Multivariate statistics, function

PPR is a linear projection-based method with nonlinear basis functions and can be described with the same three-layer network representation as a BPN (see Fig. 16). Originally proposed by Friedman and Stuetzle (1981), it is a nonlinear multivariate statistical technique suitable for analyzing high-dimensional data, Again, the general input-output relationship is again given by Eq. (22). In PPR, the basis functions 9m can adapt their shape to provide the best fit to the available data. 

Discriminant Analysis (DA) is a multivariate statistical method that generates a set of classification functions that can be used to predict into which of two or more categories an observation is most likely to fall, based on a certain combination of input variables. DA may be more effective than regression for relating groundwater age to major ion hydrochemistry and well construction because it can account for complex, non-continuous relationships between age and each individual variable used in the algorithm while inherently coping with uncertainty in the age values used for... 

Multivariate statistical analysis using classes of variables and calculating discriminant functions as linear combinations of the variables that maximize the inter-class variance and minimize the intra-class variance. Volume 2(2). 

Quantitative FTIR data were combined with chemical and petrographic data for the 24 vitrinite concentrates and subjected to bivariate and multivariate statistical analyses in order to identify the effects of coal ification on the aliphatic and aromatic functional groups. 

Another problem that has been tackled by multivariate statistical methods is the characterization of the solvation capability of organic solvents based on empirical parameters of solvent polarity (see Chapter 7). Since such empirical parameters of solvent polarity are derived from carefully selected, strongly solvent-dependent reference processes, they are molecular-microscopic parameters. The polarity of solvents thus defined cannot be described by macroscopic, bulk solvent characteristics such as relative permittivities, refractive indices, etc., or functions thereof. For the quantitative correlation of solvent-dependent processes with solvent polarities, a large variety of empirical parameters of solvent polarity have been introduced (see Chapter 7). While some solvent polarity parameters are defined to describe an individual, more specific solute/solvent interaetion, others do not separate specific solute/solvent interactions and are referred to as general solvent polarity scales. Consequently, single- and multi-parameter correlation equations have been developed for the description of all kinds of solvent effects, and the question arises as to how many empirical parameters are really necessary for the correlation analysis of solvent-dependent processes such as chemical equilibria, reaction rates, or absorption spectra. 

A wide variety of different theoretical [e.g. Kirkwood function) and empirical cf. Chapter 7) parameters of solvent polarity have successfully been tested using multivariate statistical methods in order to model the solvent-induced changes in keto/enol equilibria [134],... 

An analysis of the t(30) values, using multivariate statistical methods, has been carried out by Chastrette et al. [193]. According to this analysis, the x(30) values of non-HBD solvents are measures of the dipolarity and polarizability as well as the cohesion of the solvents. Another analysis of x(30) values in terms of functions of the dielectric constant sf and refractive index ( d) of forty non-HBD solvents has been given by Bekarek et al he emphasizes the predominant influence of the f(fir) term on the iix(30) parameter of those solvents [194]. For further correlations of the x(30) values with other empirical parameters of solvent polarity, see Section 7.6. 

Besides the descriptive values, it is also interesting to know the correlations between the two groups of variables (rXi,Yj)- The multivariate statistical methods for this data matrix are Canonical Correlation Analysis (CCA) to investigate the relationship between both sets of variables, and Multivariate Regression with a view to predicting the values of the response variables in the Y-block in function of the variables in the X-block, using a mathematical model. 

Gaussian process. The rotation rate co(t) constitutes a Gaussian process if the multivariate density function for w(ti)... w(t ) has Gausaan form. If this is so, all cumulants beyond o vanish and the process w is statistically determined by its mean (which is zero) and first self-correlation alone. In this case... 

Principal Components Analysis (PCA) is a multivariable statistical technique that can extract the strong correlations of a data set through a set of empirical orthogonal functions. Its historic origins may be traced back to the works of Beltrami in Italy (1873) and Jordan in Prance (1874) who independently formulated the singular value decomposition (SVD) of a square matrix. However, the first practical application of PCA may be attributed to Pearson s work in biology [226] following which it became a standard multivariate statistical technique [3, 121, 126, 128]. 

A useful method to test the overall impact a subject has on the parameters is to first perform principal component analysis (PCA) on the estimated model parameters. PCA is a multivariate statistical method the object of which is to take a set of p-variables, (Xi,X2,. ..Xn = X and find linear functions of X to produce a new set of uncorrelated variables Z, Z2,. .. Zn such that Zi contains the largest amount of variability, Z2 contains the second largest, etc. 

Principal components analysis (PGA), a multivariate statistical technique, was used for data reduction and pattern recognition [8]. PGA represents the variation present in many variables using a small number of principal components (PCs). PCA functions by finding a new set of axes, which more efficiently describe the variance in the data. Samples are no longer described by their intensities in... 

The adaptive least squares (ALS) method [396, 585 — 588] is a modification of discriminant analysis which separates several activity classes e.g. data ordered by a rating score) by a single discriminant function. The method has been compared with ordinary regression analysis, linear discriminant analysis, and other multivariate statistical approaches in most cases the ALS approach was found to be superior to categorize any numbers of classes of ordered data. ORMUCS (ordered multicate-gorial classification using simplex technique) [589] is an ALS-related approach which... 

Data generated with the EOS are elaborated by Exploratory Data Analysis (EDA) software, a written-in-house software package based on MATLAB [22]. The EDA software includes the usual (univariate or multivariate) descriptive statistics functions among which Principal Component Analysis (PCA) [23], with the additional utilities for easy data manipulation (e.g. data sub sampling, data set fusion) and plots customization. 

This is a multivariate statistical technique, similar to factor analysis, which attempts to represent the inter-relationships within a set of variables, its first step is to identify the linear function of the variables with the largest variance, so that this newly created artificial (or latent) variable represents as much as possible of the variability of the original data, it then chooses a second linear function, independent of the first principal component, which explains as much as possible of the remaining variability. This process continues until one chooses to call a halt, the idea being to obtain some economy in the representation of data. 

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