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Total sum of squares formula about torchmetrics HOT 3 CLOSED

vadim0x60 commented on May 28, 2024

Total sum of squares formula

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Comments (3)

github-actions commented on May 28, 2024

Hi! thanks for your contribution!, great first issue!

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SkafteNicki commented on May 28, 2024

Hi @vadim0x60, thanks for opening this issue.
The two formulations are equal:

$$TSS = \sum_{i=1}^n (y_i - \bar{y})^2$$

substitute the definition of

$$\bar{y} = \frac{\sum_{i=1}^{n} y_i}{n}$$

into the expression we get

$$TSS = \sum_{i=1}^n (y_i - \frac{1}{n}\sum_{i=1}^{n} y_i)^2$$

then expand the square term

$$TSS = \sum_{i=1}^{n} \left( y_i^2 - \frac{2y_i}{n} \sum_{i=1}^{n} y_i + \frac{1}{n^2} \sum_{i=1}^{n} y_i^2 \right)$$

Next, distributing the summation across the terms:

$$TSS = \sum_{i=1}^{n} y_i^2 - \frac{2}{n}\sum_{i=1}^{n} y_i^2 + \frac{1}{n}\sum_{i=1}^{n} y_i^2$$

And finally, simplifying the expression:

$$TSS = \sum_{i=1}^{n} y_i^2 - \frac{1}{n}(\sum_{i=1}^{n} y_i)^2$$

the last term we can rewrite again by using the definition of the average (in reverse order)

$$TSS = \sum_{i=1}^{n} y_i^2 - \frac{n^2 \times \bar{y}^2}{n}$$

expand again

$$TSS = \sum_{i=1}^{n} y_i^2 - \frac{n^2 \times \bar{y} \times \bar{y}}{n}$$

split

$$TSS = \sum_{i=1}^{n} y_i^2 - \frac{n^2 \times \bar{y}}{n} \times \bar{y}$$

simplify

$$TSS = \sum_{i=1}^{n} y_i^2 - n \times \bar{y} \times \bar{y}$$

again use the definition of the average to convert into the expression we want

$$TSS = \sum_{i=1}^{n} y_i^2 - \left( \sum_{i=1}^{n} y_i \right) \bar{y}$$

Hopefully this convince you that the two expressions are equal. The reason for why we use one over the other is kind of lost on me right now, but if I remember correctly one expression is a bit more numerical stable than the other, which is why we have chosen it.

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vadim0x60 commented on May 28, 2024

Thanks for taking the time to lay this out!

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