Multivariate correlation analysis

For this purpose the well-known univariate correlation analysis was changed to the more general multivariate case [GEISS and EINAX, 1991 1996]. Multivariate correlation analysis enables inclusion of all interactions within the variables and the exclusion of the share of the variance resulting from the variable noise. 

The computation of the multivariate autocorrelation function (MACF) is useful if the simultaneous consideration of all measured variables and their interactions is of interest. The mathematical fundamentals of multivariate correlation analysis are described in detail in Section 6.6.3. The computed multivariate autocorrelation function Rxx according to Eqs. 6-30-6-37 is demonstrated in Fig. 9-6. The periodically encountered... 

Winiwarter, S., Bonham, N. M., Ax, F., Hallberg, A., Lennemas, H., Karlen, A. Correlation of human jejunal permeability (in vivo) of drugs with experimentally and theoretically derived parameters. A multivariate data analysis approach. J. Med. Chem. 1998, 41, 4939-4949. 

Scaling is a very important operation in multivariate data analysis and we will treat the issues of scaling and normalisation in much more detail in Chapter 31. It should be noted that scaling has no impact (except when the log transform is used) on the correlation coefficient and that the Mahalanobis distance is also scale-invariant because the C matrix contains covariance (related to correlation) and variances (related to standard deviation). 

An important aspect of all methods to be discussed concerns the choice of the model complexity, i.e., choosing the right number of factors. This is especially relevant if the relations are developed for predictive purposes. Building validated predictive models for quantitative relations based on multiple predictors is known as multivariate calibration. The latter subject is of such importance in chemo-metrics that it will be treated separately in the next chapter (Chapter 36). The techniques considered in this chapter comprise Procrustes analysis (Section 35.2), canonical correlation analysis (Section 35.3), multivariate linear regression... 

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. 

Multivariate chemometric techniques have subsequently broadened the arsenal of tools that can be applied in QSAR. These include, among others. Multivariate ANOVA [9], Simplex optimization (Section 26.2.2), cluster analysis (Chapter 30) and various factor analytic methods such as principal components analysis (Chapter 31), discriminant analysis (Section 33.2.2) and canonical correlation analysis (Section 35.3). An advantage of multivariate methods is that they can be applied in... 

Alternatively, NIR spectroscopy has been applied to relate NIR data to mechanical properties [4], A multivariate data analysis was performed on a series of commercial ethene copolymers with 1-butene and 1-octene. For the density correlation, a coefficient of determination better than 99% was obtained, whereas this was 97.7% for the flexural modulus, and only 85% for the tensile strength. 

Differentiation of vapor responses of the colloidal crystal film was accomplished with spectral measurements of the shape changes of the diffraction peak. Selectivity of response was obtained by applying multivariate data analysis to correlate these spectral changes to the effects of species of different chemical nature and to establish the identity and concentration of species. 

Correlations are inherent in chemical processes even where it can be assumed that there is no correlation among the data. Principal component analysis (PCA) transforms a set of correlated variables into a new set of uncorrelated ones, known as principal components, and is an effective tool in multivariate data analysis. In the last section we describe a method that combines PCA and the steady-state data reconciliation model to provide sharper, and less confounding, statistical tests for gross errors. 

The distance between object points is considered as an inverse similarity of the objects. This similarity depends on the variables used and on the distance measure applied. The distances between the objects can be collected in a distance matrk. Most used is the euclidean distance, which is the commonly used distance, extended to more than two or three dimensions. Other distance measures (city block distance, correlation coefficient) can be applied of special importance is the mahalanobis distance which considers the spatial distribution of the object points (the correlation between the variables). Based on the Mahalanobis distance, multivariate outliers can be identified. The Mahalanobis distance is based on the covariance matrix of X this matrix plays a central role in multivariate data analysis and should be estimated by appropriate methods—mostly robust methods are adequate. 

Winiwarter S, Bonham NM, Ax F, Hallberg A, Lennernas H and Karlen A (1998) Correlation of Human Jejunal Permeability (in Vivo) of Drugs With Experimentally and Theoretically Derived Parameters A Multivariate Data Analysis Approach. J Med Chem 41 pp 4939 1949. 

Parshad, H., Frydenvang, K., Eiljefors, T., and Earsen, C.S. Correlation of aqueous solubility of salts of benzylamine with experimentally and theoretically derived parameters. A multivariate data analysis approach, Int. J. Pharmaceut., 237(1-2) 193-207, 2002. 

The a and n constants of substituents are often useful when correlated to biological activity in the statistical procedure known as multivariate regression analysis. As is well known from pharmacological testing of various drug series, such correlations can be either linear or parabolic. The linear relationship is described by the equation... 

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