A data scientist is implementing Principal Component Analysis (PCA) from scratch to reduce the dimensionality of a dataset with highly correlated features. After standardizing the data and computing the covariance matrix, the next crucial step involves an eigendecomposition. What is the primary significance of the eigenvectors derived from the covariance matrix in the context of PCA?
They are primarily used to calculate the inverse of the covariance matrix, a necessary step for data whitening or sphering transformations.
They provide a direct scalar measure of the total variance explained by each of the original features in the dataset.
They determine the optimal learning rate for a gradient descent algorithm that is used to iteratively find the principal components.
They represent the principal components, which are the new orthogonal axes of the feature space that point in the directions of maximum variance.
The correct answer is that the eigenvectors of the covariance matrix represent the principal components. In Principal Component Analysis (PCA), the goal is to find a new set of orthogonal axes where the data has the highest variance. These axes are the principal components, and they are mathematically defined by the eigenvectors of the data's covariance matrix. The corresponding eigenvalues indicate the amount of variance captured by each of these components.
Data whitening, or sphering, is a different preprocessing technique that transforms the data so its covariance matrix becomes the identity matrix. While it also uses the covariance matrix, its purpose is to decorrelate the variables and give them unit variance, which is not the primary goal of the eigenvectors in PCA.
While some variations of PCA can be solved using iterative methods like gradient descent, the standard analytical approach relies on eigendecomposition. In those iterative methods, the eigenvectors are the solution being sought, not a parameter for tuning the algorithm like a learning rate.
This option confuses eigenvectors with eigenvalues. It is the eigenvalues that are the scalar values quantifying the amount of variance captured along the direction of each corresponding eigenvector (principal component), not the eigenvectors themselves. Furthermore, this variance relates to the new components, not the original features.
Ask Bash
Bash is our AI bot, trained to help you pass your exam. AI Generated Content may display inaccurate information, always double-check anything important.
Why are eigenvectors chosen as the principal components in PCA?
Open an interactive chat with Bash
What is the role of eigenvalues in PCA, and how are they related to eigenvectors?
Open an interactive chat with Bash
How is the covariance matrix related to eigendecomposition in PCA?