NDF estimation methods

The formal definition of the NDF given in Eq. (4.11) is mathematically consistent, but difficult to implement in practice. It is therefore useful to define methods for estimating the NDF from a single realization of the granular flow. Note that mathematically a statistical estimate is a random variable, and thus should not be confused with the NDF, which is deterministic. In order to distinguish the estimated NDF from n, we will denote the estimate by h. Thus, for example, if the estimate is unbiased then (n) = n, where the expected value is taken with respect to the multi-particle joint PDF / defined in Eq. (4.7). 

Because the particles are discrete, the probability that a particle is located at a given point X is null. Thus, in order to have a finite sample of particles to estimate the NDE, we need to introduce a kernel density function hwix) centered at x with bandwidth W. Eor example, a constant kernel density function defined by 

Note that h is essentially a volume average over all particles in a neighborhood of radius W around x. The statistical properties of the NDF estimator can be found in terms of the multi-particle joint PDF. For example, the mean of the NDF estimator is 

This would occur, for example, when all particles are uniformly distributed. In this limit, the normalized standard deviation of the estimator will scale like 

Note that, just like h, the time average It is a random variable. Thus, it is not identical to the NDF. Instead its expected value is 

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