  In this paper we theoretically investigate underlying assumptions that have
been used for designing adaptive particle swarm optimization algorithms in the
past years. We relate these assumptions to the movement patterns of particles
controlled by coefficient values (inertia weight and acceleration coefficient)
and introduce three factors, namely the autocorrelation of the particle
positions, the average movement distance of the particle in each iteration, and
the focus of the search, that describe these movement patterns. We show how
these factors represent movement patterns of a particle within a swarm and how
they are affected by particle coefficients (i.e., inertia weight and
acceleration coefficients). We derive equations that provide exact coefficient
values to guarantee achieving a desired movement pattern defined by these three
factors within a swarm. We then relate these movements to the searching
capability of particles and provide guideline for designing potentially
successful adaptive methods to control coefficients in particle swarm. Finally,
we propose a new simple time adaptive particle swarm and compare its results
with previous adaptive particle swarm approaches. Our experiments show that the
theoretical findings indeed provide a beneficial guideline for successful
adaptation of the coefficients in the particle swarm optimization algorithm.
