Aizerman's potential function

Aizerman's potential function

The Aizerman potential function is a continuously differentiable, unimodal function described by parameters, which is genuinely greater than zero at every point in its domain of definition. It describes an area of ​​activity of a phenomenon that cannot be clearly delimited.

This function is used, among other things, as a membership function in the fuzzy pattern classification in order to parametrically describe the fuzziness of objects and classes. The parameters of the function are easy to interpret. It can be easily adapted to the distribution of compactly lying objects in the feature space if it is calculated as a class membership function from example objects by monitored learning .

Parametric description

Aizerman potential function with left and right-hand different parameters

Effect of parameter d on the steepness of the Aizerman potential function

The function can by parameters , , , be described analytically: ${\displaystyle x_{0}}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle d}$

${\displaystyle \mu (x;x_{0},b,c,d)={\frac {1}{1+\left({\frac {1}{b}}-1\right)({\frac {|x-x_{0}|}{c}})^{d}}}}$.

• Extension: parameters ${\displaystyle c}$

The parameter describes the area of ​​action of the phenomenon that is viewed as sharply. ${\displaystyle c}$

• Edge membership: parameters ${\displaystyle b}$

The parameter describes the existing affiliation of the appearance at the points . ${\displaystyle b}$ ${\displaystyle x_{0}\pm c}$

The points can be viewed as the edge of the expression. ${\displaystyle x_{0}\pm c}$

• Course of the membership function: parameters ${\displaystyle d}$

This parameter describes the steadily falling course of the membership function towards the edge.

• Location information: parameters ${\displaystyle x_{0}}$
  

  Class description using the Aizermann potential function. The class is rotated to the two main axes in the two-dimensional feature space.

The point marks the representative of a fuzzy set (class). He can z. B. be viewed as the focus of a class or as the arithmetic mean of all objects forming a class. ${\displaystyle x_{0}}$

The flexibility is increased by different parameters of the left and right branch of the function.

${\displaystyle \mu (x,b_{lr},c_{lr},d_{lr})={\frac {1}{1+{\frac {1}{2}}(1+\operatorname {sgn}(x_{0}-x))({\frac {1}{b_{l}}}-1)({\frac {|x-x_{0}|}{c_{l}}})^{d_{l}}+{\frac {1}{2}}(1+\operatorname {sgn}(x-x_{0}))({\frac {1}{b_{r}}}-1)({\frac {|x_{0}-x|}{c_{r}}})^{d_{r}}}}}$.

Individual evidence

1. ↑ MA Aizerman, Emmanuel M. Braverman, LI Rozoner: Theoretical foundations of the potential function method in pattern recognition learning . In: Automation and Remote Control . tape 25 , 1964, pp. 821-837 .
2. ↑ М. А. Айзерман, Э. М. Браверман, Л. И. Розоноэр: Метод потенциальных функций в теории обучения машин . Наука, 1970.
3. ↑ SF Bocklisch: Process analysis with fuzzy procedures. Verlag Technik, Berlin 1987, ISBN 3-341-00211-1 .