Sigmoidal function is any function with the following properties:

  • Monotonically increasing
  • Differentiable
  • Bounded

Sigmoidal function is useful in fuzzy logic, where it gives a value between 0 and 1 to tell how is the membership of a member to a set, with 1 means the full membership and 0 means not a member.

Usually is a measure (e.g. length) between two quantity, and , where they are the hard boundaries. In other words, implies membership quality =0 and implies =1. Between the boundaries, membership quality of measure is determined by a sigmoidal function.

Examples of sigmoidal functions are:

  • where
  • with , here and
    • this one is called the “unipolar sigmoidal function”
  • with
    • this one is called the “bipolar sigmoidal function”
    • with
  • signum function: if and if
  • saturation function: when , when and when
    • the op-amp

Reference

MathWorks’ Documentation on Fuzzy logic toolbox: Membership functions, http://www.mathworks.com/access/helpdesk/help/toolbox/fuzzy/fp608.html