PCA (Principal Component Analysis) 3D

Question 1

PCA (Principal Component Analysis) is primarily a technique for…

Accepted Answer

Reducing the number of dimensions while preserving variance structure — PCA projects high-dimensional data onto fewer axes that capture most of its variance, reducing dimensions while preserving structure.

Question 2

The principal components are the directions of…

Accepted Answer

Maximum variance, each orthogonal to all previously found components — PC1 captures the most variance; each subsequent PC captures the most remaining variance while being orthogonal to the previous ones.

Question 3

Principal components are computed as the…

Accepted Answer

Eigenvectors of the sample covariance matrix, ranked by eigenvalue — Diagonalizing the covariance matrix yields eigenvectors (the PCs) and eigenvalues (the variance captured along each).

Question 4

The eigenvalue associated with a principal component represents…

Accepted Answer

The variance captured along that component direction — Larger eigenvalues mean more variance along that direction; the explained-variance ratio is each eigenvalue over their sum.

Question 5

Before applying PCA, you should usually…

Accepted Answer

Standardize each feature to zero mean and unit variance — PCA is variance-based; without scaling, a feature measured in large units would dominate the components regardless of true importance. Centering is mandatory.

Question 6

The 'explained variance ratio' helps you…

Accepted Answer

Decide how many principal components to retain for the analysis — Summing the ratios of kept components tells you how much information is retained, guiding the dimensionality choice (often via a scree plot).

Question 7

PCA is a LINEAR method, which means it…

Accepted Answer

Cannot capture curved or non-linear manifold structure in the data — PCA finds linear combinations of features; curved/manifold structure may need non-linear techniques for visualization.

Question 8

A useful side effect of PCA is that the new components are…

Accepted Answer

Linearly uncorrelated with each other due to component orthogonality — Because PCs are orthogonal, the transformed features are uncorrelated — helpful for models sensitive to multicollinearity.

Question 9

A 'scree plot' shows…

Accepted Answer

Eigenvalues — The scree plot's elbow indicates where additional components add little variance, suggesting how many to retain.

Question 10

PCA can be computed efficiently via…

Accepted Answer

Singular Value Decomposition — SVD factorizes the data directly into components and singular values, a numerically stable way to perform PCA without forming the covariance matrix.

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