Question 5.E.4.18: It was demonstrated in Example 5.4.3 that y is linearly corr......
It was demonstrated in Example 5.4.3 that y is linearly correlated with x in the sense that y ≈ β_0e + β_1x if and only if the standardization vectors z_x and z_y are “close” in the sense that they are almost on the same line in ℜ^n. Explain why simply measuring ||z_x − z_y||_2 does not always gauge the degree of linear correlation.
If y is negatively correlated to x, then z_x = −z_y, but ||z_x − z_y||_2 = 2\sqrt{n} gives no indication of the fact that z_x and z_y are on the same line. Continuity therefore dictates that when y ≈ β_0e + β_1x with β_1 < 0, then z_x ≈ −z_y, but ||z_x − z_y||_2 ≈ 2\sqrt{n} gives no hint that z_x and z_y are almost on the same line. If we want to use norms to gauge linear correlation, we should use
min \{||z_x − z_y||_2 , ||z_x + z_y||_2\}.