Abstract
High-dimensional data can be challenging to analyze. They are difficult to visualize, need extensive computer resources, and often require special statistical methodology. Fortunately, in many practical applications, high-dimensional data have most of their variation in a lower-dimensional space that can be found using dimension reduction techniques. There are many methods designed for dimension reduction, and in this chapter we will study two closely related techniques, factor analysis and principal components analysis, often called PCA.
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Notes
- 1.
The normalized eigenvalues are determined only up to sign so they could multiplied by − 1 to become \((-0.71,-0.71)\) and (0. 71, −0. 71).
- 2.
As mentioned previously, the eigenvectors are determined only up to a sign reversal, since multiplication by − 1 would not change the spanned space or the norm. Thus, we could instead say the eigenvector has only negative values, but this would not change the interpretation.
- 3.
The graph would, of course, be everywhere increasing if \(\mathbf{o}_{2}\) were multiplied by − 1.
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Ruppert, D., Matteson, D.S. (2015). Factor Models and Principal Components. In: Statistics and Data Analysis for Financial Engineering. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2614-5_18
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DOI: https://doi.org/10.1007/978-1-4939-2614-5_18
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