Evolving Fuzzy Modeling for MANETs Using Lightweight Online Unsupervised Learning

Abstract

The study presents a methodology for evolving fuzzy modeling tasks in Mobile Ad hoc Networks (MANETs) based on distributed data-driven fuzzy clustering and reasoning. The fuzzy clustering is exploited for the purpose of learning fuzzy inference rules online. That calls for one-pass Lightweight Evolving Fuzzy Clustering Method (LEFCM) suitable for deploying on mobile devices with constrained resources in MANETs. There is no standard method to determine the optimal number of fuzzy rules and most of the fuzzy systems still apply the trial and error method, unsuitable for online modeling tasks. The proposed methodology addresses the issues of uncertainties, simplicity and speed to run in non-intrusive way. It estimates online the number of clusters and their centers in the input data space, accordingly the fuzzy rules, by online adaptation of the LEFCM threshold value that affects the number of clusters. Adaptation is based on the combination of geometrical and statistical analyses, as well as on incorporating a multidimensional fuzzy membership degree into the clustering process. The proposed LEFCM is proven by using traditional cluster validity indexes and tested on real data sets.

Appendix

1.1 Math Details (Sect. 2.2 – Step7 in LEFCM Description)

• 1. X₁ is the first three-dimensional input vector with coordinates x₁ y₁ z₁. It becomes the first cluster center Cc₁. X₂-new three-dimensional input vector with coordinates x₂ y₂ z₂

• 2. D—Euclidean distances between current example and already created cluster centers

$$ D = \sqrt {(x_{2} - x_{1} )^{2} + (y_{2} - y_{1} )^{2} + (z_{2} - z_{1} )^{2} } $$

• 3. d_XxY-projection of D in XY space

$$ d_{X \times Y} = \sqrt {(x_{2} - x_{1} )^{2} + (y_{2} - y_{1} )^{2} } $$

• 4. $\begin{aligned} &{\text{if x}}_{ 1} > {\text{x}}_{ 2}\longrightarrow {\text{x}}_{ 2} = {\text{x}}_{ 1} ,{\text{x}}_{ 1}= {\text{x}}_{ 2} \\ &{\text{if y}}_{ 1} > {\text{y}}_{ 2}\longrightarrow {\text{y}}_{ 2} = {\text{y}}_{ 1} ,{\text{y}}_{ 1}= {\text{y}}_{ 2} \\ &{\text{if z}}_{ 1} > {\text{z}}_{ 2}\longrightarrow {\text{z}}_{ 2} = {\text{z}}_{ 1} ,{\text{z}}_{ 1}= {\text{z}}_{ 2}\end{aligned}$

• 5. β—angle between D and X × Y—projection (see Fig. 2)

$$ tg\beta = {\frac{{(z_{2} - z_{1} )}}{{d_{X \times Y} }}}\,\beta = atg{\frac{{(z_{2} - z_{1} )}}{{d_{X \times Y} }}} $$

• 6. $\sin \beta = {\frac{{(z_{2} - z_{1} )}}{D}}\,D = {\frac{{(z_{2} - z_{1} )}}{\sin \beta }}$

• 7. $z_{1} = {\frac{{z_{2} (d_{X \times Y} - S)}}{D}}$

• 8. $\alpha = a\cos {\frac{{(x_{2} - x_{1} )}}{{d_{X \times Y} }}}$

• 9. $S_{X \times Y}$—the projection of S in XY space

$$ \begin{gathered} S_{X \times Y} = S\cos \beta \hfill \\ x_{1} = x_{2} - S_{X \times Y} \cos \alpha \hfill \\ y_{1} = y_{2} - S_{X \times Y} \sin \alpha \hfill \\ \end{gathered} $$

1.2 Generalization in q-dimensional space

The Euclidean distance between q-dimensional input vectors P = (p₁, p₂, …,p_q) and Q = (q₁, q₂, …,q_q) in Euclidean q-space, is defined as:

$$ D = \sqrt {(p_{1} - q_{1} )^{2} + \cdots + (p_{q} - q_{q} )^{2} } = \sqrt {\sum\limits_{i = 1}^{q} {\left( {p_{i} - q_{i} } \right)^{2} } } $$

By decomposing D and S in all projections in q-space, the angles β_n and $\alpha_{{p_{i} \times p_{j} }}$are obtained, as well as ${\text{d}}_{{p_{i} \times p_{j} }}$ and ${\text{S}}_{{p_{i} \times p_{j} }}$. Thus, the new radius and the center coordinates of updated sphere in the q-dimensional space are calculated.