# Difference between weighted average regression and locally weighted regression?

I was reading about Locally weighted regression in paper written on Locally Weighted Learning by CHRISTOPHER G. ATKESON1, ANDREW W. MOORE and STEFAN SCHAAL that came up in Artificial intelligence review. . But I could not understand it fully. Especially the difference they say between the Distance weighted averaging and local weighted regression.

The equation to find the $\hat y$ which is the prediction got with respect to the training set given.

In general,

C(q) = $\sum_{i=0}^n\biggr(\big (\hat y - y_i )^2 \times K(d(x_i, q)) \biggr)$

For Distance weighted averaging,

$$\hat y = \frac {\sum y_i K(d(x_i, q)} {K(d(x_i, q)}$$

For locally weighted regression,

$$\hat y = x_i ^ T \beta$$

I am clear that I am missing something while understanding these two as I am not able to understand the physical interpretation diagram they have given,

for distance weighted averaging it is

for Locally weighted averaging it is

Can some one explain why the author says in case of locally weighted averaging the string can pull in such a way that the line can translate and rotate and whereas in weighted averaging it can only move up or down ?