Hi Laxman,
Principal Component Analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.
If you're looking for a conceptual explanation of how the PCA algorithm is implemented, here's a brief overview:
- Standardize the data: PCA is sensitive to the variances of the initial variables. Therefore, it's common practice to standardize the dataset such that each feature has a mean of 0 and a standard deviation of 1.
- Compute the covariance matrix: Calculate the covariance matrix of the standardized data to understand how the features vary together.
- Calculate the eigenvalues and eigenvectors: Decompose the covariance matrix into its eigenvalues and eigenvectors. The eigenvectors represent the directions of maximum variance, and the eigenvalues represent the magnitude of the variance in the directions of the eigenvectors.
- Sort eigenvalues and their corresponding eigenvectors: Rank the eigenvectors in order of decreasing eigenvalues. This ranking is important because it tells you the relative significance of each eigenvector in explaining the variance of the dataset.
- Select a subset of the eigenvectors: Choose the top k eigenvectors that correspond to the k largest eigenvalues, where k is the number of dimensions you want to keep in your reduced dataset.
- Transform the original dataset: Project the original dataset onto the space spanned by the selected eigenvectors. This is done by multiplying the standardized data by the matrix of the selected eigenvectors. The result is a new dataset with reduced dimensions.
Each step is critical in ensuring that the PCA algorithm accurately captures the most significant features of the original dataset in a reduced dimensionality space.
