The Fuzzy Mortality Model based on Quaternion Theory

Abstract

The mortality models are of fundamental importance in many areas, such as the pension plans, the care of the elderly, the provision of health service, etc. In the paper, we propose a new class of mortality models based on a fuzzy version of the well-known Lee–Carter model (1992). Theoretical backgrounds are based on the algebraic approach to fuzzy numbers (Ishikawa, 1997, Kosiński, Prokopowicz, Ślęzak, 2003, Rossa, Socha, Szymański, 2015, Szymański, Rossa, 2014). The essential idea in our approach focuses on representing a membership function of a fuzzy number as an element of quaternion algebra. If the membership function μ(z) of a fuzzy number is strictly monotonic on two disjoint intervals, then it can be decomposed into strictly decreasing and strictly increasing functions Φ(z), Ψ(z), and the inverse functions f(u)=Φ−1(u) and g(u)=Ψ−1(u), u ∈ [0, 1] can be found. Thus, the membership function μ(z) can be represented by means of a complex-valued function f(u) + ig(u), where i is an imaginary unit. Then the pair (f, g) is a quaternion. The quaternion-valued, square integrable functions form a tool for constructing the new class of mortality models.