4
$\begingroup$

given this equation:

$$A^t_j=f(k^i_1 \sum_{i=0,i\neq j}^n A^{t-1}_i W_{ij}+k^j_2 A^{t-1}_j)$$

and these two infos:

Notation: What is the scope of a sum?

https://people.richland.edu/james/lecture/m116/sequences/sequences.html (bottom)

It seems to me that the subscript determines the scope of sigma, e.g.:

+Aj: $$\sum_{i,j}^n A_i + A_j = \sum_{i,j}^n (A_i + A_j)$$

+A: $$\sum_{i,j}^n A_i + A = (\sum_{i,j}^n A_i) + A = A+\sum_{i,j}^n A_i$$

However, in the article* where I got this equation from they "sigma" Ai * W and then add k * Aj to the sum - ie doing b) instead of a).

Is what they did still valid and not wrong?

*Stylios, C. D. and Groumpos, P. P. (2004) "Modeling Complex Systems Using Fuzzy Cognitive Maps", IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART A: SYSTEMS AND HUMANS, VOL. 34, NO. 1, JANUARY 2004, pp. 155-162.

Edit (29-Mai-17): FYI, I got lucky this time because they provided example calculations. But this is not always the case. In fact, I have a bunch of other papers where I am left puzzled. This is why I kind of hope they did it wrong since otherwise you technically need to contact every author and ask for the intended meaning of his sigma equation.

$\endgroup$
1
$\begingroup$

A word of warning. The subscript $i,j$ in

\begin{align*} \sum_{i,j} A_i+A_j \end{align*}

does not tell anything about the scope of the sigma operator $\Sigma$.

The scope of the sigma operator $\Sigma$ is solely defined via arithmetic precedence rules. The scope is given by the expression that follows immediately the $\Sigma$ and is valid respecting the arithmetic precedence rules up to an operator with precedence level equal to '$+$' or up to the end if no such operator follows.

This implies that \begin{align*} \sum_{i,j}A_i+A_j=\sum_{i,j}(A_i)+A_j=\left(\sum_{i,j}A_i\right)+A_j \end{align*}

It also shows the calculation using \begin{align*} A^t_j&=f(k^i_1 \sum_{i=0,i\neq j}^n A^{t-1}_i W_{ij}+k^j_2 A^{t-1}_j)\\ &=f(k^i_1 \sum_{i=0,i\neq j}^n \left(A^{t-1}_i W_{ij}\right)+k^j_2 A^{t-1}_j) \end{align*} in the referred paper is valid.

Hint: You might find chapter 2: Sums in Concrete Mathematics by R.L. Graham, D.E. Knuth and O. Patashnik helpful. It provides a thorough introduction in the usage of sums.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.