Similarity/Distance Measures — Two Vector Pairs

(i) x = (0, -1, 0, 1), y = (1, 0, -1, 0) — Cosine & Correlation

Cosine similarity:
$$\cos(\mathbf{x},\mathbf{y}) = \frac{\mathbf{x}\cdot\mathbf{y}}{\|\mathbf{x}\|\|\mathbf{y}\|}$$

• Dot product: $(0)(1)+(-1)(0)+(0)(-1)+(1)(0) = 0$
• $\|\mathbf{x}\| = \sqrt{0^2+(-1)^2+0^2+1^2} = \sqrt{2}$
• $\|\mathbf{y}\| = \sqrt{1^2+0^2+(-1)^2+0^2} = \sqrt{2}$

$$\cos(\mathbf{x},\mathbf{y}) = \frac{0}{\sqrt{2}\cdot\sqrt{2}} = \boxed{0}$$

Correlation:
Mean $\bar{x} = \dfrac{0-1+0+1}{4} = 0$, mean $\bar{y} = \dfrac{1+0-1+0}{4} = 0$

Since both means are already 0, $(x_i-\bar{x}) = x_i$ and $(y_i-\bar{y})=y_i$:

$$r = \frac{\sum x_i y_i}{\sqrt{\sum x_i^2 \cdot \sum y_i^2}} = \frac{0}{\sqrt{2 \times 2}} = \boxed{0}$$

Both cosine and correlation are 0 — x and y are orthogonal/uncorrelated.

(ii) x = (0, 1, 0, 1), y = (1, 0, 1, 0) — Euclidean & SMC

Euclidean distance:
$$d(\mathbf{x},\mathbf{y}) = \sqrt{(0-1)^2+(1-0)^2+(0-1)^2+(1-0)^2} = \sqrt{1+1+1+1} = \sqrt{4} = \boxed{2}$$

Simple Matching Coefficient (SMC) — counts matches including 0-0 (used for symmetric binary attributes):
$$SMC = \frac{f_{11}+f_{00}}{f_{11}+f_{00}+f_{01}+f_{10}}$$

$f_{11}=0,\ f_{00}=0,\ f_{01}=2,\ f_{10}=2$

$$SMC = \frac{0+0}{0+0+2+2} = \boxed{0}$$

Summary Table

Both vector pairs turn out to be complete "opposites" of each other bit-wise, which is why every measure returns either 0 (no similarity/correlation) or the maximum possible distance.