ). For such analyses it is assumed that bgp can be represented as a linear combination of the total species mass concentrations, M-j, viz.. 

In Sect. 4.1 wc will discuss the method of linear statistical discriminant analysis. Here, however, some comments are given in advance. 

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