All MCQs

The ratio expresses how much of the dataset’s total variance is represented by one component. Ratios across all full components sum to one, aside from numerical rounding.

Each PCA eigenvalue measures the amount of sample variance along its eigenvector direction. Larger eigenvalues identify more informative variance directions.

PCA based on raw covariance can be dominated by variables measured on larger scales. Standardization allows relative patterns across variables to influence the components more comparably.

Subtracting each feature mean places the cloud’s center at the origin. The components then reflect covariance and directions of deviation from the average observation.

The covariance matrix summarizes how centered variables vary together. Its eigenvectors identify directions of variance, and its eigenvalues quantify the corresponding amounts.

Orthogonality prevents later components from repeating the same linear direction as earlier ones. For centered data, the resulting component scores have zero sample covariance.

PCA creates each component by weighting and summing the original variables. The weights define a new direction in feature space.

PCA chooses the first direction along which centered observations vary most. This direction captures the greatest possible variance among unit-length linear projections.

SVD is defined for general m-by-n matrices and does not require symmetry. This makes it broadly useful for data tables, images, and document-term matrices.

Large discarded singular values would remove substantial structure from the matrix. Small discarded values usually imply a more accurate low-rank approximation.

Dominant singular components capture the strongest directions of variation or energy in the matrix. Discarding weaker components produces a lower-dimensional representation.

A low-rank approximation reconstructs the matrix from a subset of singular components. It preserves major structure while reducing storage and complexity.