THE HYPERBOLIC TANGENT DISTRIBUTION FAMILY

A general description and analysis of the hyperbolic tangent distribut ion family are presented. Analytical formulae are given for both the c umulative and frequency functions of the distribution, as well as for the ordinary moments of whole order. A linear transformation of the si ze coordinate axis is applied by means of which the distribution funct ion can be located appropriately over the interval between the observe d smallest and largest sizes of particles during the fitting procedure . Including the parameters of this transformation, the distribution fa mily possesses four parameters, which allows for effective fitting to experimental size distribution data. A two-level fitting procedure has been developed which was tested by using simulated noisy data. A numb er of distribution functions (log-normal, Rosin-Rammler, and beta dist ributions) may also be described well by hyperbolic tangent distributi on functions, so that it provides a convenient method for comparing me asurement data described quantitatively in different ways. (C) 1998 El sevier Science S.A. All rights reserved.