8 In particular, for two paired data sets, a CCA identifies paired directions such that the projection of the first data set along the first direction is maximally correlated with the projection of the second data set along the second direction. A canonical correlation analysis (CCA) is a very general technique for quantifying relationships between two sets of variables, and most parametric tests of significance are essentially special cases of CCA. For example, in an engineering application one may wish to find a connection between a set of processing variables (controlled by the engineer) and a set of output variables, the latter characterizing a product. In many cases, one is interested in finding the correlations between two sets of variables.
5 There are also several extensions of PCA that are useful if, for example, one wishes to emphasize some variables over others, such as weighted PCA, 3, 6 or if a non-linear model of the data is appropriate, such as generalized PCA. 3 It is widely applied in pattern classification and is used, for example, in such diverse fields as drug discovery 4 and face recognition. One of the most useful such strategies is the principal component analysis (PCA), 2 a multivariate technique in which a linear projection is used to transform data into a smaller set of uncorrelated variables. 1 For this purpose, several different strategies are utilized to highlight significant correlations among the relevant variables. One goal of data analytics is the effective dimensional reduction of large, high-dimensional data sets by the identification of a few low-dimensional axes that are most important. Finally, we describe how this approach facilitates experimental planning and process control. We demonstrate that our approach leads to a substantial enhancement of correlations, as illustrated by two experimental applications of substantial interest to the materials science community, namely: (1) determining the interdependence of processing and microstructural variables associated with doped polycrystalline aluminas, and (2) relating microstructural decriptors to the electrical and optoelectronic properties of thin-film solar cells based on CuInSe 2 absorbers. With this in mind, we describe here a versatile, Monte-Carlo based methodology that is useful in identifying non-linear functions of the variables that lead to strong input/output correlations.
One shortcoming of the canonical correlation analysis, however, is that it provides only a linear combination of variables that maximizes these correlations. It is especially useful in data analytics as a dimensional reduction strategy that simplifies a complex, multidimensional parameter space by identifying a relatively few combinations of variables that are maximally correlated. A canonical correlation analysis is a generic parametric model used in the statistical analysis of data involving interrelated or interdependent input and output variables.
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