Resolution of the Wicksell's equation by Minimum Distance Estimation

Abstract

The estimation of the grain size in granular materials is usually performed by 2Dobservations. Unfolding the grain size distribution from apparent 2D sizes is commonly referred as the corpuscle problem. For spherical particles, the distribution of the apparent size can be related to that of the actual size thanks to the Wicksell’s equation. The Saltikov method, which is based on Wicksell’s equation, is the most widely used method for resolving corpuscle problems. This method is recursive and works on the finite histogram of the grain size. In this paper, we propose an algorithm based on a minimizing procedure to numerically solve the Wicksell’s equation, assuming a parametric model for the distribution (e.g. lognormal distribution). This algorithm is applied on real material and the results are compared to those found using Saltikov or Saltikov-basedstereology techniques. A criterion is proposed for choosing the number of bins in the Saltikov method. The accuracy of the proposed algorithm, depending on the sample size, is studied.

Defining and applying a new approximation for the parametric probability densities of spherical particle profile sizes

“…Each line corresponds 1000 simulated samples, resulting in 1000 estimates for each entry. ML estimates were computed using our approximation, MoM by using the known moments and (Baddeley and Jensen, 2004, p. 37), MDE by (Depriester and Kubler, 2019), Saltykov method by (Gulbin, 2008) using q = 20 size classes. when an initial guess for the values becomes further from an estimate (Depriester and Kubler, 2019), the estimation using ML and our approximation was also numerically more efficient, as was the single density computation.…”

“…Our approach has a different application area than those of Depriester and Kubler (2019). We approximate the probability density which facilitates ML, while Depriester and Kubler (2019) approximated the distribution function which facilitates MDE.…”

“…Gulbin, 2008). However, minimum distance estimation (MDE) was recently suggested as a powerful technique for modelling a distribution function of particle profiles, which is useful for parameter estimation in medium and large samples of spherical particle profiles (Depriester and Kubler, 2019). Also, the method of moments (MoM) has been described for spherical particle profiles (Goldsmith, 1967), although used very seldom.…”

Maximum likelihood estimation of parameters of spherical particle size distributions from profile size measurements and its application for small samples

“…For spherical particle profiles, however, the moments are known, and the method of moments (MoM) is sometimes used (Goldsmith, 1967). The stereological use of minimum distance estimation (MDE) has also been recently described (Depriester and Kubler, 2019). Still, the maximum likelihood (ML) method is 'the most popular technique for deriving estimators' (Casella and Berger, 2002, p. 315), providing point estimators that are asymptotically unbiased and of lowest variance under mild assumptions and for large sample sizes.…”

“…One of limitations of the parametric methods is that the estimates typically have a little bias, even when all assumptions hold, while the commonly applied methods follow demands from the biological sciences which include unbiasedness. However, the strict unbiasedness seldom applies to the geosciences (Gulbin, 2008;Lopez-Sanchez and Llana-Fúnez, 2016;Durand et al, 2006b), and to some material studies (Depriester and Kubler, 2019), where it is traditional to approximate convex non-spherical particles of irregular shape via spheres, estimating the conventional 'effective diameter' which includes the bias. Another problem is the complicated numerical calculation of the probability density of the particle profiles' probability density, which involves an improper integral even when the particles are spherical (Wicksell, 1925).…”