Output error vs equation error

We are learning about the parity space approach in class and we were given this figure to demonstrate the output error and equation error. [![Output and equation error][1]][1]

My question is what are these 2 representations supposed to mean? Are they equivalent to each other? If yes, how?

I saw these 2 forms referred to in another place (a paper by Tomita, Yutaka et al), but couldn't understand that paper at all. Can someone help to explain it in this specific context? [1]: https://i.sstatic.net/tRPcE.png

• How did your lecturer explain it? What notes did they give? Commented Oct 23, 2023 at 8:38
• They said that the residual can be calculated as $r=y-\hat{y}$ in the output error form and that equals $r=y-\frac{\hat{B}}{\hat{A}}u$, we can then multiply both sides by $\hat{A}$ and we get $\hat{A}r=\hat{A}y - \hat{B}u$ which is the equation corresponding to the equation error form. I am not sure if I misunderstood something that was explained though. Commented Oct 23, 2023 at 9:51