First, a model is presented that is able to produce a valid distribution function for a single sediment sample.
The truncation to the unit circle of the bivariate normal distribution implicitly defines a valid distribution on the sphere.
In order to have a valid distribution, the PDF of GLD must satisfy the following conditions: (i) For all x over the allowed domain, f(x) ≥ 0, and (ii) ( {int}_{Q(0)}^{Q(1)}f(x) dx=1.
A valid distribution for the fluxes v1, v2, v3 and v4 is then obtained.
Second, it can be shown that the functions whose integrals we seek have no singularities in the region of interest as long as supp p ( x t, y t ) ⊆ ℵ, for any valid distribution p ( x t, y t ), where 'supp' denotes support.
However, the previous equation holds only if the two-dimensional function (45) is a valid Wigner distribution.
Unlike the existing sigma point generation methodologies in engineering, where negative probability weights may occur, we develop an algorithm capable of generating sample points that always form a valid probability distribution while still allowing the user to sample using a random number generator.
At each edge addition, a probability of deg v)/ C is assigned for all vertices v in [0,…, t−1], and (deg v +1)/ C for the node t itself, where C is a normalizing factor to make a valid probability distribution, and the degree includes all edges previously introduced.
This empirical choice makes large violations exponentially unlikely, and C is a normalization constant to make p(f) a valid probability distribution, i.e. all entries in the discrete joint table sum up to 1. (Alternative methods of representing a systematic solution of ε e,i,j would also have been possible).
We therefore chose a more rigorous approach, based on the choice of control subjects who were almost certain to be seronegative, yielding a valid "background" OD distribution; we also chose a curve largely covering this distribution and made a conservative choice of cutoff (error risk below 1/1000 for the first "positive" value).
Within the spray model, both distributions must be calculated at every control volume at every time-step where the spray is present and valid distributions must be guaranteed.
