What does it mean to have a Sigma on top of an expression

[helium]

I'm reading through this book https://www.amazon.com/Hands-Machin...1571958438&sprefix=hands+on+ma,aps,641&sr=8-3 and trying to decipher the following equation (I uploaded a image of it). I haven't seen a Sigma on top of another expression like this before and I'm not quite sure if I understand how to intepret it. Does anyone know what exactly this means?

Thanks for the help!

Note: I've also tried to add the Latex for this equation from this online notebook, but I'm not sure if it displays correctly here

[MATH]r_j = \dfrac{\displaystyle \sum\limits_{\textstyle {i=1 \atop \hat{y}_j^{(i)} \ne y^{(i)}}}^{m}{w^{(i)}}}{\displaystyle \sum\limits_{i=1}^{m}{w^{(i)}}} \quad \text{where }\hat{y}_j^{(i)}\text{ is the }j^{\text{th}}\text{ predictor's prediction for the }i^{\text{th}}\text{ instance.}[/MATH]

 

[tkhunny]

It means the same as what it generally means. Add up a bunch of stuff.

 

[Steven G]

You compute the numerator, you compute the denominator and then divide. It does not matter how the numerator and/or the denominator is written. If it has sigma (sum) notation it is what it is.

 

[Dr.Peterson]

I'm reading through this book https://www.amazon.com/Hands-Machine-Learning-Scikit-Learn-TensorFlow/dp/1492032646/ref=sr_1_3?crid=3UROI9PR9DS7D&keywords=hands+on+machine+learning+with+scikit-learn+and+tensorflow&qid=1571958438&sprefix=hands+on+ma,aps,641&sr=8-3 and trying to decipher the following equation (I uploaded a image of it). I haven't seen a Sigma on top of another expression like this before and I'm not quite sure if I understand how to intepret it. Does anyone know what exactly this means?

I think you're referring to a sigma with a bunch of stuff under it, not to it being in a fraction. This appears to be a hybrid notation where they are combining the notation that means "for i from 1 to m" with the notation in which the set containing index values, or some condition describing that set, is put under the sigma. Examples of both forms (but not this hybrid) are found here.

I read it as the sum over i from 1 to m, but excluding those values of i for which [MATH]\hat{y}_j^{(i)} = y^{(i)}[/MATH].

 

[helium]

I read it as the sum over i from 1 to m, but excluding those values of i for which [MATH]^y(i)j=y(i)[/MATH]

Ah, I think that makes sense! That is how I would interpret it as well and after checking the book that is most likely the intention. Thanks!