Abstract
Given a p-dimensional random variable X, Principal Components Analysis defines its optimal representation in a lower dimensional space. In this article we assume that X is distributed according to a Mixture of two Multivariate Normal Distributions and we project it onto an optimal vector space. We propose an original combination of principal components and linear discriminant analysis where the area under the ROC curve appears as the link between both methods. We represent X in terms of a small number of independent factors with maximum contribution to the area under the ROC curve of an optimal linear discriminant function. A practical example illustrates how these factors describe the differences between two categories in a simple classification problem.
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Notes
- 1.
Class conditional distributions
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Receiver Operating Characteristic
- 3.
But not necessarily orthogonal
- 4.
remember that \(\mathbf{B} = (\Sigma _{0} + \Sigma _{1})\)
- 5.
This makes our proposal similar to the methodology applied in Caprihan (2008).
- 6.
This is less than angle formed by the hands of a clock at five past twelve.
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Cuevas-Covarrubias, C. (2013). Principal Components Analysis for a Gaussian Mixture. In: Lausen, B., Van den Poel, D., Ultsch, A. (eds) Algorithms from and for Nature and Life. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Cham. https://doi.org/10.1007/978-3-319-00035-0_17
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DOI: https://doi.org/10.1007/978-3-319-00035-0_17
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