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I'm curently studing the topic of Possibilistic shell clustering and I have a hard time with understanding the concept of variable $\eta$. Can somebody in simpe words explain me what exectly this variable is responsible for. The objective function which uses $\eta$ looks like that $$ J = \sum_{j=1}^{C}\sum_{i=1}^{N}u_{ij}^md_{ij}^2 + \sum_{j=1}^{C}\eta_{j}\sum_{i=1}^{N}(1-u_{ij})^m $$

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1 Answer 1

$\eta$ is the bandwidth and controls the dependence of $u_{ij}$ on $d_{ij}$. ($u_{ij}$ is the degree of membership of the $i$'th point in the $j$'th cluster while $d_{ij}$ is the distance.)

You can think of it as a measure of the radius of the cluster. Formally,

$\eta_j \equiv K \dfrac{\sum_{i=1}^N u_{ij}^m d_{ij}^2 }{\sum_{i=1}^N u_{ij}^m}$, where $m>1$ is the fuzzifying weighting exponent, and $K$ is typically chosen to be 1.

For reference see A possibilistic approach to clustering [pdf]

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