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I learned about PCA and how to find the principal components via eigenvectors/values. Now for the following problems my professor says that "Feature 2 is constant and can hence be ignored, so you can produce a scatter plot of feature 1 versus 3, and then you can see the main axes:"

Problems (ignore the Shepard diagrams)

So I should be able to draw a scatter plot and then see the main axes (eigenvectors) of the data. I must severely misunderstand something because when I draw the scatter plot I don't see the connection between the solution and the plot.

How to see (no computation of eigenvectors/values) the principal components in such simple plots?

Solutions for checking the result: Solutions

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  • $\begingroup$ When you standardize the data in each coordinate and approximate the scatterplot with an ellipse, the main axis of the ellipse describes the first PC and its minor axis describes the second PC. See the figure just above the "Application" section of my post at stats.stackexchange.com/a/71303/919 and read the discussion of a "45 degree angle" that precedes it. (The lines shown in that figure are the regression lines, not the PCs.) $\endgroup$ – whuber Jan 27 '19 at 14:23
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First, you de-mean your data, now new $(x,z)$ coordinates will be $(-2,-2),(1,-1),(-1,1),(2,2)$. If you plot this, you'll see that lines $x=z$ and $x=-z$ are indeed principal axes. $x=z$ describes larger amount of variance in the data since it fits the points $(-2,-2),(2,2)$, so a vector in the form $[1,1]$ is your first principal axis, and $[1,-1]$ or $[-1,1]$ (it doesn't matter if you negate your PCs or not) is your second. When normalized, you get the answers in $4.1$. The $4.2$ and $4.3$ contains only the dot product results (after de-meaning).

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