In busenurk/rinfis: Intuitionistic Fuzzy Inference Package

knitr::opts_chunk$set(
  collapse = TRUE,
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library(rinfis)

Introduction

In this vignette, we explain how the ifcm function in the rinfis package works.

Fuzzy c-means clustering algorithm is proposed by Bezdek et al.$^{[3]}$ Intuitionistic fuzzy c-means clustering algorithm is proposed with different versions. In this package Chaira et al.(2006) and Chaira (2011) are taken as references.

Intuitionistic fuzzy set is an extension of classical fuzzy set. The advantage of intuitionistic fuzzy set is to use the hesitation degree. The membership function is always not accurately defined due to the personal error and so intuitionistic fuzzy set gives better result where uncertainty in the form of hesitation degree is used.$^{[1]}$

An intuitionistic fuzzy set in $X$ is given by:

$$A={x,\mu_{A}(x),v_{A}(X)|x\in X}$$

where $\mu_{A}(x)\to [0,1]$, $v_{A}(x)\to [0,1]$ are the membership and non-membership degrees with the condition $0\leq \mu_{A}(x)+v_{A}(x)\leq1$.

The hesitation degree $\pi_{A}(x)$ is given by:

$$\pi_{A}(x)=1-\mu_{A}(x)-v_{A}(x)$$ where $0\leq\pi_{A}(x)\leq 1$.

The type of fuzzy generator determines how to calculate the non-membership degree. In ifcm function, Yager and Sugeno type fuzzy generator can be selected by fgen parameter.

Fuzzy clustering with Yager's Intuitionistic Fuzzy Generator (Default)

If fgen="Yager", intuitionistic fuzzy generator is created from Yager generating function.

$A_{\lambda}^{IFS}={x,\mu_{A}(x),(1-\mu_{A}(x)^{\alpha})^{1/\alpha}|x\in X}$

and the hesitation degree is given by:

$$\pi_{A}(x)=1-\mu_{A}(x)-(1-\mu_{A}(x)^{\alpha})^{1/\alpha}$$ In ifcm function the default of $\alpha$ corresponds to 0.85.$^{[1]}$

Fuzzy clustering with Sugeno's Intuitionistic Fuzzy Generator

If fgen="Sugeno", intuitionistic fuzzy generator is created from Sugeno generating function.

$A_{\lambda}^{IFS}={x,\mu_{A}(x),\frac{1-\mu_{A}(x)}{1+\lambda\mu_{A}(x)}|x\in X}$

with $\lambda\geq0$.

Thus the hesitation degree is given by:

$$\pi_{A}(x)=1-\mu_{A}(x)-\frac{1-\mu_{A}(x)}{1+\lambda\mu_{A}(x)}$$ In ifcm function the default of $\lambda$ corresponds to 2.$^{[2]}$

Example

References

[1] Chaira, T. (2011) A novel intuitionistic fuzzy C means clustering algorithm and its application to medical images, Applied Soft Computing, Vol. 11(2), 1711-1717. 📃

[2] Chaira, T., Ray, A. K., Salvetti, O. (2006) Intuitionistic fuzzy C means clustering in medical images, Advances in Pattern Recognition, 226-230. 📃

[3] Bezdek, J. C., Hall, L.A., Clark, L.P. (1993) Review of MR segmentation technique in pattern recognition, Medical Physics, Vol. 10 (20), 33-48.📃

busenurk/rinfis documentation built on Dec. 19, 2021, 12:46 p.m.