(New page: The PCA diagonalizes the maximum likelihood estimate of the covariance matrix <math>C=\frac{1}{n} \sum_{i=1}^{n} \vec{x_i}{x_i}^T</math> by solving the eigenvalue equation <math>C\vec{x...) |
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| + | The PCA, or Principal Component Analysis finds an orthonormal basis that best represents the data. | ||
The PCA diagonalizes the maximum likelihood estimate of the covariance matrix | The PCA diagonalizes the maximum likelihood estimate of the covariance matrix | ||
| − | <math>C=\frac{1}{n} \sum_{i=1}^{n} \vec{x_i}{x_i}^T</math> | + | <math>C=\frac{1}{n} \sum_{i=1}^{n} \vec{x_i}\vec{x_i}^T</math> |
by solving the eigenvalue equation | by solving the eigenvalue equation | ||
| − | <math>C\vec{ | + | <math>C\vec{e} = \lambda \vec{e}</math> |
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| + | The solutions to these equations are eigenvalues <math>\lambda_1 \ge \lambda_@ \ge \cdots \ge \lambda_m</math>. Often only <math>k \lt m </math> eigenvalues will have a nonzero value, meaning that the inherent dimensionality of the data is k, being n-k dimensions noise. | ||
Revision as of 00:05, 18 April 2008
The PCA, or Principal Component Analysis finds an orthonormal basis that best represents the data. The PCA diagonalizes the maximum likelihood estimate of the covariance matrix
$ C=\frac{1}{n} \sum_{i=1}^{n} \vec{x_i}\vec{x_i}^T $
by solving the eigenvalue equation
$ C\vec{e} = \lambda \vec{e} $
The solutions to these equations are eigenvalues $ \lambda_1 \ge \lambda_@ \ge \cdots \ge \lambda_m $. Often only $ k \lt m $ eigenvalues will have a nonzero value, meaning that the inherent dimensionality of the data is k, being n-k dimensions noise.
