Principal Components Decomposition Algorithm

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A Principal Components Decomposition Algorithm is a matrix decomposition algorithm that can be applied by a PCA system (to solve a PCA task to return principal components of a matrix).



References

2014

2012

  • http://en.wikipedia.org/wiki/Principal_component_analysis
    • QUOTE: Principal component analysis (PCA) is a mathematical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. The number of principal components is less than or equal to the number of original variables. This transformation is defined in such a way that the first principal component has the largest possible variance (that is, accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it be orthogonal to (i.e., uncorrelated with) the preceding components. Principal components are guaranteed to be independent only if the data set is jointly normally distributed. PCA is sensitive to the relative scaling of the original variables. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT), the Hotelling transform or proper orthogonal decomposition (POD).

      PCA was invented in 1901 by Karl Pearson.[1] Now it is mostly used as a tool in exploratory data analysis and for making predictive models. PCA can be done by eigenvalue decomposition of a data covariance (or correlation) matrix or singular value decomposition of a data matrix, usually after mean centering (and normalizing or using Z-scores) the data matrix for each attribute.[2] The results of a PCA are usually discussed in terms of component scores, sometimes called factor scores (the transformed variable values corresponding to a particular data point), and loadings (the weight by which each standardized original variable should be multiplied to get the component score).[3]

      PCA is the simplest of the true eigenvector-based multivariate analyses. Often, its operation can be thought of as revealing the internal structure of the data in a way that best explains the variance in the data. If a multivariate dataset is visualised as a set of coordinates in a high-dimensional data space (1 axis per variable), PCA can supply the user with a lower-dimensional picture, a "shadow" of this object when viewed from its (in some sense) most informative viewpoint. This is done by using only the first few principal components so that the dimensionality of the transformed data is reduced.

      PCA is closely related to factor analysis. Factor analysis typically incorporates more domain specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix.

  1. Pearson, K. (1901). "On Lines and Planes of Closest Fit to Systems of Points in Space" (PDF). Philosophical Magazine 2 (6): 559–572. http://stat.smmu.edu.cn/history/pearson1901.pdf. 
  2. Abdi. H., & Williams, L.J. (2010). "Principal component analysis.". Wiley Interdisciplinary Reviews: Computational Statistics, 2: 433–459. 
  3. Shaw P.J.A. (2003) Multivariate statistics for the Environmental Sciences, Hodder-Arnold. ISBN 0-340-80763-6.[page needed]

2011

2009

  • http://www.statistics.com/resources/glossary/p/pca.php
    • QUOTE: The purpose of principal component analysis is to derive a small number of linear combinations (principal components) of a set of variables that retain as much of the information in the original variables as possible. This technique is often used when there are large numbers of variables, and you wish to reduce them to a smaller number of variable combinations by combining similar variables (ones that contain much the same information).

      Principal components are linear combinations of variables that retain maximal amount of information about the variables. The term "maximal amount of information" here means the best least-square fit, or, in other words, maximal ability to explain variance of the original data.

      In technical terms, a principal component for a given set of N-dimensional data, is a linear combination of the original variables with coefficients equal to the components of an eigenvector of the correlation or covariance matrix. Principal components are usually sorted by descending order of the eigenvalues - i.e. the first principal component corresponds to the eigenvector with the maximal eigenvalue.


2006

2002

1999

1901

  • (Pearson, 1901) ⇒ K. Pearson. (1901). “On Lines and Planes of Closest Fit to Systems of Points in Space" In: Philosophical Magazine, 2(11). doi:10.1080/14786440109462720.