The correct answer is A.

Each Principal Component (PC) is indeed chosen to have the maximum possible variance given all earlier PCs. This is because the goal of Principal Component Analysis (PCA) is to capture as much of the variability in the data as possible. The first PC is the linear combination of the original variables that explains the largest amount of variance in the data. Each subsequent PC is chosen to explain the maximum amount of variance that is orthogonal (uncorrelated) to the variance explained by the previous PCs. This ensures that each PC is uncorrelated with all earlier PCs and captures a unique aspect of the variability in the data. Therefore, each PC is chosen to maximize the variance given all earlier PCs, which makes choice A the correct answer.

Choice B is incorrect. Principal components are not chosen to have the maximum possible variance given all later PCs. Instead, each PC is selected based on its ability to account for as much of the variability in the data as possible, given all earlier PCs.

Choice C is incorrect. The statement contradicts the fundamental principle of principal component analysis (PCA). In PCA, each PC is chosen to maximize variance and not minimize it. Therefore, a PC would not be selected to have the minimum possible variance given all earlier PCs.

Choice D is incorrect. This choice incorrectly suggests that each subsequent PC should minimize variance considering later PCs which goes against the basic premise of PCA where each subsequent PC should maximize remaining variance after accounting for previous ones.

Things to Remember

• Principal Component Analysis (PCA) is a dimensionality reduction technique used to simplify complex datasets by transforming them into a set of linearly uncorrelated variables called principal components.
• The first principal component captures the most variance in the data, with each subsequent component capturing as much of the remaining variance as possible.
• PCA is sensitive to the scale of the variables, so it is important to standardize the data.