Either way, we have many supposedly distinct patterns. In our case, since n is smaller than our number of variables, the most non-zero eigenvectors that the PCA will return is n. n>k, then the PCA will return k distinct eigenvectors. If the number of observations is much larger than the number of variables in the dataset, i.e. The dimensions associated with equation (1) are as follows: Now comes the more difficult part: interpreting them. x: the rotated data or your PCs (The columns of U in the equation above)Īnd that’s it! You have the results of the PCA.rotation: the loading matrix whose columns are the eigenvectors (W in the equation above).sdev: the standard deviations of the principal components (if you square them, you get the eigenvalues of the covariance/correlation matrix).Prcomp returns three sets of results in a list that we have called “PCA”: There are various other inputs into the function, listed here, that can be included if necessary. The first input into the function is your data matrix and the second input is used to declare if your dataset should be scaled to have a unit variance before the PCA is conducted. You can perform a PCA on this dataset with a single function in R, prcomp. Where X is the dataset, W is the weighting matrix, whose columns are the key patterns in the data, and U is the matrix whose columns are the resulting principal components (PCs). In matrix notation, we will denote the PCA analysis formula as: This dataset contains 4968 columns, or variables, each of which is a grid cell over the U.S., and 533 rows, each of which is a yearly observation. One such dataset that will be used as an example is the Living Blended Drought Atlas (LBDA) which is a reconstruction of the Palmer Drought Severity Index (PDSI) over the contiguous United States from 1473-2005. Often times, you will perform a PCA on large datasets that contain many variables and/or many observations.
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The goal of this post is to elaborate on how to proceed after you have conducted a PCA and to address some common questions and concerns associated with the method. This post is meant to be a continuation of Dave Gold’s introductory post on Principal Component Analysis, which is an excellent explanation on how to conduct a PCA and visualize the principal components.
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