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Home > Principal components analysis

PCA is also called the Karhunen-Loève transform (named after Kari Karhunen and Michel Loève) or the Hotelling transform (in honor of Harold Hotelling). PCA has the speciality of being the optimal linear transformation for keeping the subspace that has largest variance. However this comes at the price of greater computational requirement, e.g. if compared to the discrete cosine transform. Unlike other linear transforms, the PCA does not have a fixed set of basis vectors. Its basis vectors depend on the data set.

Assuming zero empirical mean (the empirical mean of the distribution has been subtracted away from the data set), the principal component w₁ of a dataset x can be defined as:

(See arg max for the notation.) With the first components, the -th component can be found by subtracting the first principal components from x:

and by substituting this as the new dataset to find a principal component in

.

A simpler way to calculate the components w_i uses the empirical covariance matrix of x, the measurement vector. By finding the eigenvalues and eigenvectors of the covariance matrix, we find that the eigenvectors with the largest eigenvalues correspond to the dimensions that have the strongest correlation in the dataset. The original measurements are finally projected onto the reduced vector space. Note that the eigenvectors X are actually the columns of the matrix V, where X=ULV′ is the singular value decompositionIn linear algebra the singular value decomposition (SVD is a factorization of a rectangular real or complex matrix analogous to the diagonalization of symmetric or Hermitian square matrices using a basis of eigenvectors (see spectral theorem). Suppose M i of X.

PCA is equivalent to empirical orthogonal functionsIn statistics and signal processing, the method of empirical orthogonal functions is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from the data. The i''th basis function is chosen to be orthogonal to (EOF).

PCA is a popular technique in pattern recognitionFor the William Gibson novel, see Pattern Recognition (novel). Pattern recognition (also known as classification or pattern classification) is a field within the area of computer science and can be defined as "the act of taking in raw data and taking an a. However, PCA is not optimized for class separability. An alternative is the linear discriminant analysis , which does take this into account. PCA optimally minimizes reconstruction error under the L2 normIn mathematics, the Lp and spaces are spaces of p-power integrable functions and corresponding sequence spaces''. They form an important class of examples of Banach spaces in functional analysis, and of topological vector spaces. See also root mean square.

1 Algorithm details

Following is a detailed English description of PCA using the covariance method. Suppose you have n data vectors of d dimensions each, and you want to project your data into a k dimensional subspace.

Find the basis vectors

1. Organize your data into column vectors, so you end up with a matrix, D.
2. Find the empirical mean along each dimension, so you end up with a empirical mean vector, M.
3. Subtract the empirical mean vector M from each column of the data matrix D. Store mean-subtracted data matrix in S.
4. Find the empirical covariance matrix C of S. .
5. Compute and sort by decreasing eigenvalue, the eigenvectors V of C.
6. Save the mean vector M. Save the first k columns of V as P. P will have dimension .

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Topics: Principal Components Analysis Matrix Data Pca Empirical Vector...

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