Why do we maximize the variance during the analysis of the main components?

Question

I am trying to read the PCA and saw that the goal is to maximize dispersion. I don’t quite understand why. Any explanations / pointers to other related topics would be helpful.

Answer 1

Difference is a measure of the "variability" of the data that you have. The potential number of components is infinite, so you want to “squeeze” the very information in each component of the final set that you create.

If for exaggeration you had to choose one main component, you would like it to take into account the great variability: therefore, search for maximum variance, so that one component collects the most “uniqueness” of the data set.

Answer 2

Note that PCA does not actually increase the variance of your data. Rather, it rotates the dataset in such a way as to align the directions in which it is most common with major axes. This allows you to remove those sizes for which the data is almost equal. This reduces the dimensionality of the data, while maintaining the dispersion (or distribution) between the points as close as possible to the original.

Answer 3

Maximizing the variances of the component vectors coincides with maximizing the "uniqueness" of these vectors. Thus, you are vectors as far as possible from each other. Thus, if you use only the first N component vectors, you collect more space with highly changing vectors than with similar vectors. Think about what the core component means.

Take, for example, a situation where you have two lines orthogonal in three-dimensional space. You can completely capture the environment with these orthogonal lines than two lines that are parallel (or almost parallel). When applied to very high dimensional states using very few vectors, this becomes a much more important relationship between supported vectors. In the sense of linear algebra, you want independent lines to be created using PCA, otherwise some of these lines will be redundant.

See PDF from the Princeton Department of CS for an explanation.