Gradient computations and geometrical meaning of importance measures

Reliability importance is a scalar which is a partial derivative of system reliability function with respect to component reliability variable. Gradient is a vector consisting of partial derivatives of one continuous function with respect to all variables and gradient direction points in the direction of the greatest rate of the function increase. In order to analyze the direction that the system performance rises most quickly, this paper studies the gradient computations and geometrical meaning of importance measures. The expressions and physical meaning of importance measures are described at first. Then, the representations of importance measures in gradient are introduced, and relationships between importance measures and gradient are analyzed. Thirdly, the characteristics and geometrical meaning of gradient representations of importance measures are discussed. Finally, numerical examples in typical systems are demonstrated to verify the representation methods of importance measures in gradient.