I am reading Xie and Beni's well-known paper on a fuzzy cluster validity function. In it, they use an alternative sigma notation for summation. I have seen it only infrequently and I'd love some clarification on precisely how to read it. For context, here's the relevant section of their paper:

Consider a fuzzy c-partition of the data set $X=\{X_j;j=1,2,..., n\}$ with $V_i(i=1,2,...,c)$ the centroid of each cluster and $\mu_{ij}(i=1,2,...,c, j=1,2,...,n)$ as the fuzzy membership of data point $j$ (also called vector $j$) belonging to class $i$.

Definition 1: $d_{ij}=\mu_{ij}||X_j-V_i||$, is called the fuzzy deviation of $X_j$ from class $i$.

Note that $||\cdot||$ is the usual Euclidean norm. Thus $d_{ij}$ is just the Euclidean distance between $X_j$ and $V_i$ weighted by the fuzzy membership of data point $j$ belonging to class $i$.

Definition 2: $n_i = \sum_x {_j}\mu_{ij}$ is the fuzzy number of vectors in or fuzzy cardinality of class $i$.

Note that $\sum_x {_i}n_i=n$, where $n$ is a "hard" number, e.g., the total number of data points in $X$. In the extreme case, when the partition is hard, $n_i$ becomes exactly the number of vectors in class $i$.

Definition 3: For each class $i$, the summation of the squares of fuzzy deviation of each data point, denoted by $\sigma_i$, is called the variation of class $i$, that is: $\sigma_i = \sum_x {_j}(d_{ij})^2 = (d_{i1})^2 + (d_{i2})^2 + ... + (d_{in})^2$ . The summation of the variations of all classes, denoted by $\sigma$, is called the total variation of data set $X$ with respect to the fuzzy c-partition, i.e., $\sigma = \sum_x{_i}\sigma_i=\sum_x{_i}\sum_x{_j}(d_{ij})^2$.

Could the first summation in Definition 2 be rewritten as $\sum_{j=1}^n\mu_i{_j}$?

Why use the alternative notation?


Xie, X.L., Beni, G., 1991. A validity measure for fuzzy clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence 13, 841–847. doi:10.1109/34.85677



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