  We analyze the noise in macro-particle methods used in plasma physics and
fluid dynamics, leading to approaches for minimizing the total error, focusing
on electrostatic models in one dimension. We describe kernel density estimation
for continuous values of the spatial variable, $x$, and compute the covariance
matrix for uniform true density. The band width of the covariance matrix is
related to the width of the kernel. We find the presence of constant negative
terms in the elements of the covariance matrix both on and off-diagonal. These
negative correlations are related to the fact that the total number of
particles is fixed at each time step. The effect of these negative correlations
on the electric field, computed by Gauss's law, is that the noise in the
electric field is related to a process called the Ornstein-Uhlenbeck bridge.
  For non-constant density still with continuous $x$, we analyze the total
error in the density estimation and discuss it in terms of bias-variance
optimization (BVO). For some characteristic length $l$ and kernel width $h$,
having too few particles within $h$ leads to too much variance; for $h$ large
relative to $l$, there is too much smoothing of the density. The optimum
between these two limits is found by BVO.
  We repeat the analysis for $x$ discretized on a grid, connecting particle
shapes and estimation kernels. If the particle shape satisfies a sum rule, the
charge deposited on the grid is conserved exactly. Particle shapes expressed as
the convolution of a kernel with another kernel that satisfies the sum rule
also obey the sum rule. This property holds for kernels of arbitrary width,
including widths that are not integer multiples of the grid spacing.
  We show good agreement with numerical computations.
