Dynamic Max Count

This contains the ideas and notes for a Dynamic Max Count (Dynamic Max-in-time) aggregate operator

Concept

Instead of using Hyper-buckets that have constant densities which can not be updated reasonably using the MaxCountProgramNotes ideas, we propose a probabilistic method where by we define a probability density function in each hyper-bucket. In a sense we are not trying to minimize skew in the bucket creation process, but recognizing and modeling skew in each bucket. The added advantage to this concept is that a region equivalent to a hyper-bucket containing no points may be excluded from the index. Consequently the algorithm searches a smaller space. For example if it becomes necessary to shrink the size of a hyper-bucket to latex2($10~unit^{6}$) size in a latex2($10000~unit^6$) space we will have latex2($1 \times 10^18$) buckets. and points concentrated in specific inEach probability density will need the following properties:

1. Parameters that define the distribution e.g.
   1. Center location
   2. Spatial size
   3. Standard deviation
   4. A measure of symmetry or skew

2. A multi-dimensional probability function preferably a function that uses types functions as parameters e.g. latex2($$p(x_u(t),x_l(t),y_u(t),y_l(t)[,z_u(t),z_l(t)])$$)

3. A theory to update, delete or insert points and the distributions based on changes to points.

Based on this last item, we must maintain a database of 4-dimensional points that we index using 4-dimensional, probability buckets.