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'''Fuzzy clustering by Local Approximation of MEmberships (FLAME)''' is a novel [[data clustering]] algorithm that defines clusters in the dense parts of a dataset and perform cluster assignment solely based on the '''neighborhood relationships''' among objects. The key feature of this algorithm is that, the neighborhood relationships among neighboring objects in the feature space is used to constrain the memberships of neighboring objects in the fuzzy membership space.
The FLAME algorithm is mainly divided into three steps:
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### All outliers are assigned with fixed and full membership to the outlier group;
### The rest are assigned with equal memberships to all clusters and the outlier group;
## Then the fuzzy memberships of all type 3 objects are updated by a converging iterative procedure called '''Local/Neighborhood Approximation of Fuzzy Memberships''', in which the fuzzy membership of each object is updated by a linear combination of the fuzzy memberships of its nearest neighbors.
# Cluster construction from fuzzy memberships in two possible ways:
## One-to-one object-cluster assignment, to assign each object to the cluster in which it has the highest membership;
## One-to-multiple object-clusters assignment, to assign each object to the cluster in which it has a membership higher than a threshold.
==The Optimization Problem in FLAME==
The '''Local/Neighborhood Approximation of Fuzzy Memberships''' is a procedure to minimize the '''Local/Neighborhood Approximation Error (LAE/NAE)''' defined as the following:
::<math>
E(\{\boldsymbol{p}\})=\sum_{\boldsymbol{x}\in\boldsymbol{X}} \bigg\| \boldsymbol{p(x)}-\sum_{ \boldsymbol{y \in \mathcal{N}(x)} } w_{\boldsymbol{xy}} \boldsymbol{p(y)} \bigg\|^2
</math>
where <math>\boldsymbol{X}</math> is the set of all type 3 objects, <math>\boldsymbol{p(x)}</math> is the fuzzy membership vector of object <math>\boldsymbol{x}</math>, <math>\mathcal{N}(x)</math> is the set of nearest neighbors of <math>\boldsymbol{x}</math>, and <math>w_{\boldsymbol{xy}}</math> are the coefficients reflecting the relative proximities of the nearest neighbors.
The NAE can be minimized by solving the following linear equations with unique solution which is the unique global minimum of NAE with value zero:
::<math>
p_k(\boldsymbol{x})-\sum_{\boldsymbol{y\in \mathcal{N}(x)}} w_{ \boldsymbol{xy} } p_k(\boldsymbol{y}) = 0, \quad\forall{\boldsymbol{x}\in \boldsymbol{X} },\quad k=1,...,M
</math>
where <math>M</math> is the number of CSOs plus one (for the outlier group). The following iterative procedure can be used to solve these linear equations:
::<math>
{\boldsymbol{p}^{t+1}(\boldsymbol{x})} = \sum_{ \boldsymbol{y\in \mathcal{N}(x)} }
w_{\boldsymbol{xy}} {\boldsymbol{p}^t (\boldsymbol{y})}
</math>
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