Dimensional Analysis by Matrix Transformation

The classic technique to determine dimensionless numbers, described above, is cumbersome to use in cases where the list of related physical quantities becomes large. Pawlowski [8] developed a matrix transformation technique that offers a systematic approach to the generation of II-sets. 

Biot Bi hi T h heat transfer coefficient k thermal conductivity 

Grashof Gr pATgP V1 Gr = pATGa P fluid expansion coefficient 

Nahme-Griffith Na aATBr a Viscosity temperature dependence 

Mass Biot hpyil D hm mass transfer coefficient D mass diffusivity 

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Dimensional analysis can be simplified by arranging all relevant variables from the relevance list in a matrix form, with a subsequent transformation yielding the required dimensionless numbers. The dimensional matrix consists of a square core matrix and a residual... 

Fisher suggested to transform the multivariate observations x to another coordinate system that enhances the separation of the samples belonging to each class tt [74]. Fisher s discriminant analysis (FDA) is optimal in terms of maximizing the separation among the set of classes. Suppose that there is a set of n = ni + U2 + + rig) m-dimensional (number of process variables) samples xi, , x belonging to classes tt, i = 1, , g. The total scatter of data points (St) consists of two types of scatter, within-class scatter Sw and hetween-class scatter Sb- The objective of the transformation proposed by Fisher is to maximize S while minimizing Sw Fisher s approach does not require that the populations have Normal distributions, but it implicitly assumes that the population covariance matrices are equal, because a pooled estimate of the common covariance matrix (S ) is used (Eq. 3.45). 

The main advantage of the ACE algorithm is the diversity of possible transformations. For collinear data, a previous reduction of dimensionality of the X matrix is to be recommended, for example, by means of principal component analysis. 

The objective of a principal component analysis (PCA) is to transform a number of correlated variables into a smaller set of new, uncorrelated variables (factors or latent variables). The first few factors should then explain most of the relevant variation in the data set. To allow this reduction in dimensionality, the variables are characterized by a partial correlation. The new variables can then be generated through a linear combination of the original variables, i.e. the original matrix X is then the product of a score matrix P and the transpose ( ) of the loading matrix A ... 

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