  In this paper a method of obtaining smooth analytical estimates of
probability densities, radial distribution functions and potentials of mean
force from sampled data in a statistically controlled fashion is presented. The
approach is general and can be applied to any density of a single random
variable. The method outlined here avoids the use of histograms, which require
the specification of a physical parameter (bin size) and tend to give noisy
results. The technique is an extension of the Berg-Harris method [B.A. Berg and
R.C. Harris, Comp. Phys. Comm. 179, 443 (2008)], which is typically inaccurate
for radial distribution functions and potentials of mean force due to a
non-uniform Jacobian factor. In addition, the standard method often requires a
large number of Fourier modes to represent radial distribution functions, which
tends to lead to oscillatory fits. It is shown that the issues of poor sampling
due to a Jacobian factor can be resolved using a biased resampling scheme,
while the requirement of a large number of Fourier modes is mitigated through
an automated piecewise construction approach. The method is demonstrated by
analyzing the radial distribution functions in an energy-discretized water
model. In addition, the fitting procedure is illustrated on three more
applications for which the original Berg-Harris method is not suitable, namely,
a random variable with a discontinuous probability density, a density with long
tails, and the distribution of the first arrival times of a diffusing particle
to a sphere, which has both long tails and short-time structure. In all cases,
the resampled, piecewise analytical fit outperforms the histogram and the
original Berg-Harris method.
