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Mathematical Examination of Principle Component Analysis Abstract Principle component analysis (PCA) is a useful technique for reducing the dimensionality of a data set. In general, PCA is useful when data sets have high dimensionality. High dimensionality data sets can make it difficult to find patterns within data and to express these patterns (similarities or differences), because simple graphical representations are usually not possible with high dimensional data sets. Therefore, PCA is a technique used to discern patterns in high dimensionality and then reduce or compress the original data set into a number of principle components, such that the greatest variance by any projection of the data is represented by the first principle component and the next principle component represents the second greatest variance, and so on. Thus PCA makes use of these principle components by eliminating the higher order principle components that represent the least variation in the data, and only analyzing the main principle components without much loss of information. By reducing the data set with PCA, the large majority of information contained in the original data set is retained and the reduction of dimensions allows for “easier” analysis. In conclusion, PCA is an optimal linear transformation that keeps a subspace, of the original high dimensionality data, that has the largest variance, i.e., the large majority of the information contained in the original data. This technique has found applications in computer vision, image compression, genetic micro-array research, and other fields. Research will be conducted on the mathematics behind PCA and some of it’s applications, using the math word processor program LaTex to display the results of this research. Biography
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