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Principal Component Analysis

The principal component analysis is that mathematical procedure which uses an orthogonal transformation for converting a set of observations which have possibly correlated values into the set of values containing uncorrelated variable. which are also called as principal components. The number of original variables is either equal to or less than the number of original variables. The transformation is for definition is such a way that the first of the principal component gets highest variance possible. Each succeeding component also gets the highest variance which is possible within the constraint that it is orthogonal to its preceding components. If the data is jointly and normally distributed, there is guarantee of principal components being independent. The PCA is also sensitive to the relative scaling of all the original variables. The name varies with the field of application with various names given like the hostelling transform or the proper orthogonal decomposition, and the Kurhunrn Loeve Transform.

Karl Pearson is attributed with the discovery of PCA in the year 1901. It finds its usage now in various exploratory data analysis and also for making predictive models. The PCA is done by Eigen value decomposition of the covariance matrix of the data and singular value decomposition of the matrix of data, which is done after entering mean for the data of each attribute.

The mathematical definition of a PCA is the orthogonal linear transformation which transforms the data to a new coordinate system in a way that the greatest variance(by any type of projection) of the data lies on the first coordinate, which is called the first principle component, second greatest variance, and so on.

In a data matrix X_Tthat has zero empirical mean and each N row representing a different repletion of the experiment and each of the M columns giving a particular datum., the single value of X is X= WΣV_T. Here the M x M matrix is the matrix of eigenvectors while the matrix is the M x N. Rectangular and diagonal matrix which contains the nonnegative real numbers on the diagonal. The N x N matrix V is formed by the matrix of eigenvectors X_TX . The PCA dimension preserving dimensionality is given by -



Generalizations

Non linear generalizations- The modern methods of non linear dimensionality reduction are applied to theoretical and algorithmic roots of PCA and K means. The original idea of Pearson was to take the straight line or a plane which can be bet fitted to set of data points.

Multilinear generalizations- In the multilinear subspace learning, PCA are generalized for the MPCA or the multilinear PCA which extracts the features directly from the tensor representation. Solving of MPCA is done performing the PCA in all modes of the tensors. Application of MPCA has been found in gait and face recognition.

Higher order-the N way principal component analysis is performed by models such as tucker decomposition, multiple factor analysis, and PARAFEC, co inertia analysis, STATSIS or DISTASIS.

Weighted PCA robustness- PCA is the mathematically optima method and is also sensitive to outliners of the data which produce the large errors that PCA tries to remove. Hence, usually the outliners are removed before computing the PCA.

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