PCA is mostly used as a tool in exploratory data analysis and for making predictive models. It’s often used to instant vert x pdf genetic distance and relatedness between populations. 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.

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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. CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. PCA can be thought of as fitting an n-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small, and by omitting that axis and its corresponding principal component from our representation of the dataset, we lose only a commensurately small amount of information. To find the axes of the ellipsoid, we must first subtract the mean of each variable from the dataset to center the data around the origin. Then, we compute the covariance matrix of the data, and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix.

Then we must normalize each of the orthogonal eigenvectors to become unit vectors. This procedure is sensitive to the scaling of the data, and there is no consensus as to how to best scale the data to obtain optimal results. The quantity to be maximised can be recognised as a Rayleigh quotient. It turns out that this gives the remaining eigenvectors of XTX, with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the loading vectors are eigenvectors of XTX. W is a p-by-p matrix whose columns are the eigenvectors of XTX. The transpose of W is sometimes called the whitening or sphering transformation.

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