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Update terms for ESS, TSS, and SSR to specify uncentered vs centered?

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2/5
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52/100
Issue type
Documentation
Clarity
Clearly specified
Activity status
Stale
Tech stack
tex
Domain
documentation

Research direction

Locate the lecture text containing the displayed TSS, SSR, and ESS definitions, using the wording from this issue and the original discussion in pull request #217. Update the terminology as proposed, including the centered-case note, then verify that the rendered equations and note are correct.

Written by the indexing model from the issue text.

Description

I think a more commonly seen definitions of (centered) TSS, ESS and SSR are given here where the decomposition only holds when $X$ includes intercept term.

So, I think it might be clearer if we replace

Here are some more standard definitions:

* The **total sum of squares** is $:=  \| y \|^2$.
* The **sum of squared residuals** is $:= \| \hat u \|^2$.
* The **explained sum of squares** is $:= \| \hat y \|^2$.

> TSS = ESS + SSR

with

Define:

* The (uncentered) **total sum of squares** (TSS) is $:=  \| y \|^2$.
* The (uncentered) **sum of squared residuals** (SSR) is $:= \| \hat u \|^2$.
* The **explained sum of squares** (ESS) is $:= \| \hat y \|^2$.

 We have the relationship:

$$
\text{TSS} = \text{ESS} + \text{SSR}
$$

```{note}
For the centered case, see [here](https://en.wikipedia.org/wiki/Explained_sum_of_squares).
```

Please let me know your thoughts on this!

Originally posted by @HumphreyYang in https://github.com/QuantEcon/lecture-python-advanced.myst/pull/217#discussion_r2250749514

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