# What does removing the first principal component from data signify?

The first principal component is the axis along which the data varies the most. So, what happens if I remove that while retaining all the remaining components? I am guessing that the data kind of coalesces together, but I am not sure.

• Remove it from what?
– mkt
Mar 15, 2018 at 14:45
• You can always ignore the first PC, meaning simply not use it. Then you'll be focusing on whatever the others tell you. A fairly common example is that you have various measures with dimension length all fed into a PCA. Then the first PC is sometimes interpretable as an overall summary of size and the others convey information on what is left, optimistically shape and pessimistically measurement error. Mar 15, 2018 at 14:51
• In classic, linear PCA you can and may remove any, not only first, PC not only conceptually but also actually. This amounts to removing the corresponding term from the linear combination of components as variable predictors. Mar 15, 2018 at 15:12
• Removing a dimension from a data cloud, such as removing 1st PC of it, amounts to projecting data points onto the (hyper)plane perpendicular to the axis of that dimension. Imagine as example that your data is spheroid in 3D space. The PC1 is the spheroids main axis. Removing it is the projecting onto the plane which that axis pierces at 90 degree angle. Then, you are left with spherical data cloud lying in that plane. Mar 16, 2018 at 8:35
• Your question 'Will the data points plotted on the new axes now look more uniform'? is a puzzling one. By definition, what you are doing is ignoring (/removing) the axis of most variance in the data. Whether that is a good or a bad idea depends on your data and your question. If you are more specific about your problem, then you might get answers that can explain things more intuitively.
– mkt
Mar 16, 2018 at 8:39

## 1 Answer

Removing a dimension from a data cloud, such as removing 1st PC of it, amounts to projecting data points onto the (hyper)plane perpendicular to the axis of that dimension. Imagine as example that your data is spheroid in 3D space. The PC1 is the spheroids main axis. Removing it is the projecting onto the plane which that axis pierces at 90 degree angle. Then, you are left with spherical data cloud lying in that plane.

• I've copied this comment by @ttnphns as a community wiki answer because the comment is, more or less, an answer to this question. We have a dramatic gap between answers and questions. At least part of the problem is that some questions are answered in comments: if comments which answered the question were answers instead, we would have fewer unanswered questions.
– mkt
Aug 23, 2018 at 11:03